Research Article The Pricing of Asian Options in Uncertain...

17
Research Article The Pricing of Asian Options in Uncertain Volatility Model Yulian Fan and Huadong Zhang School of Science, North China University of Technology, Beijing 100144, China Correspondence should be addressed to Yulian Fan; [email protected] Received 2 December 2013; Accepted 29 April 2014; Published 16 June 2014 Academic Editor: Xiaodong Lin Copyright © 2014 Y. Fan and H. Zhang. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper studies the pricing of Asian options when the volatility of the underlying asset is uncertain. We use the nonlinear Feynman-Kac formula in the G-expectation theory to get the two-dimensional nonlinear PDEs. For the arithmetic average fixed strike Asian options, the nonlinear PDEs can be transferred to linear PDEs. For the arithmetic average floating strike Asian options, we use a dimension reduction technique to transfer the two-dimensional nonlinear PDEs to one-dimensional nonlinear PDEs. en we introduce the applicable numerical computation methods for these two classes of PDEs and analyze the performance of the numerical algorithms. 1. Introduction Model uncertainty aversion has shown to have important consequences in price behavior in capital markets [1–3] and macroeconomics [4, 5]. Denis and Martini [6] prove, in the model uncertainty situation the value of any bounded contingent claim can be expressed as the supremum of its expectation over some set of martingale measures. In finance, there is an important case of model uncer- tainty called volatility uncertainty in which the uncertainty comes from the volatility coefficient. A major difficulty here is that the probabilities are mutually singular. is type of uncertainty is initially studied by Avellaneda et al. [7] and Lyons [8] for the superhedging of European options with payoffs depending only on the basic assets’ terminal values. In this case the values of the derivatives should satisfy the (Black-Scholes-Barenblatt) BSB equations. e correspond- ing discrete-time case has been studied in Delbaen [9]. But for the path-dependent options, the difficulty is dramatically increased. In this paper, we are going to study the pricing of Asian options in the uncertain volatility model. As far as we know, there is little literature on this problem. In the model without volatility uncertainty, there is a vast literature on pricing of Asian options. In the geometric Brownian motion model, Asian options pricing methods are studied extensively. Geman and Yor [10] develop a dimensional Laplace transform method to price Asian options (see also [11]). Kemna and Vorst [12] use Monte Carlo simulation with a specific variance reduction method to compute the prices of fixed strike average-rate options. Ingersoll [13], and Wilmott et al. [14] prove that the two-dimensional PDE for a floating strike Asian option can be reduced to a one-dimensional PDE. Rogers and Shi [15] formulate a one-dimensional PDE that can model the prices of both floating and fixed strike Asian options and propose a numerical technique to compute tight lower bound on European average-rate options’ prices. Barraquand and Pudet [16] describe a forward shooting grid algorithm and prove that it is unconditionally convergent. In Zvan et al. [17] and Zvan et al. [18], a flux limiter is used to retain accuracy while preventing oscillations. Simon et al. [19] propose an easy computable upper bound for the prices of arithmetic Asian options. Kaas et al. [20] and Dhaene et al. [21] compute the sharp lower and upper bounds for the prices of arithmetic Asian options by using the method of “comonotonic approximations.” Veˇ ceˇ r[22], and Zhang [23, 24] propose various PDE methods. In Marcozzi [25], the first-order hyperbolic term is discretized by using a first- order upwind type method, resulting in at most first-order accuracy. A related approach based on a combination of the (weighted essentially nonoscillatory) WENO discretization and the grid stretching is used for Asian options in Oosterlee et al. [26]. Linetsky [27] investigates a spectral expansion approach. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 786391, 16 pages http://dx.doi.org/10.1155/2014/786391

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Research ArticleThe Pricing of Asian Options in Uncertain Volatility Model

Yulian Fan and Huadong Zhang

School of Science North China University of Technology Beijing 100144 China

Correspondence should be addressed to Yulian Fan fanylncuteducn

Received 2 December 2013 Accepted 29 April 2014 Published 16 June 2014

Academic Editor Xiaodong Lin

Copyright copy 2014 Y Fan and H ZhangThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies the pricing of Asian options when the volatility of the underlying asset is uncertain We use the nonlinearFeynman-Kac formula in the G-expectation theory to get the two-dimensional nonlinear PDEs For the arithmetic average fixedstrike Asian options the nonlinear PDEs can be transferred to linear PDEs For the arithmetic average floating strike Asian optionswe use a dimension reduction technique to transfer the two-dimensional nonlinear PDEs to one-dimensional nonlinear PDEsThen we introduce the applicable numerical computation methods for these two classes of PDEs and analyze the performance ofthe numerical algorithms

1 Introduction

Model uncertainty aversion has shown to have importantconsequences in price behavior in capital markets [1ndash3] andmacroeconomics [4 5] Denis and Martini [6] prove inthe model uncertainty situation the value of any boundedcontingent claim can be expressed as the supremum of itsexpectation over some set of martingale measures

In finance there is an important case of model uncer-tainty called volatility uncertainty in which the uncertaintycomes from the volatility coefficient A major difficulty hereis that the probabilities are mutually singular This type ofuncertainty is initially studied by Avellaneda et al [7] andLyons [8] for the superhedging of European options withpayoffs depending only on the basic assetsrsquo terminal valuesIn this case the values of the derivatives should satisfy the(Black-Scholes-Barenblatt) BSB equations The correspond-ing discrete-time case has been studied in Delbaen [9] Butfor the path-dependent options the difficulty is dramaticallyincreased In this paper we are going to study the pricing ofAsian options in the uncertain volatility model As far as weknow there is little literature on this problem

In the model without volatility uncertainty there is a vastliterature on pricing of Asian options

In the geometric Brownian motion model Asian optionspricing methods are studied extensively Geman and Yor[10] develop a dimensional Laplace transform method to

price Asian options (see also [11]) Kemna and Vorst [12] useMonte Carlo simulation with a specific variance reductionmethod to compute the prices of fixed strike average-rateoptions Ingersoll [13] and Wilmott et al [14] prove thatthe two-dimensional PDE for a floating strike Asian optioncan be reduced to a one-dimensional PDE Rogers and Shi[15] formulate a one-dimensional PDE that can model theprices of both floating and fixed strike Asian options andpropose a numerical technique to compute tight lower boundon European average-rate optionsrsquo prices Barraquand andPudet [16] describe a forward shooting grid algorithm andprove that it is unconditionally convergent In Zvan et al[17] and Zvan et al [18] a flux limiter is used to retainaccuracy while preventing oscillations Simon et al [19]propose an easy computable upper bound for the prices ofarithmetic Asian options Kaas et al [20] and Dhaene etal [21] compute the sharp lower and upper bounds for theprices of arithmetic Asian options by using the method ofldquocomonotonic approximationsrdquo Vecer [22] and Zhang [2324] propose various PDE methods In Marcozzi [25] thefirst-order hyperbolic term is discretized by using a first-order upwind type method resulting in at most first-orderaccuracy A related approach based on a combination of the(weighted essentially nonoscillatory) WENO discretizationand the grid stretching is used for Asian options in Oosterleeet al [26] Linetsky [27] investigates a spectral expansionapproach

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 786391 16 pageshttpdxdoiorg1011552014786391

2 Mathematical Problems in Engineering

In the jump diffusion models methods for pricing Asianoptions are still under development Fusai [28] obtains asimple expression for the double transform of the prices ofcontinuously monitored Asian options by means of Fourierand Laplace transform DrsquoHalluin et al [29] propose a semi-Lagrangian method to price the continuously observed fixedstrike Asian options Cai and Kou [30] consider the pricingof Asian options for the double exponential jump diffusionmodel Bayraktar and Xing [31] obtain a fast numericalapproximation scheme and analyze the performance of thenumerical algorithm

In this paper we use the stochastic processes driven by119866-Brownian motion to describe the stocksrsquo price processes119866-Brownian motion is introduced in Peng [32ndash34] and ithas the properties that its increments are zero-mean inde-pendent and stationary and can be proved to be 119866-normallydistributed The 119866-normal distribution random variableshave no mean uncertainty but can have variance uncertaintyThe variance uncertainty of the 119866-normal distribution isthe source of the volatility uncertainty of the stocksrsquo priceprocesses

The paper is organized as follows In Section 2 we usethe nonlinear Feynman-Kac formula in the 119866-expectationtheory to get the two-dimensional nonlinear PDEs satisfiedby the values of the Asian options For the arithmeticaverage fixed strike Asian options we prove the convexityof the optionsrsquo values with respect to the stock prices andtransfer the nonlinear PDEs to linear PDEswith the uncertainvolatilities being replaced by the maximum volatilities Forthe arithmetic average floating strike Asian options weuse a dimension reduction technique to transfer the two-dimensional nonlinear PDEs to one-dimensional nonlinearPDEs In Section 3 we introduce the applicable numericalcomputation methods for these two classes of PDEs and givethe numerical results

2 The Pricing of Asian Options in UncertainVolatility Model

21 119866-Brownian Motion and 119866-Expectation In this subsec-tion we will introduce the 119866-Brownian motion and 119866-expectation defined in Peng [33ndash36]

Let Ω be a given set and let H be a linear space of realvalued functions defined on Ω We suppose that H satisfies119888 isin H for each constant 119888 and |119883| isin H if 119883 isin H The spaceH can be considered as the space of random variables

A sublinear expectation E onH is a functional E H rarr

R satisfying the following properties for all 119883119884 isin H wehave

(a) Monotonicity If 119883 ge 119884 then E[119883] ge E[119884]

(b) Constant preserving E[119888] = 119888 for 119888 isin R

(c) Sub-additivity E[119883 + 119884] le E[119883] + E[119884]

(d) Positive homogeneity E[120582119883] = 120582E[119883] forall120582 ge 0

The triple (ΩHE) is called a sublinear expectationspace

Definition 1 A 119889-dimensional process (119861119905)119905ge0

on a sublinearexpectation space (ΩHE) is called a119866-Brownianmotion ifthe following properties are satisfied

(i) 1198610(120596) = 0

(ii) For each 119905 119904 ge 0 the increment 119861119905+119904

minus 119861119905is

119873(0 times Σ)-distributed and is independent from(119861

1199051

1198611199052

119861119905119899

) for each 119899 isin 119873 and 0 le 1199051le sdot sdot sdot le

119905119899le 119905

Remark 2 Just as in the classical situation the increments of119866-Brownian motion (119861

119905+119904minus 119861

119905)119904ge0

are independent fromΩ119905

Let 120587119873

119905= 119905

119873

0 119905

119873

1 119905

119873

119873 such that 0 = 119905

119873

0lt 119905

119873

1lt sdot sdot sdot lt

119905119873

119873= 119905 for 119873 = 1 2 be a sequence of partitions of [0 119905]

with

120583 (120587119873

119905) = max 1003816100381610038161003816119905119894+1 minus 119905

119894

1003816100381610038161003816 119894 = 0 1 119873 minus 1 (1)

Thequadratic variation process of 1-dimensional119866-Brownianmotion (119861

119905)119905ge0

with 1198611

119889

== 119873(0 times [1205902 120590

2]) is defined as

⟨119861⟩119905= lim

120583(120587119873

119905)rarr0

119873minus1

sum

119895=0

(119861119905119873

119895+1

minus 119861119905119873

119895

)2

= 1198612

119905minus 2int

119905

0

119861119904119889119861

119904 (2)

(⟨119861⟩119905)119905ge0

is an increasing process with ⟨119861⟩0= 0

LetΩ = 119862119889

0(R+

) be the space of allR119889-valued continuouspaths (120596

119905)119905isinR+ with 120596

0= 0 equipped with the distance

120588 (1205961 120596

2) =

infin

sum

119894=1

2minus119894

[(max119905isin[0119894]

10038161003816100381610038161003816120596

1

119905minus 120596

2

119905

10038161003816100381610038161003816) and 1] (3)

For each fixed 119879 isin [0infin) we setΩ119879= 120596

sdotand119879 120596 isin Ω

From Peng [36] the distribution of the quadratic vari-ation process of the 119866-Brownian motion has the followingproperties

Lemma 3 For each fixed 119904 119905 ge 0 ⟨119861⟩119904+119905

minus ⟨119861⟩119904is identically

distributed with ⟨119861⟩119905and independent from Ω

119904

Theorem 4 ⟨119861⟩119905is119873([120590

2119905 120590

2119905] times 0)-distributed that is for

each 120593 isin 119862119897119871119894119901

(R)

E [120593 (⟨119861⟩119905)] = sup

1205902leVle1205902

120593 (V119905) (4)

The sublinear expectation E in the above theorem is called119866-expectation defined on 119871

119894119901(Ω) via the following procedure

Let

119871119894119901(Ω

119879) = 120601 (119861

1199051and119879

119861119905119899and119879

) 119899 isin N

1199051 119905

119899isin [0infin) 120601 isin 119862

119897119871119894119901(R

119889times119899)

(5)

with 119861119905= 120596

119905 119905 isin [0infin) for 120596 isin Ω and

119871119894119901(Ω) =

infin

⋃

119899=1

119871119894119901(Ω

119899) (6)

Mathematical Problems in Engineering 3

Firstly construct a sequence of 119889-dimensional randomvectors (120585)infin

119894=1on a sublinear expectation space (Ω H E) such

that 120585119894is 119866-normal distributed and 120585

119894+1is independent from

(1205851 120585

119894) for each 119894 = 1 2 For each119883 isin 119871

119894119901(Ω) with

119883 = 120601 (1198611199051

minus 1198611199050

1198611199052

minus 1198611199051

119861119905119899

minus 119861119905119899minus1

) (7)

some 120601 isin 119862119897119871119894119901

(119877119889times119899

) and 0 = 1199050lt 119905

1lt sdot sdot sdot lt 119905

119899lt infin set

E [120601 (1198611199051

minus 1198611199050

1198611199052

minus 1198611199051

119861119905119899

minus 119861119905119899minus1

)]

= E [120601 (radic1199051minus 119905

01205851 radic119905

119899minus 119905

119899minus1120585119899)]

(8)

And the related conditional expectation of

119883 = 120601 (1198611199051

1198611199052

minus 1198611199051

119861119905119899

minus 119861119905119899minus1

) (9)

underΩ119905119895

is defined by

E [119883 | Ω119905119895

] = 120595 (1198611199051

119861119905119895

minus 119861119905119895minus1

) (10)

where

120595 (1199091 119909

119895)

= E [120601 (1199091 119909

119895 radic119905

119895+1minus 119905

119895120585119895+1

radic119905119899minus 119905

119899minus1120585119899)]

(11)

E[sdot] consistently defines a sublinear expectation on 119871119894119901(Ω)

and (119861119905)119905ge0

is a 119866-Brownian motionWe let 119871119901

119866(Ω

119905119896

) 119901 ge 1 denote the completion of

119871119894119901(Ω

119905119896

) = 120593 (1198611199051and119879

119861119905119899and119879

) 119899 isin N

1199051 119905

119899isin [0infin) 120593 isin 119862

119897119871119894119901(R

119889times119899)

(12)

under the norm 119883119901= (E[|119883|

119901])

1119901

22 The PDEs We assume the security market is definedon a sublinear expectation space (ΩH E) which openscontinuously and offers a constant riskless interest rate 119903 toall borrowers and lenders And we also assume there are notransaction costs andor taxes Suppose that the price process(119878

119905)119905ge0

of some risky asset is given by

119889119878] = 120583119878]119889] + 119878]119889119861] 1198780= 119878 (13)

where 119861 is the 119866-Brownian motion with 1205902= minusE[minus1198612

1] and

1205902

= E[1198612

1] and 120583 and are constants Hence (119878])]ge0 is a

geometric 119866-Brownian motion

119878] = 119878119905exp (120583 (] minus 119905)

minus1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905))

(14)

Define

119868] = int

]

0

119878120591119889120591 (15)

Then 119868] satisfies the following stochastic differential equation

119889119868] = 119878]119889] 1198680= 0 (16)

Let119883] = (119878] 119868])119879 so

119889119883] = 120583119883119883]119889] +

119883119883]119889119861] 119883

0= 119909 = (119878 0) (17)

where 120583119883 and

119883 are 2 times 2matrixes

120583119883

= (120583 0

1 0)

119883= (

0

0 0) (18)

The typical Asian options in this security market are thefollowing four types

the arithmetic average fixed strike call option withfinal payoff

max (119868119879

119879minus 119870 0) (19)

the corresponding arithmetic average fixed strike putoption with payoff

max (119870 minus119868119879

119879 0) (20)

the arithmetic average floating strike call option withfinal payoff

max(119878119879minus

119868119879

119879 0) (21)

and the corresponding arithmetic average floatingstrike put option with payoff

max(119868119879

119879minus 119878

119879 0) (22)

Let 119881(119905 119878119905 119868

119905) denote the price of one of the options at

time 119905 Take the arithmetic average fixed strike call option forexample we know 119881(119879 119878

119879 119868

119879) = max(119868

119879119879 minus 119870 0)

Here we still assume the market operates in a risk neutralworld [7]

Peng [35] introduces the axiomatic conditions satisfied bythe valuation mechanism In the sublinear expectation space(ΩH E) E

119905[sdot] is a filtration consistent valuation operator

By this valuation mechanism we have

119881 (119905 119878119905 119868

119905) = E [119890

minus119903(119879minus119905)119881 (119879 119878

119879 119868

119879) | Ω

119905] (23)

Here we assume the riskless interest rate is 119903Since we assume the market is risk neutral we have 120583 =

119903 It is easy to verify that the discounted prices of the stocksatisfy the valuation mechanism E

119905[sdot]

119905ge0 that is

119878119905= E

119905[119890

minus119903(]minus119905)119878]] 0 le 119905 le ] le 119879 (24)

4 Mathematical Problems in Engineering

Let us assume

119889119883119905119909

] = 120583119883119883

119905119909

] 119889] + 119883119883

119905119909

] 119889119861] 119883119905= 119909 = (119878 119868) (25)

with

120583119883

= (119903 0

1 0)

119883= (

0

0 0) (26)

The following result is a corollary of the nonlinearFeynman-Kac formula of Peng [36]

Corollary 5 Let

119884119905119909

119904= E [119890

minus119903(119879minus119904)Φ(119883

119905119909

119879)

+ int

119879

119904

119890minus119903(]minus119904)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119879

119904

119890minus119903(]minus119904)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩] | Ω119904]

(27)

where Φ R2rarr R is a given Lipschitz function and 119891 119892

R2times R rarr R are given Lipschitz functions that is |120601(119909 119910) minus

120601(1199091015840 119910

1015840)| le 119870(|119909 minus 119909

1015840| + |119910 minus 119910

1015840|) for each 119909 119909

1015840isin R2 119910 1199101015840

isin

R 120601 = 119891 and 119892 And denote

119906 (119905 119909) = 119884119905119909

119905 (28)

Then

119906 (119905 119909) = E[119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575)

+ int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]]

(29)

is a viscosity solution of the following PDE

120597119905119906 + sup

1205902le1205902le1205902

(1

2⟨119863

2119906 (119905 119909)

119883119909

119883119909⟩ + 119892 (119909 119906)) 120590

2

+ ⟨119863119906 (119905 119909) 120583119883119909⟩ minus 119903119906 + 119891 (119909 119906) = 0

119906 (119879 119909) = Φ (119909)

(30)

Proof The proof of expression (29) is the same as that ofPeng [36] By the proof of Pengrsquos theorem of the nonlinearFeynman-Kac formula and the boundedness of 119890minus119903(119879minus119905) wecan easily prove that 119906 is a Lipschitz function that is

10038161003816100381610038161003816119906 (119905 119909) minus 119906 (119905 119909

1015840)10038161003816100381610038161003816le 119862

10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816 (31)

E [10038161003816100381610038161003816119884

119905119909

119904

10038161003816100381610038161003816] le 119862 (1 + |119909|) (32)

Let us see the continuity of 119906with respect to 119905 Firstly we have

|119906 (119905 119909) minus 119906 (119905 + 120575 119909)|

le E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

+ E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

(33)

Replace 119890minus119903120575 with its Taylorrsquos expansion in the first term on

the right side of (33) we have

E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

= E [10038161003816100381610038161003816(1 minus 119903120575 + (119903120575)

2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)

minus 119906 (119905 + 120575 119909)10038161003816100381610038161003816]

= E [10038161003816100381610038161003816119906 (119905 + 120575119883

119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

+ (minus119903120575 + (119903120575)2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)10038161003816100381610038161003816]

le E [10038161003816100381610038161003816119883

119905119909

119905+120575minus 119909

10038161003816100381610038161003816]

+ E [10038161003816100381610038161003816(minus119903120575 + (119903120575)

2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)10038161003816100381610038161003816]

(34)

By the following formulas [36]

E [10038161003816100381610038161003816119883

119905119909

119905+120575minus 119909

10038161003816100381610038161003816

2

] le 119862 (1 + 1199092) 120575

E [10038161003816100381610038161003816119883

119905119909

119905+120575

10038161003816100381610038161003816

2

] le 119862 (1 + 1199092)

(35)

and (32) we have

E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

le 1198621015840(1 + |119909|

2)12

(12057512

+ 120575 +1205752

2sdot sdot sdot )

(36)

Mathematical Problems in Engineering 5

For the second term on the right side of (33) we have

E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

= E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

times (119891 (119883119905119909

] 119884119905119909

] ) minus 119891 (119909 119884119905119909

119905)) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

times (119892 (119883119905119909

] 119884119905119909

] ) minus 119892 (119909 119884119905119909

119905)) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

+ E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119909 119884119905119909

119905) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119909 119884119905119909

119905) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

le 119862 (1 + |119909|) 120575

(37)

Therefore 119906 is continuous in 119905Now we are going to prove that 119906(119905 119909) is a viscosity

solution of PDE (30) For fixed (119905 119909) isin (0 119879) times R2 let 120595 isin

11986223

119887((0 119879) times R2

) such that 120595 ge 119906 and 120595(119905 119909) = 119906(119905 119909) By(29) and Taylorrsquos expansion for 120575 isin (0 119879 minus 119905)

0 le E[119890minus119903120575

120595 (119905 + 120575119883119905119909

119905+120575) minus 120595 (119905 119909)

+ int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]]

le E [ minus 119903120595 (119905 119909) 120575 + 120597119905120595 (119905 119909) 120575

+ ⟨119863120595 (119905 119909) 120583119883119909⟩ 120575

+ ⟨1

2119863

2120595 (119905 119909)

119883119909

119883119909⟩ (⟨119861⟩

119905+120575minus ⟨119861⟩

119905)

+ 119891 (119909 120595) 120575 + 119892 (119909 120595) (⟨119861⟩119905+120575

minus ⟨119861⟩119905)]

+ 119862 (1 + |119909| + |119909|2+ |119909|

3) 120575

32

le (120597119905120595 + sup

1205902le1205902le1205902

(1

2⟨119863

2120595 (119905 119909)

119883119909

119883119909⟩

+119892 (119909 120595) ) 1205902+ ⟨119863120595 (119905 119909) 120583

119883119909⟩

minus119903120595 (119905 119909) + 119891 (119909 120595))120575

+ 119862 (1 + |119909| + |119909|2+ |119909|

3) 120575

32

(38)

It is easy to check that

120597119905120595 + sup

1205902le1205902le1205902

(1

2⟨119863

2120595 (119905 119909)

119883119909

119883119909⟩ + 119892 (119909 120595)) 120590

2

+ ⟨119863120595 (119905 119909) 120583119883119909⟩ minus 119903120595 (119905 119909) + 119891 (119909 120595) ge 0

(39)

Thus 119906 is a viscosity subsolution of (30) Similarly we canprove that 119906 is a viscosity supersolution of (30)

For the arithmetic average Asian options we have

119881 (119905 119909) = 119881 (119905 119878 119868) = E [119890minus119903(119879minus119905)

119881 (119879 119878119879 119868

119879)] (40)

Then by Corollary 5 119881(119905 119909) satisfies the following PDE

120597119905119881 (119905 119909) + sup

1205902le1205902le1205902

1

2⟨119863

2119881 (119905 119909)

119883119909

119883119909⟩120590

2

+ ⟨119863119881 (119905 119909) 120583119883119909⟩ minus 119903119881 (119905 119909) = 0

119881 (119879 119909) = 120593 (119879 119909)

(41)

By (18) the above PDE is

120597119905119881 (119905 119878 119868) + sup

1205902le1205902le1205902

1

2211987821205902119881119878119878

(119905 119878 119868)

+ 119903119878119881119878(119905 119878 119868) + 119878119881

119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

119881 (119879 119878 119868) = 120593 (119879 119878 119868)

(42)

where120593 (119879 119878 119868)

=

max ( 119868

119879minus 119870 0) for the arithmetic average

fixed strike call option

max (119870 minus119868

119879 0) for the arithmetic average

fixed strike put option

max (119878 minus119868

119879 0) for the arithmetic average

floating strike call option

max ( 119868

119879minus 119878 0) for the arithmetic average

floating strike put option(43)

6 Mathematical Problems in Engineering

23 The Arithmetic Average Fixed Strike Asian Call OptionLet us see the arithmetic average fixed strikeAsian call optionFrom (14) we have

119878119905119878

] = 119878 exp 120583 (] minus 119905) minus1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905)

(44)

Then by the definition of 119868] we have

119868119905119868

] = 119868 + 119878int

]

119905

exp 120583 (120591 minus 119905) minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591

(45)

Since

119881 (119905 119909) = 119881 (119905 119878 119868) = E [119890minus119903(119879minus119905)

119881 (119879 119878119879 119868

119879)]

= E [119890minus119903(119879minus119905)

(119868119879

119879minus 119870)

+

]

119881 (119905 119878 119868) = E[119890minus119903(119879minus119905)

times (((119868 + 119878int

]

119905

exp 120583 (120591 minus 119905)

minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591)

times119879minus1) minus 119870)

+

]

(46)

By the convexities of function 119891(119909) = (119909 minus 119896)+ and the

sublinear expectation we can say

119881(119905 1205821198781+ (1 minus 120582) 119878

2 119868) le 120582119881 (119905 119878

1 119868) + (1 minus 120582)119881 (119905 119878

2 119868)

(47)

that is 119881(119905 119878 119868) is convex about 119878 So in the numericalcomputation the second order difference of 119881 is positiveThen the numerical solutions of (43) are equivalent to thoseof PDE

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

119881 (119879 119878 119868) = (119868

119879minus 119870)

+

(48)

Theoretically the above PDE is defined on domain 119863 =

(119905 119878 119868) isin [0 119879] times [0infin) times [0infin) But generally closedform solutions cannot be found we have to find numericalsolutions We have no interest in finding the solution in the

entire infinite domain Therefore we consider the domain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119868

lowast)

Now we determine the boundary conditions of (48) If119868 ge 119870119879 that is 119868119879 ge 119870 for the positiveness of 119878] we have119868119879119879 ge 119870 Therefore the final payoff on the option is certain

to be positive At time 119879 this payoff can be expressed as

119868119879

119879minus 119870 =

119868

119879minus 119870 +

1

119879int

119879

119905

119878]119889] (49)

This payoff can also be obtained by using the followingself-financing duplicating portfolio strategy Assume that aninvestor commits (119868119879minus119870)119890

minus119903(119879minus119905) into riskless bonds in orderto secure the return promised by the first part of the rightof (49) that is (119868119879 minus 119870) at time 119879 If the investor is to becertain of accruing the return promised by the second part ofthe right of (49) he is obliged to transfer a certain portionof stock (equal to (1119879)119890

minus119903(119879minus])Δ]) to a riskless bond in every

time interval (] ] + Δ]) Overall this strategy requires a sumequal to (119868119879 minus 119870)119890

minus119903(119879minus119905) plus the portion of a stock whichcan be expressed as follows

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (50)

Then the price of the option when 119868119905ge 119870119879must be equal to

119881 (119905 119878 119868) = (119868

119879minus 119870) 119890

minus119903(119879minus119905)+

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (51)

The above discussion is similar to that in the model with-out volatility uncertainty Equation (51) obviously satisfies(48) So we only need to find the solution of (48) in thedomain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119870119879)

On the boundary 119868 = 119870119879 the boundary condition from(51) can be written as

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (52)

We note that if 119878119905= 0 (13) tells us that 119878(]) is also equal

to zero for ] isin [119905 119879] and 119868(119879) is thus equal to 119868(119905) By (46)we can write the following boundary condition

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

(53)

We need another boundary condition at 119878 = 119878max =

Slowast Hugger [37] gives the boundary condition at 119878 = 119878maxin the model without volatility uncertainty By the samediscussion we can get the same boundary condition inuncertain volatility model

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+Smax119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

(54)

Mathematical Problems in Engineering 7

So the values of the arithmetic average fixed strike Asiancall option satisfy

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

V (119879 119878 119868) = (119868

119879minus 119870)

+

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+

119878max119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

(55)

For the arithmetic average fixed strike Asian put optionwith terminal payoff 119901(119879 119878

119879 119868

119879) = max(119870 minus 119868

119879119879 0) we will

value this option by the following put-call parityIt is easily to see that we have

max(119868119879

119879minus 119870 0) = max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889] minus 119870

(56)

So

119888 (119905 119878119905 119868

119905)

= 119890minus119903(119879minus119905)

E119905[max (119868

119879

119879minus 119870 0)]

= 119890minus119903(119879minus119905)

E119905[max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889]]

minus 119890minus119903(119879minus119905)

119870

= 119901 (119905 119878119905 119868

119905) minus 119890

minus119903(119879minus119905)119870 +

1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889]

+ 119890minus119903(119879minus119905)

119864120590

119905[1

119879int

119879

119905

119878]119889]]

(57)

119890minus119903(119879minus119905)

119864120590

119905[(1119879) int

119879

119905119878]119889]] is the time 119905 value of time 119879

claim (1119879) int119879

119905119878]119889] where the stock has certain volatility

120590 To get claim (1119879) int119879

119905119878]119889] at 119879 let us see the replicating

strategy in any time interval (] ] + Δ]) transfer a certainportion (1119879)119890

minus119903(119879minus])Δ] of stock 119878] to a riskless bond This

strategy ensures the final payoff of (1119879) int119879

119905119878]119889] The stocks

that should be transferred are

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (58)

Then the time 119905 value of (1119879) int119879

119905119878]119889] equals (1119903119879)(1 minus

119890minus119903(119879minus119905)

)119878119905 So the put-call parity can be written as

119888 (119905 119878 119868) + 119870119890minus119903(119879minus119905)

= 119901 (119905 119878 119868) +1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889] +1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

119905

(59)

24 The Arithmetic Average Floating Strike Asian Call OptionFor the arithmetic average floating strike Asian call (put)option the PDE (43) still applies with terminal condition

119881 (119879 119878 119868) = (119878 minus119868

119879)

+

(119881 (119879 119878 119868) = (119868

119879minus 119878)

+

) (60)

In Ingersoll [13] where there is no volatility uncertaintydue to the linear homogeneity of the PDE and its terminalcondition their PDE can be reduced to a one-dimensionalPDE In our uncertain volatility model the PDE (43) is notlinear homogeneous anymore But we can still reduce PDE(43) to a one-dimensional PDE by the following argument

For the arithmetic average floating strike Asian calloption we know

119881 (119905 119878119905 119868

119905) = E

119905[119890

minus119903(119879minus119905)(119878

119879minus

119868119879

119879)

+

]

= E119905[

[

119890minus119903(119879minus119905)

(119878119879minus

119868119905+ int

119879

119905119878]119889]

119879)

+

]

]

(61)

Denote

119872]119905= exp(120583 (] minus 119905) minus

1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905))

(62)

Then by (14) we have 119878] = 119878119905119872

]119905and 119878

119905= 119878]119872

119905

] So

119868119905= int

119905

0

119878]119889] = int

119905

0

119878119905119872

]119905119889] = 119878

119905int

119905

0

119872]119905119889]

119881 (119905 119878119905 119868

119905)

= E119905[

[

119890minus119903(119879minus119905)

(119878119905119872

119879

119905minus

119878119905int119905

0119872

]119905119889] + 119878

119905int119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905E

119905[

[

119890minus119903(119879minus119905)

(119872119879

119905minus

int119905

0119872

]119905119889] + int

119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905119867(119905 119877

119905)

(63)with 119877

119905= 119868

119905119878

119905

Then the function119867(119905 119877) satisfies the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(64)

which is a one-dimensional PDE

8 Mathematical Problems in Engineering

For the European style Asian option we must imposeboundary conditions both at 119877 = 0 and as 119877 rarr infin Theboundary condition as 119877 rarr infin is simple The only way that119877 can tend to infinity is that 119878 tends to zero In this case theoption will not be exercised and so

lim119877rarrinfin

119867(119905 119877) = 0 (65)

For computational purpose we truncate the asset region(0 +infin) into (0 119877max) where 119877max is sufficiently large toensure the accuracy of the solution If 119877 = 0 the PDEdegenerates into the first-order hyperbolic PDE

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 (66)

So119867(119905 119877) satisfies

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (1 minus119877

119879)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = 0

(67)

By the same argument for the arithmetic average floatingstrike Asian put option its prices could be calculated by

119881 (119905 119878 119868) = 119878119867 (119905 119877) (68)

with119867(119905 119877) satisfying the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (119877

119879minus 1)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = (119877max119879

minus exp minus119903 (119879 minus 119905))

(69)

Generally the put-call parity will not hold any more inthe uncertain volatility model Let us see the following argu-ment The mature payoff of a portfolio with one Europeanarithmetic average floating strike Asian call holding long andone put holding short is

119878119879max (1 minus

119877119879

119879 0) minus 119878

119879max (119877

119879

119879minus 1 0)

= 119878119879minus

119877119879119878119879

119879

(70)

The value of this portfolio is equal to one consisting ofone stock and a financial product whose payoff is minus119877

119879119878119879119879

The values of the financial product with final payoff minus119877119879119878119879119879

satisfy the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(71)

with terminal condition 119867(119879 119877119879) = minus119877

119879119878119879119879 We seek a

solution of (71) of the form

119867(119905 119877) = 119886119905+ 119887

119905119877 (72)

with 119886119879

= 0 and 119887119879

= minus1119879 Substituting (72) into (71) andsatisfying the boundary conditions we find that

119886119905= minus

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119887

119905= minus

1

119879119890minus119903(119879minus119905)

(73)

For

119878119879max (1 minus

119877119879

119879 0) = 119878

119879max (119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879

(74)

by the valuation mechanism in space (ΩH E) we take thediscounted 119866-conditional expectation on both sides and get

E119905[119890

minus119903(119879minus119905)119878119879max(1 minus

119877119879

119879 0)]

= 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879]

le 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0)]

+ 119890minus119903(119879minus119905)

E119905[119878

119879minus

119877119879119878119879

119879]

(75)

That is

119888 (119905 119878 119868) le 119901 (119905 119878 119868) + 119878 minus119878

119903119879(1 minus 119890

minus119903(119879minus119905))

minus1

119879119890minus119903(119879minus119905)

int

119905

0

119878120591119889120591

(76)

where 119888(119905 119878 119868) is the time 119905 price of the arithmetic averagefloating strike Asian call option and 119901(119905 119878 119868) is the time119905 price of the arithmetic average floating strike Asian putoption

3 Numerical Computation

In the model without volatility uncertainty Asian optionsare much more difficult to value than regular stock optionsStandard techniques tend to be impractical inaccurate orslow For example traditional binomial lattice methodsrequire such enormous amounts of computermemory for thenecessity of keeping track of every possible path throughout

Mathematical Problems in Engineering 9

the tree that they are effectively unusable Monte Carlosimulation works well for European style options (see [12])but is relatively slow Sun [38] proposed to use the weightedupwind method [39] to price Asian options but he didnot give numerical results The Asian option pricing in theuncertain volatility model will be even more difficult than inthe model without volatility uncertainty

For the arithmetic average fixed strike Asian call optionsdue to the convexity of the value 119881(119905 119878 119868) with respect tovariable 119878 (43) can be written as (48) which are the equationssatisfied by the arithmetic average fixed strike Asian calloptions with certain volatility 120590

Zvan et al [17] point out that the Crank-Nicolson schemein conjunction with the van Leer flux limiter method canbe used to numerically value Asian options with certainvolatilities The van Leer limiter has the property that it istotal variation diminishing and thus produces oscillation freesecond order accurate solutions But the van Leer limiter isnonlinear so the solutions should be obtained by using fullNewton iteration at each time level In this paper we will usethe alternating direction implicit (ADI) methods (see Duffy[40])

The arithmetic average fixed strike Asian put options canbe valued by the put-call parity

For the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69))These PDEs are one-dimensional in space And they are not linear equations butHJB equations in some sense For such PDEs we will use thefully implicit positive coefficient finite different method [41]to solve them We will see that this method can ensure thenumerical solutions converge to the viscosity solutions andis accurate efficient and quick to be implemented

All the calculations were performed byMatlab 2009 on acomputer with 183GHz hard disk and 512MB memory

31 Numerical Computation of Fixed Strike Asian OptionsAn equivalent formulation of (48) in terms of the average119860

119905= (1119905) int

119905

0119878120591119889120591 is given in Barraquand and Pudet [16] in

certain volatility model with volatility 120590 = 120590

120597119905119881 (119905 119878 119860) +

1

2211987821205902119881119878119878

(119905 119878 119860) + 119903119878119881119878(119905 119878 119860)

+119878 minus 119860

119905119881119860(119905 119878 119860) minus 119903119881 (119905 119878 119860) = 0

119881 (119879 119878 119860) = (119860 minus 119870)+

(77)

Equation (77) is convection dominated in the 119860 direc-tion because there is no diffusion effect in this dimensionBarraquand and Pudet [16] find that an explicit centrallyweighted scheme for (77) is unstable In particular theconvective term in the 119860 dimension becomes very large as119905 rarr 0 Barraquand and Pudet also note that implicit cen-trally weighted schemes will generally produce unsatisfactoryresults because of the numerical diffusion introduced by thisfirst order accurate in time scheme

Zvan et al [17] point out that under the conditionof convection dominated solutions generated by using a

centrally weighted scheme on the convective term for (77)cannot be ensured to be free of oscillations And solutionsgenerated by the first order upstream weighting scheme onthe convective termare no longer oscillatory but their profilesare too diffuse To produce oscillation free solutions withoutthe excessive diffusion of first order upstream weightingZvan et al [17] suggest the nonlinear van Leer flux limiter inconjunction with the Crank-Nicolson scheme Since the fluxlimiter is nonlinear the solutions should be obtained by usingfull Newton iteration at each time level

The ADI method pioneered in the United States by Dou-glas Rachford Peaceman Gunn and others has a numberof advantages First explicit difference methods are rarelyused to solve initial boundary value problems because oftheir poor stability Implicit methods have superior stabilityproperties but unfortunately they are difficult to solve in twoand more dimensions Consequently ADI methods becomean alternative because they can be programmed by solvinga simple trigonal system of equations For the convection-dominated problems we could use the exponentially fittedschemes in each leg of the approximate scheme to ensure thestability of the results [40] Since Barraquand and Pudet [16]have used the forward shooting grid method and Zvan et al[17] have used the van Leer flux limiter we will test the ADImethod in this paper by comparing our results with theirs

For convenience let us first convert (55) to be thefollowing forward equation in time by substituting 119905 with120591 = 119879 minus 119905

minus 120597120591119881 (120591 119878 119868) +

1

2211987821205902119881119878119878

(120591 119878 119868) + 119903119878119881119878(120591 119878 119868)

+ 119878119881119868(120591 119878 119868) minus 119903119881 (120591 119878 119868) = 0

119881 (0 119878 119868) = (119868

119879minus 119870)

+

119881 (120591 0 119868) = 119890minus119903120591

(119868

119879minus 119870)

+

119881 (119905 119878max 119868) = max ( 119868

119879minus 119870) 119890

minus119903120591+

119878max119903119879

(1 minus 119890minus119903120591

) 0

119881 (120591 119878 119870119879) =1

119903119879(1 minus 119890

minus119903120591) 119878

(78)

We let 120591119899 119899 = 0 1 119873 be a set of portion points in

[0 119879] satisfying 0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 and use even

partition of time that is 120591119899= 119899Δ120591 119899 = 0 1 119873 Define the

partitions of space variables as 0 = 1198780lt 119878

1lt sdot sdot sdot lt 119878

119872= 119878max

and 0 = 1198680lt 119868

1lt sdot sdot sdot lt 119868

119869= 119868max

With ADI we march from time level 119899 to time level119899 + 12 and then from time level 119899 + 12 to time level 119899 + 1In this case we use exponential fitting in all space variables

10 Mathematical Problems in Engineering

and implicit Euler in time The first leg is given by thescheme

minus

119881119899+12

119894119895minus 119881

119899

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899

119894119895+1minus 119881

119899

119894119895minus1

Δ119868119895+1

+ Δ119868119895

minus 119903119881119899+12

119894119895= 0

(79)

with

120588119894119895

=119903119878

119894Δ119878

119894+1

2coth(

119903119878119894Δ119878

119894+1

12059022119878

2

119894

) coth (119909) =1198902119909

+ 1

1198902119909 minus 1

(80)

Let

120572119894= minus

120588119894119895

Δ119878119894Δ119878

119894+1

120573119894= minus

120588119894119895

(Δ119878119894+1

)2minus

119903119878119894

Δ119878119894+1

120574119894= minus 120572

119894minus 120573

119894+ 119903

(81)

Then the scheme (79) can be written as

120572119894119881

119899+12

119894minus1119895+ (120574

119894+

2

Δ120591)119881

119899+12

119894119895+ 120573

119894119881

119899+12

119894+1119895

=2

Δ120591119881

119899

119894119895+

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895+1minus

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895minus1

(82)

We let

119881119899

119895= [119881

119899

1119895 119881

119899

2119895 119881

119899

119872minus1119895]119879

119881119899+12

119895= [119881

119899+12

1119895 119881

119899+12

2119895 119881

119899+12

119872minus1119895]119879

119891119899+12

119895= [120572

1119881

119899+12

0119895 0 0 120573

119872minus1119881

119899+12

119872119895]119879

119903119899

119895= [(

2

Δ120591119881

119899

1119895+

1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895+1

minus1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895minus1)

(2

Δ120591119881

119899

119872minus1119895+

119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895+1

minus119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895minus1)]

119879

1198601=

[[[

[

1205741

1205731

1205722

1205742

1205732

d d d120572119872minus1

120574119872minus1

]]]

](119872minus1)times(119872minus1)

(83)

Then we can write (82) as the following equivalent matrixform

1198601119881

119899+12

119895+ 119891

119899+12

119895= 119903

119899

119895 (84)

The second leg of the discretized PDE in (78) is given bythe scheme

minus

119881119899+1

119894119895minus 119881

119899+12

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899+1

119894119895+1minus 119881

119899+1

119894119895

Δ119868119895+1

minus 119903119881119899+12

119894119895= 0

(85)

Let

120575119895=

2

Δ120591+

119878119894

Δ119868119895+1

120598119895= minus

119878119894

Δ119868119895+1

(86)

Then the scheme (85) can be written as

120575119895119881

119899+1

119894119895minus 120598

119895119881

119899+1

119894119895+1= minus 120572

119894119881

119899+12

119894minus1119895minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119895

minus 120573119894119881

119899+12

119894+1119895

(87)

Mathematical Problems in Engineering 11

300

300

250

200

200

150

100

100

100

50

50

0

0 0

Call

valu

e

sI

Figure 1 Fixed strike Asian call value when = 08 120590 = 05 119903 =

010 119879 = 1 and 119870 = 100

We let

119881119899+1

119894= [119881

119899+1

1198940 119881

119899+1

1198941 119881

119899+1

119894119869minus1]119879

119881119899+12

119894= [119881

119899+12

1198940 119881

119899+12

1198941 119881

119899+12

119894119869minus1]119879

119865119894= [0 0 0 120598

119869minus1119881

119899+1

119894119869]119879

119869

119877119899+12

119894= [(minus120572

119894119881

119899+12

119894minus10minus(120574

119894minus

2

Δ120591)119881

119899+12

1198940minus120573

119894119881

119899+12

119894+10)

(minus120572119894119881

119899+12

119894minus1119869minus1minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119869minus1minus 120573

119894119881

119899+12

119894+1119869minus1)]

119879

1198602=

[[[

[

1205750

minus1205980

0 1205751

minus1205981

d d minus120598119869minus2

120575119869minus1

]]]

]119869times119869

(88)

The matrix form of (87) is

1198602119881

119899+1

119894+ 119865

119899+1

119894= 119877

119899+12

119894 (89)

Thematrix equations (84) and (89) can be solved by using LUdecomposition

The goal of the computation is to find the fair price of theAsian option at emission that is119881(119878

0 119868

0 0)Weneed tomake

sense of any value 119868 isin [0 119868max] and stock price 119878 isin [0 119878max] attime 119905 = 0 There are no problems with the stock prices butthe integral at time zero can only be 0 for continuous averagesthat is 119868

0= 0

Figure 1 is the results we obtained by using the abovediscretization scheme with = 08 120590 = 05 119903 = 010 119879 = 1and 119870 = 100 The point marked by lowast is the price 119881(0 119878 0)

of the Asian option We choose 119878max = 3119870 to ensure thedesirable accuracy

Table 1 shows the convergence of the results with therefinement of the grid From (78) the values of variable 119868

lie in the interval [0 119870119879] where 119870119879 is the boundary and0 is the point where the continuous averages Asian options

are valued so there is no point that is less important on theinterval [0 119870119879] Hence we use uniform grids in direction 119868

In our uncertain volatility model the pricing PDEs (48)are similar to the PDEs in certain volatility model withvolatility 120590 = 120590 Barraquand and Pudet [16] and Zvan etal [17] give numerical results of Asian options with certainvolatilities In the uncertain volatility model when =

08 120590 = 05 the Asian option PDE is the same as that ofthe Asian option with certain volatility 120590 = 04 When theparameters 119903 = 010 119879 = 1 119870 = 95 and grid Δ119868 =

011875 ΔS = 285 the price of the Asian call option wecalculate is 1382 The results with the same parameters ofBarraquand and Pudet [16] and Zvan et al [17] are 13825 and13721 respectively

Table 2 is the comparison of our results with those ofZvan et al [17] Z F and V in Table 2 refer to the resultof Zvan et al [17] Our results are denoted by Asian fixedstrike which calculated with Δ120591 = 001 002 004 Δ119868 =

0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Since119868max = 119870119879 the partitions in 119868 direction are related with thematurity 119879 The execution time is much less than theirs Thegrid spacing was chosen to achieve an accuracy of at least010 of 119878 Our results are close to that of Zvan et al [17] andthe difference is less than 010 of 119878

Our calculation procedure is different from Zvan et al[17] as well as Barraquand and Pudet [16] in the followingfour aspects First our pricing PDE (78) is in terms of therunning sum 119868

119905 Zvan et al [17] and Barraquand and Pudet

[16] use the pricing PDE

minus 120597120591119881 (120591 119878 119860) +

1

211987821205902119881119878119878

(120591 119878 119860) + 119903119878119881119878(120591 119878 119860)

+119878 minus 119860

119905119881119860(120591 119878 119860) minus 119903119881 (120591 119878 119860) = 0

(90)

where 119860119905is defined as 119860

119905= 119868

119905119905 But PDE (78) and (90) are

equivalent for pricing fixed strike European Asian optionsSecond since the PDEs are different PDE (78) and (90) musthave different boundary conditions We do not know theboundary conditions used by Zvan et al [17] The forwardshooting grid method used by Barraquand and Pudet [16]proceeds without any boundary conditions Third our PDE(78) is defined on the domain 119863 = (119905 119878 119868) 0 le 119905 le 119879 0 le

119878 le 119878max 0 le 119868 le 119870119879 and 119868max = 119870119879 is the boundary and119868 = 0 is the point where the options are valued So there isno point that is less important on the interval [0 119870119879] Henceour numerical schemes use uniform grid spacing while Zvanet al [17] use nonuniform grid spacing of 41 times 45 on domain[0 300] times [0 300] To achieve an accuracy of at least 01 of119878 our grids are much finer than Zvan et al [17]

32 Numerical Computation of Floating Strike Asian OptionsTo value the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69)) for 119867(119905 119877) andthe values of the options will be 119881(119905 119878 119868) = 119878119867(119905 119877) with119877 = 119868119878 Actually what we really care about is the optionvalues at time 119905 = 0 while from the definition of 119868

119905 it is

obvious that 1198680= 0 and therefore 119877

0= 0 so our goal of the

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

2 Mathematical Problems in Engineering

In the jump diffusion models methods for pricing Asianoptions are still under development Fusai [28] obtains asimple expression for the double transform of the prices ofcontinuously monitored Asian options by means of Fourierand Laplace transform DrsquoHalluin et al [29] propose a semi-Lagrangian method to price the continuously observed fixedstrike Asian options Cai and Kou [30] consider the pricingof Asian options for the double exponential jump diffusionmodel Bayraktar and Xing [31] obtain a fast numericalapproximation scheme and analyze the performance of thenumerical algorithm

In this paper we use the stochastic processes driven by119866-Brownian motion to describe the stocksrsquo price processes119866-Brownian motion is introduced in Peng [32ndash34] and ithas the properties that its increments are zero-mean inde-pendent and stationary and can be proved to be 119866-normallydistributed The 119866-normal distribution random variableshave no mean uncertainty but can have variance uncertaintyThe variance uncertainty of the 119866-normal distribution isthe source of the volatility uncertainty of the stocksrsquo priceprocesses

The paper is organized as follows In Section 2 we usethe nonlinear Feynman-Kac formula in the 119866-expectationtheory to get the two-dimensional nonlinear PDEs satisfiedby the values of the Asian options For the arithmeticaverage fixed strike Asian options we prove the convexityof the optionsrsquo values with respect to the stock prices andtransfer the nonlinear PDEs to linear PDEswith the uncertainvolatilities being replaced by the maximum volatilities Forthe arithmetic average floating strike Asian options weuse a dimension reduction technique to transfer the two-dimensional nonlinear PDEs to one-dimensional nonlinearPDEs In Section 3 we introduce the applicable numericalcomputation methods for these two classes of PDEs and givethe numerical results

2 The Pricing of Asian Options in UncertainVolatility Model

21 119866-Brownian Motion and 119866-Expectation In this subsec-tion we will introduce the 119866-Brownian motion and 119866-expectation defined in Peng [33ndash36]

Let Ω be a given set and let H be a linear space of realvalued functions defined on Ω We suppose that H satisfies119888 isin H for each constant 119888 and |119883| isin H if 119883 isin H The spaceH can be considered as the space of random variables

A sublinear expectation E onH is a functional E H rarr

R satisfying the following properties for all 119883119884 isin H wehave

(a) Monotonicity If 119883 ge 119884 then E[119883] ge E[119884]

(b) Constant preserving E[119888] = 119888 for 119888 isin R

(c) Sub-additivity E[119883 + 119884] le E[119883] + E[119884]

(d) Positive homogeneity E[120582119883] = 120582E[119883] forall120582 ge 0

The triple (ΩHE) is called a sublinear expectationspace

Definition 1 A 119889-dimensional process (119861119905)119905ge0

on a sublinearexpectation space (ΩHE) is called a119866-Brownianmotion ifthe following properties are satisfied

(i) 1198610(120596) = 0

(ii) For each 119905 119904 ge 0 the increment 119861119905+119904

minus 119861119905is

119873(0 times Σ)-distributed and is independent from(119861

1199051

1198611199052

119861119905119899

) for each 119899 isin 119873 and 0 le 1199051le sdot sdot sdot le

119905119899le 119905

Remark 2 Just as in the classical situation the increments of119866-Brownian motion (119861

119905+119904minus 119861

119905)119904ge0

are independent fromΩ119905

Let 120587119873

119905= 119905

119873

0 119905

119873

1 119905

119873

119873 such that 0 = 119905

119873

0lt 119905

119873

1lt sdot sdot sdot lt

119905119873

119873= 119905 for 119873 = 1 2 be a sequence of partitions of [0 119905]

with

120583 (120587119873

119905) = max 1003816100381610038161003816119905119894+1 minus 119905

119894

1003816100381610038161003816 119894 = 0 1 119873 minus 1 (1)

Thequadratic variation process of 1-dimensional119866-Brownianmotion (119861

119905)119905ge0

with 1198611

119889

== 119873(0 times [1205902 120590

2]) is defined as

⟨119861⟩119905= lim

120583(120587119873

119905)rarr0

119873minus1

sum

119895=0

(119861119905119873

119895+1

minus 119861119905119873

119895

)2

= 1198612

119905minus 2int

119905

0

119861119904119889119861

119904 (2)

(⟨119861⟩119905)119905ge0

is an increasing process with ⟨119861⟩0= 0

LetΩ = 119862119889

0(R+

) be the space of allR119889-valued continuouspaths (120596

119905)119905isinR+ with 120596

0= 0 equipped with the distance

120588 (1205961 120596

2) =

infin

sum

119894=1

2minus119894

[(max119905isin[0119894]

10038161003816100381610038161003816120596

1

119905minus 120596

2

119905

10038161003816100381610038161003816) and 1] (3)

For each fixed 119879 isin [0infin) we setΩ119879= 120596

sdotand119879 120596 isin Ω

From Peng [36] the distribution of the quadratic vari-ation process of the 119866-Brownian motion has the followingproperties

Lemma 3 For each fixed 119904 119905 ge 0 ⟨119861⟩119904+119905

minus ⟨119861⟩119904is identically

distributed with ⟨119861⟩119905and independent from Ω

119904

Theorem 4 ⟨119861⟩119905is119873([120590

2119905 120590

2119905] times 0)-distributed that is for

each 120593 isin 119862119897119871119894119901

(R)

E [120593 (⟨119861⟩119905)] = sup

1205902leVle1205902

120593 (V119905) (4)

The sublinear expectation E in the above theorem is called119866-expectation defined on 119871

119894119901(Ω) via the following procedure

Let

119871119894119901(Ω

119879) = 120601 (119861

1199051and119879

119861119905119899and119879

) 119899 isin N

1199051 119905

119899isin [0infin) 120601 isin 119862

119897119871119894119901(R

119889times119899)

(5)

with 119861119905= 120596

119905 119905 isin [0infin) for 120596 isin Ω and

119871119894119901(Ω) =

infin

⋃

119899=1

119871119894119901(Ω

119899) (6)

Mathematical Problems in Engineering 3

Firstly construct a sequence of 119889-dimensional randomvectors (120585)infin

119894=1on a sublinear expectation space (Ω H E) such

that 120585119894is 119866-normal distributed and 120585

119894+1is independent from

(1205851 120585

119894) for each 119894 = 1 2 For each119883 isin 119871

119894119901(Ω) with

119883 = 120601 (1198611199051

minus 1198611199050

1198611199052

minus 1198611199051

119861119905119899

minus 119861119905119899minus1

) (7)

some 120601 isin 119862119897119871119894119901

(119877119889times119899

) and 0 = 1199050lt 119905

1lt sdot sdot sdot lt 119905

119899lt infin set

E [120601 (1198611199051

minus 1198611199050

1198611199052

minus 1198611199051

119861119905119899

minus 119861119905119899minus1

)]

= E [120601 (radic1199051minus 119905

01205851 radic119905

119899minus 119905

119899minus1120585119899)]

(8)

And the related conditional expectation of

119883 = 120601 (1198611199051

1198611199052

minus 1198611199051

119861119905119899

minus 119861119905119899minus1

) (9)

underΩ119905119895

is defined by

E [119883 | Ω119905119895

] = 120595 (1198611199051

119861119905119895

minus 119861119905119895minus1

) (10)

where

120595 (1199091 119909

119895)

= E [120601 (1199091 119909

119895 radic119905

119895+1minus 119905

119895120585119895+1

radic119905119899minus 119905

119899minus1120585119899)]

(11)

E[sdot] consistently defines a sublinear expectation on 119871119894119901(Ω)

and (119861119905)119905ge0

is a 119866-Brownian motionWe let 119871119901

119866(Ω

119905119896

) 119901 ge 1 denote the completion of

119871119894119901(Ω

119905119896

) = 120593 (1198611199051and119879

119861119905119899and119879

) 119899 isin N

1199051 119905

119899isin [0infin) 120593 isin 119862

119897119871119894119901(R

119889times119899)

(12)

under the norm 119883119901= (E[|119883|

119901])

1119901

22 The PDEs We assume the security market is definedon a sublinear expectation space (ΩH E) which openscontinuously and offers a constant riskless interest rate 119903 toall borrowers and lenders And we also assume there are notransaction costs andor taxes Suppose that the price process(119878

119905)119905ge0

of some risky asset is given by

119889119878] = 120583119878]119889] + 119878]119889119861] 1198780= 119878 (13)

where 119861 is the 119866-Brownian motion with 1205902= minusE[minus1198612

1] and

1205902

= E[1198612

1] and 120583 and are constants Hence (119878])]ge0 is a

geometric 119866-Brownian motion

119878] = 119878119905exp (120583 (] minus 119905)

minus1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905))

(14)

Define

119868] = int

]

0

119878120591119889120591 (15)

Then 119868] satisfies the following stochastic differential equation

119889119868] = 119878]119889] 1198680= 0 (16)

Let119883] = (119878] 119868])119879 so

119889119883] = 120583119883119883]119889] +

119883119883]119889119861] 119883

0= 119909 = (119878 0) (17)

where 120583119883 and

119883 are 2 times 2matrixes

120583119883

= (120583 0

1 0)

119883= (

0

0 0) (18)

The typical Asian options in this security market are thefollowing four types

the arithmetic average fixed strike call option withfinal payoff

max (119868119879

119879minus 119870 0) (19)

the corresponding arithmetic average fixed strike putoption with payoff

max (119870 minus119868119879

119879 0) (20)

the arithmetic average floating strike call option withfinal payoff

max(119878119879minus

119868119879

119879 0) (21)

and the corresponding arithmetic average floatingstrike put option with payoff

max(119868119879

119879minus 119878

119879 0) (22)

Let 119881(119905 119878119905 119868

119905) denote the price of one of the options at

time 119905 Take the arithmetic average fixed strike call option forexample we know 119881(119879 119878

119879 119868

119879) = max(119868

119879119879 minus 119870 0)

Here we still assume the market operates in a risk neutralworld [7]

Peng [35] introduces the axiomatic conditions satisfied bythe valuation mechanism In the sublinear expectation space(ΩH E) E

119905[sdot] is a filtration consistent valuation operator

By this valuation mechanism we have

119881 (119905 119878119905 119868

119905) = E [119890

minus119903(119879minus119905)119881 (119879 119878

119879 119868

119879) | Ω

119905] (23)

Here we assume the riskless interest rate is 119903Since we assume the market is risk neutral we have 120583 =

119903 It is easy to verify that the discounted prices of the stocksatisfy the valuation mechanism E

119905[sdot]

119905ge0 that is

119878119905= E

119905[119890

minus119903(]minus119905)119878]] 0 le 119905 le ] le 119879 (24)

4 Mathematical Problems in Engineering

Let us assume

119889119883119905119909

] = 120583119883119883

119905119909

] 119889] + 119883119883

119905119909

] 119889119861] 119883119905= 119909 = (119878 119868) (25)

with

120583119883

= (119903 0

1 0)

119883= (

0

0 0) (26)

The following result is a corollary of the nonlinearFeynman-Kac formula of Peng [36]

Corollary 5 Let

119884119905119909

119904= E [119890

minus119903(119879minus119904)Φ(119883

119905119909

119879)

+ int

119879

119904

119890minus119903(]minus119904)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119879

119904

119890minus119903(]minus119904)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩] | Ω119904]

(27)

where Φ R2rarr R is a given Lipschitz function and 119891 119892

R2times R rarr R are given Lipschitz functions that is |120601(119909 119910) minus

120601(1199091015840 119910

1015840)| le 119870(|119909 minus 119909

1015840| + |119910 minus 119910

1015840|) for each 119909 119909

1015840isin R2 119910 1199101015840

isin

R 120601 = 119891 and 119892 And denote

119906 (119905 119909) = 119884119905119909

119905 (28)

Then

119906 (119905 119909) = E[119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575)

+ int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]]

(29)

is a viscosity solution of the following PDE

120597119905119906 + sup

1205902le1205902le1205902

(1

2⟨119863

2119906 (119905 119909)

119883119909

119883119909⟩ + 119892 (119909 119906)) 120590

2

+ ⟨119863119906 (119905 119909) 120583119883119909⟩ minus 119903119906 + 119891 (119909 119906) = 0

119906 (119879 119909) = Φ (119909)

(30)

Proof The proof of expression (29) is the same as that ofPeng [36] By the proof of Pengrsquos theorem of the nonlinearFeynman-Kac formula and the boundedness of 119890minus119903(119879minus119905) wecan easily prove that 119906 is a Lipschitz function that is

10038161003816100381610038161003816119906 (119905 119909) minus 119906 (119905 119909

1015840)10038161003816100381610038161003816le 119862

10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816 (31)

E [10038161003816100381610038161003816119884

119905119909

119904

10038161003816100381610038161003816] le 119862 (1 + |119909|) (32)

Let us see the continuity of 119906with respect to 119905 Firstly we have

|119906 (119905 119909) minus 119906 (119905 + 120575 119909)|

le E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

+ E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

(33)

Replace 119890minus119903120575 with its Taylorrsquos expansion in the first term on

the right side of (33) we have

E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

= E [10038161003816100381610038161003816(1 minus 119903120575 + (119903120575)

2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)

minus 119906 (119905 + 120575 119909)10038161003816100381610038161003816]

= E [10038161003816100381610038161003816119906 (119905 + 120575119883

119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

+ (minus119903120575 + (119903120575)2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)10038161003816100381610038161003816]

le E [10038161003816100381610038161003816119883

119905119909

119905+120575minus 119909

10038161003816100381610038161003816]

+ E [10038161003816100381610038161003816(minus119903120575 + (119903120575)

2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)10038161003816100381610038161003816]

(34)

By the following formulas [36]

E [10038161003816100381610038161003816119883

119905119909

119905+120575minus 119909

10038161003816100381610038161003816

2

] le 119862 (1 + 1199092) 120575

E [10038161003816100381610038161003816119883

119905119909

119905+120575

10038161003816100381610038161003816

2

] le 119862 (1 + 1199092)

(35)

and (32) we have

E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

le 1198621015840(1 + |119909|

2)12

(12057512

+ 120575 +1205752

2sdot sdot sdot )

(36)

Mathematical Problems in Engineering 5

For the second term on the right side of (33) we have

E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

= E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

times (119891 (119883119905119909

] 119884119905119909

] ) minus 119891 (119909 119884119905119909

119905)) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

times (119892 (119883119905119909

] 119884119905119909

] ) minus 119892 (119909 119884119905119909

119905)) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

+ E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119909 119884119905119909

119905) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119909 119884119905119909

119905) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

le 119862 (1 + |119909|) 120575

(37)

Therefore 119906 is continuous in 119905Now we are going to prove that 119906(119905 119909) is a viscosity

solution of PDE (30) For fixed (119905 119909) isin (0 119879) times R2 let 120595 isin

11986223

119887((0 119879) times R2

) such that 120595 ge 119906 and 120595(119905 119909) = 119906(119905 119909) By(29) and Taylorrsquos expansion for 120575 isin (0 119879 minus 119905)

0 le E[119890minus119903120575

120595 (119905 + 120575119883119905119909

119905+120575) minus 120595 (119905 119909)

+ int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]]

le E [ minus 119903120595 (119905 119909) 120575 + 120597119905120595 (119905 119909) 120575

+ ⟨119863120595 (119905 119909) 120583119883119909⟩ 120575

+ ⟨1

2119863

2120595 (119905 119909)

119883119909

119883119909⟩ (⟨119861⟩

119905+120575minus ⟨119861⟩

119905)

+ 119891 (119909 120595) 120575 + 119892 (119909 120595) (⟨119861⟩119905+120575

minus ⟨119861⟩119905)]

+ 119862 (1 + |119909| + |119909|2+ |119909|

3) 120575

32

le (120597119905120595 + sup

1205902le1205902le1205902

(1

2⟨119863

2120595 (119905 119909)

119883119909

119883119909⟩

+119892 (119909 120595) ) 1205902+ ⟨119863120595 (119905 119909) 120583

119883119909⟩

minus119903120595 (119905 119909) + 119891 (119909 120595))120575

+ 119862 (1 + |119909| + |119909|2+ |119909|

3) 120575

32

(38)

It is easy to check that

120597119905120595 + sup

1205902le1205902le1205902

(1

2⟨119863

2120595 (119905 119909)

119883119909

119883119909⟩ + 119892 (119909 120595)) 120590

2

+ ⟨119863120595 (119905 119909) 120583119883119909⟩ minus 119903120595 (119905 119909) + 119891 (119909 120595) ge 0

(39)

Thus 119906 is a viscosity subsolution of (30) Similarly we canprove that 119906 is a viscosity supersolution of (30)

For the arithmetic average Asian options we have

119881 (119905 119909) = 119881 (119905 119878 119868) = E [119890minus119903(119879minus119905)

119881 (119879 119878119879 119868

119879)] (40)

Then by Corollary 5 119881(119905 119909) satisfies the following PDE

120597119905119881 (119905 119909) + sup

1205902le1205902le1205902

1

2⟨119863

2119881 (119905 119909)

119883119909

119883119909⟩120590

2

+ ⟨119863119881 (119905 119909) 120583119883119909⟩ minus 119903119881 (119905 119909) = 0

119881 (119879 119909) = 120593 (119879 119909)

(41)

By (18) the above PDE is

120597119905119881 (119905 119878 119868) + sup

1205902le1205902le1205902

1

2211987821205902119881119878119878

(119905 119878 119868)

+ 119903119878119881119878(119905 119878 119868) + 119878119881

119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

119881 (119879 119878 119868) = 120593 (119879 119878 119868)

(42)

where120593 (119879 119878 119868)

=

max ( 119868

119879minus 119870 0) for the arithmetic average

fixed strike call option

max (119870 minus119868

119879 0) for the arithmetic average

fixed strike put option

max (119878 minus119868

119879 0) for the arithmetic average

floating strike call option

max ( 119868

119879minus 119878 0) for the arithmetic average

floating strike put option(43)

6 Mathematical Problems in Engineering

23 The Arithmetic Average Fixed Strike Asian Call OptionLet us see the arithmetic average fixed strikeAsian call optionFrom (14) we have

119878119905119878

] = 119878 exp 120583 (] minus 119905) minus1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905)

(44)

Then by the definition of 119868] we have

119868119905119868

] = 119868 + 119878int

]

119905

exp 120583 (120591 minus 119905) minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591

(45)

Since

119881 (119905 119909) = 119881 (119905 119878 119868) = E [119890minus119903(119879minus119905)

119881 (119879 119878119879 119868

119879)]

= E [119890minus119903(119879minus119905)

(119868119879

119879minus 119870)

+

]

119881 (119905 119878 119868) = E[119890minus119903(119879minus119905)

times (((119868 + 119878int

]

119905

exp 120583 (120591 minus 119905)

minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591)

times119879minus1) minus 119870)

+

]

(46)

By the convexities of function 119891(119909) = (119909 minus 119896)+ and the

sublinear expectation we can say

119881(119905 1205821198781+ (1 minus 120582) 119878

2 119868) le 120582119881 (119905 119878

1 119868) + (1 minus 120582)119881 (119905 119878

2 119868)

(47)

that is 119881(119905 119878 119868) is convex about 119878 So in the numericalcomputation the second order difference of 119881 is positiveThen the numerical solutions of (43) are equivalent to thoseof PDE

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

119881 (119879 119878 119868) = (119868

119879minus 119870)

+

(48)

Theoretically the above PDE is defined on domain 119863 =

(119905 119878 119868) isin [0 119879] times [0infin) times [0infin) But generally closedform solutions cannot be found we have to find numericalsolutions We have no interest in finding the solution in the

entire infinite domain Therefore we consider the domain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119868

lowast)

Now we determine the boundary conditions of (48) If119868 ge 119870119879 that is 119868119879 ge 119870 for the positiveness of 119878] we have119868119879119879 ge 119870 Therefore the final payoff on the option is certain

to be positive At time 119879 this payoff can be expressed as

119868119879

119879minus 119870 =

119868

119879minus 119870 +

1

119879int

119879

119905

119878]119889] (49)

This payoff can also be obtained by using the followingself-financing duplicating portfolio strategy Assume that aninvestor commits (119868119879minus119870)119890

minus119903(119879minus119905) into riskless bonds in orderto secure the return promised by the first part of the rightof (49) that is (119868119879 minus 119870) at time 119879 If the investor is to becertain of accruing the return promised by the second part ofthe right of (49) he is obliged to transfer a certain portionof stock (equal to (1119879)119890

minus119903(119879minus])Δ]) to a riskless bond in every

time interval (] ] + Δ]) Overall this strategy requires a sumequal to (119868119879 minus 119870)119890

minus119903(119879minus119905) plus the portion of a stock whichcan be expressed as follows

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (50)

Then the price of the option when 119868119905ge 119870119879must be equal to

119881 (119905 119878 119868) = (119868

119879minus 119870) 119890

minus119903(119879minus119905)+

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (51)

The above discussion is similar to that in the model with-out volatility uncertainty Equation (51) obviously satisfies(48) So we only need to find the solution of (48) in thedomain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119870119879)

On the boundary 119868 = 119870119879 the boundary condition from(51) can be written as

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (52)

We note that if 119878119905= 0 (13) tells us that 119878(]) is also equal

to zero for ] isin [119905 119879] and 119868(119879) is thus equal to 119868(119905) By (46)we can write the following boundary condition

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

(53)

We need another boundary condition at 119878 = 119878max =

Slowast Hugger [37] gives the boundary condition at 119878 = 119878maxin the model without volatility uncertainty By the samediscussion we can get the same boundary condition inuncertain volatility model

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+Smax119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

(54)

Mathematical Problems in Engineering 7

So the values of the arithmetic average fixed strike Asiancall option satisfy

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

V (119879 119878 119868) = (119868

119879minus 119870)

+

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+

119878max119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

(55)

For the arithmetic average fixed strike Asian put optionwith terminal payoff 119901(119879 119878

119879 119868

119879) = max(119870 minus 119868

119879119879 0) we will

value this option by the following put-call parityIt is easily to see that we have

max(119868119879

119879minus 119870 0) = max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889] minus 119870

(56)

So

119888 (119905 119878119905 119868

119905)

= 119890minus119903(119879minus119905)

E119905[max (119868

119879

119879minus 119870 0)]

= 119890minus119903(119879minus119905)

E119905[max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889]]

minus 119890minus119903(119879minus119905)

119870

= 119901 (119905 119878119905 119868

119905) minus 119890

minus119903(119879minus119905)119870 +

1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889]

+ 119890minus119903(119879minus119905)

119864120590

119905[1

119879int

119879

119905

119878]119889]]

(57)

119890minus119903(119879minus119905)

119864120590

119905[(1119879) int

119879

119905119878]119889]] is the time 119905 value of time 119879

claim (1119879) int119879

119905119878]119889] where the stock has certain volatility

120590 To get claim (1119879) int119879

119905119878]119889] at 119879 let us see the replicating

strategy in any time interval (] ] + Δ]) transfer a certainportion (1119879)119890

minus119903(119879minus])Δ] of stock 119878] to a riskless bond This

strategy ensures the final payoff of (1119879) int119879

119905119878]119889] The stocks

that should be transferred are

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (58)

Then the time 119905 value of (1119879) int119879

119905119878]119889] equals (1119903119879)(1 minus

119890minus119903(119879minus119905)

)119878119905 So the put-call parity can be written as

119888 (119905 119878 119868) + 119870119890minus119903(119879minus119905)

= 119901 (119905 119878 119868) +1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889] +1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

119905

(59)

24 The Arithmetic Average Floating Strike Asian Call OptionFor the arithmetic average floating strike Asian call (put)option the PDE (43) still applies with terminal condition

119881 (119879 119878 119868) = (119878 minus119868

119879)

+

(119881 (119879 119878 119868) = (119868

119879minus 119878)

+

) (60)

In Ingersoll [13] where there is no volatility uncertaintydue to the linear homogeneity of the PDE and its terminalcondition their PDE can be reduced to a one-dimensionalPDE In our uncertain volatility model the PDE (43) is notlinear homogeneous anymore But we can still reduce PDE(43) to a one-dimensional PDE by the following argument

For the arithmetic average floating strike Asian calloption we know

119881 (119905 119878119905 119868

119905) = E

119905[119890

minus119903(119879minus119905)(119878

119879minus

119868119879

119879)

+

]

= E119905[

[

119890minus119903(119879minus119905)

(119878119879minus

119868119905+ int

119879

119905119878]119889]

119879)

+

]

]

(61)

Denote

119872]119905= exp(120583 (] minus 119905) minus

1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905))

(62)

Then by (14) we have 119878] = 119878119905119872

]119905and 119878

119905= 119878]119872

119905

] So

119868119905= int

119905

0

119878]119889] = int

119905

0

119878119905119872

]119905119889] = 119878

119905int

119905

0

119872]119905119889]

119881 (119905 119878119905 119868

119905)

= E119905[

[

119890minus119903(119879minus119905)

(119878119905119872

119879

119905minus

119878119905int119905

0119872

]119905119889] + 119878

119905int119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905E

119905[

[

119890minus119903(119879minus119905)

(119872119879

119905minus

int119905

0119872

]119905119889] + int

119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905119867(119905 119877

119905)

(63)with 119877

119905= 119868

119905119878

119905

Then the function119867(119905 119877) satisfies the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(64)

which is a one-dimensional PDE

8 Mathematical Problems in Engineering

For the European style Asian option we must imposeboundary conditions both at 119877 = 0 and as 119877 rarr infin Theboundary condition as 119877 rarr infin is simple The only way that119877 can tend to infinity is that 119878 tends to zero In this case theoption will not be exercised and so

lim119877rarrinfin

119867(119905 119877) = 0 (65)

For computational purpose we truncate the asset region(0 +infin) into (0 119877max) where 119877max is sufficiently large toensure the accuracy of the solution If 119877 = 0 the PDEdegenerates into the first-order hyperbolic PDE

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 (66)

So119867(119905 119877) satisfies

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (1 minus119877

119879)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = 0

(67)

By the same argument for the arithmetic average floatingstrike Asian put option its prices could be calculated by

119881 (119905 119878 119868) = 119878119867 (119905 119877) (68)

with119867(119905 119877) satisfying the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (119877

119879minus 1)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = (119877max119879

minus exp minus119903 (119879 minus 119905))

(69)

Generally the put-call parity will not hold any more inthe uncertain volatility model Let us see the following argu-ment The mature payoff of a portfolio with one Europeanarithmetic average floating strike Asian call holding long andone put holding short is

119878119879max (1 minus

119877119879

119879 0) minus 119878

119879max (119877

119879

119879minus 1 0)

= 119878119879minus

119877119879119878119879

119879

(70)

The value of this portfolio is equal to one consisting ofone stock and a financial product whose payoff is minus119877

119879119878119879119879

The values of the financial product with final payoff minus119877119879119878119879119879

satisfy the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(71)

with terminal condition 119867(119879 119877119879) = minus119877

119879119878119879119879 We seek a

solution of (71) of the form

119867(119905 119877) = 119886119905+ 119887

119905119877 (72)

with 119886119879

= 0 and 119887119879

= minus1119879 Substituting (72) into (71) andsatisfying the boundary conditions we find that

119886119905= minus

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119887

119905= minus

1

119879119890minus119903(119879minus119905)

(73)

For

119878119879max (1 minus

119877119879

119879 0) = 119878

119879max (119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879

(74)

by the valuation mechanism in space (ΩH E) we take thediscounted 119866-conditional expectation on both sides and get

E119905[119890

minus119903(119879minus119905)119878119879max(1 minus

119877119879

119879 0)]

= 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879]

le 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0)]

+ 119890minus119903(119879minus119905)

E119905[119878

119879minus

119877119879119878119879

119879]

(75)

That is

119888 (119905 119878 119868) le 119901 (119905 119878 119868) + 119878 minus119878

119903119879(1 minus 119890

minus119903(119879minus119905))

minus1

119879119890minus119903(119879minus119905)

int

119905

0

119878120591119889120591

(76)

where 119888(119905 119878 119868) is the time 119905 price of the arithmetic averagefloating strike Asian call option and 119901(119905 119878 119868) is the time119905 price of the arithmetic average floating strike Asian putoption

3 Numerical Computation

In the model without volatility uncertainty Asian optionsare much more difficult to value than regular stock optionsStandard techniques tend to be impractical inaccurate orslow For example traditional binomial lattice methodsrequire such enormous amounts of computermemory for thenecessity of keeping track of every possible path throughout

Mathematical Problems in Engineering 9

the tree that they are effectively unusable Monte Carlosimulation works well for European style options (see [12])but is relatively slow Sun [38] proposed to use the weightedupwind method [39] to price Asian options but he didnot give numerical results The Asian option pricing in theuncertain volatility model will be even more difficult than inthe model without volatility uncertainty

For the arithmetic average fixed strike Asian call optionsdue to the convexity of the value 119881(119905 119878 119868) with respect tovariable 119878 (43) can be written as (48) which are the equationssatisfied by the arithmetic average fixed strike Asian calloptions with certain volatility 120590

Zvan et al [17] point out that the Crank-Nicolson schemein conjunction with the van Leer flux limiter method canbe used to numerically value Asian options with certainvolatilities The van Leer limiter has the property that it istotal variation diminishing and thus produces oscillation freesecond order accurate solutions But the van Leer limiter isnonlinear so the solutions should be obtained by using fullNewton iteration at each time level In this paper we will usethe alternating direction implicit (ADI) methods (see Duffy[40])

The arithmetic average fixed strike Asian put options canbe valued by the put-call parity

For the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69))These PDEs are one-dimensional in space And they are not linear equations butHJB equations in some sense For such PDEs we will use thefully implicit positive coefficient finite different method [41]to solve them We will see that this method can ensure thenumerical solutions converge to the viscosity solutions andis accurate efficient and quick to be implemented

All the calculations were performed byMatlab 2009 on acomputer with 183GHz hard disk and 512MB memory

31 Numerical Computation of Fixed Strike Asian OptionsAn equivalent formulation of (48) in terms of the average119860

119905= (1119905) int

119905

0119878120591119889120591 is given in Barraquand and Pudet [16] in

certain volatility model with volatility 120590 = 120590

120597119905119881 (119905 119878 119860) +

1

2211987821205902119881119878119878

(119905 119878 119860) + 119903119878119881119878(119905 119878 119860)

+119878 minus 119860

119905119881119860(119905 119878 119860) minus 119903119881 (119905 119878 119860) = 0

119881 (119879 119878 119860) = (119860 minus 119870)+

(77)

Equation (77) is convection dominated in the 119860 direc-tion because there is no diffusion effect in this dimensionBarraquand and Pudet [16] find that an explicit centrallyweighted scheme for (77) is unstable In particular theconvective term in the 119860 dimension becomes very large as119905 rarr 0 Barraquand and Pudet also note that implicit cen-trally weighted schemes will generally produce unsatisfactoryresults because of the numerical diffusion introduced by thisfirst order accurate in time scheme

Zvan et al [17] point out that under the conditionof convection dominated solutions generated by using a

centrally weighted scheme on the convective term for (77)cannot be ensured to be free of oscillations And solutionsgenerated by the first order upstream weighting scheme onthe convective termare no longer oscillatory but their profilesare too diffuse To produce oscillation free solutions withoutthe excessive diffusion of first order upstream weightingZvan et al [17] suggest the nonlinear van Leer flux limiter inconjunction with the Crank-Nicolson scheme Since the fluxlimiter is nonlinear the solutions should be obtained by usingfull Newton iteration at each time level

The ADI method pioneered in the United States by Dou-glas Rachford Peaceman Gunn and others has a numberof advantages First explicit difference methods are rarelyused to solve initial boundary value problems because oftheir poor stability Implicit methods have superior stabilityproperties but unfortunately they are difficult to solve in twoand more dimensions Consequently ADI methods becomean alternative because they can be programmed by solvinga simple trigonal system of equations For the convection-dominated problems we could use the exponentially fittedschemes in each leg of the approximate scheme to ensure thestability of the results [40] Since Barraquand and Pudet [16]have used the forward shooting grid method and Zvan et al[17] have used the van Leer flux limiter we will test the ADImethod in this paper by comparing our results with theirs

For convenience let us first convert (55) to be thefollowing forward equation in time by substituting 119905 with120591 = 119879 minus 119905

minus 120597120591119881 (120591 119878 119868) +

1

2211987821205902119881119878119878

(120591 119878 119868) + 119903119878119881119878(120591 119878 119868)

+ 119878119881119868(120591 119878 119868) minus 119903119881 (120591 119878 119868) = 0

119881 (0 119878 119868) = (119868

119879minus 119870)

+

119881 (120591 0 119868) = 119890minus119903120591

(119868

119879minus 119870)

+

119881 (119905 119878max 119868) = max ( 119868

119879minus 119870) 119890

minus119903120591+

119878max119903119879

(1 minus 119890minus119903120591

) 0

119881 (120591 119878 119870119879) =1

119903119879(1 minus 119890

minus119903120591) 119878

(78)

We let 120591119899 119899 = 0 1 119873 be a set of portion points in

[0 119879] satisfying 0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 and use even

partition of time that is 120591119899= 119899Δ120591 119899 = 0 1 119873 Define the

partitions of space variables as 0 = 1198780lt 119878

1lt sdot sdot sdot lt 119878

119872= 119878max

and 0 = 1198680lt 119868

1lt sdot sdot sdot lt 119868

119869= 119868max

With ADI we march from time level 119899 to time level119899 + 12 and then from time level 119899 + 12 to time level 119899 + 1In this case we use exponential fitting in all space variables

10 Mathematical Problems in Engineering

and implicit Euler in time The first leg is given by thescheme

minus

119881119899+12

119894119895minus 119881

119899

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899

119894119895+1minus 119881

119899

119894119895minus1

Δ119868119895+1

+ Δ119868119895

minus 119903119881119899+12

119894119895= 0

(79)

with

120588119894119895

=119903119878

119894Δ119878

119894+1

2coth(

119903119878119894Δ119878

119894+1

12059022119878

2

119894

) coth (119909) =1198902119909

+ 1

1198902119909 minus 1

(80)

Let

120572119894= minus

120588119894119895

Δ119878119894Δ119878

119894+1

120573119894= minus

120588119894119895

(Δ119878119894+1

)2minus

119903119878119894

Δ119878119894+1

120574119894= minus 120572

119894minus 120573

119894+ 119903

(81)

Then the scheme (79) can be written as

120572119894119881

119899+12

119894minus1119895+ (120574

119894+

2

Δ120591)119881

119899+12

119894119895+ 120573

119894119881

119899+12

119894+1119895

=2

Δ120591119881

119899

119894119895+

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895+1minus

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895minus1

(82)

We let

119881119899

119895= [119881

119899

1119895 119881

119899

2119895 119881

119899

119872minus1119895]119879

119881119899+12

119895= [119881

119899+12

1119895 119881

119899+12

2119895 119881

119899+12

119872minus1119895]119879

119891119899+12

119895= [120572

1119881

119899+12

0119895 0 0 120573

119872minus1119881

119899+12

119872119895]119879

119903119899

119895= [(

2

Δ120591119881

119899

1119895+

1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895+1

minus1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895minus1)

(2

Δ120591119881

119899

119872minus1119895+

119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895+1

minus119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895minus1)]

119879

1198601=

[[[

[

1205741

1205731

1205722

1205742

1205732

d d d120572119872minus1

120574119872minus1

]]]

](119872minus1)times(119872minus1)

(83)

Then we can write (82) as the following equivalent matrixform

1198601119881

119899+12

119895+ 119891

119899+12

119895= 119903

119899

119895 (84)

The second leg of the discretized PDE in (78) is given bythe scheme

minus

119881119899+1

119894119895minus 119881

119899+12

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899+1

119894119895+1minus 119881

119899+1

119894119895

Δ119868119895+1

minus 119903119881119899+12

119894119895= 0

(85)

Let

120575119895=

2

Δ120591+

119878119894

Δ119868119895+1

120598119895= minus

119878119894

Δ119868119895+1

(86)

Then the scheme (85) can be written as

120575119895119881

119899+1

119894119895minus 120598

119895119881

119899+1

119894119895+1= minus 120572

119894119881

119899+12

119894minus1119895minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119895

minus 120573119894119881

119899+12

119894+1119895

(87)

Mathematical Problems in Engineering 11

300

300

250

200

200

150

100

100

100

50

50

0

0 0

Call

valu

e

sI

Figure 1 Fixed strike Asian call value when = 08 120590 = 05 119903 =

010 119879 = 1 and 119870 = 100

We let

119881119899+1

119894= [119881

119899+1

1198940 119881

119899+1

1198941 119881

119899+1

119894119869minus1]119879

119881119899+12

119894= [119881

119899+12

1198940 119881

119899+12

1198941 119881

119899+12

119894119869minus1]119879

119865119894= [0 0 0 120598

119869minus1119881

119899+1

119894119869]119879

119869

119877119899+12

119894= [(minus120572

119894119881

119899+12

119894minus10minus(120574

119894minus

2

Δ120591)119881

119899+12

1198940minus120573

119894119881

119899+12

119894+10)

(minus120572119894119881

119899+12

119894minus1119869minus1minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119869minus1minus 120573

119894119881

119899+12

119894+1119869minus1)]

119879

1198602=

[[[

[

1205750

minus1205980

0 1205751

minus1205981

d d minus120598119869minus2

120575119869minus1

]]]

]119869times119869

(88)

The matrix form of (87) is

1198602119881

119899+1

119894+ 119865

119899+1

119894= 119877

119899+12

119894 (89)

Thematrix equations (84) and (89) can be solved by using LUdecomposition

The goal of the computation is to find the fair price of theAsian option at emission that is119881(119878

0 119868

0 0)Weneed tomake

sense of any value 119868 isin [0 119868max] and stock price 119878 isin [0 119878max] attime 119905 = 0 There are no problems with the stock prices butthe integral at time zero can only be 0 for continuous averagesthat is 119868

0= 0

Figure 1 is the results we obtained by using the abovediscretization scheme with = 08 120590 = 05 119903 = 010 119879 = 1and 119870 = 100 The point marked by lowast is the price 119881(0 119878 0)

of the Asian option We choose 119878max = 3119870 to ensure thedesirable accuracy

Table 1 shows the convergence of the results with therefinement of the grid From (78) the values of variable 119868

lie in the interval [0 119870119879] where 119870119879 is the boundary and0 is the point where the continuous averages Asian options

are valued so there is no point that is less important on theinterval [0 119870119879] Hence we use uniform grids in direction 119868

In our uncertain volatility model the pricing PDEs (48)are similar to the PDEs in certain volatility model withvolatility 120590 = 120590 Barraquand and Pudet [16] and Zvan etal [17] give numerical results of Asian options with certainvolatilities In the uncertain volatility model when =

08 120590 = 05 the Asian option PDE is the same as that ofthe Asian option with certain volatility 120590 = 04 When theparameters 119903 = 010 119879 = 1 119870 = 95 and grid Δ119868 =

011875 ΔS = 285 the price of the Asian call option wecalculate is 1382 The results with the same parameters ofBarraquand and Pudet [16] and Zvan et al [17] are 13825 and13721 respectively

Table 2 is the comparison of our results with those ofZvan et al [17] Z F and V in Table 2 refer to the resultof Zvan et al [17] Our results are denoted by Asian fixedstrike which calculated with Δ120591 = 001 002 004 Δ119868 =

0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Since119868max = 119870119879 the partitions in 119868 direction are related with thematurity 119879 The execution time is much less than theirs Thegrid spacing was chosen to achieve an accuracy of at least010 of 119878 Our results are close to that of Zvan et al [17] andthe difference is less than 010 of 119878

Our calculation procedure is different from Zvan et al[17] as well as Barraquand and Pudet [16] in the followingfour aspects First our pricing PDE (78) is in terms of therunning sum 119868

119905 Zvan et al [17] and Barraquand and Pudet

[16] use the pricing PDE

minus 120597120591119881 (120591 119878 119860) +

1

211987821205902119881119878119878

(120591 119878 119860) + 119903119878119881119878(120591 119878 119860)

+119878 minus 119860

119905119881119860(120591 119878 119860) minus 119903119881 (120591 119878 119860) = 0

(90)

where 119860119905is defined as 119860

119905= 119868

119905119905 But PDE (78) and (90) are

equivalent for pricing fixed strike European Asian optionsSecond since the PDEs are different PDE (78) and (90) musthave different boundary conditions We do not know theboundary conditions used by Zvan et al [17] The forwardshooting grid method used by Barraquand and Pudet [16]proceeds without any boundary conditions Third our PDE(78) is defined on the domain 119863 = (119905 119878 119868) 0 le 119905 le 119879 0 le

119878 le 119878max 0 le 119868 le 119870119879 and 119868max = 119870119879 is the boundary and119868 = 0 is the point where the options are valued So there isno point that is less important on the interval [0 119870119879] Henceour numerical schemes use uniform grid spacing while Zvanet al [17] use nonuniform grid spacing of 41 times 45 on domain[0 300] times [0 300] To achieve an accuracy of at least 01 of119878 our grids are much finer than Zvan et al [17]

32 Numerical Computation of Floating Strike Asian OptionsTo value the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69)) for 119867(119905 119877) andthe values of the options will be 119881(119905 119878 119868) = 119878119867(119905 119877) with119877 = 119868119878 Actually what we really care about is the optionvalues at time 119905 = 0 while from the definition of 119868

119905 it is

obvious that 1198680= 0 and therefore 119877

0= 0 so our goal of the

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

Mathematical Problems in Engineering 3

Firstly construct a sequence of 119889-dimensional randomvectors (120585)infin

119894=1on a sublinear expectation space (Ω H E) such

that 120585119894is 119866-normal distributed and 120585

119894+1is independent from

(1205851 120585

119894) for each 119894 = 1 2 For each119883 isin 119871

119894119901(Ω) with

119883 = 120601 (1198611199051

minus 1198611199050

1198611199052

minus 1198611199051

119861119905119899

minus 119861119905119899minus1

) (7)

some 120601 isin 119862119897119871119894119901

(119877119889times119899

) and 0 = 1199050lt 119905

1lt sdot sdot sdot lt 119905

119899lt infin set

E [120601 (1198611199051

minus 1198611199050

1198611199052

minus 1198611199051

119861119905119899

minus 119861119905119899minus1

)]

= E [120601 (radic1199051minus 119905

01205851 radic119905

119899minus 119905

119899minus1120585119899)]

(8)

And the related conditional expectation of

119883 = 120601 (1198611199051

1198611199052

minus 1198611199051

119861119905119899

minus 119861119905119899minus1

) (9)

underΩ119905119895

is defined by

E [119883 | Ω119905119895

] = 120595 (1198611199051

119861119905119895

minus 119861119905119895minus1

) (10)

where

120595 (1199091 119909

119895)

= E [120601 (1199091 119909

119895 radic119905

119895+1minus 119905

119895120585119895+1

radic119905119899minus 119905

119899minus1120585119899)]

(11)

E[sdot] consistently defines a sublinear expectation on 119871119894119901(Ω)

and (119861119905)119905ge0

is a 119866-Brownian motionWe let 119871119901

119866(Ω

119905119896

) 119901 ge 1 denote the completion of

119871119894119901(Ω

119905119896

) = 120593 (1198611199051and119879

119861119905119899and119879

) 119899 isin N

1199051 119905

119899isin [0infin) 120593 isin 119862

119897119871119894119901(R

119889times119899)

(12)

under the norm 119883119901= (E[|119883|

119901])

1119901

22 The PDEs We assume the security market is definedon a sublinear expectation space (ΩH E) which openscontinuously and offers a constant riskless interest rate 119903 toall borrowers and lenders And we also assume there are notransaction costs andor taxes Suppose that the price process(119878

119905)119905ge0

of some risky asset is given by

119889119878] = 120583119878]119889] + 119878]119889119861] 1198780= 119878 (13)

where 119861 is the 119866-Brownian motion with 1205902= minusE[minus1198612

1] and

1205902

= E[1198612

1] and 120583 and are constants Hence (119878])]ge0 is a

geometric 119866-Brownian motion

119878] = 119878119905exp (120583 (] minus 119905)

minus1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905))

(14)

Define

119868] = int

]

0

119878120591119889120591 (15)

Then 119868] satisfies the following stochastic differential equation

119889119868] = 119878]119889] 1198680= 0 (16)

Let119883] = (119878] 119868])119879 so

119889119883] = 120583119883119883]119889] +

119883119883]119889119861] 119883

0= 119909 = (119878 0) (17)

where 120583119883 and

119883 are 2 times 2matrixes

120583119883

= (120583 0

1 0)

119883= (

0

0 0) (18)

The typical Asian options in this security market are thefollowing four types

the arithmetic average fixed strike call option withfinal payoff

max (119868119879

119879minus 119870 0) (19)

the corresponding arithmetic average fixed strike putoption with payoff

max (119870 minus119868119879

119879 0) (20)

the arithmetic average floating strike call option withfinal payoff

max(119878119879minus

119868119879

119879 0) (21)

and the corresponding arithmetic average floatingstrike put option with payoff

max(119868119879

119879minus 119878

119879 0) (22)

Let 119881(119905 119878119905 119868

119905) denote the price of one of the options at

time 119905 Take the arithmetic average fixed strike call option forexample we know 119881(119879 119878

119879 119868

119879) = max(119868

119879119879 minus 119870 0)

Here we still assume the market operates in a risk neutralworld [7]

Peng [35] introduces the axiomatic conditions satisfied bythe valuation mechanism In the sublinear expectation space(ΩH E) E

119905[sdot] is a filtration consistent valuation operator

By this valuation mechanism we have

119881 (119905 119878119905 119868

119905) = E [119890

minus119903(119879minus119905)119881 (119879 119878

119879 119868

119879) | Ω

119905] (23)

Here we assume the riskless interest rate is 119903Since we assume the market is risk neutral we have 120583 =

119903 It is easy to verify that the discounted prices of the stocksatisfy the valuation mechanism E

119905[sdot]

119905ge0 that is

119878119905= E

119905[119890

minus119903(]minus119905)119878]] 0 le 119905 le ] le 119879 (24)

4 Mathematical Problems in Engineering

Let us assume

119889119883119905119909

] = 120583119883119883

119905119909

] 119889] + 119883119883

119905119909

] 119889119861] 119883119905= 119909 = (119878 119868) (25)

with

120583119883

= (119903 0

1 0)

119883= (

0

0 0) (26)

The following result is a corollary of the nonlinearFeynman-Kac formula of Peng [36]

Corollary 5 Let

119884119905119909

119904= E [119890

minus119903(119879minus119904)Φ(119883

119905119909

119879)

+ int

119879

119904

119890minus119903(]minus119904)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119879

119904

119890minus119903(]minus119904)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩] | Ω119904]

(27)

where Φ R2rarr R is a given Lipschitz function and 119891 119892

R2times R rarr R are given Lipschitz functions that is |120601(119909 119910) minus

120601(1199091015840 119910

1015840)| le 119870(|119909 minus 119909

1015840| + |119910 minus 119910

1015840|) for each 119909 119909

1015840isin R2 119910 1199101015840

isin

R 120601 = 119891 and 119892 And denote

119906 (119905 119909) = 119884119905119909

119905 (28)

Then

119906 (119905 119909) = E[119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575)

+ int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]]

(29)

is a viscosity solution of the following PDE

120597119905119906 + sup

1205902le1205902le1205902

(1

2⟨119863

2119906 (119905 119909)

119883119909

119883119909⟩ + 119892 (119909 119906)) 120590

2

+ ⟨119863119906 (119905 119909) 120583119883119909⟩ minus 119903119906 + 119891 (119909 119906) = 0

119906 (119879 119909) = Φ (119909)

(30)

Proof The proof of expression (29) is the same as that ofPeng [36] By the proof of Pengrsquos theorem of the nonlinearFeynman-Kac formula and the boundedness of 119890minus119903(119879minus119905) wecan easily prove that 119906 is a Lipschitz function that is

10038161003816100381610038161003816119906 (119905 119909) minus 119906 (119905 119909

1015840)10038161003816100381610038161003816le 119862

10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816 (31)

E [10038161003816100381610038161003816119884

119905119909

119904

10038161003816100381610038161003816] le 119862 (1 + |119909|) (32)

Let us see the continuity of 119906with respect to 119905 Firstly we have

|119906 (119905 119909) minus 119906 (119905 + 120575 119909)|

le E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

+ E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

(33)

Replace 119890minus119903120575 with its Taylorrsquos expansion in the first term on

the right side of (33) we have

E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

= E [10038161003816100381610038161003816(1 minus 119903120575 + (119903120575)

2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)

minus 119906 (119905 + 120575 119909)10038161003816100381610038161003816]

= E [10038161003816100381610038161003816119906 (119905 + 120575119883

119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

+ (minus119903120575 + (119903120575)2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)10038161003816100381610038161003816]

le E [10038161003816100381610038161003816119883

119905119909

119905+120575minus 119909

10038161003816100381610038161003816]

+ E [10038161003816100381610038161003816(minus119903120575 + (119903120575)

2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)10038161003816100381610038161003816]

(34)

By the following formulas [36]

E [10038161003816100381610038161003816119883

119905119909

119905+120575minus 119909

10038161003816100381610038161003816

2

] le 119862 (1 + 1199092) 120575

E [10038161003816100381610038161003816119883

119905119909

119905+120575

10038161003816100381610038161003816

2

] le 119862 (1 + 1199092)

(35)

and (32) we have

E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

le 1198621015840(1 + |119909|

2)12

(12057512

+ 120575 +1205752

2sdot sdot sdot )

(36)

Mathematical Problems in Engineering 5

For the second term on the right side of (33) we have

E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

= E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

times (119891 (119883119905119909

] 119884119905119909

] ) minus 119891 (119909 119884119905119909

119905)) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

times (119892 (119883119905119909

] 119884119905119909

] ) minus 119892 (119909 119884119905119909

119905)) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

+ E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119909 119884119905119909

119905) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119909 119884119905119909

119905) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

le 119862 (1 + |119909|) 120575

(37)

Therefore 119906 is continuous in 119905Now we are going to prove that 119906(119905 119909) is a viscosity

solution of PDE (30) For fixed (119905 119909) isin (0 119879) times R2 let 120595 isin

11986223

119887((0 119879) times R2

) such that 120595 ge 119906 and 120595(119905 119909) = 119906(119905 119909) By(29) and Taylorrsquos expansion for 120575 isin (0 119879 minus 119905)

0 le E[119890minus119903120575

120595 (119905 + 120575119883119905119909

119905+120575) minus 120595 (119905 119909)

+ int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]]

le E [ minus 119903120595 (119905 119909) 120575 + 120597119905120595 (119905 119909) 120575

+ ⟨119863120595 (119905 119909) 120583119883119909⟩ 120575

+ ⟨1

2119863

2120595 (119905 119909)

119883119909

119883119909⟩ (⟨119861⟩

119905+120575minus ⟨119861⟩

119905)

+ 119891 (119909 120595) 120575 + 119892 (119909 120595) (⟨119861⟩119905+120575

minus ⟨119861⟩119905)]

+ 119862 (1 + |119909| + |119909|2+ |119909|

3) 120575

32

le (120597119905120595 + sup

1205902le1205902le1205902

(1

2⟨119863

2120595 (119905 119909)

119883119909

119883119909⟩

+119892 (119909 120595) ) 1205902+ ⟨119863120595 (119905 119909) 120583

119883119909⟩

minus119903120595 (119905 119909) + 119891 (119909 120595))120575

+ 119862 (1 + |119909| + |119909|2+ |119909|

3) 120575

32

(38)

It is easy to check that

120597119905120595 + sup

1205902le1205902le1205902

(1

2⟨119863

2120595 (119905 119909)

119883119909

119883119909⟩ + 119892 (119909 120595)) 120590

2

+ ⟨119863120595 (119905 119909) 120583119883119909⟩ minus 119903120595 (119905 119909) + 119891 (119909 120595) ge 0

(39)

Thus 119906 is a viscosity subsolution of (30) Similarly we canprove that 119906 is a viscosity supersolution of (30)

For the arithmetic average Asian options we have

119881 (119905 119909) = 119881 (119905 119878 119868) = E [119890minus119903(119879minus119905)

119881 (119879 119878119879 119868

119879)] (40)

Then by Corollary 5 119881(119905 119909) satisfies the following PDE

120597119905119881 (119905 119909) + sup

1205902le1205902le1205902

1

2⟨119863

2119881 (119905 119909)

119883119909

119883119909⟩120590

2

+ ⟨119863119881 (119905 119909) 120583119883119909⟩ minus 119903119881 (119905 119909) = 0

119881 (119879 119909) = 120593 (119879 119909)

(41)

By (18) the above PDE is

120597119905119881 (119905 119878 119868) + sup

1205902le1205902le1205902

1

2211987821205902119881119878119878

(119905 119878 119868)

+ 119903119878119881119878(119905 119878 119868) + 119878119881

119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

119881 (119879 119878 119868) = 120593 (119879 119878 119868)

(42)

where120593 (119879 119878 119868)

=

max ( 119868

119879minus 119870 0) for the arithmetic average

fixed strike call option

max (119870 minus119868

119879 0) for the arithmetic average

fixed strike put option

max (119878 minus119868

119879 0) for the arithmetic average

floating strike call option

max ( 119868

119879minus 119878 0) for the arithmetic average

floating strike put option(43)

6 Mathematical Problems in Engineering

23 The Arithmetic Average Fixed Strike Asian Call OptionLet us see the arithmetic average fixed strikeAsian call optionFrom (14) we have

119878119905119878

] = 119878 exp 120583 (] minus 119905) minus1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905)

(44)

Then by the definition of 119868] we have

119868119905119868

] = 119868 + 119878int

]

119905

exp 120583 (120591 minus 119905) minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591

(45)

Since

119881 (119905 119909) = 119881 (119905 119878 119868) = E [119890minus119903(119879minus119905)

119881 (119879 119878119879 119868

119879)]

= E [119890minus119903(119879minus119905)

(119868119879

119879minus 119870)

+

]

119881 (119905 119878 119868) = E[119890minus119903(119879minus119905)

times (((119868 + 119878int

]

119905

exp 120583 (120591 minus 119905)

minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591)

times119879minus1) minus 119870)

+

]

(46)

By the convexities of function 119891(119909) = (119909 minus 119896)+ and the

sublinear expectation we can say

119881(119905 1205821198781+ (1 minus 120582) 119878

2 119868) le 120582119881 (119905 119878

1 119868) + (1 minus 120582)119881 (119905 119878

2 119868)

(47)

that is 119881(119905 119878 119868) is convex about 119878 So in the numericalcomputation the second order difference of 119881 is positiveThen the numerical solutions of (43) are equivalent to thoseof PDE

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

119881 (119879 119878 119868) = (119868

119879minus 119870)

+

(48)

Theoretically the above PDE is defined on domain 119863 =

(119905 119878 119868) isin [0 119879] times [0infin) times [0infin) But generally closedform solutions cannot be found we have to find numericalsolutions We have no interest in finding the solution in the

entire infinite domain Therefore we consider the domain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119868

lowast)

Now we determine the boundary conditions of (48) If119868 ge 119870119879 that is 119868119879 ge 119870 for the positiveness of 119878] we have119868119879119879 ge 119870 Therefore the final payoff on the option is certain

to be positive At time 119879 this payoff can be expressed as

119868119879

119879minus 119870 =

119868

119879minus 119870 +

1

119879int

119879

119905

119878]119889] (49)

This payoff can also be obtained by using the followingself-financing duplicating portfolio strategy Assume that aninvestor commits (119868119879minus119870)119890

minus119903(119879minus119905) into riskless bonds in orderto secure the return promised by the first part of the rightof (49) that is (119868119879 minus 119870) at time 119879 If the investor is to becertain of accruing the return promised by the second part ofthe right of (49) he is obliged to transfer a certain portionof stock (equal to (1119879)119890

minus119903(119879minus])Δ]) to a riskless bond in every

time interval (] ] + Δ]) Overall this strategy requires a sumequal to (119868119879 minus 119870)119890

minus119903(119879minus119905) plus the portion of a stock whichcan be expressed as follows

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (50)

Then the price of the option when 119868119905ge 119870119879must be equal to

119881 (119905 119878 119868) = (119868

119879minus 119870) 119890

minus119903(119879minus119905)+

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (51)

The above discussion is similar to that in the model with-out volatility uncertainty Equation (51) obviously satisfies(48) So we only need to find the solution of (48) in thedomain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119870119879)

On the boundary 119868 = 119870119879 the boundary condition from(51) can be written as

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (52)

We note that if 119878119905= 0 (13) tells us that 119878(]) is also equal

to zero for ] isin [119905 119879] and 119868(119879) is thus equal to 119868(119905) By (46)we can write the following boundary condition

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

(53)

We need another boundary condition at 119878 = 119878max =

Slowast Hugger [37] gives the boundary condition at 119878 = 119878maxin the model without volatility uncertainty By the samediscussion we can get the same boundary condition inuncertain volatility model

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+Smax119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

(54)

Mathematical Problems in Engineering 7

So the values of the arithmetic average fixed strike Asiancall option satisfy

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

V (119879 119878 119868) = (119868

119879minus 119870)

+

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+

119878max119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

(55)

For the arithmetic average fixed strike Asian put optionwith terminal payoff 119901(119879 119878

119879 119868

119879) = max(119870 minus 119868

119879119879 0) we will

value this option by the following put-call parityIt is easily to see that we have

max(119868119879

119879minus 119870 0) = max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889] minus 119870

(56)

So

119888 (119905 119878119905 119868

119905)

= 119890minus119903(119879minus119905)

E119905[max (119868

119879

119879minus 119870 0)]

= 119890minus119903(119879minus119905)

E119905[max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889]]

minus 119890minus119903(119879minus119905)

119870

= 119901 (119905 119878119905 119868

119905) minus 119890

minus119903(119879minus119905)119870 +

1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889]

+ 119890minus119903(119879minus119905)

119864120590

119905[1

119879int

119879

119905

119878]119889]]

(57)

119890minus119903(119879minus119905)

119864120590

119905[(1119879) int

119879

119905119878]119889]] is the time 119905 value of time 119879

claim (1119879) int119879

119905119878]119889] where the stock has certain volatility

120590 To get claim (1119879) int119879

119905119878]119889] at 119879 let us see the replicating

strategy in any time interval (] ] + Δ]) transfer a certainportion (1119879)119890

minus119903(119879minus])Δ] of stock 119878] to a riskless bond This

strategy ensures the final payoff of (1119879) int119879

119905119878]119889] The stocks

that should be transferred are

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (58)

Then the time 119905 value of (1119879) int119879

119905119878]119889] equals (1119903119879)(1 minus

119890minus119903(119879minus119905)

)119878119905 So the put-call parity can be written as

119888 (119905 119878 119868) + 119870119890minus119903(119879minus119905)

= 119901 (119905 119878 119868) +1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889] +1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

119905

(59)

24 The Arithmetic Average Floating Strike Asian Call OptionFor the arithmetic average floating strike Asian call (put)option the PDE (43) still applies with terminal condition

119881 (119879 119878 119868) = (119878 minus119868

119879)

+

(119881 (119879 119878 119868) = (119868

119879minus 119878)

+

) (60)

In Ingersoll [13] where there is no volatility uncertaintydue to the linear homogeneity of the PDE and its terminalcondition their PDE can be reduced to a one-dimensionalPDE In our uncertain volatility model the PDE (43) is notlinear homogeneous anymore But we can still reduce PDE(43) to a one-dimensional PDE by the following argument

For the arithmetic average floating strike Asian calloption we know

119881 (119905 119878119905 119868

119905) = E

119905[119890

minus119903(119879minus119905)(119878

119879minus

119868119879

119879)

+

]

= E119905[

[

119890minus119903(119879minus119905)

(119878119879minus

119868119905+ int

119879

119905119878]119889]

119879)

+

]

]

(61)

Denote

119872]119905= exp(120583 (] minus 119905) minus

1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905))

(62)

Then by (14) we have 119878] = 119878119905119872

]119905and 119878

119905= 119878]119872

119905

] So

119868119905= int

119905

0

119878]119889] = int

119905

0

119878119905119872

]119905119889] = 119878

119905int

119905

0

119872]119905119889]

119881 (119905 119878119905 119868

119905)

= E119905[

[

119890minus119903(119879minus119905)

(119878119905119872

119879

119905minus

119878119905int119905

0119872

]119905119889] + 119878

119905int119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905E

119905[

[

119890minus119903(119879minus119905)

(119872119879

119905minus

int119905

0119872

]119905119889] + int

119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905119867(119905 119877

119905)

(63)with 119877

119905= 119868

119905119878

119905

Then the function119867(119905 119877) satisfies the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(64)

which is a one-dimensional PDE

8 Mathematical Problems in Engineering

For the European style Asian option we must imposeboundary conditions both at 119877 = 0 and as 119877 rarr infin Theboundary condition as 119877 rarr infin is simple The only way that119877 can tend to infinity is that 119878 tends to zero In this case theoption will not be exercised and so

lim119877rarrinfin

119867(119905 119877) = 0 (65)

For computational purpose we truncate the asset region(0 +infin) into (0 119877max) where 119877max is sufficiently large toensure the accuracy of the solution If 119877 = 0 the PDEdegenerates into the first-order hyperbolic PDE

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 (66)

So119867(119905 119877) satisfies

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (1 minus119877

119879)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = 0

(67)

By the same argument for the arithmetic average floatingstrike Asian put option its prices could be calculated by

119881 (119905 119878 119868) = 119878119867 (119905 119877) (68)

with119867(119905 119877) satisfying the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (119877

119879minus 1)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = (119877max119879

minus exp minus119903 (119879 minus 119905))

(69)

Generally the put-call parity will not hold any more inthe uncertain volatility model Let us see the following argu-ment The mature payoff of a portfolio with one Europeanarithmetic average floating strike Asian call holding long andone put holding short is

119878119879max (1 minus

119877119879

119879 0) minus 119878

119879max (119877

119879

119879minus 1 0)

= 119878119879minus

119877119879119878119879

119879

(70)

The value of this portfolio is equal to one consisting ofone stock and a financial product whose payoff is minus119877

119879119878119879119879

The values of the financial product with final payoff minus119877119879119878119879119879

satisfy the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(71)

with terminal condition 119867(119879 119877119879) = minus119877

119879119878119879119879 We seek a

solution of (71) of the form

119867(119905 119877) = 119886119905+ 119887

119905119877 (72)

with 119886119879

= 0 and 119887119879

= minus1119879 Substituting (72) into (71) andsatisfying the boundary conditions we find that

119886119905= minus

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119887

119905= minus

1

119879119890minus119903(119879minus119905)

(73)

For

119878119879max (1 minus

119877119879

119879 0) = 119878

119879max (119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879

(74)

by the valuation mechanism in space (ΩH E) we take thediscounted 119866-conditional expectation on both sides and get

E119905[119890

minus119903(119879minus119905)119878119879max(1 minus

119877119879

119879 0)]

= 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879]

le 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0)]

+ 119890minus119903(119879minus119905)

E119905[119878

119879minus

119877119879119878119879

119879]

(75)

That is

119888 (119905 119878 119868) le 119901 (119905 119878 119868) + 119878 minus119878

119903119879(1 minus 119890

minus119903(119879minus119905))

minus1

119879119890minus119903(119879minus119905)

int

119905

0

119878120591119889120591

(76)

where 119888(119905 119878 119868) is the time 119905 price of the arithmetic averagefloating strike Asian call option and 119901(119905 119878 119868) is the time119905 price of the arithmetic average floating strike Asian putoption

3 Numerical Computation

In the model without volatility uncertainty Asian optionsare much more difficult to value than regular stock optionsStandard techniques tend to be impractical inaccurate orslow For example traditional binomial lattice methodsrequire such enormous amounts of computermemory for thenecessity of keeping track of every possible path throughout

Mathematical Problems in Engineering 9

the tree that they are effectively unusable Monte Carlosimulation works well for European style options (see [12])but is relatively slow Sun [38] proposed to use the weightedupwind method [39] to price Asian options but he didnot give numerical results The Asian option pricing in theuncertain volatility model will be even more difficult than inthe model without volatility uncertainty

For the arithmetic average fixed strike Asian call optionsdue to the convexity of the value 119881(119905 119878 119868) with respect tovariable 119878 (43) can be written as (48) which are the equationssatisfied by the arithmetic average fixed strike Asian calloptions with certain volatility 120590

Zvan et al [17] point out that the Crank-Nicolson schemein conjunction with the van Leer flux limiter method canbe used to numerically value Asian options with certainvolatilities The van Leer limiter has the property that it istotal variation diminishing and thus produces oscillation freesecond order accurate solutions But the van Leer limiter isnonlinear so the solutions should be obtained by using fullNewton iteration at each time level In this paper we will usethe alternating direction implicit (ADI) methods (see Duffy[40])

The arithmetic average fixed strike Asian put options canbe valued by the put-call parity

For the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69))These PDEs are one-dimensional in space And they are not linear equations butHJB equations in some sense For such PDEs we will use thefully implicit positive coefficient finite different method [41]to solve them We will see that this method can ensure thenumerical solutions converge to the viscosity solutions andis accurate efficient and quick to be implemented

All the calculations were performed byMatlab 2009 on acomputer with 183GHz hard disk and 512MB memory

31 Numerical Computation of Fixed Strike Asian OptionsAn equivalent formulation of (48) in terms of the average119860

119905= (1119905) int

119905

0119878120591119889120591 is given in Barraquand and Pudet [16] in

certain volatility model with volatility 120590 = 120590

120597119905119881 (119905 119878 119860) +

1

2211987821205902119881119878119878

(119905 119878 119860) + 119903119878119881119878(119905 119878 119860)

+119878 minus 119860

119905119881119860(119905 119878 119860) minus 119903119881 (119905 119878 119860) = 0

119881 (119879 119878 119860) = (119860 minus 119870)+

(77)

Equation (77) is convection dominated in the 119860 direc-tion because there is no diffusion effect in this dimensionBarraquand and Pudet [16] find that an explicit centrallyweighted scheme for (77) is unstable In particular theconvective term in the 119860 dimension becomes very large as119905 rarr 0 Barraquand and Pudet also note that implicit cen-trally weighted schemes will generally produce unsatisfactoryresults because of the numerical diffusion introduced by thisfirst order accurate in time scheme

Zvan et al [17] point out that under the conditionof convection dominated solutions generated by using a

centrally weighted scheme on the convective term for (77)cannot be ensured to be free of oscillations And solutionsgenerated by the first order upstream weighting scheme onthe convective termare no longer oscillatory but their profilesare too diffuse To produce oscillation free solutions withoutthe excessive diffusion of first order upstream weightingZvan et al [17] suggest the nonlinear van Leer flux limiter inconjunction with the Crank-Nicolson scheme Since the fluxlimiter is nonlinear the solutions should be obtained by usingfull Newton iteration at each time level

The ADI method pioneered in the United States by Dou-glas Rachford Peaceman Gunn and others has a numberof advantages First explicit difference methods are rarelyused to solve initial boundary value problems because oftheir poor stability Implicit methods have superior stabilityproperties but unfortunately they are difficult to solve in twoand more dimensions Consequently ADI methods becomean alternative because they can be programmed by solvinga simple trigonal system of equations For the convection-dominated problems we could use the exponentially fittedschemes in each leg of the approximate scheme to ensure thestability of the results [40] Since Barraquand and Pudet [16]have used the forward shooting grid method and Zvan et al[17] have used the van Leer flux limiter we will test the ADImethod in this paper by comparing our results with theirs

For convenience let us first convert (55) to be thefollowing forward equation in time by substituting 119905 with120591 = 119879 minus 119905

minus 120597120591119881 (120591 119878 119868) +

1

2211987821205902119881119878119878

(120591 119878 119868) + 119903119878119881119878(120591 119878 119868)

+ 119878119881119868(120591 119878 119868) minus 119903119881 (120591 119878 119868) = 0

119881 (0 119878 119868) = (119868

119879minus 119870)

+

119881 (120591 0 119868) = 119890minus119903120591

(119868

119879minus 119870)

+

119881 (119905 119878max 119868) = max ( 119868

119879minus 119870) 119890

minus119903120591+

119878max119903119879

(1 minus 119890minus119903120591

) 0

119881 (120591 119878 119870119879) =1

119903119879(1 minus 119890

minus119903120591) 119878

(78)

We let 120591119899 119899 = 0 1 119873 be a set of portion points in

[0 119879] satisfying 0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 and use even

partition of time that is 120591119899= 119899Δ120591 119899 = 0 1 119873 Define the

partitions of space variables as 0 = 1198780lt 119878

1lt sdot sdot sdot lt 119878

119872= 119878max

and 0 = 1198680lt 119868

1lt sdot sdot sdot lt 119868

119869= 119868max

With ADI we march from time level 119899 to time level119899 + 12 and then from time level 119899 + 12 to time level 119899 + 1In this case we use exponential fitting in all space variables

10 Mathematical Problems in Engineering

and implicit Euler in time The first leg is given by thescheme

minus

119881119899+12

119894119895minus 119881

119899

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899

119894119895+1minus 119881

119899

119894119895minus1

Δ119868119895+1

+ Δ119868119895

minus 119903119881119899+12

119894119895= 0

(79)

with

120588119894119895

=119903119878

119894Δ119878

119894+1

2coth(

119903119878119894Δ119878

119894+1

12059022119878

2

119894

) coth (119909) =1198902119909

+ 1

1198902119909 minus 1

(80)

Let

120572119894= minus

120588119894119895

Δ119878119894Δ119878

119894+1

120573119894= minus

120588119894119895

(Δ119878119894+1

)2minus

119903119878119894

Δ119878119894+1

120574119894= minus 120572

119894minus 120573

119894+ 119903

(81)

Then the scheme (79) can be written as

120572119894119881

119899+12

119894minus1119895+ (120574

119894+

2

Δ120591)119881

119899+12

119894119895+ 120573

119894119881

119899+12

119894+1119895

=2

Δ120591119881

119899

119894119895+

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895+1minus

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895minus1

(82)

We let

119881119899

119895= [119881

119899

1119895 119881

119899

2119895 119881

119899

119872minus1119895]119879

119881119899+12

119895= [119881

119899+12

1119895 119881

119899+12

2119895 119881

119899+12

119872minus1119895]119879

119891119899+12

119895= [120572

1119881

119899+12

0119895 0 0 120573

119872minus1119881

119899+12

119872119895]119879

119903119899

119895= [(

2

Δ120591119881

119899

1119895+

1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895+1

minus1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895minus1)

(2

Δ120591119881

119899

119872minus1119895+

119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895+1

minus119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895minus1)]

119879

1198601=

[[[

[

1205741

1205731

1205722

1205742

1205732

d d d120572119872minus1

120574119872minus1

]]]

](119872minus1)times(119872minus1)

(83)

Then we can write (82) as the following equivalent matrixform

1198601119881

119899+12

119895+ 119891

119899+12

119895= 119903

119899

119895 (84)

The second leg of the discretized PDE in (78) is given bythe scheme

minus

119881119899+1

119894119895minus 119881

119899+12

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899+1

119894119895+1minus 119881

119899+1

119894119895

Δ119868119895+1

minus 119903119881119899+12

119894119895= 0

(85)

Let

120575119895=

2

Δ120591+

119878119894

Δ119868119895+1

120598119895= minus

119878119894

Δ119868119895+1

(86)

Then the scheme (85) can be written as

120575119895119881

119899+1

119894119895minus 120598

119895119881

119899+1

119894119895+1= minus 120572

119894119881

119899+12

119894minus1119895minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119895

minus 120573119894119881

119899+12

119894+1119895

(87)

Mathematical Problems in Engineering 11

300

300

250

200

200

150

100

100

100

50

50

0

0 0

Call

valu

e

sI

Figure 1 Fixed strike Asian call value when = 08 120590 = 05 119903 =

010 119879 = 1 and 119870 = 100

We let

119881119899+1

119894= [119881

119899+1

1198940 119881

119899+1

1198941 119881

119899+1

119894119869minus1]119879

119881119899+12

119894= [119881

119899+12

1198940 119881

119899+12

1198941 119881

119899+12

119894119869minus1]119879

119865119894= [0 0 0 120598

119869minus1119881

119899+1

119894119869]119879

119869

119877119899+12

119894= [(minus120572

119894119881

119899+12

119894minus10minus(120574

119894minus

2

Δ120591)119881

119899+12

1198940minus120573

119894119881

119899+12

119894+10)

(minus120572119894119881

119899+12

119894minus1119869minus1minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119869minus1minus 120573

119894119881

119899+12

119894+1119869minus1)]

119879

1198602=

[[[

[

1205750

minus1205980

0 1205751

minus1205981

d d minus120598119869minus2

120575119869minus1

]]]

]119869times119869

(88)

The matrix form of (87) is

1198602119881

119899+1

119894+ 119865

119899+1

119894= 119877

119899+12

119894 (89)

Thematrix equations (84) and (89) can be solved by using LUdecomposition

The goal of the computation is to find the fair price of theAsian option at emission that is119881(119878

0 119868

0 0)Weneed tomake

sense of any value 119868 isin [0 119868max] and stock price 119878 isin [0 119878max] attime 119905 = 0 There are no problems with the stock prices butthe integral at time zero can only be 0 for continuous averagesthat is 119868

0= 0

Figure 1 is the results we obtained by using the abovediscretization scheme with = 08 120590 = 05 119903 = 010 119879 = 1and 119870 = 100 The point marked by lowast is the price 119881(0 119878 0)

of the Asian option We choose 119878max = 3119870 to ensure thedesirable accuracy

Table 1 shows the convergence of the results with therefinement of the grid From (78) the values of variable 119868

lie in the interval [0 119870119879] where 119870119879 is the boundary and0 is the point where the continuous averages Asian options

are valued so there is no point that is less important on theinterval [0 119870119879] Hence we use uniform grids in direction 119868

In our uncertain volatility model the pricing PDEs (48)are similar to the PDEs in certain volatility model withvolatility 120590 = 120590 Barraquand and Pudet [16] and Zvan etal [17] give numerical results of Asian options with certainvolatilities In the uncertain volatility model when =

08 120590 = 05 the Asian option PDE is the same as that ofthe Asian option with certain volatility 120590 = 04 When theparameters 119903 = 010 119879 = 1 119870 = 95 and grid Δ119868 =

011875 ΔS = 285 the price of the Asian call option wecalculate is 1382 The results with the same parameters ofBarraquand and Pudet [16] and Zvan et al [17] are 13825 and13721 respectively

Table 2 is the comparison of our results with those ofZvan et al [17] Z F and V in Table 2 refer to the resultof Zvan et al [17] Our results are denoted by Asian fixedstrike which calculated with Δ120591 = 001 002 004 Δ119868 =

0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Since119868max = 119870119879 the partitions in 119868 direction are related with thematurity 119879 The execution time is much less than theirs Thegrid spacing was chosen to achieve an accuracy of at least010 of 119878 Our results are close to that of Zvan et al [17] andthe difference is less than 010 of 119878

Our calculation procedure is different from Zvan et al[17] as well as Barraquand and Pudet [16] in the followingfour aspects First our pricing PDE (78) is in terms of therunning sum 119868

119905 Zvan et al [17] and Barraquand and Pudet

[16] use the pricing PDE

minus 120597120591119881 (120591 119878 119860) +

1

211987821205902119881119878119878

(120591 119878 119860) + 119903119878119881119878(120591 119878 119860)

+119878 minus 119860

119905119881119860(120591 119878 119860) minus 119903119881 (120591 119878 119860) = 0

(90)

where 119860119905is defined as 119860

119905= 119868

119905119905 But PDE (78) and (90) are

equivalent for pricing fixed strike European Asian optionsSecond since the PDEs are different PDE (78) and (90) musthave different boundary conditions We do not know theboundary conditions used by Zvan et al [17] The forwardshooting grid method used by Barraquand and Pudet [16]proceeds without any boundary conditions Third our PDE(78) is defined on the domain 119863 = (119905 119878 119868) 0 le 119905 le 119879 0 le

119878 le 119878max 0 le 119868 le 119870119879 and 119868max = 119870119879 is the boundary and119868 = 0 is the point where the options are valued So there isno point that is less important on the interval [0 119870119879] Henceour numerical schemes use uniform grid spacing while Zvanet al [17] use nonuniform grid spacing of 41 times 45 on domain[0 300] times [0 300] To achieve an accuracy of at least 01 of119878 our grids are much finer than Zvan et al [17]

32 Numerical Computation of Floating Strike Asian OptionsTo value the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69)) for 119867(119905 119877) andthe values of the options will be 119881(119905 119878 119868) = 119878119867(119905 119877) with119877 = 119868119878 Actually what we really care about is the optionvalues at time 119905 = 0 while from the definition of 119868

119905 it is

obvious that 1198680= 0 and therefore 119877

0= 0 so our goal of the

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

4 Mathematical Problems in Engineering

Let us assume

119889119883119905119909

] = 120583119883119883

119905119909

] 119889] + 119883119883

119905119909

] 119889119861] 119883119905= 119909 = (119878 119868) (25)

with

120583119883

= (119903 0

1 0)

119883= (

0

0 0) (26)

The following result is a corollary of the nonlinearFeynman-Kac formula of Peng [36]

Corollary 5 Let

119884119905119909

119904= E [119890

minus119903(119879minus119904)Φ(119883

119905119909

119879)

+ int

119879

119904

119890minus119903(]minus119904)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119879

119904

119890minus119903(]minus119904)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩] | Ω119904]

(27)

where Φ R2rarr R is a given Lipschitz function and 119891 119892

R2times R rarr R are given Lipschitz functions that is |120601(119909 119910) minus

120601(1199091015840 119910

1015840)| le 119870(|119909 minus 119909

1015840| + |119910 minus 119910

1015840|) for each 119909 119909

1015840isin R2 119910 1199101015840

isin

R 120601 = 119891 and 119892 And denote

119906 (119905 119909) = 119884119905119909

119905 (28)

Then

119906 (119905 119909) = E[119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575)

+ int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]]

(29)

is a viscosity solution of the following PDE

120597119905119906 + sup

1205902le1205902le1205902

(1

2⟨119863

2119906 (119905 119909)

119883119909

119883119909⟩ + 119892 (119909 119906)) 120590

2

+ ⟨119863119906 (119905 119909) 120583119883119909⟩ minus 119903119906 + 119891 (119909 119906) = 0

119906 (119879 119909) = Φ (119909)

(30)

Proof The proof of expression (29) is the same as that ofPeng [36] By the proof of Pengrsquos theorem of the nonlinearFeynman-Kac formula and the boundedness of 119890minus119903(119879minus119905) wecan easily prove that 119906 is a Lipschitz function that is

10038161003816100381610038161003816119906 (119905 119909) minus 119906 (119905 119909

1015840)10038161003816100381610038161003816le 119862

10038161003816100381610038161003816119909 minus 119909

101584010038161003816100381610038161003816 (31)

E [10038161003816100381610038161003816119884

119905119909

119904

10038161003816100381610038161003816] le 119862 (1 + |119909|) (32)

Let us see the continuity of 119906with respect to 119905 Firstly we have

|119906 (119905 119909) minus 119906 (119905 + 120575 119909)|

le E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

+ E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

(33)

Replace 119890minus119903120575 with its Taylorrsquos expansion in the first term on

the right side of (33) we have

E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

= E [10038161003816100381610038161003816(1 minus 119903120575 + (119903120575)

2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)

minus 119906 (119905 + 120575 119909)10038161003816100381610038161003816]

= E [10038161003816100381610038161003816119906 (119905 + 120575119883

119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

+ (minus119903120575 + (119903120575)2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)10038161003816100381610038161003816]

le E [10038161003816100381610038161003816119883

119905119909

119905+120575minus 119909

10038161003816100381610038161003816]

+ E [10038161003816100381610038161003816(minus119903120575 + (119903120575)

2+ sdot sdot sdot ) 119906 (119905 + 120575119883

119905119909

119905+120575)10038161003816100381610038161003816]

(34)

By the following formulas [36]

E [10038161003816100381610038161003816119883

119905119909

119905+120575minus 119909

10038161003816100381610038161003816

2

] le 119862 (1 + 1199092) 120575

E [10038161003816100381610038161003816119883

119905119909

119905+120575

10038161003816100381610038161003816

2

] le 119862 (1 + 1199092)

(35)

and (32) we have

E [10038161003816100381610038161003816119890minus119903120575

119906 (119905 + 120575119883119905119909

119905+120575) minus 119906 (119905 + 120575 119909)

10038161003816100381610038161003816]

le 1198621015840(1 + |119909|

2)12

(12057512

+ 120575 +1205752

2sdot sdot sdot )

(36)

Mathematical Problems in Engineering 5

For the second term on the right side of (33) we have

E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

= E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

times (119891 (119883119905119909

] 119884119905119909

] ) minus 119891 (119909 119884119905119909

119905)) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

times (119892 (119883119905119909

] 119884119905119909

] ) minus 119892 (119909 119884119905119909

119905)) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

+ E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119909 119884119905119909

119905) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119909 119884119905119909

119905) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

le 119862 (1 + |119909|) 120575

(37)

Therefore 119906 is continuous in 119905Now we are going to prove that 119906(119905 119909) is a viscosity

solution of PDE (30) For fixed (119905 119909) isin (0 119879) times R2 let 120595 isin

11986223

119887((0 119879) times R2

) such that 120595 ge 119906 and 120595(119905 119909) = 119906(119905 119909) By(29) and Taylorrsquos expansion for 120575 isin (0 119879 minus 119905)

0 le E[119890minus119903120575

120595 (119905 + 120575119883119905119909

119905+120575) minus 120595 (119905 119909)

+ int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]]

le E [ minus 119903120595 (119905 119909) 120575 + 120597119905120595 (119905 119909) 120575

+ ⟨119863120595 (119905 119909) 120583119883119909⟩ 120575

+ ⟨1

2119863

2120595 (119905 119909)

119883119909

119883119909⟩ (⟨119861⟩

119905+120575minus ⟨119861⟩

119905)

+ 119891 (119909 120595) 120575 + 119892 (119909 120595) (⟨119861⟩119905+120575

minus ⟨119861⟩119905)]

+ 119862 (1 + |119909| + |119909|2+ |119909|

3) 120575

32

le (120597119905120595 + sup

1205902le1205902le1205902

(1

2⟨119863

2120595 (119905 119909)

119883119909

119883119909⟩

+119892 (119909 120595) ) 1205902+ ⟨119863120595 (119905 119909) 120583

119883119909⟩

minus119903120595 (119905 119909) + 119891 (119909 120595))120575

+ 119862 (1 + |119909| + |119909|2+ |119909|

3) 120575

32

(38)

It is easy to check that

120597119905120595 + sup

1205902le1205902le1205902

(1

2⟨119863

2120595 (119905 119909)

119883119909

119883119909⟩ + 119892 (119909 120595)) 120590

2

+ ⟨119863120595 (119905 119909) 120583119883119909⟩ minus 119903120595 (119905 119909) + 119891 (119909 120595) ge 0

(39)

Thus 119906 is a viscosity subsolution of (30) Similarly we canprove that 119906 is a viscosity supersolution of (30)

For the arithmetic average Asian options we have

119881 (119905 119909) = 119881 (119905 119878 119868) = E [119890minus119903(119879minus119905)

119881 (119879 119878119879 119868

119879)] (40)

Then by Corollary 5 119881(119905 119909) satisfies the following PDE

120597119905119881 (119905 119909) + sup

1205902le1205902le1205902

1

2⟨119863

2119881 (119905 119909)

119883119909

119883119909⟩120590

2

+ ⟨119863119881 (119905 119909) 120583119883119909⟩ minus 119903119881 (119905 119909) = 0

119881 (119879 119909) = 120593 (119879 119909)

(41)

By (18) the above PDE is

120597119905119881 (119905 119878 119868) + sup

1205902le1205902le1205902

1

2211987821205902119881119878119878

(119905 119878 119868)

+ 119903119878119881119878(119905 119878 119868) + 119878119881

119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

119881 (119879 119878 119868) = 120593 (119879 119878 119868)

(42)

where120593 (119879 119878 119868)

=

max ( 119868

119879minus 119870 0) for the arithmetic average

fixed strike call option

max (119870 minus119868

119879 0) for the arithmetic average

fixed strike put option

max (119878 minus119868

119879 0) for the arithmetic average

floating strike call option

max ( 119868

119879minus 119878 0) for the arithmetic average

floating strike put option(43)

6 Mathematical Problems in Engineering

23 The Arithmetic Average Fixed Strike Asian Call OptionLet us see the arithmetic average fixed strikeAsian call optionFrom (14) we have

119878119905119878

] = 119878 exp 120583 (] minus 119905) minus1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905)

(44)

Then by the definition of 119868] we have

119868119905119868

] = 119868 + 119878int

]

119905

exp 120583 (120591 minus 119905) minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591

(45)

Since

119881 (119905 119909) = 119881 (119905 119878 119868) = E [119890minus119903(119879minus119905)

119881 (119879 119878119879 119868

119879)]

= E [119890minus119903(119879minus119905)

(119868119879

119879minus 119870)

+

]

119881 (119905 119878 119868) = E[119890minus119903(119879minus119905)

times (((119868 + 119878int

]

119905

exp 120583 (120591 minus 119905)

minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591)

times119879minus1) minus 119870)

+

]

(46)

By the convexities of function 119891(119909) = (119909 minus 119896)+ and the

sublinear expectation we can say

119881(119905 1205821198781+ (1 minus 120582) 119878

2 119868) le 120582119881 (119905 119878

1 119868) + (1 minus 120582)119881 (119905 119878

2 119868)

(47)

that is 119881(119905 119878 119868) is convex about 119878 So in the numericalcomputation the second order difference of 119881 is positiveThen the numerical solutions of (43) are equivalent to thoseof PDE

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

119881 (119879 119878 119868) = (119868

119879minus 119870)

+

(48)

Theoretically the above PDE is defined on domain 119863 =

(119905 119878 119868) isin [0 119879] times [0infin) times [0infin) But generally closedform solutions cannot be found we have to find numericalsolutions We have no interest in finding the solution in the

entire infinite domain Therefore we consider the domain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119868

lowast)

Now we determine the boundary conditions of (48) If119868 ge 119870119879 that is 119868119879 ge 119870 for the positiveness of 119878] we have119868119879119879 ge 119870 Therefore the final payoff on the option is certain

to be positive At time 119879 this payoff can be expressed as

119868119879

119879minus 119870 =

119868

119879minus 119870 +

1

119879int

119879

119905

119878]119889] (49)

This payoff can also be obtained by using the followingself-financing duplicating portfolio strategy Assume that aninvestor commits (119868119879minus119870)119890

minus119903(119879minus119905) into riskless bonds in orderto secure the return promised by the first part of the rightof (49) that is (119868119879 minus 119870) at time 119879 If the investor is to becertain of accruing the return promised by the second part ofthe right of (49) he is obliged to transfer a certain portionof stock (equal to (1119879)119890

minus119903(119879minus])Δ]) to a riskless bond in every

time interval (] ] + Δ]) Overall this strategy requires a sumequal to (119868119879 minus 119870)119890

minus119903(119879minus119905) plus the portion of a stock whichcan be expressed as follows

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (50)

Then the price of the option when 119868119905ge 119870119879must be equal to

119881 (119905 119878 119868) = (119868

119879minus 119870) 119890

minus119903(119879minus119905)+

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (51)

The above discussion is similar to that in the model with-out volatility uncertainty Equation (51) obviously satisfies(48) So we only need to find the solution of (48) in thedomain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119870119879)

On the boundary 119868 = 119870119879 the boundary condition from(51) can be written as

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (52)

We note that if 119878119905= 0 (13) tells us that 119878(]) is also equal

to zero for ] isin [119905 119879] and 119868(119879) is thus equal to 119868(119905) By (46)we can write the following boundary condition

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

(53)

We need another boundary condition at 119878 = 119878max =

Slowast Hugger [37] gives the boundary condition at 119878 = 119878maxin the model without volatility uncertainty By the samediscussion we can get the same boundary condition inuncertain volatility model

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+Smax119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

(54)

Mathematical Problems in Engineering 7

So the values of the arithmetic average fixed strike Asiancall option satisfy

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

V (119879 119878 119868) = (119868

119879minus 119870)

+

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+

119878max119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

(55)

For the arithmetic average fixed strike Asian put optionwith terminal payoff 119901(119879 119878

119879 119868

119879) = max(119870 minus 119868

119879119879 0) we will

value this option by the following put-call parityIt is easily to see that we have

max(119868119879

119879minus 119870 0) = max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889] minus 119870

(56)

So

119888 (119905 119878119905 119868

119905)

= 119890minus119903(119879minus119905)

E119905[max (119868

119879

119879minus 119870 0)]

= 119890minus119903(119879minus119905)

E119905[max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889]]

minus 119890minus119903(119879minus119905)

119870

= 119901 (119905 119878119905 119868

119905) minus 119890

minus119903(119879minus119905)119870 +

1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889]

+ 119890minus119903(119879minus119905)

119864120590

119905[1

119879int

119879

119905

119878]119889]]

(57)

119890minus119903(119879minus119905)

119864120590

119905[(1119879) int

119879

119905119878]119889]] is the time 119905 value of time 119879

claim (1119879) int119879

119905119878]119889] where the stock has certain volatility

120590 To get claim (1119879) int119879

119905119878]119889] at 119879 let us see the replicating

strategy in any time interval (] ] + Δ]) transfer a certainportion (1119879)119890

minus119903(119879minus])Δ] of stock 119878] to a riskless bond This

strategy ensures the final payoff of (1119879) int119879

119905119878]119889] The stocks

that should be transferred are

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (58)

Then the time 119905 value of (1119879) int119879

119905119878]119889] equals (1119903119879)(1 minus

119890minus119903(119879minus119905)

)119878119905 So the put-call parity can be written as

119888 (119905 119878 119868) + 119870119890minus119903(119879minus119905)

= 119901 (119905 119878 119868) +1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889] +1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

119905

(59)

24 The Arithmetic Average Floating Strike Asian Call OptionFor the arithmetic average floating strike Asian call (put)option the PDE (43) still applies with terminal condition

119881 (119879 119878 119868) = (119878 minus119868

119879)

+

(119881 (119879 119878 119868) = (119868

119879minus 119878)

+

) (60)

In Ingersoll [13] where there is no volatility uncertaintydue to the linear homogeneity of the PDE and its terminalcondition their PDE can be reduced to a one-dimensionalPDE In our uncertain volatility model the PDE (43) is notlinear homogeneous anymore But we can still reduce PDE(43) to a one-dimensional PDE by the following argument

For the arithmetic average floating strike Asian calloption we know

119881 (119905 119878119905 119868

119905) = E

119905[119890

minus119903(119879minus119905)(119878

119879minus

119868119879

119879)

+

]

= E119905[

[

119890minus119903(119879minus119905)

(119878119879minus

119868119905+ int

119879

119905119878]119889]

119879)

+

]

]

(61)

Denote

119872]119905= exp(120583 (] minus 119905) minus

1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905))

(62)

Then by (14) we have 119878] = 119878119905119872

]119905and 119878

119905= 119878]119872

119905

] So

119868119905= int

119905

0

119878]119889] = int

119905

0

119878119905119872

]119905119889] = 119878

119905int

119905

0

119872]119905119889]

119881 (119905 119878119905 119868

119905)

= E119905[

[

119890minus119903(119879minus119905)

(119878119905119872

119879

119905minus

119878119905int119905

0119872

]119905119889] + 119878

119905int119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905E

119905[

[

119890minus119903(119879minus119905)

(119872119879

119905minus

int119905

0119872

]119905119889] + int

119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905119867(119905 119877

119905)

(63)with 119877

119905= 119868

119905119878

119905

Then the function119867(119905 119877) satisfies the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(64)

which is a one-dimensional PDE

8 Mathematical Problems in Engineering

For the European style Asian option we must imposeboundary conditions both at 119877 = 0 and as 119877 rarr infin Theboundary condition as 119877 rarr infin is simple The only way that119877 can tend to infinity is that 119878 tends to zero In this case theoption will not be exercised and so

lim119877rarrinfin

119867(119905 119877) = 0 (65)

For computational purpose we truncate the asset region(0 +infin) into (0 119877max) where 119877max is sufficiently large toensure the accuracy of the solution If 119877 = 0 the PDEdegenerates into the first-order hyperbolic PDE

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 (66)

So119867(119905 119877) satisfies

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (1 minus119877

119879)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = 0

(67)

By the same argument for the arithmetic average floatingstrike Asian put option its prices could be calculated by

119881 (119905 119878 119868) = 119878119867 (119905 119877) (68)

with119867(119905 119877) satisfying the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (119877

119879minus 1)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = (119877max119879

minus exp minus119903 (119879 minus 119905))

(69)

Generally the put-call parity will not hold any more inthe uncertain volatility model Let us see the following argu-ment The mature payoff of a portfolio with one Europeanarithmetic average floating strike Asian call holding long andone put holding short is

119878119879max (1 minus

119877119879

119879 0) minus 119878

119879max (119877

119879

119879minus 1 0)

= 119878119879minus

119877119879119878119879

119879

(70)

The value of this portfolio is equal to one consisting ofone stock and a financial product whose payoff is minus119877

119879119878119879119879

The values of the financial product with final payoff minus119877119879119878119879119879

satisfy the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(71)

with terminal condition 119867(119879 119877119879) = minus119877

119879119878119879119879 We seek a

solution of (71) of the form

119867(119905 119877) = 119886119905+ 119887

119905119877 (72)

with 119886119879

= 0 and 119887119879

= minus1119879 Substituting (72) into (71) andsatisfying the boundary conditions we find that

119886119905= minus

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119887

119905= minus

1

119879119890minus119903(119879minus119905)

(73)

For

119878119879max (1 minus

119877119879

119879 0) = 119878

119879max (119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879

(74)

by the valuation mechanism in space (ΩH E) we take thediscounted 119866-conditional expectation on both sides and get

E119905[119890

minus119903(119879minus119905)119878119879max(1 minus

119877119879

119879 0)]

= 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879]

le 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0)]

+ 119890minus119903(119879minus119905)

E119905[119878

119879minus

119877119879119878119879

119879]

(75)

That is

119888 (119905 119878 119868) le 119901 (119905 119878 119868) + 119878 minus119878

119903119879(1 minus 119890

minus119903(119879minus119905))

minus1

119879119890minus119903(119879minus119905)

int

119905

0

119878120591119889120591

(76)

where 119888(119905 119878 119868) is the time 119905 price of the arithmetic averagefloating strike Asian call option and 119901(119905 119878 119868) is the time119905 price of the arithmetic average floating strike Asian putoption

3 Numerical Computation

In the model without volatility uncertainty Asian optionsare much more difficult to value than regular stock optionsStandard techniques tend to be impractical inaccurate orslow For example traditional binomial lattice methodsrequire such enormous amounts of computermemory for thenecessity of keeping track of every possible path throughout

Mathematical Problems in Engineering 9

the tree that they are effectively unusable Monte Carlosimulation works well for European style options (see [12])but is relatively slow Sun [38] proposed to use the weightedupwind method [39] to price Asian options but he didnot give numerical results The Asian option pricing in theuncertain volatility model will be even more difficult than inthe model without volatility uncertainty

For the arithmetic average fixed strike Asian call optionsdue to the convexity of the value 119881(119905 119878 119868) with respect tovariable 119878 (43) can be written as (48) which are the equationssatisfied by the arithmetic average fixed strike Asian calloptions with certain volatility 120590

Zvan et al [17] point out that the Crank-Nicolson schemein conjunction with the van Leer flux limiter method canbe used to numerically value Asian options with certainvolatilities The van Leer limiter has the property that it istotal variation diminishing and thus produces oscillation freesecond order accurate solutions But the van Leer limiter isnonlinear so the solutions should be obtained by using fullNewton iteration at each time level In this paper we will usethe alternating direction implicit (ADI) methods (see Duffy[40])

The arithmetic average fixed strike Asian put options canbe valued by the put-call parity

For the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69))These PDEs are one-dimensional in space And they are not linear equations butHJB equations in some sense For such PDEs we will use thefully implicit positive coefficient finite different method [41]to solve them We will see that this method can ensure thenumerical solutions converge to the viscosity solutions andis accurate efficient and quick to be implemented

All the calculations were performed byMatlab 2009 on acomputer with 183GHz hard disk and 512MB memory

31 Numerical Computation of Fixed Strike Asian OptionsAn equivalent formulation of (48) in terms of the average119860

119905= (1119905) int

119905

0119878120591119889120591 is given in Barraquand and Pudet [16] in

certain volatility model with volatility 120590 = 120590

120597119905119881 (119905 119878 119860) +

1

2211987821205902119881119878119878

(119905 119878 119860) + 119903119878119881119878(119905 119878 119860)

+119878 minus 119860

119905119881119860(119905 119878 119860) minus 119903119881 (119905 119878 119860) = 0

119881 (119879 119878 119860) = (119860 minus 119870)+

(77)

Equation (77) is convection dominated in the 119860 direc-tion because there is no diffusion effect in this dimensionBarraquand and Pudet [16] find that an explicit centrallyweighted scheme for (77) is unstable In particular theconvective term in the 119860 dimension becomes very large as119905 rarr 0 Barraquand and Pudet also note that implicit cen-trally weighted schemes will generally produce unsatisfactoryresults because of the numerical diffusion introduced by thisfirst order accurate in time scheme

Zvan et al [17] point out that under the conditionof convection dominated solutions generated by using a

centrally weighted scheme on the convective term for (77)cannot be ensured to be free of oscillations And solutionsgenerated by the first order upstream weighting scheme onthe convective termare no longer oscillatory but their profilesare too diffuse To produce oscillation free solutions withoutthe excessive diffusion of first order upstream weightingZvan et al [17] suggest the nonlinear van Leer flux limiter inconjunction with the Crank-Nicolson scheme Since the fluxlimiter is nonlinear the solutions should be obtained by usingfull Newton iteration at each time level

The ADI method pioneered in the United States by Dou-glas Rachford Peaceman Gunn and others has a numberof advantages First explicit difference methods are rarelyused to solve initial boundary value problems because oftheir poor stability Implicit methods have superior stabilityproperties but unfortunately they are difficult to solve in twoand more dimensions Consequently ADI methods becomean alternative because they can be programmed by solvinga simple trigonal system of equations For the convection-dominated problems we could use the exponentially fittedschemes in each leg of the approximate scheme to ensure thestability of the results [40] Since Barraquand and Pudet [16]have used the forward shooting grid method and Zvan et al[17] have used the van Leer flux limiter we will test the ADImethod in this paper by comparing our results with theirs

For convenience let us first convert (55) to be thefollowing forward equation in time by substituting 119905 with120591 = 119879 minus 119905

minus 120597120591119881 (120591 119878 119868) +

1

2211987821205902119881119878119878

(120591 119878 119868) + 119903119878119881119878(120591 119878 119868)

+ 119878119881119868(120591 119878 119868) minus 119903119881 (120591 119878 119868) = 0

119881 (0 119878 119868) = (119868

119879minus 119870)

+

119881 (120591 0 119868) = 119890minus119903120591

(119868

119879minus 119870)

+

119881 (119905 119878max 119868) = max ( 119868

119879minus 119870) 119890

minus119903120591+

119878max119903119879

(1 minus 119890minus119903120591

) 0

119881 (120591 119878 119870119879) =1

119903119879(1 minus 119890

minus119903120591) 119878

(78)

We let 120591119899 119899 = 0 1 119873 be a set of portion points in

[0 119879] satisfying 0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 and use even

partition of time that is 120591119899= 119899Δ120591 119899 = 0 1 119873 Define the

partitions of space variables as 0 = 1198780lt 119878

1lt sdot sdot sdot lt 119878

119872= 119878max

and 0 = 1198680lt 119868

1lt sdot sdot sdot lt 119868

119869= 119868max

With ADI we march from time level 119899 to time level119899 + 12 and then from time level 119899 + 12 to time level 119899 + 1In this case we use exponential fitting in all space variables

10 Mathematical Problems in Engineering

and implicit Euler in time The first leg is given by thescheme

minus

119881119899+12

119894119895minus 119881

119899

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899

119894119895+1minus 119881

119899

119894119895minus1

Δ119868119895+1

+ Δ119868119895

minus 119903119881119899+12

119894119895= 0

(79)

with

120588119894119895

=119903119878

119894Δ119878

119894+1

2coth(

119903119878119894Δ119878

119894+1

12059022119878

2

119894

) coth (119909) =1198902119909

+ 1

1198902119909 minus 1

(80)

Let

120572119894= minus

120588119894119895

Δ119878119894Δ119878

119894+1

120573119894= minus

120588119894119895

(Δ119878119894+1

)2minus

119903119878119894

Δ119878119894+1

120574119894= minus 120572

119894minus 120573

119894+ 119903

(81)

Then the scheme (79) can be written as

120572119894119881

119899+12

119894minus1119895+ (120574

119894+

2

Δ120591)119881

119899+12

119894119895+ 120573

119894119881

119899+12

119894+1119895

=2

Δ120591119881

119899

119894119895+

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895+1minus

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895minus1

(82)

We let

119881119899

119895= [119881

119899

1119895 119881

119899

2119895 119881

119899

119872minus1119895]119879

119881119899+12

119895= [119881

119899+12

1119895 119881

119899+12

2119895 119881

119899+12

119872minus1119895]119879

119891119899+12

119895= [120572

1119881

119899+12

0119895 0 0 120573

119872minus1119881

119899+12

119872119895]119879

119903119899

119895= [(

2

Δ120591119881

119899

1119895+

1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895+1

minus1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895minus1)

(2

Δ120591119881

119899

119872minus1119895+

119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895+1

minus119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895minus1)]

119879

1198601=

[[[

[

1205741

1205731

1205722

1205742

1205732

d d d120572119872minus1

120574119872minus1

]]]

](119872minus1)times(119872minus1)

(83)

Then we can write (82) as the following equivalent matrixform

1198601119881

119899+12

119895+ 119891

119899+12

119895= 119903

119899

119895 (84)

The second leg of the discretized PDE in (78) is given bythe scheme

minus

119881119899+1

119894119895minus 119881

119899+12

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899+1

119894119895+1minus 119881

119899+1

119894119895

Δ119868119895+1

minus 119903119881119899+12

119894119895= 0

(85)

Let

120575119895=

2

Δ120591+

119878119894

Δ119868119895+1

120598119895= minus

119878119894

Δ119868119895+1

(86)

Then the scheme (85) can be written as

120575119895119881

119899+1

119894119895minus 120598

119895119881

119899+1

119894119895+1= minus 120572

119894119881

119899+12

119894minus1119895minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119895

minus 120573119894119881

119899+12

119894+1119895

(87)

Mathematical Problems in Engineering 11

300

300

250

200

200

150

100

100

100

50

50

0

0 0

Call

valu

e

sI

Figure 1 Fixed strike Asian call value when = 08 120590 = 05 119903 =

010 119879 = 1 and 119870 = 100

We let

119881119899+1

119894= [119881

119899+1

1198940 119881

119899+1

1198941 119881

119899+1

119894119869minus1]119879

119881119899+12

119894= [119881

119899+12

1198940 119881

119899+12

1198941 119881

119899+12

119894119869minus1]119879

119865119894= [0 0 0 120598

119869minus1119881

119899+1

119894119869]119879

119869

119877119899+12

119894= [(minus120572

119894119881

119899+12

119894minus10minus(120574

119894minus

2

Δ120591)119881

119899+12

1198940minus120573

119894119881

119899+12

119894+10)

(minus120572119894119881

119899+12

119894minus1119869minus1minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119869minus1minus 120573

119894119881

119899+12

119894+1119869minus1)]

119879

1198602=

[[[

[

1205750

minus1205980

0 1205751

minus1205981

d d minus120598119869minus2

120575119869minus1

]]]

]119869times119869

(88)

The matrix form of (87) is

1198602119881

119899+1

119894+ 119865

119899+1

119894= 119877

119899+12

119894 (89)

Thematrix equations (84) and (89) can be solved by using LUdecomposition

The goal of the computation is to find the fair price of theAsian option at emission that is119881(119878

0 119868

0 0)Weneed tomake

sense of any value 119868 isin [0 119868max] and stock price 119878 isin [0 119878max] attime 119905 = 0 There are no problems with the stock prices butthe integral at time zero can only be 0 for continuous averagesthat is 119868

0= 0

Figure 1 is the results we obtained by using the abovediscretization scheme with = 08 120590 = 05 119903 = 010 119879 = 1and 119870 = 100 The point marked by lowast is the price 119881(0 119878 0)

of the Asian option We choose 119878max = 3119870 to ensure thedesirable accuracy

Table 1 shows the convergence of the results with therefinement of the grid From (78) the values of variable 119868

lie in the interval [0 119870119879] where 119870119879 is the boundary and0 is the point where the continuous averages Asian options

are valued so there is no point that is less important on theinterval [0 119870119879] Hence we use uniform grids in direction 119868

In our uncertain volatility model the pricing PDEs (48)are similar to the PDEs in certain volatility model withvolatility 120590 = 120590 Barraquand and Pudet [16] and Zvan etal [17] give numerical results of Asian options with certainvolatilities In the uncertain volatility model when =

08 120590 = 05 the Asian option PDE is the same as that ofthe Asian option with certain volatility 120590 = 04 When theparameters 119903 = 010 119879 = 1 119870 = 95 and grid Δ119868 =

011875 ΔS = 285 the price of the Asian call option wecalculate is 1382 The results with the same parameters ofBarraquand and Pudet [16] and Zvan et al [17] are 13825 and13721 respectively

Table 2 is the comparison of our results with those ofZvan et al [17] Z F and V in Table 2 refer to the resultof Zvan et al [17] Our results are denoted by Asian fixedstrike which calculated with Δ120591 = 001 002 004 Δ119868 =

0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Since119868max = 119870119879 the partitions in 119868 direction are related with thematurity 119879 The execution time is much less than theirs Thegrid spacing was chosen to achieve an accuracy of at least010 of 119878 Our results are close to that of Zvan et al [17] andthe difference is less than 010 of 119878

Our calculation procedure is different from Zvan et al[17] as well as Barraquand and Pudet [16] in the followingfour aspects First our pricing PDE (78) is in terms of therunning sum 119868

119905 Zvan et al [17] and Barraquand and Pudet

[16] use the pricing PDE

minus 120597120591119881 (120591 119878 119860) +

1

211987821205902119881119878119878

(120591 119878 119860) + 119903119878119881119878(120591 119878 119860)

+119878 minus 119860

119905119881119860(120591 119878 119860) minus 119903119881 (120591 119878 119860) = 0

(90)

where 119860119905is defined as 119860

119905= 119868

119905119905 But PDE (78) and (90) are

equivalent for pricing fixed strike European Asian optionsSecond since the PDEs are different PDE (78) and (90) musthave different boundary conditions We do not know theboundary conditions used by Zvan et al [17] The forwardshooting grid method used by Barraquand and Pudet [16]proceeds without any boundary conditions Third our PDE(78) is defined on the domain 119863 = (119905 119878 119868) 0 le 119905 le 119879 0 le

119878 le 119878max 0 le 119868 le 119870119879 and 119868max = 119870119879 is the boundary and119868 = 0 is the point where the options are valued So there isno point that is less important on the interval [0 119870119879] Henceour numerical schemes use uniform grid spacing while Zvanet al [17] use nonuniform grid spacing of 41 times 45 on domain[0 300] times [0 300] To achieve an accuracy of at least 01 of119878 our grids are much finer than Zvan et al [17]

32 Numerical Computation of Floating Strike Asian OptionsTo value the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69)) for 119867(119905 119877) andthe values of the options will be 119881(119905 119878 119868) = 119878119867(119905 119877) with119877 = 119868119878 Actually what we really care about is the optionvalues at time 119905 = 0 while from the definition of 119868

119905 it is

obvious that 1198680= 0 and therefore 119877

0= 0 so our goal of the

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

Mathematical Problems in Engineering 5

For the second term on the right side of (33) we have

E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

= E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

times (119891 (119883119905119909

] 119884119905119909

] ) minus 119891 (119909 119884119905119909

119905)) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

times (119892 (119883119905119909

] 119884119905119909

] ) minus 119892 (119909 119884119905119909

119905)) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

+ E[

100381610038161003816100381610038161003816100381610038161003816

int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119909 119884119905119909

119905) 119889]

+ int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119909 119884119905119909

119905) 119889⟨119861⟩]

100381610038161003816100381610038161003816100381610038161003816

]

le 119862 (1 + |119909|) 120575

(37)

Therefore 119906 is continuous in 119905Now we are going to prove that 119906(119905 119909) is a viscosity

solution of PDE (30) For fixed (119905 119909) isin (0 119879) times R2 let 120595 isin

11986223

119887((0 119879) times R2

) such that 120595 ge 119906 and 120595(119905 119909) = 119906(119905 119909) By(29) and Taylorrsquos expansion for 120575 isin (0 119879 minus 119905)

0 le E[119890minus119903120575

120595 (119905 + 120575119883119905119909

119905+120575) minus 120595 (119905 119909)

+ int

119905+120575

119905

119890minus119903(]minus119905)

119891 (119883119905119909

] 119884119905119909

] ) 119889]

+int

119905+120575

119905

119890minus119903(]minus119905)

119892 (119883119905119909

] 119884119905119909

] ) 119889⟨119861⟩]]

le E [ minus 119903120595 (119905 119909) 120575 + 120597119905120595 (119905 119909) 120575

+ ⟨119863120595 (119905 119909) 120583119883119909⟩ 120575

+ ⟨1

2119863

2120595 (119905 119909)

119883119909

119883119909⟩ (⟨119861⟩

119905+120575minus ⟨119861⟩

119905)

+ 119891 (119909 120595) 120575 + 119892 (119909 120595) (⟨119861⟩119905+120575

minus ⟨119861⟩119905)]

+ 119862 (1 + |119909| + |119909|2+ |119909|

3) 120575

32

le (120597119905120595 + sup

1205902le1205902le1205902

(1

2⟨119863

2120595 (119905 119909)

119883119909

119883119909⟩

+119892 (119909 120595) ) 1205902+ ⟨119863120595 (119905 119909) 120583

119883119909⟩

minus119903120595 (119905 119909) + 119891 (119909 120595))120575

+ 119862 (1 + |119909| + |119909|2+ |119909|

3) 120575

32

(38)

It is easy to check that

120597119905120595 + sup

1205902le1205902le1205902

(1

2⟨119863

2120595 (119905 119909)

119883119909

119883119909⟩ + 119892 (119909 120595)) 120590

2

+ ⟨119863120595 (119905 119909) 120583119883119909⟩ minus 119903120595 (119905 119909) + 119891 (119909 120595) ge 0

(39)

Thus 119906 is a viscosity subsolution of (30) Similarly we canprove that 119906 is a viscosity supersolution of (30)

For the arithmetic average Asian options we have

119881 (119905 119909) = 119881 (119905 119878 119868) = E [119890minus119903(119879minus119905)

119881 (119879 119878119879 119868

119879)] (40)

Then by Corollary 5 119881(119905 119909) satisfies the following PDE

120597119905119881 (119905 119909) + sup

1205902le1205902le1205902

1

2⟨119863

2119881 (119905 119909)

119883119909

119883119909⟩120590

2

+ ⟨119863119881 (119905 119909) 120583119883119909⟩ minus 119903119881 (119905 119909) = 0

119881 (119879 119909) = 120593 (119879 119909)

(41)

By (18) the above PDE is

120597119905119881 (119905 119878 119868) + sup

1205902le1205902le1205902

1

2211987821205902119881119878119878

(119905 119878 119868)

+ 119903119878119881119878(119905 119878 119868) + 119878119881

119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

119881 (119879 119878 119868) = 120593 (119879 119878 119868)

(42)

where120593 (119879 119878 119868)

=

max ( 119868

119879minus 119870 0) for the arithmetic average

fixed strike call option

max (119870 minus119868

119879 0) for the arithmetic average

fixed strike put option

max (119878 minus119868

119879 0) for the arithmetic average

floating strike call option

max ( 119868

119879minus 119878 0) for the arithmetic average

floating strike put option(43)

6 Mathematical Problems in Engineering

23 The Arithmetic Average Fixed Strike Asian Call OptionLet us see the arithmetic average fixed strikeAsian call optionFrom (14) we have

119878119905119878

] = 119878 exp 120583 (] minus 119905) minus1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905)

(44)

Then by the definition of 119868] we have

119868119905119868

] = 119868 + 119878int

]

119905

exp 120583 (120591 minus 119905) minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591

(45)

Since

119881 (119905 119909) = 119881 (119905 119878 119868) = E [119890minus119903(119879minus119905)

119881 (119879 119878119879 119868

119879)]

= E [119890minus119903(119879minus119905)

(119868119879

119879minus 119870)

+

]

119881 (119905 119878 119868) = E[119890minus119903(119879minus119905)

times (((119868 + 119878int

]

119905

exp 120583 (120591 minus 119905)

minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591)

times119879minus1) minus 119870)

+

]

(46)

By the convexities of function 119891(119909) = (119909 minus 119896)+ and the

sublinear expectation we can say

119881(119905 1205821198781+ (1 minus 120582) 119878

2 119868) le 120582119881 (119905 119878

1 119868) + (1 minus 120582)119881 (119905 119878

2 119868)

(47)

that is 119881(119905 119878 119868) is convex about 119878 So in the numericalcomputation the second order difference of 119881 is positiveThen the numerical solutions of (43) are equivalent to thoseof PDE

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

119881 (119879 119878 119868) = (119868

119879minus 119870)

+

(48)

Theoretically the above PDE is defined on domain 119863 =

(119905 119878 119868) isin [0 119879] times [0infin) times [0infin) But generally closedform solutions cannot be found we have to find numericalsolutions We have no interest in finding the solution in the

entire infinite domain Therefore we consider the domain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119868

lowast)

Now we determine the boundary conditions of (48) If119868 ge 119870119879 that is 119868119879 ge 119870 for the positiveness of 119878] we have119868119879119879 ge 119870 Therefore the final payoff on the option is certain

to be positive At time 119879 this payoff can be expressed as

119868119879

119879minus 119870 =

119868

119879minus 119870 +

1

119879int

119879

119905

119878]119889] (49)

This payoff can also be obtained by using the followingself-financing duplicating portfolio strategy Assume that aninvestor commits (119868119879minus119870)119890

minus119903(119879minus119905) into riskless bonds in orderto secure the return promised by the first part of the rightof (49) that is (119868119879 minus 119870) at time 119879 If the investor is to becertain of accruing the return promised by the second part ofthe right of (49) he is obliged to transfer a certain portionof stock (equal to (1119879)119890

minus119903(119879minus])Δ]) to a riskless bond in every

time interval (] ] + Δ]) Overall this strategy requires a sumequal to (119868119879 minus 119870)119890

minus119903(119879minus119905) plus the portion of a stock whichcan be expressed as follows

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (50)

Then the price of the option when 119868119905ge 119870119879must be equal to

119881 (119905 119878 119868) = (119868

119879minus 119870) 119890

minus119903(119879minus119905)+

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (51)

The above discussion is similar to that in the model with-out volatility uncertainty Equation (51) obviously satisfies(48) So we only need to find the solution of (48) in thedomain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119870119879)

On the boundary 119868 = 119870119879 the boundary condition from(51) can be written as

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (52)

We note that if 119878119905= 0 (13) tells us that 119878(]) is also equal

to zero for ] isin [119905 119879] and 119868(119879) is thus equal to 119868(119905) By (46)we can write the following boundary condition

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

(53)

We need another boundary condition at 119878 = 119878max =

Slowast Hugger [37] gives the boundary condition at 119878 = 119878maxin the model without volatility uncertainty By the samediscussion we can get the same boundary condition inuncertain volatility model

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+Smax119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

(54)

Mathematical Problems in Engineering 7

So the values of the arithmetic average fixed strike Asiancall option satisfy

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

V (119879 119878 119868) = (119868

119879minus 119870)

+

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+

119878max119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

(55)

For the arithmetic average fixed strike Asian put optionwith terminal payoff 119901(119879 119878

119879 119868

119879) = max(119870 minus 119868

119879119879 0) we will

value this option by the following put-call parityIt is easily to see that we have

max(119868119879

119879minus 119870 0) = max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889] minus 119870

(56)

So

119888 (119905 119878119905 119868

119905)

= 119890minus119903(119879minus119905)

E119905[max (119868

119879

119879minus 119870 0)]

= 119890minus119903(119879minus119905)

E119905[max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889]]

minus 119890minus119903(119879minus119905)

119870

= 119901 (119905 119878119905 119868

119905) minus 119890

minus119903(119879minus119905)119870 +

1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889]

+ 119890minus119903(119879minus119905)

119864120590

119905[1

119879int

119879

119905

119878]119889]]

(57)

119890minus119903(119879minus119905)

119864120590

119905[(1119879) int

119879

119905119878]119889]] is the time 119905 value of time 119879

claim (1119879) int119879

119905119878]119889] where the stock has certain volatility

120590 To get claim (1119879) int119879

119905119878]119889] at 119879 let us see the replicating

strategy in any time interval (] ] + Δ]) transfer a certainportion (1119879)119890

minus119903(119879minus])Δ] of stock 119878] to a riskless bond This

strategy ensures the final payoff of (1119879) int119879

119905119878]119889] The stocks

that should be transferred are

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (58)

Then the time 119905 value of (1119879) int119879

119905119878]119889] equals (1119903119879)(1 minus

119890minus119903(119879minus119905)

)119878119905 So the put-call parity can be written as

119888 (119905 119878 119868) + 119870119890minus119903(119879minus119905)

= 119901 (119905 119878 119868) +1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889] +1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

119905

(59)

24 The Arithmetic Average Floating Strike Asian Call OptionFor the arithmetic average floating strike Asian call (put)option the PDE (43) still applies with terminal condition

119881 (119879 119878 119868) = (119878 minus119868

119879)

+

(119881 (119879 119878 119868) = (119868

119879minus 119878)

+

) (60)

In Ingersoll [13] where there is no volatility uncertaintydue to the linear homogeneity of the PDE and its terminalcondition their PDE can be reduced to a one-dimensionalPDE In our uncertain volatility model the PDE (43) is notlinear homogeneous anymore But we can still reduce PDE(43) to a one-dimensional PDE by the following argument

For the arithmetic average floating strike Asian calloption we know

119881 (119905 119878119905 119868

119905) = E

119905[119890

minus119903(119879minus119905)(119878

119879minus

119868119879

119879)

+

]

= E119905[

[

119890minus119903(119879minus119905)

(119878119879minus

119868119905+ int

119879

119905119878]119889]

119879)

+

]

]

(61)

Denote

119872]119905= exp(120583 (] minus 119905) minus

1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905))

(62)

Then by (14) we have 119878] = 119878119905119872

]119905and 119878

119905= 119878]119872

119905

] So

119868119905= int

119905

0

119878]119889] = int

119905

0

119878119905119872

]119905119889] = 119878

119905int

119905

0

119872]119905119889]

119881 (119905 119878119905 119868

119905)

= E119905[

[

119890minus119903(119879minus119905)

(119878119905119872

119879

119905minus

119878119905int119905

0119872

]119905119889] + 119878

119905int119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905E

119905[

[

119890minus119903(119879minus119905)

(119872119879

119905minus

int119905

0119872

]119905119889] + int

119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905119867(119905 119877

119905)

(63)with 119877

119905= 119868

119905119878

119905

Then the function119867(119905 119877) satisfies the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(64)

which is a one-dimensional PDE

8 Mathematical Problems in Engineering

For the European style Asian option we must imposeboundary conditions both at 119877 = 0 and as 119877 rarr infin Theboundary condition as 119877 rarr infin is simple The only way that119877 can tend to infinity is that 119878 tends to zero In this case theoption will not be exercised and so

lim119877rarrinfin

119867(119905 119877) = 0 (65)

For computational purpose we truncate the asset region(0 +infin) into (0 119877max) where 119877max is sufficiently large toensure the accuracy of the solution If 119877 = 0 the PDEdegenerates into the first-order hyperbolic PDE

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 (66)

So119867(119905 119877) satisfies

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (1 minus119877

119879)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = 0

(67)

By the same argument for the arithmetic average floatingstrike Asian put option its prices could be calculated by

119881 (119905 119878 119868) = 119878119867 (119905 119877) (68)

with119867(119905 119877) satisfying the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (119877

119879minus 1)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = (119877max119879

minus exp minus119903 (119879 minus 119905))

(69)

Generally the put-call parity will not hold any more inthe uncertain volatility model Let us see the following argu-ment The mature payoff of a portfolio with one Europeanarithmetic average floating strike Asian call holding long andone put holding short is

119878119879max (1 minus

119877119879

119879 0) minus 119878

119879max (119877

119879

119879minus 1 0)

= 119878119879minus

119877119879119878119879

119879

(70)

The value of this portfolio is equal to one consisting ofone stock and a financial product whose payoff is minus119877

119879119878119879119879

The values of the financial product with final payoff minus119877119879119878119879119879

satisfy the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(71)

with terminal condition 119867(119879 119877119879) = minus119877

119879119878119879119879 We seek a

solution of (71) of the form

119867(119905 119877) = 119886119905+ 119887

119905119877 (72)

with 119886119879

= 0 and 119887119879

= minus1119879 Substituting (72) into (71) andsatisfying the boundary conditions we find that

119886119905= minus

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119887

119905= minus

1

119879119890minus119903(119879minus119905)

(73)

For

119878119879max (1 minus

119877119879

119879 0) = 119878

119879max (119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879

(74)

by the valuation mechanism in space (ΩH E) we take thediscounted 119866-conditional expectation on both sides and get

E119905[119890

minus119903(119879minus119905)119878119879max(1 minus

119877119879

119879 0)]

= 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879]

le 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0)]

+ 119890minus119903(119879minus119905)

E119905[119878

119879minus

119877119879119878119879

119879]

(75)

That is

119888 (119905 119878 119868) le 119901 (119905 119878 119868) + 119878 minus119878

119903119879(1 minus 119890

minus119903(119879minus119905))

minus1

119879119890minus119903(119879minus119905)

int

119905

0

119878120591119889120591

(76)

where 119888(119905 119878 119868) is the time 119905 price of the arithmetic averagefloating strike Asian call option and 119901(119905 119878 119868) is the time119905 price of the arithmetic average floating strike Asian putoption

3 Numerical Computation

In the model without volatility uncertainty Asian optionsare much more difficult to value than regular stock optionsStandard techniques tend to be impractical inaccurate orslow For example traditional binomial lattice methodsrequire such enormous amounts of computermemory for thenecessity of keeping track of every possible path throughout

Mathematical Problems in Engineering 9

the tree that they are effectively unusable Monte Carlosimulation works well for European style options (see [12])but is relatively slow Sun [38] proposed to use the weightedupwind method [39] to price Asian options but he didnot give numerical results The Asian option pricing in theuncertain volatility model will be even more difficult than inthe model without volatility uncertainty

For the arithmetic average fixed strike Asian call optionsdue to the convexity of the value 119881(119905 119878 119868) with respect tovariable 119878 (43) can be written as (48) which are the equationssatisfied by the arithmetic average fixed strike Asian calloptions with certain volatility 120590

Zvan et al [17] point out that the Crank-Nicolson schemein conjunction with the van Leer flux limiter method canbe used to numerically value Asian options with certainvolatilities The van Leer limiter has the property that it istotal variation diminishing and thus produces oscillation freesecond order accurate solutions But the van Leer limiter isnonlinear so the solutions should be obtained by using fullNewton iteration at each time level In this paper we will usethe alternating direction implicit (ADI) methods (see Duffy[40])

The arithmetic average fixed strike Asian put options canbe valued by the put-call parity

For the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69))These PDEs are one-dimensional in space And they are not linear equations butHJB equations in some sense For such PDEs we will use thefully implicit positive coefficient finite different method [41]to solve them We will see that this method can ensure thenumerical solutions converge to the viscosity solutions andis accurate efficient and quick to be implemented

All the calculations were performed byMatlab 2009 on acomputer with 183GHz hard disk and 512MB memory

31 Numerical Computation of Fixed Strike Asian OptionsAn equivalent formulation of (48) in terms of the average119860

119905= (1119905) int

119905

0119878120591119889120591 is given in Barraquand and Pudet [16] in

certain volatility model with volatility 120590 = 120590

120597119905119881 (119905 119878 119860) +

1

2211987821205902119881119878119878

(119905 119878 119860) + 119903119878119881119878(119905 119878 119860)

+119878 minus 119860

119905119881119860(119905 119878 119860) minus 119903119881 (119905 119878 119860) = 0

119881 (119879 119878 119860) = (119860 minus 119870)+

(77)

Equation (77) is convection dominated in the 119860 direc-tion because there is no diffusion effect in this dimensionBarraquand and Pudet [16] find that an explicit centrallyweighted scheme for (77) is unstable In particular theconvective term in the 119860 dimension becomes very large as119905 rarr 0 Barraquand and Pudet also note that implicit cen-trally weighted schemes will generally produce unsatisfactoryresults because of the numerical diffusion introduced by thisfirst order accurate in time scheme

Zvan et al [17] point out that under the conditionof convection dominated solutions generated by using a

centrally weighted scheme on the convective term for (77)cannot be ensured to be free of oscillations And solutionsgenerated by the first order upstream weighting scheme onthe convective termare no longer oscillatory but their profilesare too diffuse To produce oscillation free solutions withoutthe excessive diffusion of first order upstream weightingZvan et al [17] suggest the nonlinear van Leer flux limiter inconjunction with the Crank-Nicolson scheme Since the fluxlimiter is nonlinear the solutions should be obtained by usingfull Newton iteration at each time level

The ADI method pioneered in the United States by Dou-glas Rachford Peaceman Gunn and others has a numberof advantages First explicit difference methods are rarelyused to solve initial boundary value problems because oftheir poor stability Implicit methods have superior stabilityproperties but unfortunately they are difficult to solve in twoand more dimensions Consequently ADI methods becomean alternative because they can be programmed by solvinga simple trigonal system of equations For the convection-dominated problems we could use the exponentially fittedschemes in each leg of the approximate scheme to ensure thestability of the results [40] Since Barraquand and Pudet [16]have used the forward shooting grid method and Zvan et al[17] have used the van Leer flux limiter we will test the ADImethod in this paper by comparing our results with theirs

For convenience let us first convert (55) to be thefollowing forward equation in time by substituting 119905 with120591 = 119879 minus 119905

minus 120597120591119881 (120591 119878 119868) +

1

2211987821205902119881119878119878

(120591 119878 119868) + 119903119878119881119878(120591 119878 119868)

+ 119878119881119868(120591 119878 119868) minus 119903119881 (120591 119878 119868) = 0

119881 (0 119878 119868) = (119868

119879minus 119870)

+

119881 (120591 0 119868) = 119890minus119903120591

(119868

119879minus 119870)

+

119881 (119905 119878max 119868) = max ( 119868

119879minus 119870) 119890

minus119903120591+

119878max119903119879

(1 minus 119890minus119903120591

) 0

119881 (120591 119878 119870119879) =1

119903119879(1 minus 119890

minus119903120591) 119878

(78)

We let 120591119899 119899 = 0 1 119873 be a set of portion points in

[0 119879] satisfying 0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 and use even

partition of time that is 120591119899= 119899Δ120591 119899 = 0 1 119873 Define the

partitions of space variables as 0 = 1198780lt 119878

1lt sdot sdot sdot lt 119878

119872= 119878max

and 0 = 1198680lt 119868

1lt sdot sdot sdot lt 119868

119869= 119868max

With ADI we march from time level 119899 to time level119899 + 12 and then from time level 119899 + 12 to time level 119899 + 1In this case we use exponential fitting in all space variables

10 Mathematical Problems in Engineering

and implicit Euler in time The first leg is given by thescheme

minus

119881119899+12

119894119895minus 119881

119899

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899

119894119895+1minus 119881

119899

119894119895minus1

Δ119868119895+1

+ Δ119868119895

minus 119903119881119899+12

119894119895= 0

(79)

with

120588119894119895

=119903119878

119894Δ119878

119894+1

2coth(

119903119878119894Δ119878

119894+1

12059022119878

2

119894

) coth (119909) =1198902119909

+ 1

1198902119909 minus 1

(80)

Let

120572119894= minus

120588119894119895

Δ119878119894Δ119878

119894+1

120573119894= minus

120588119894119895

(Δ119878119894+1

)2minus

119903119878119894

Δ119878119894+1

120574119894= minus 120572

119894minus 120573

119894+ 119903

(81)

Then the scheme (79) can be written as

120572119894119881

119899+12

119894minus1119895+ (120574

119894+

2

Δ120591)119881

119899+12

119894119895+ 120573

119894119881

119899+12

119894+1119895

=2

Δ120591119881

119899

119894119895+

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895+1minus

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895minus1

(82)

We let

119881119899

119895= [119881

119899

1119895 119881

119899

2119895 119881

119899

119872minus1119895]119879

119881119899+12

119895= [119881

119899+12

1119895 119881

119899+12

2119895 119881

119899+12

119872minus1119895]119879

119891119899+12

119895= [120572

1119881

119899+12

0119895 0 0 120573

119872minus1119881

119899+12

119872119895]119879

119903119899

119895= [(

2

Δ120591119881

119899

1119895+

1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895+1

minus1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895minus1)

(2

Δ120591119881

119899

119872minus1119895+

119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895+1

minus119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895minus1)]

119879

1198601=

[[[

[

1205741

1205731

1205722

1205742

1205732

d d d120572119872minus1

120574119872minus1

]]]

](119872minus1)times(119872minus1)

(83)

Then we can write (82) as the following equivalent matrixform

1198601119881

119899+12

119895+ 119891

119899+12

119895= 119903

119899

119895 (84)

The second leg of the discretized PDE in (78) is given bythe scheme

minus

119881119899+1

119894119895minus 119881

119899+12

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899+1

119894119895+1minus 119881

119899+1

119894119895

Δ119868119895+1

minus 119903119881119899+12

119894119895= 0

(85)

Let

120575119895=

2

Δ120591+

119878119894

Δ119868119895+1

120598119895= minus

119878119894

Δ119868119895+1

(86)

Then the scheme (85) can be written as

120575119895119881

119899+1

119894119895minus 120598

119895119881

119899+1

119894119895+1= minus 120572

119894119881

119899+12

119894minus1119895minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119895

minus 120573119894119881

119899+12

119894+1119895

(87)

Mathematical Problems in Engineering 11

300

300

250

200

200

150

100

100

100

50

50

0

0 0

Call

valu

e

sI

Figure 1 Fixed strike Asian call value when = 08 120590 = 05 119903 =

010 119879 = 1 and 119870 = 100

We let

119881119899+1

119894= [119881

119899+1

1198940 119881

119899+1

1198941 119881

119899+1

119894119869minus1]119879

119881119899+12

119894= [119881

119899+12

1198940 119881

119899+12

1198941 119881

119899+12

119894119869minus1]119879

119865119894= [0 0 0 120598

119869minus1119881

119899+1

119894119869]119879

119869

119877119899+12

119894= [(minus120572

119894119881

119899+12

119894minus10minus(120574

119894minus

2

Δ120591)119881

119899+12

1198940minus120573

119894119881

119899+12

119894+10)

(minus120572119894119881

119899+12

119894minus1119869minus1minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119869minus1minus 120573

119894119881

119899+12

119894+1119869minus1)]

119879

1198602=

[[[

[

1205750

minus1205980

0 1205751

minus1205981

d d minus120598119869minus2

120575119869minus1

]]]

]119869times119869

(88)

The matrix form of (87) is

1198602119881

119899+1

119894+ 119865

119899+1

119894= 119877

119899+12

119894 (89)

Thematrix equations (84) and (89) can be solved by using LUdecomposition

The goal of the computation is to find the fair price of theAsian option at emission that is119881(119878

0 119868

0 0)Weneed tomake

sense of any value 119868 isin [0 119868max] and stock price 119878 isin [0 119878max] attime 119905 = 0 There are no problems with the stock prices butthe integral at time zero can only be 0 for continuous averagesthat is 119868

0= 0

Figure 1 is the results we obtained by using the abovediscretization scheme with = 08 120590 = 05 119903 = 010 119879 = 1and 119870 = 100 The point marked by lowast is the price 119881(0 119878 0)

of the Asian option We choose 119878max = 3119870 to ensure thedesirable accuracy

Table 1 shows the convergence of the results with therefinement of the grid From (78) the values of variable 119868

lie in the interval [0 119870119879] where 119870119879 is the boundary and0 is the point where the continuous averages Asian options

are valued so there is no point that is less important on theinterval [0 119870119879] Hence we use uniform grids in direction 119868

In our uncertain volatility model the pricing PDEs (48)are similar to the PDEs in certain volatility model withvolatility 120590 = 120590 Barraquand and Pudet [16] and Zvan etal [17] give numerical results of Asian options with certainvolatilities In the uncertain volatility model when =

08 120590 = 05 the Asian option PDE is the same as that ofthe Asian option with certain volatility 120590 = 04 When theparameters 119903 = 010 119879 = 1 119870 = 95 and grid Δ119868 =

011875 ΔS = 285 the price of the Asian call option wecalculate is 1382 The results with the same parameters ofBarraquand and Pudet [16] and Zvan et al [17] are 13825 and13721 respectively

Table 2 is the comparison of our results with those ofZvan et al [17] Z F and V in Table 2 refer to the resultof Zvan et al [17] Our results are denoted by Asian fixedstrike which calculated with Δ120591 = 001 002 004 Δ119868 =

0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Since119868max = 119870119879 the partitions in 119868 direction are related with thematurity 119879 The execution time is much less than theirs Thegrid spacing was chosen to achieve an accuracy of at least010 of 119878 Our results are close to that of Zvan et al [17] andthe difference is less than 010 of 119878

Our calculation procedure is different from Zvan et al[17] as well as Barraquand and Pudet [16] in the followingfour aspects First our pricing PDE (78) is in terms of therunning sum 119868

119905 Zvan et al [17] and Barraquand and Pudet

[16] use the pricing PDE

minus 120597120591119881 (120591 119878 119860) +

1

211987821205902119881119878119878

(120591 119878 119860) + 119903119878119881119878(120591 119878 119860)

+119878 minus 119860

119905119881119860(120591 119878 119860) minus 119903119881 (120591 119878 119860) = 0

(90)

where 119860119905is defined as 119860

119905= 119868

119905119905 But PDE (78) and (90) are

equivalent for pricing fixed strike European Asian optionsSecond since the PDEs are different PDE (78) and (90) musthave different boundary conditions We do not know theboundary conditions used by Zvan et al [17] The forwardshooting grid method used by Barraquand and Pudet [16]proceeds without any boundary conditions Third our PDE(78) is defined on the domain 119863 = (119905 119878 119868) 0 le 119905 le 119879 0 le

119878 le 119878max 0 le 119868 le 119870119879 and 119868max = 119870119879 is the boundary and119868 = 0 is the point where the options are valued So there isno point that is less important on the interval [0 119870119879] Henceour numerical schemes use uniform grid spacing while Zvanet al [17] use nonuniform grid spacing of 41 times 45 on domain[0 300] times [0 300] To achieve an accuracy of at least 01 of119878 our grids are much finer than Zvan et al [17]

32 Numerical Computation of Floating Strike Asian OptionsTo value the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69)) for 119867(119905 119877) andthe values of the options will be 119881(119905 119878 119868) = 119878119867(119905 119877) with119877 = 119868119878 Actually what we really care about is the optionvalues at time 119905 = 0 while from the definition of 119868

119905 it is

obvious that 1198680= 0 and therefore 119877

0= 0 so our goal of the

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Page 6: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

6 Mathematical Problems in Engineering

23 The Arithmetic Average Fixed Strike Asian Call OptionLet us see the arithmetic average fixed strikeAsian call optionFrom (14) we have

119878119905119878

] = 119878 exp 120583 (] minus 119905) minus1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905)

(44)

Then by the definition of 119868] we have

119868119905119868

] = 119868 + 119878int

]

119905

exp 120583 (120591 minus 119905) minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591

(45)

Since

119881 (119905 119909) = 119881 (119905 119878 119868) = E [119890minus119903(119879minus119905)

119881 (119879 119878119879 119868

119879)]

= E [119890minus119903(119879minus119905)

(119868119879

119879minus 119870)

+

]

119881 (119905 119878 119868) = E[119890minus119903(119879minus119905)

times (((119868 + 119878int

]

119905

exp 120583 (120591 minus 119905)

minus1

22(⟨119861⟩

120591minus ⟨119861⟩

119905)

+ (119861120591minus 119861

119905) 119889120591)

times119879minus1) minus 119870)

+

]

(46)

By the convexities of function 119891(119909) = (119909 minus 119896)+ and the

sublinear expectation we can say

119881(119905 1205821198781+ (1 minus 120582) 119878

2 119868) le 120582119881 (119905 119878

1 119868) + (1 minus 120582)119881 (119905 119878

2 119868)

(47)

that is 119881(119905 119878 119868) is convex about 119878 So in the numericalcomputation the second order difference of 119881 is positiveThen the numerical solutions of (43) are equivalent to thoseof PDE

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

119881 (119879 119878 119868) = (119868

119879minus 119870)

+

(48)

Theoretically the above PDE is defined on domain 119863 =

(119905 119878 119868) isin [0 119879] times [0infin) times [0infin) But generally closedform solutions cannot be found we have to find numericalsolutions We have no interest in finding the solution in the

entire infinite domain Therefore we consider the domain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119868

lowast)

Now we determine the boundary conditions of (48) If119868 ge 119870119879 that is 119868119879 ge 119870 for the positiveness of 119878] we have119868119879119879 ge 119870 Therefore the final payoff on the option is certain

to be positive At time 119879 this payoff can be expressed as

119868119879

119879minus 119870 =

119868

119879minus 119870 +

1

119879int

119879

119905

119878]119889] (49)

This payoff can also be obtained by using the followingself-financing duplicating portfolio strategy Assume that aninvestor commits (119868119879minus119870)119890

minus119903(119879minus119905) into riskless bonds in orderto secure the return promised by the first part of the rightof (49) that is (119868119879 minus 119870) at time 119879 If the investor is to becertain of accruing the return promised by the second part ofthe right of (49) he is obliged to transfer a certain portionof stock (equal to (1119879)119890

minus119903(119879minus])Δ]) to a riskless bond in every

time interval (] ] + Δ]) Overall this strategy requires a sumequal to (119868119879 minus 119870)119890

minus119903(119879minus119905) plus the portion of a stock whichcan be expressed as follows

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (50)

Then the price of the option when 119868119905ge 119870119879must be equal to

119881 (119905 119878 119868) = (119868

119879minus 119870) 119890

minus119903(119879minus119905)+

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (51)

The above discussion is similar to that in the model with-out volatility uncertainty Equation (51) obviously satisfies(48) So we only need to find the solution of (48) in thedomain119863

lowast= (119905 119878 119868) isin [0 119879] times [0 119878

lowast) times [0 119870119879)

On the boundary 119868 = 119870119879 the boundary condition from(51) can be written as

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878 (52)

We note that if 119878119905= 0 (13) tells us that 119878(]) is also equal

to zero for ] isin [119905 119879] and 119868(119879) is thus equal to 119868(119905) By (46)we can write the following boundary condition

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

(53)

We need another boundary condition at 119878 = 119878max =

Slowast Hugger [37] gives the boundary condition at 119878 = 119878maxin the model without volatility uncertainty By the samediscussion we can get the same boundary condition inuncertain volatility model

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+Smax119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

(54)

Mathematical Problems in Engineering 7

So the values of the arithmetic average fixed strike Asiancall option satisfy

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

V (119879 119878 119868) = (119868

119879minus 119870)

+

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+

119878max119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

(55)

For the arithmetic average fixed strike Asian put optionwith terminal payoff 119901(119879 119878

119879 119868

119879) = max(119870 minus 119868

119879119879 0) we will

value this option by the following put-call parityIt is easily to see that we have

max(119868119879

119879minus 119870 0) = max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889] minus 119870

(56)

So

119888 (119905 119878119905 119868

119905)

= 119890minus119903(119879minus119905)

E119905[max (119868

119879

119879minus 119870 0)]

= 119890minus119903(119879minus119905)

E119905[max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889]]

minus 119890minus119903(119879minus119905)

119870

= 119901 (119905 119878119905 119868

119905) minus 119890

minus119903(119879minus119905)119870 +

1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889]

+ 119890minus119903(119879minus119905)

119864120590

119905[1

119879int

119879

119905

119878]119889]]

(57)

119890minus119903(119879minus119905)

119864120590

119905[(1119879) int

119879

119905119878]119889]] is the time 119905 value of time 119879

claim (1119879) int119879

119905119878]119889] where the stock has certain volatility

120590 To get claim (1119879) int119879

119905119878]119889] at 119879 let us see the replicating

strategy in any time interval (] ] + Δ]) transfer a certainportion (1119879)119890

minus119903(119879minus])Δ] of stock 119878] to a riskless bond This

strategy ensures the final payoff of (1119879) int119879

119905119878]119889] The stocks

that should be transferred are

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (58)

Then the time 119905 value of (1119879) int119879

119905119878]119889] equals (1119903119879)(1 minus

119890minus119903(119879minus119905)

)119878119905 So the put-call parity can be written as

119888 (119905 119878 119868) + 119870119890minus119903(119879minus119905)

= 119901 (119905 119878 119868) +1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889] +1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

119905

(59)

24 The Arithmetic Average Floating Strike Asian Call OptionFor the arithmetic average floating strike Asian call (put)option the PDE (43) still applies with terminal condition

119881 (119879 119878 119868) = (119878 minus119868

119879)

+

(119881 (119879 119878 119868) = (119868

119879minus 119878)

+

) (60)

In Ingersoll [13] where there is no volatility uncertaintydue to the linear homogeneity of the PDE and its terminalcondition their PDE can be reduced to a one-dimensionalPDE In our uncertain volatility model the PDE (43) is notlinear homogeneous anymore But we can still reduce PDE(43) to a one-dimensional PDE by the following argument

For the arithmetic average floating strike Asian calloption we know

119881 (119905 119878119905 119868

119905) = E

119905[119890

minus119903(119879minus119905)(119878

119879minus

119868119879

119879)

+

]

= E119905[

[

119890minus119903(119879minus119905)

(119878119879minus

119868119905+ int

119879

119905119878]119889]

119879)

+

]

]

(61)

Denote

119872]119905= exp(120583 (] minus 119905) minus

1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905))

(62)

Then by (14) we have 119878] = 119878119905119872

]119905and 119878

119905= 119878]119872

119905

] So

119868119905= int

119905

0

119878]119889] = int

119905

0

119878119905119872

]119905119889] = 119878

119905int

119905

0

119872]119905119889]

119881 (119905 119878119905 119868

119905)

= E119905[

[

119890minus119903(119879minus119905)

(119878119905119872

119879

119905minus

119878119905int119905

0119872

]119905119889] + 119878

119905int119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905E

119905[

[

119890minus119903(119879minus119905)

(119872119879

119905minus

int119905

0119872

]119905119889] + int

119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905119867(119905 119877

119905)

(63)with 119877

119905= 119868

119905119878

119905

Then the function119867(119905 119877) satisfies the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(64)

which is a one-dimensional PDE

8 Mathematical Problems in Engineering

For the European style Asian option we must imposeboundary conditions both at 119877 = 0 and as 119877 rarr infin Theboundary condition as 119877 rarr infin is simple The only way that119877 can tend to infinity is that 119878 tends to zero In this case theoption will not be exercised and so

lim119877rarrinfin

119867(119905 119877) = 0 (65)

For computational purpose we truncate the asset region(0 +infin) into (0 119877max) where 119877max is sufficiently large toensure the accuracy of the solution If 119877 = 0 the PDEdegenerates into the first-order hyperbolic PDE

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 (66)

So119867(119905 119877) satisfies

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (1 minus119877

119879)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = 0

(67)

By the same argument for the arithmetic average floatingstrike Asian put option its prices could be calculated by

119881 (119905 119878 119868) = 119878119867 (119905 119877) (68)

with119867(119905 119877) satisfying the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (119877

119879minus 1)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = (119877max119879

minus exp minus119903 (119879 minus 119905))

(69)

Generally the put-call parity will not hold any more inthe uncertain volatility model Let us see the following argu-ment The mature payoff of a portfolio with one Europeanarithmetic average floating strike Asian call holding long andone put holding short is

119878119879max (1 minus

119877119879

119879 0) minus 119878

119879max (119877

119879

119879minus 1 0)

= 119878119879minus

119877119879119878119879

119879

(70)

The value of this portfolio is equal to one consisting ofone stock and a financial product whose payoff is minus119877

119879119878119879119879

The values of the financial product with final payoff minus119877119879119878119879119879

satisfy the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(71)

with terminal condition 119867(119879 119877119879) = minus119877

119879119878119879119879 We seek a

solution of (71) of the form

119867(119905 119877) = 119886119905+ 119887

119905119877 (72)

with 119886119879

= 0 and 119887119879

= minus1119879 Substituting (72) into (71) andsatisfying the boundary conditions we find that

119886119905= minus

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119887

119905= minus

1

119879119890minus119903(119879minus119905)

(73)

For

119878119879max (1 minus

119877119879

119879 0) = 119878

119879max (119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879

(74)

by the valuation mechanism in space (ΩH E) we take thediscounted 119866-conditional expectation on both sides and get

E119905[119890

minus119903(119879minus119905)119878119879max(1 minus

119877119879

119879 0)]

= 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879]

le 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0)]

+ 119890minus119903(119879minus119905)

E119905[119878

119879minus

119877119879119878119879

119879]

(75)

That is

119888 (119905 119878 119868) le 119901 (119905 119878 119868) + 119878 minus119878

119903119879(1 minus 119890

minus119903(119879minus119905))

minus1

119879119890minus119903(119879minus119905)

int

119905

0

119878120591119889120591

(76)

where 119888(119905 119878 119868) is the time 119905 price of the arithmetic averagefloating strike Asian call option and 119901(119905 119878 119868) is the time119905 price of the arithmetic average floating strike Asian putoption

3 Numerical Computation

In the model without volatility uncertainty Asian optionsare much more difficult to value than regular stock optionsStandard techniques tend to be impractical inaccurate orslow For example traditional binomial lattice methodsrequire such enormous amounts of computermemory for thenecessity of keeping track of every possible path throughout

Mathematical Problems in Engineering 9

the tree that they are effectively unusable Monte Carlosimulation works well for European style options (see [12])but is relatively slow Sun [38] proposed to use the weightedupwind method [39] to price Asian options but he didnot give numerical results The Asian option pricing in theuncertain volatility model will be even more difficult than inthe model without volatility uncertainty

For the arithmetic average fixed strike Asian call optionsdue to the convexity of the value 119881(119905 119878 119868) with respect tovariable 119878 (43) can be written as (48) which are the equationssatisfied by the arithmetic average fixed strike Asian calloptions with certain volatility 120590

Zvan et al [17] point out that the Crank-Nicolson schemein conjunction with the van Leer flux limiter method canbe used to numerically value Asian options with certainvolatilities The van Leer limiter has the property that it istotal variation diminishing and thus produces oscillation freesecond order accurate solutions But the van Leer limiter isnonlinear so the solutions should be obtained by using fullNewton iteration at each time level In this paper we will usethe alternating direction implicit (ADI) methods (see Duffy[40])

The arithmetic average fixed strike Asian put options canbe valued by the put-call parity

For the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69))These PDEs are one-dimensional in space And they are not linear equations butHJB equations in some sense For such PDEs we will use thefully implicit positive coefficient finite different method [41]to solve them We will see that this method can ensure thenumerical solutions converge to the viscosity solutions andis accurate efficient and quick to be implemented

All the calculations were performed byMatlab 2009 on acomputer with 183GHz hard disk and 512MB memory

31 Numerical Computation of Fixed Strike Asian OptionsAn equivalent formulation of (48) in terms of the average119860

119905= (1119905) int

119905

0119878120591119889120591 is given in Barraquand and Pudet [16] in

certain volatility model with volatility 120590 = 120590

120597119905119881 (119905 119878 119860) +

1

2211987821205902119881119878119878

(119905 119878 119860) + 119903119878119881119878(119905 119878 119860)

+119878 minus 119860

119905119881119860(119905 119878 119860) minus 119903119881 (119905 119878 119860) = 0

119881 (119879 119878 119860) = (119860 minus 119870)+

(77)

Equation (77) is convection dominated in the 119860 direc-tion because there is no diffusion effect in this dimensionBarraquand and Pudet [16] find that an explicit centrallyweighted scheme for (77) is unstable In particular theconvective term in the 119860 dimension becomes very large as119905 rarr 0 Barraquand and Pudet also note that implicit cen-trally weighted schemes will generally produce unsatisfactoryresults because of the numerical diffusion introduced by thisfirst order accurate in time scheme

Zvan et al [17] point out that under the conditionof convection dominated solutions generated by using a

centrally weighted scheme on the convective term for (77)cannot be ensured to be free of oscillations And solutionsgenerated by the first order upstream weighting scheme onthe convective termare no longer oscillatory but their profilesare too diffuse To produce oscillation free solutions withoutthe excessive diffusion of first order upstream weightingZvan et al [17] suggest the nonlinear van Leer flux limiter inconjunction with the Crank-Nicolson scheme Since the fluxlimiter is nonlinear the solutions should be obtained by usingfull Newton iteration at each time level

The ADI method pioneered in the United States by Dou-glas Rachford Peaceman Gunn and others has a numberof advantages First explicit difference methods are rarelyused to solve initial boundary value problems because oftheir poor stability Implicit methods have superior stabilityproperties but unfortunately they are difficult to solve in twoand more dimensions Consequently ADI methods becomean alternative because they can be programmed by solvinga simple trigonal system of equations For the convection-dominated problems we could use the exponentially fittedschemes in each leg of the approximate scheme to ensure thestability of the results [40] Since Barraquand and Pudet [16]have used the forward shooting grid method and Zvan et al[17] have used the van Leer flux limiter we will test the ADImethod in this paper by comparing our results with theirs

For convenience let us first convert (55) to be thefollowing forward equation in time by substituting 119905 with120591 = 119879 minus 119905

minus 120597120591119881 (120591 119878 119868) +

1

2211987821205902119881119878119878

(120591 119878 119868) + 119903119878119881119878(120591 119878 119868)

+ 119878119881119868(120591 119878 119868) minus 119903119881 (120591 119878 119868) = 0

119881 (0 119878 119868) = (119868

119879minus 119870)

+

119881 (120591 0 119868) = 119890minus119903120591

(119868

119879minus 119870)

+

119881 (119905 119878max 119868) = max ( 119868

119879minus 119870) 119890

minus119903120591+

119878max119903119879

(1 minus 119890minus119903120591

) 0

119881 (120591 119878 119870119879) =1

119903119879(1 minus 119890

minus119903120591) 119878

(78)

We let 120591119899 119899 = 0 1 119873 be a set of portion points in

[0 119879] satisfying 0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 and use even

partition of time that is 120591119899= 119899Δ120591 119899 = 0 1 119873 Define the

partitions of space variables as 0 = 1198780lt 119878

1lt sdot sdot sdot lt 119878

119872= 119878max

and 0 = 1198680lt 119868

1lt sdot sdot sdot lt 119868

119869= 119868max

With ADI we march from time level 119899 to time level119899 + 12 and then from time level 119899 + 12 to time level 119899 + 1In this case we use exponential fitting in all space variables

10 Mathematical Problems in Engineering

and implicit Euler in time The first leg is given by thescheme

minus

119881119899+12

119894119895minus 119881

119899

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899

119894119895+1minus 119881

119899

119894119895minus1

Δ119868119895+1

+ Δ119868119895

minus 119903119881119899+12

119894119895= 0

(79)

with

120588119894119895

=119903119878

119894Δ119878

119894+1

2coth(

119903119878119894Δ119878

119894+1

12059022119878

2

119894

) coth (119909) =1198902119909

+ 1

1198902119909 minus 1

(80)

Let

120572119894= minus

120588119894119895

Δ119878119894Δ119878

119894+1

120573119894= minus

120588119894119895

(Δ119878119894+1

)2minus

119903119878119894

Δ119878119894+1

120574119894= minus 120572

119894minus 120573

119894+ 119903

(81)

Then the scheme (79) can be written as

120572119894119881

119899+12

119894minus1119895+ (120574

119894+

2

Δ120591)119881

119899+12

119894119895+ 120573

119894119881

119899+12

119894+1119895

=2

Δ120591119881

119899

119894119895+

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895+1minus

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895minus1

(82)

We let

119881119899

119895= [119881

119899

1119895 119881

119899

2119895 119881

119899

119872minus1119895]119879

119881119899+12

119895= [119881

119899+12

1119895 119881

119899+12

2119895 119881

119899+12

119872minus1119895]119879

119891119899+12

119895= [120572

1119881

119899+12

0119895 0 0 120573

119872minus1119881

119899+12

119872119895]119879

119903119899

119895= [(

2

Δ120591119881

119899

1119895+

1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895+1

minus1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895minus1)

(2

Δ120591119881

119899

119872minus1119895+

119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895+1

minus119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895minus1)]

119879

1198601=

[[[

[

1205741

1205731

1205722

1205742

1205732

d d d120572119872minus1

120574119872minus1

]]]

](119872minus1)times(119872minus1)

(83)

Then we can write (82) as the following equivalent matrixform

1198601119881

119899+12

119895+ 119891

119899+12

119895= 119903

119899

119895 (84)

The second leg of the discretized PDE in (78) is given bythe scheme

minus

119881119899+1

119894119895minus 119881

119899+12

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899+1

119894119895+1minus 119881

119899+1

119894119895

Δ119868119895+1

minus 119903119881119899+12

119894119895= 0

(85)

Let

120575119895=

2

Δ120591+

119878119894

Δ119868119895+1

120598119895= minus

119878119894

Δ119868119895+1

(86)

Then the scheme (85) can be written as

120575119895119881

119899+1

119894119895minus 120598

119895119881

119899+1

119894119895+1= minus 120572

119894119881

119899+12

119894minus1119895minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119895

minus 120573119894119881

119899+12

119894+1119895

(87)

Mathematical Problems in Engineering 11

300

300

250

200

200

150

100

100

100

50

50

0

0 0

Call

valu

e

sI

Figure 1 Fixed strike Asian call value when = 08 120590 = 05 119903 =

010 119879 = 1 and 119870 = 100

We let

119881119899+1

119894= [119881

119899+1

1198940 119881

119899+1

1198941 119881

119899+1

119894119869minus1]119879

119881119899+12

119894= [119881

119899+12

1198940 119881

119899+12

1198941 119881

119899+12

119894119869minus1]119879

119865119894= [0 0 0 120598

119869minus1119881

119899+1

119894119869]119879

119869

119877119899+12

119894= [(minus120572

119894119881

119899+12

119894minus10minus(120574

119894minus

2

Δ120591)119881

119899+12

1198940minus120573

119894119881

119899+12

119894+10)

(minus120572119894119881

119899+12

119894minus1119869minus1minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119869minus1minus 120573

119894119881

119899+12

119894+1119869minus1)]

119879

1198602=

[[[

[

1205750

minus1205980

0 1205751

minus1205981

d d minus120598119869minus2

120575119869minus1

]]]

]119869times119869

(88)

The matrix form of (87) is

1198602119881

119899+1

119894+ 119865

119899+1

119894= 119877

119899+12

119894 (89)

Thematrix equations (84) and (89) can be solved by using LUdecomposition

The goal of the computation is to find the fair price of theAsian option at emission that is119881(119878

0 119868

0 0)Weneed tomake

sense of any value 119868 isin [0 119868max] and stock price 119878 isin [0 119878max] attime 119905 = 0 There are no problems with the stock prices butthe integral at time zero can only be 0 for continuous averagesthat is 119868

0= 0

Figure 1 is the results we obtained by using the abovediscretization scheme with = 08 120590 = 05 119903 = 010 119879 = 1and 119870 = 100 The point marked by lowast is the price 119881(0 119878 0)

of the Asian option We choose 119878max = 3119870 to ensure thedesirable accuracy

Table 1 shows the convergence of the results with therefinement of the grid From (78) the values of variable 119868

lie in the interval [0 119870119879] where 119870119879 is the boundary and0 is the point where the continuous averages Asian options

are valued so there is no point that is less important on theinterval [0 119870119879] Hence we use uniform grids in direction 119868

In our uncertain volatility model the pricing PDEs (48)are similar to the PDEs in certain volatility model withvolatility 120590 = 120590 Barraquand and Pudet [16] and Zvan etal [17] give numerical results of Asian options with certainvolatilities In the uncertain volatility model when =

08 120590 = 05 the Asian option PDE is the same as that ofthe Asian option with certain volatility 120590 = 04 When theparameters 119903 = 010 119879 = 1 119870 = 95 and grid Δ119868 =

011875 ΔS = 285 the price of the Asian call option wecalculate is 1382 The results with the same parameters ofBarraquand and Pudet [16] and Zvan et al [17] are 13825 and13721 respectively

Table 2 is the comparison of our results with those ofZvan et al [17] Z F and V in Table 2 refer to the resultof Zvan et al [17] Our results are denoted by Asian fixedstrike which calculated with Δ120591 = 001 002 004 Δ119868 =

0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Since119868max = 119870119879 the partitions in 119868 direction are related with thematurity 119879 The execution time is much less than theirs Thegrid spacing was chosen to achieve an accuracy of at least010 of 119878 Our results are close to that of Zvan et al [17] andthe difference is less than 010 of 119878

Our calculation procedure is different from Zvan et al[17] as well as Barraquand and Pudet [16] in the followingfour aspects First our pricing PDE (78) is in terms of therunning sum 119868

119905 Zvan et al [17] and Barraquand and Pudet

[16] use the pricing PDE

minus 120597120591119881 (120591 119878 119860) +

1

211987821205902119881119878119878

(120591 119878 119860) + 119903119878119881119878(120591 119878 119860)

+119878 minus 119860

119905119881119860(120591 119878 119860) minus 119903119881 (120591 119878 119860) = 0

(90)

where 119860119905is defined as 119860

119905= 119868

119905119905 But PDE (78) and (90) are

equivalent for pricing fixed strike European Asian optionsSecond since the PDEs are different PDE (78) and (90) musthave different boundary conditions We do not know theboundary conditions used by Zvan et al [17] The forwardshooting grid method used by Barraquand and Pudet [16]proceeds without any boundary conditions Third our PDE(78) is defined on the domain 119863 = (119905 119878 119868) 0 le 119905 le 119879 0 le

119878 le 119878max 0 le 119868 le 119870119879 and 119868max = 119870119879 is the boundary and119868 = 0 is the point where the options are valued So there isno point that is less important on the interval [0 119870119879] Henceour numerical schemes use uniform grid spacing while Zvanet al [17] use nonuniform grid spacing of 41 times 45 on domain[0 300] times [0 300] To achieve an accuracy of at least 01 of119878 our grids are much finer than Zvan et al [17]

32 Numerical Computation of Floating Strike Asian OptionsTo value the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69)) for 119867(119905 119877) andthe values of the options will be 119881(119905 119878 119868) = 119878119867(119905 119877) with119877 = 119868119878 Actually what we really care about is the optionvalues at time 119905 = 0 while from the definition of 119868

119905 it is

obvious that 1198680= 0 and therefore 119877

0= 0 so our goal of the

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

Mathematical Problems in Engineering 7

So the values of the arithmetic average fixed strike Asiancall option satisfy

120597119905119881 (119905 119878 119868) +

1

2211987821205902119881119878119878

(119905 119878 119868) + 119903119878119881119878(119905 119878 119868)

+ 119878119881119868(119905 119878 119868) minus 119903119881 (119905 119878 119868) = 0

V (119879 119878 119868) = (119868

119879minus 119870)

+

119881 (119905 0 119868) = 119890minus119903(119879minus119905)

(119868

119879minus 119870)

+

119881 (119905 119878max 119868)

= max ( 119868

119879minus 119870) 119890

minus119903(119879minus119905)+

119878max119903119879

(1 minus 119890minus119903(119879minus119905)

) 0

119881 (119905 119878 119870119879) =1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

(55)

For the arithmetic average fixed strike Asian put optionwith terminal payoff 119901(119879 119878

119879 119868

119879) = max(119870 minus 119868

119879119879 0) we will

value this option by the following put-call parityIt is easily to see that we have

max(119868119879

119879minus 119870 0) = max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889] minus 119870

(56)

So

119888 (119905 119878119905 119868

119905)

= 119890minus119903(119879minus119905)

E119905[max (119868

119879

119879minus 119870 0)]

= 119890minus119903(119879minus119905)

E119905[max (119870 minus

119868119879

119879 0) +

1

119879int

119879

0

119878]119889]]

minus 119890minus119903(119879minus119905)

119870

= 119901 (119905 119878119905 119868

119905) minus 119890

minus119903(119879minus119905)119870 +

1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889]

+ 119890minus119903(119879minus119905)

119864120590

119905[1

119879int

119879

119905

119878]119889]]

(57)

119890minus119903(119879minus119905)

119864120590

119905[(1119879) int

119879

119905119878]119889]] is the time 119905 value of time 119879

claim (1119879) int119879

119905119878]119889] where the stock has certain volatility

120590 To get claim (1119879) int119879

119905119878]119889] at 119879 let us see the replicating

strategy in any time interval (] ] + Δ]) transfer a certainportion (1119879)119890

minus119903(119879minus])Δ] of stock 119878] to a riskless bond This

strategy ensures the final payoff of (1119879) int119879

119905119878]119889] The stocks

that should be transferred are

int

119879

119905

1

119879119890minus119903(119879minus])

119889] =1

119903119879(1 minus 119890

minus119903(119879minus119905)) (58)

Then the time 119905 value of (1119879) int119879

119905119878]119889] equals (1119903119879)(1 minus

119890minus119903(119879minus119905)

)119878119905 So the put-call parity can be written as

119888 (119905 119878 119868) + 119870119890minus119903(119879minus119905)

= 119901 (119905 119878 119868) +1

119879119890minus119903(119879minus119905)

int

119905

0

119878]119889] +1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119878

119905

(59)

24 The Arithmetic Average Floating Strike Asian Call OptionFor the arithmetic average floating strike Asian call (put)option the PDE (43) still applies with terminal condition

119881 (119879 119878 119868) = (119878 minus119868

119879)

+

(119881 (119879 119878 119868) = (119868

119879minus 119878)

+

) (60)

In Ingersoll [13] where there is no volatility uncertaintydue to the linear homogeneity of the PDE and its terminalcondition their PDE can be reduced to a one-dimensionalPDE In our uncertain volatility model the PDE (43) is notlinear homogeneous anymore But we can still reduce PDE(43) to a one-dimensional PDE by the following argument

For the arithmetic average floating strike Asian calloption we know

119881 (119905 119878119905 119868

119905) = E

119905[119890

minus119903(119879minus119905)(119878

119879minus

119868119879

119879)

+

]

= E119905[

[

119890minus119903(119879minus119905)

(119878119879minus

119868119905+ int

119879

119905119878]119889]

119879)

+

]

]

(61)

Denote

119872]119905= exp(120583 (] minus 119905) minus

1

22(⟨119861⟩] minus ⟨119861⟩

119905) + (119861] minus 119861

119905))

(62)

Then by (14) we have 119878] = 119878119905119872

]119905and 119878

119905= 119878]119872

119905

] So

119868119905= int

119905

0

119878]119889] = int

119905

0

119878119905119872

]119905119889] = 119878

119905int

119905

0

119872]119905119889]

119881 (119905 119878119905 119868

119905)

= E119905[

[

119890minus119903(119879minus119905)

(119878119905119872

119879

119905minus

119878119905int119905

0119872

]119905119889] + 119878

119905int119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905E

119905[

[

119890minus119903(119879minus119905)

(119872119879

119905minus

int119905

0119872

]119905119889] + int

119879

119905119872

]119905119889]

119879)

+

]

]

= 119878119905119867(119905 119877

119905)

(63)with 119877

119905= 119868

119905119878

119905

Then the function119867(119905 119877) satisfies the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(64)

which is a one-dimensional PDE

8 Mathematical Problems in Engineering

For the European style Asian option we must imposeboundary conditions both at 119877 = 0 and as 119877 rarr infin Theboundary condition as 119877 rarr infin is simple The only way that119877 can tend to infinity is that 119878 tends to zero In this case theoption will not be exercised and so

lim119877rarrinfin

119867(119905 119877) = 0 (65)

For computational purpose we truncate the asset region(0 +infin) into (0 119877max) where 119877max is sufficiently large toensure the accuracy of the solution If 119877 = 0 the PDEdegenerates into the first-order hyperbolic PDE

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 (66)

So119867(119905 119877) satisfies

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (1 minus119877

119879)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = 0

(67)

By the same argument for the arithmetic average floatingstrike Asian put option its prices could be calculated by

119881 (119905 119878 119868) = 119878119867 (119905 119877) (68)

with119867(119905 119877) satisfying the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (119877

119879minus 1)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = (119877max119879

minus exp minus119903 (119879 minus 119905))

(69)

Generally the put-call parity will not hold any more inthe uncertain volatility model Let us see the following argu-ment The mature payoff of a portfolio with one Europeanarithmetic average floating strike Asian call holding long andone put holding short is

119878119879max (1 minus

119877119879

119879 0) minus 119878

119879max (119877

119879

119879minus 1 0)

= 119878119879minus

119877119879119878119879

119879

(70)

The value of this portfolio is equal to one consisting ofone stock and a financial product whose payoff is minus119877

119879119878119879119879

The values of the financial product with final payoff minus119877119879119878119879119879

satisfy the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(71)

with terminal condition 119867(119879 119877119879) = minus119877

119879119878119879119879 We seek a

solution of (71) of the form

119867(119905 119877) = 119886119905+ 119887

119905119877 (72)

with 119886119879

= 0 and 119887119879

= minus1119879 Substituting (72) into (71) andsatisfying the boundary conditions we find that

119886119905= minus

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119887

119905= minus

1

119879119890minus119903(119879minus119905)

(73)

For

119878119879max (1 minus

119877119879

119879 0) = 119878

119879max (119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879

(74)

by the valuation mechanism in space (ΩH E) we take thediscounted 119866-conditional expectation on both sides and get

E119905[119890

minus119903(119879minus119905)119878119879max(1 minus

119877119879

119879 0)]

= 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879]

le 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0)]

+ 119890minus119903(119879minus119905)

E119905[119878

119879minus

119877119879119878119879

119879]

(75)

That is

119888 (119905 119878 119868) le 119901 (119905 119878 119868) + 119878 minus119878

119903119879(1 minus 119890

minus119903(119879minus119905))

minus1

119879119890minus119903(119879minus119905)

int

119905

0

119878120591119889120591

(76)

where 119888(119905 119878 119868) is the time 119905 price of the arithmetic averagefloating strike Asian call option and 119901(119905 119878 119868) is the time119905 price of the arithmetic average floating strike Asian putoption

3 Numerical Computation

In the model without volatility uncertainty Asian optionsare much more difficult to value than regular stock optionsStandard techniques tend to be impractical inaccurate orslow For example traditional binomial lattice methodsrequire such enormous amounts of computermemory for thenecessity of keeping track of every possible path throughout

Mathematical Problems in Engineering 9

the tree that they are effectively unusable Monte Carlosimulation works well for European style options (see [12])but is relatively slow Sun [38] proposed to use the weightedupwind method [39] to price Asian options but he didnot give numerical results The Asian option pricing in theuncertain volatility model will be even more difficult than inthe model without volatility uncertainty

For the arithmetic average fixed strike Asian call optionsdue to the convexity of the value 119881(119905 119878 119868) with respect tovariable 119878 (43) can be written as (48) which are the equationssatisfied by the arithmetic average fixed strike Asian calloptions with certain volatility 120590

Zvan et al [17] point out that the Crank-Nicolson schemein conjunction with the van Leer flux limiter method canbe used to numerically value Asian options with certainvolatilities The van Leer limiter has the property that it istotal variation diminishing and thus produces oscillation freesecond order accurate solutions But the van Leer limiter isnonlinear so the solutions should be obtained by using fullNewton iteration at each time level In this paper we will usethe alternating direction implicit (ADI) methods (see Duffy[40])

The arithmetic average fixed strike Asian put options canbe valued by the put-call parity

For the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69))These PDEs are one-dimensional in space And they are not linear equations butHJB equations in some sense For such PDEs we will use thefully implicit positive coefficient finite different method [41]to solve them We will see that this method can ensure thenumerical solutions converge to the viscosity solutions andis accurate efficient and quick to be implemented

All the calculations were performed byMatlab 2009 on acomputer with 183GHz hard disk and 512MB memory

31 Numerical Computation of Fixed Strike Asian OptionsAn equivalent formulation of (48) in terms of the average119860

119905= (1119905) int

119905

0119878120591119889120591 is given in Barraquand and Pudet [16] in

certain volatility model with volatility 120590 = 120590

120597119905119881 (119905 119878 119860) +

1

2211987821205902119881119878119878

(119905 119878 119860) + 119903119878119881119878(119905 119878 119860)

+119878 minus 119860

119905119881119860(119905 119878 119860) minus 119903119881 (119905 119878 119860) = 0

119881 (119879 119878 119860) = (119860 minus 119870)+

(77)

Equation (77) is convection dominated in the 119860 direc-tion because there is no diffusion effect in this dimensionBarraquand and Pudet [16] find that an explicit centrallyweighted scheme for (77) is unstable In particular theconvective term in the 119860 dimension becomes very large as119905 rarr 0 Barraquand and Pudet also note that implicit cen-trally weighted schemes will generally produce unsatisfactoryresults because of the numerical diffusion introduced by thisfirst order accurate in time scheme

Zvan et al [17] point out that under the conditionof convection dominated solutions generated by using a

centrally weighted scheme on the convective term for (77)cannot be ensured to be free of oscillations And solutionsgenerated by the first order upstream weighting scheme onthe convective termare no longer oscillatory but their profilesare too diffuse To produce oscillation free solutions withoutthe excessive diffusion of first order upstream weightingZvan et al [17] suggest the nonlinear van Leer flux limiter inconjunction with the Crank-Nicolson scheme Since the fluxlimiter is nonlinear the solutions should be obtained by usingfull Newton iteration at each time level

The ADI method pioneered in the United States by Dou-glas Rachford Peaceman Gunn and others has a numberof advantages First explicit difference methods are rarelyused to solve initial boundary value problems because oftheir poor stability Implicit methods have superior stabilityproperties but unfortunately they are difficult to solve in twoand more dimensions Consequently ADI methods becomean alternative because they can be programmed by solvinga simple trigonal system of equations For the convection-dominated problems we could use the exponentially fittedschemes in each leg of the approximate scheme to ensure thestability of the results [40] Since Barraquand and Pudet [16]have used the forward shooting grid method and Zvan et al[17] have used the van Leer flux limiter we will test the ADImethod in this paper by comparing our results with theirs

For convenience let us first convert (55) to be thefollowing forward equation in time by substituting 119905 with120591 = 119879 minus 119905

minus 120597120591119881 (120591 119878 119868) +

1

2211987821205902119881119878119878

(120591 119878 119868) + 119903119878119881119878(120591 119878 119868)

+ 119878119881119868(120591 119878 119868) minus 119903119881 (120591 119878 119868) = 0

119881 (0 119878 119868) = (119868

119879minus 119870)

+

119881 (120591 0 119868) = 119890minus119903120591

(119868

119879minus 119870)

+

119881 (119905 119878max 119868) = max ( 119868

119879minus 119870) 119890

minus119903120591+

119878max119903119879

(1 minus 119890minus119903120591

) 0

119881 (120591 119878 119870119879) =1

119903119879(1 minus 119890

minus119903120591) 119878

(78)

We let 120591119899 119899 = 0 1 119873 be a set of portion points in

[0 119879] satisfying 0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 and use even

partition of time that is 120591119899= 119899Δ120591 119899 = 0 1 119873 Define the

partitions of space variables as 0 = 1198780lt 119878

1lt sdot sdot sdot lt 119878

119872= 119878max

and 0 = 1198680lt 119868

1lt sdot sdot sdot lt 119868

119869= 119868max

With ADI we march from time level 119899 to time level119899 + 12 and then from time level 119899 + 12 to time level 119899 + 1In this case we use exponential fitting in all space variables

10 Mathematical Problems in Engineering

and implicit Euler in time The first leg is given by thescheme

minus

119881119899+12

119894119895minus 119881

119899

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899

119894119895+1minus 119881

119899

119894119895minus1

Δ119868119895+1

+ Δ119868119895

minus 119903119881119899+12

119894119895= 0

(79)

with

120588119894119895

=119903119878

119894Δ119878

119894+1

2coth(

119903119878119894Δ119878

119894+1

12059022119878

2

119894

) coth (119909) =1198902119909

+ 1

1198902119909 minus 1

(80)

Let

120572119894= minus

120588119894119895

Δ119878119894Δ119878

119894+1

120573119894= minus

120588119894119895

(Δ119878119894+1

)2minus

119903119878119894

Δ119878119894+1

120574119894= minus 120572

119894minus 120573

119894+ 119903

(81)

Then the scheme (79) can be written as

120572119894119881

119899+12

119894minus1119895+ (120574

119894+

2

Δ120591)119881

119899+12

119894119895+ 120573

119894119881

119899+12

119894+1119895

=2

Δ120591119881

119899

119894119895+

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895+1minus

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895minus1

(82)

We let

119881119899

119895= [119881

119899

1119895 119881

119899

2119895 119881

119899

119872minus1119895]119879

119881119899+12

119895= [119881

119899+12

1119895 119881

119899+12

2119895 119881

119899+12

119872minus1119895]119879

119891119899+12

119895= [120572

1119881

119899+12

0119895 0 0 120573

119872minus1119881

119899+12

119872119895]119879

119903119899

119895= [(

2

Δ120591119881

119899

1119895+

1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895+1

minus1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895minus1)

(2

Δ120591119881

119899

119872minus1119895+

119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895+1

minus119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895minus1)]

119879

1198601=

[[[

[

1205741

1205731

1205722

1205742

1205732

d d d120572119872minus1

120574119872minus1

]]]

](119872minus1)times(119872minus1)

(83)

Then we can write (82) as the following equivalent matrixform

1198601119881

119899+12

119895+ 119891

119899+12

119895= 119903

119899

119895 (84)

The second leg of the discretized PDE in (78) is given bythe scheme

minus

119881119899+1

119894119895minus 119881

119899+12

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899+1

119894119895+1minus 119881

119899+1

119894119895

Δ119868119895+1

minus 119903119881119899+12

119894119895= 0

(85)

Let

120575119895=

2

Δ120591+

119878119894

Δ119868119895+1

120598119895= minus

119878119894

Δ119868119895+1

(86)

Then the scheme (85) can be written as

120575119895119881

119899+1

119894119895minus 120598

119895119881

119899+1

119894119895+1= minus 120572

119894119881

119899+12

119894minus1119895minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119895

minus 120573119894119881

119899+12

119894+1119895

(87)

Mathematical Problems in Engineering 11

300

300

250

200

200

150

100

100

100

50

50

0

0 0

Call

valu

e

sI

Figure 1 Fixed strike Asian call value when = 08 120590 = 05 119903 =

010 119879 = 1 and 119870 = 100

We let

119881119899+1

119894= [119881

119899+1

1198940 119881

119899+1

1198941 119881

119899+1

119894119869minus1]119879

119881119899+12

119894= [119881

119899+12

1198940 119881

119899+12

1198941 119881

119899+12

119894119869minus1]119879

119865119894= [0 0 0 120598

119869minus1119881

119899+1

119894119869]119879

119869

119877119899+12

119894= [(minus120572

119894119881

119899+12

119894minus10minus(120574

119894minus

2

Δ120591)119881

119899+12

1198940minus120573

119894119881

119899+12

119894+10)

(minus120572119894119881

119899+12

119894minus1119869minus1minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119869minus1minus 120573

119894119881

119899+12

119894+1119869minus1)]

119879

1198602=

[[[

[

1205750

minus1205980

0 1205751

minus1205981

d d minus120598119869minus2

120575119869minus1

]]]

]119869times119869

(88)

The matrix form of (87) is

1198602119881

119899+1

119894+ 119865

119899+1

119894= 119877

119899+12

119894 (89)

Thematrix equations (84) and (89) can be solved by using LUdecomposition

The goal of the computation is to find the fair price of theAsian option at emission that is119881(119878

0 119868

0 0)Weneed tomake

sense of any value 119868 isin [0 119868max] and stock price 119878 isin [0 119878max] attime 119905 = 0 There are no problems with the stock prices butthe integral at time zero can only be 0 for continuous averagesthat is 119868

0= 0

Figure 1 is the results we obtained by using the abovediscretization scheme with = 08 120590 = 05 119903 = 010 119879 = 1and 119870 = 100 The point marked by lowast is the price 119881(0 119878 0)

of the Asian option We choose 119878max = 3119870 to ensure thedesirable accuracy

Table 1 shows the convergence of the results with therefinement of the grid From (78) the values of variable 119868

lie in the interval [0 119870119879] where 119870119879 is the boundary and0 is the point where the continuous averages Asian options

are valued so there is no point that is less important on theinterval [0 119870119879] Hence we use uniform grids in direction 119868

In our uncertain volatility model the pricing PDEs (48)are similar to the PDEs in certain volatility model withvolatility 120590 = 120590 Barraquand and Pudet [16] and Zvan etal [17] give numerical results of Asian options with certainvolatilities In the uncertain volatility model when =

08 120590 = 05 the Asian option PDE is the same as that ofthe Asian option with certain volatility 120590 = 04 When theparameters 119903 = 010 119879 = 1 119870 = 95 and grid Δ119868 =

011875 ΔS = 285 the price of the Asian call option wecalculate is 1382 The results with the same parameters ofBarraquand and Pudet [16] and Zvan et al [17] are 13825 and13721 respectively

Table 2 is the comparison of our results with those ofZvan et al [17] Z F and V in Table 2 refer to the resultof Zvan et al [17] Our results are denoted by Asian fixedstrike which calculated with Δ120591 = 001 002 004 Δ119868 =

0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Since119868max = 119870119879 the partitions in 119868 direction are related with thematurity 119879 The execution time is much less than theirs Thegrid spacing was chosen to achieve an accuracy of at least010 of 119878 Our results are close to that of Zvan et al [17] andthe difference is less than 010 of 119878

Our calculation procedure is different from Zvan et al[17] as well as Barraquand and Pudet [16] in the followingfour aspects First our pricing PDE (78) is in terms of therunning sum 119868

119905 Zvan et al [17] and Barraquand and Pudet

[16] use the pricing PDE

minus 120597120591119881 (120591 119878 119860) +

1

211987821205902119881119878119878

(120591 119878 119860) + 119903119878119881119878(120591 119878 119860)

+119878 minus 119860

119905119881119860(120591 119878 119860) minus 119903119881 (120591 119878 119860) = 0

(90)

where 119860119905is defined as 119860

119905= 119868

119905119905 But PDE (78) and (90) are

equivalent for pricing fixed strike European Asian optionsSecond since the PDEs are different PDE (78) and (90) musthave different boundary conditions We do not know theboundary conditions used by Zvan et al [17] The forwardshooting grid method used by Barraquand and Pudet [16]proceeds without any boundary conditions Third our PDE(78) is defined on the domain 119863 = (119905 119878 119868) 0 le 119905 le 119879 0 le

119878 le 119878max 0 le 119868 le 119870119879 and 119868max = 119870119879 is the boundary and119868 = 0 is the point where the options are valued So there isno point that is less important on the interval [0 119870119879] Henceour numerical schemes use uniform grid spacing while Zvanet al [17] use nonuniform grid spacing of 41 times 45 on domain[0 300] times [0 300] To achieve an accuracy of at least 01 of119878 our grids are much finer than Zvan et al [17]

32 Numerical Computation of Floating Strike Asian OptionsTo value the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69)) for 119867(119905 119877) andthe values of the options will be 119881(119905 119878 119868) = 119878119867(119905 119877) with119877 = 119868119878 Actually what we really care about is the optionvalues at time 119905 = 0 while from the definition of 119868

119905 it is

obvious that 1198680= 0 and therefore 119877

0= 0 so our goal of the

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

8 Mathematical Problems in Engineering

For the European style Asian option we must imposeboundary conditions both at 119877 = 0 and as 119877 rarr infin Theboundary condition as 119877 rarr infin is simple The only way that119877 can tend to infinity is that 119878 tends to zero In this case theoption will not be exercised and so

lim119877rarrinfin

119867(119905 119877) = 0 (65)

For computational purpose we truncate the asset region(0 +infin) into (0 119877max) where 119877max is sufficiently large toensure the accuracy of the solution If 119877 = 0 the PDEdegenerates into the first-order hyperbolic PDE

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 (66)

So119867(119905 119877) satisfies

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (1 minus119877

119879)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = 0

(67)

By the same argument for the arithmetic average floatingstrike Asian put option its prices could be calculated by

119881 (119905 119878 119868) = 119878119867 (119905 119877) (68)

with119867(119905 119877) satisfying the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

119867 (119879 119877) = (119877

119879minus 1)

+

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0

119867 (119905 119877max) = (119877max119879

minus exp minus119903 (119879 minus 119905))

(69)

Generally the put-call parity will not hold any more inthe uncertain volatility model Let us see the following argu-ment The mature payoff of a portfolio with one Europeanarithmetic average floating strike Asian call holding long andone put holding short is

119878119879max (1 minus

119877119879

119879 0) minus 119878

119879max (119877

119879

119879minus 1 0)

= 119878119879minus

119877119879119878119879

119879

(70)

The value of this portfolio is equal to one consisting ofone stock and a financial product whose payoff is minus119877

119879119878119879119879

The values of the financial product with final payoff minus119877119879119878119879119879

satisfy the following PDE

120597119905119867(119905 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(119905 119877)

+ (1 minus 119903119877)119867119877(119905 119877) = 0

(71)

with terminal condition 119867(119879 119877119879) = minus119877

119879119878119879119879 We seek a

solution of (71) of the form

119867(119905 119877) = 119886119905+ 119887

119905119877 (72)

with 119886119879

= 0 and 119887119879

= minus1119879 Substituting (72) into (71) andsatisfying the boundary conditions we find that

119886119905= minus

1

119903119879(1 minus 119890

minus119903(119879minus119905)) 119887

119905= minus

1

119879119890minus119903(119879minus119905)

(73)

For

119878119879max (1 minus

119877119879

119879 0) = 119878

119879max (119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879

(74)

by the valuation mechanism in space (ΩH E) we take thediscounted 119866-conditional expectation on both sides and get

E119905[119890

minus119903(119879minus119905)119878119879max(1 minus

119877119879

119879 0)]

= 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0) + 119878

119879minus

119877119879119878119879

119879]

le 119890minus119903(119879minus119905)

E119905[119878

119879max(119877

119879

119879minus 1 0)]

+ 119890minus119903(119879minus119905)

E119905[119878

119879minus

119877119879119878119879

119879]

(75)

That is

119888 (119905 119878 119868) le 119901 (119905 119878 119868) + 119878 minus119878

119903119879(1 minus 119890

minus119903(119879minus119905))

minus1

119879119890minus119903(119879minus119905)

int

119905

0

119878120591119889120591

(76)

where 119888(119905 119878 119868) is the time 119905 price of the arithmetic averagefloating strike Asian call option and 119901(119905 119878 119868) is the time119905 price of the arithmetic average floating strike Asian putoption

3 Numerical Computation

In the model without volatility uncertainty Asian optionsare much more difficult to value than regular stock optionsStandard techniques tend to be impractical inaccurate orslow For example traditional binomial lattice methodsrequire such enormous amounts of computermemory for thenecessity of keeping track of every possible path throughout

Mathematical Problems in Engineering 9

the tree that they are effectively unusable Monte Carlosimulation works well for European style options (see [12])but is relatively slow Sun [38] proposed to use the weightedupwind method [39] to price Asian options but he didnot give numerical results The Asian option pricing in theuncertain volatility model will be even more difficult than inthe model without volatility uncertainty

For the arithmetic average fixed strike Asian call optionsdue to the convexity of the value 119881(119905 119878 119868) with respect tovariable 119878 (43) can be written as (48) which are the equationssatisfied by the arithmetic average fixed strike Asian calloptions with certain volatility 120590

Zvan et al [17] point out that the Crank-Nicolson schemein conjunction with the van Leer flux limiter method canbe used to numerically value Asian options with certainvolatilities The van Leer limiter has the property that it istotal variation diminishing and thus produces oscillation freesecond order accurate solutions But the van Leer limiter isnonlinear so the solutions should be obtained by using fullNewton iteration at each time level In this paper we will usethe alternating direction implicit (ADI) methods (see Duffy[40])

The arithmetic average fixed strike Asian put options canbe valued by the put-call parity

For the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69))These PDEs are one-dimensional in space And they are not linear equations butHJB equations in some sense For such PDEs we will use thefully implicit positive coefficient finite different method [41]to solve them We will see that this method can ensure thenumerical solutions converge to the viscosity solutions andis accurate efficient and quick to be implemented

All the calculations were performed byMatlab 2009 on acomputer with 183GHz hard disk and 512MB memory

31 Numerical Computation of Fixed Strike Asian OptionsAn equivalent formulation of (48) in terms of the average119860

119905= (1119905) int

119905

0119878120591119889120591 is given in Barraquand and Pudet [16] in

certain volatility model with volatility 120590 = 120590

120597119905119881 (119905 119878 119860) +

1

2211987821205902119881119878119878

(119905 119878 119860) + 119903119878119881119878(119905 119878 119860)

+119878 minus 119860

119905119881119860(119905 119878 119860) minus 119903119881 (119905 119878 119860) = 0

119881 (119879 119878 119860) = (119860 minus 119870)+

(77)

Equation (77) is convection dominated in the 119860 direc-tion because there is no diffusion effect in this dimensionBarraquand and Pudet [16] find that an explicit centrallyweighted scheme for (77) is unstable In particular theconvective term in the 119860 dimension becomes very large as119905 rarr 0 Barraquand and Pudet also note that implicit cen-trally weighted schemes will generally produce unsatisfactoryresults because of the numerical diffusion introduced by thisfirst order accurate in time scheme

Zvan et al [17] point out that under the conditionof convection dominated solutions generated by using a

centrally weighted scheme on the convective term for (77)cannot be ensured to be free of oscillations And solutionsgenerated by the first order upstream weighting scheme onthe convective termare no longer oscillatory but their profilesare too diffuse To produce oscillation free solutions withoutthe excessive diffusion of first order upstream weightingZvan et al [17] suggest the nonlinear van Leer flux limiter inconjunction with the Crank-Nicolson scheme Since the fluxlimiter is nonlinear the solutions should be obtained by usingfull Newton iteration at each time level

The ADI method pioneered in the United States by Dou-glas Rachford Peaceman Gunn and others has a numberof advantages First explicit difference methods are rarelyused to solve initial boundary value problems because oftheir poor stability Implicit methods have superior stabilityproperties but unfortunately they are difficult to solve in twoand more dimensions Consequently ADI methods becomean alternative because they can be programmed by solvinga simple trigonal system of equations For the convection-dominated problems we could use the exponentially fittedschemes in each leg of the approximate scheme to ensure thestability of the results [40] Since Barraquand and Pudet [16]have used the forward shooting grid method and Zvan et al[17] have used the van Leer flux limiter we will test the ADImethod in this paper by comparing our results with theirs

For convenience let us first convert (55) to be thefollowing forward equation in time by substituting 119905 with120591 = 119879 minus 119905

minus 120597120591119881 (120591 119878 119868) +

1

2211987821205902119881119878119878

(120591 119878 119868) + 119903119878119881119878(120591 119878 119868)

+ 119878119881119868(120591 119878 119868) minus 119903119881 (120591 119878 119868) = 0

119881 (0 119878 119868) = (119868

119879minus 119870)

+

119881 (120591 0 119868) = 119890minus119903120591

(119868

119879minus 119870)

+

119881 (119905 119878max 119868) = max ( 119868

119879minus 119870) 119890

minus119903120591+

119878max119903119879

(1 minus 119890minus119903120591

) 0

119881 (120591 119878 119870119879) =1

119903119879(1 minus 119890

minus119903120591) 119878

(78)

We let 120591119899 119899 = 0 1 119873 be a set of portion points in

[0 119879] satisfying 0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 and use even

partition of time that is 120591119899= 119899Δ120591 119899 = 0 1 119873 Define the

partitions of space variables as 0 = 1198780lt 119878

1lt sdot sdot sdot lt 119878

119872= 119878max

and 0 = 1198680lt 119868

1lt sdot sdot sdot lt 119868

119869= 119868max

With ADI we march from time level 119899 to time level119899 + 12 and then from time level 119899 + 12 to time level 119899 + 1In this case we use exponential fitting in all space variables

10 Mathematical Problems in Engineering

and implicit Euler in time The first leg is given by thescheme

minus

119881119899+12

119894119895minus 119881

119899

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899

119894119895+1minus 119881

119899

119894119895minus1

Δ119868119895+1

+ Δ119868119895

minus 119903119881119899+12

119894119895= 0

(79)

with

120588119894119895

=119903119878

119894Δ119878

119894+1

2coth(

119903119878119894Δ119878

119894+1

12059022119878

2

119894

) coth (119909) =1198902119909

+ 1

1198902119909 minus 1

(80)

Let

120572119894= minus

120588119894119895

Δ119878119894Δ119878

119894+1

120573119894= minus

120588119894119895

(Δ119878119894+1

)2minus

119903119878119894

Δ119878119894+1

120574119894= minus 120572

119894minus 120573

119894+ 119903

(81)

Then the scheme (79) can be written as

120572119894119881

119899+12

119894minus1119895+ (120574

119894+

2

Δ120591)119881

119899+12

119894119895+ 120573

119894119881

119899+12

119894+1119895

=2

Δ120591119881

119899

119894119895+

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895+1minus

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895minus1

(82)

We let

119881119899

119895= [119881

119899

1119895 119881

119899

2119895 119881

119899

119872minus1119895]119879

119881119899+12

119895= [119881

119899+12

1119895 119881

119899+12

2119895 119881

119899+12

119872minus1119895]119879

119891119899+12

119895= [120572

1119881

119899+12

0119895 0 0 120573

119872minus1119881

119899+12

119872119895]119879

119903119899

119895= [(

2

Δ120591119881

119899

1119895+

1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895+1

minus1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895minus1)

(2

Δ120591119881

119899

119872minus1119895+

119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895+1

minus119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895minus1)]

119879

1198601=

[[[

[

1205741

1205731

1205722

1205742

1205732

d d d120572119872minus1

120574119872minus1

]]]

](119872minus1)times(119872minus1)

(83)

Then we can write (82) as the following equivalent matrixform

1198601119881

119899+12

119895+ 119891

119899+12

119895= 119903

119899

119895 (84)

The second leg of the discretized PDE in (78) is given bythe scheme

minus

119881119899+1

119894119895minus 119881

119899+12

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899+1

119894119895+1minus 119881

119899+1

119894119895

Δ119868119895+1

minus 119903119881119899+12

119894119895= 0

(85)

Let

120575119895=

2

Δ120591+

119878119894

Δ119868119895+1

120598119895= minus

119878119894

Δ119868119895+1

(86)

Then the scheme (85) can be written as

120575119895119881

119899+1

119894119895minus 120598

119895119881

119899+1

119894119895+1= minus 120572

119894119881

119899+12

119894minus1119895minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119895

minus 120573119894119881

119899+12

119894+1119895

(87)

Mathematical Problems in Engineering 11

300

300

250

200

200

150

100

100

100

50

50

0

0 0

Call

valu

e

sI

Figure 1 Fixed strike Asian call value when = 08 120590 = 05 119903 =

010 119879 = 1 and 119870 = 100

We let

119881119899+1

119894= [119881

119899+1

1198940 119881

119899+1

1198941 119881

119899+1

119894119869minus1]119879

119881119899+12

119894= [119881

119899+12

1198940 119881

119899+12

1198941 119881

119899+12

119894119869minus1]119879

119865119894= [0 0 0 120598

119869minus1119881

119899+1

119894119869]119879

119869

119877119899+12

119894= [(minus120572

119894119881

119899+12

119894minus10minus(120574

119894minus

2

Δ120591)119881

119899+12

1198940minus120573

119894119881

119899+12

119894+10)

(minus120572119894119881

119899+12

119894minus1119869minus1minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119869minus1minus 120573

119894119881

119899+12

119894+1119869minus1)]

119879

1198602=

[[[

[

1205750

minus1205980

0 1205751

minus1205981

d d minus120598119869minus2

120575119869minus1

]]]

]119869times119869

(88)

The matrix form of (87) is

1198602119881

119899+1

119894+ 119865

119899+1

119894= 119877

119899+12

119894 (89)

Thematrix equations (84) and (89) can be solved by using LUdecomposition

The goal of the computation is to find the fair price of theAsian option at emission that is119881(119878

0 119868

0 0)Weneed tomake

sense of any value 119868 isin [0 119868max] and stock price 119878 isin [0 119878max] attime 119905 = 0 There are no problems with the stock prices butthe integral at time zero can only be 0 for continuous averagesthat is 119868

0= 0

Figure 1 is the results we obtained by using the abovediscretization scheme with = 08 120590 = 05 119903 = 010 119879 = 1and 119870 = 100 The point marked by lowast is the price 119881(0 119878 0)

of the Asian option We choose 119878max = 3119870 to ensure thedesirable accuracy

Table 1 shows the convergence of the results with therefinement of the grid From (78) the values of variable 119868

lie in the interval [0 119870119879] where 119870119879 is the boundary and0 is the point where the continuous averages Asian options

are valued so there is no point that is less important on theinterval [0 119870119879] Hence we use uniform grids in direction 119868

In our uncertain volatility model the pricing PDEs (48)are similar to the PDEs in certain volatility model withvolatility 120590 = 120590 Barraquand and Pudet [16] and Zvan etal [17] give numerical results of Asian options with certainvolatilities In the uncertain volatility model when =

08 120590 = 05 the Asian option PDE is the same as that ofthe Asian option with certain volatility 120590 = 04 When theparameters 119903 = 010 119879 = 1 119870 = 95 and grid Δ119868 =

011875 ΔS = 285 the price of the Asian call option wecalculate is 1382 The results with the same parameters ofBarraquand and Pudet [16] and Zvan et al [17] are 13825 and13721 respectively

Table 2 is the comparison of our results with those ofZvan et al [17] Z F and V in Table 2 refer to the resultof Zvan et al [17] Our results are denoted by Asian fixedstrike which calculated with Δ120591 = 001 002 004 Δ119868 =

0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Since119868max = 119870119879 the partitions in 119868 direction are related with thematurity 119879 The execution time is much less than theirs Thegrid spacing was chosen to achieve an accuracy of at least010 of 119878 Our results are close to that of Zvan et al [17] andthe difference is less than 010 of 119878

Our calculation procedure is different from Zvan et al[17] as well as Barraquand and Pudet [16] in the followingfour aspects First our pricing PDE (78) is in terms of therunning sum 119868

119905 Zvan et al [17] and Barraquand and Pudet

[16] use the pricing PDE

minus 120597120591119881 (120591 119878 119860) +

1

211987821205902119881119878119878

(120591 119878 119860) + 119903119878119881119878(120591 119878 119860)

+119878 minus 119860

119905119881119860(120591 119878 119860) minus 119903119881 (120591 119878 119860) = 0

(90)

where 119860119905is defined as 119860

119905= 119868

119905119905 But PDE (78) and (90) are

equivalent for pricing fixed strike European Asian optionsSecond since the PDEs are different PDE (78) and (90) musthave different boundary conditions We do not know theboundary conditions used by Zvan et al [17] The forwardshooting grid method used by Barraquand and Pudet [16]proceeds without any boundary conditions Third our PDE(78) is defined on the domain 119863 = (119905 119878 119868) 0 le 119905 le 119879 0 le

119878 le 119878max 0 le 119868 le 119870119879 and 119868max = 119870119879 is the boundary and119868 = 0 is the point where the options are valued So there isno point that is less important on the interval [0 119870119879] Henceour numerical schemes use uniform grid spacing while Zvanet al [17] use nonuniform grid spacing of 41 times 45 on domain[0 300] times [0 300] To achieve an accuracy of at least 01 of119878 our grids are much finer than Zvan et al [17]

32 Numerical Computation of Floating Strike Asian OptionsTo value the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69)) for 119867(119905 119877) andthe values of the options will be 119881(119905 119878 119868) = 119878119867(119905 119877) with119877 = 119868119878 Actually what we really care about is the optionvalues at time 119905 = 0 while from the definition of 119868

119905 it is

obvious that 1198680= 0 and therefore 119877

0= 0 so our goal of the

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

Mathematical Problems in Engineering 9

the tree that they are effectively unusable Monte Carlosimulation works well for European style options (see [12])but is relatively slow Sun [38] proposed to use the weightedupwind method [39] to price Asian options but he didnot give numerical results The Asian option pricing in theuncertain volatility model will be even more difficult than inthe model without volatility uncertainty

For the arithmetic average fixed strike Asian call optionsdue to the convexity of the value 119881(119905 119878 119868) with respect tovariable 119878 (43) can be written as (48) which are the equationssatisfied by the arithmetic average fixed strike Asian calloptions with certain volatility 120590

Zvan et al [17] point out that the Crank-Nicolson schemein conjunction with the van Leer flux limiter method canbe used to numerically value Asian options with certainvolatilities The van Leer limiter has the property that it istotal variation diminishing and thus produces oscillation freesecond order accurate solutions But the van Leer limiter isnonlinear so the solutions should be obtained by using fullNewton iteration at each time level In this paper we will usethe alternating direction implicit (ADI) methods (see Duffy[40])

The arithmetic average fixed strike Asian put options canbe valued by the put-call parity

For the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69))These PDEs are one-dimensional in space And they are not linear equations butHJB equations in some sense For such PDEs we will use thefully implicit positive coefficient finite different method [41]to solve them We will see that this method can ensure thenumerical solutions converge to the viscosity solutions andis accurate efficient and quick to be implemented

All the calculations were performed byMatlab 2009 on acomputer with 183GHz hard disk and 512MB memory

31 Numerical Computation of Fixed Strike Asian OptionsAn equivalent formulation of (48) in terms of the average119860

119905= (1119905) int

119905

0119878120591119889120591 is given in Barraquand and Pudet [16] in

certain volatility model with volatility 120590 = 120590

120597119905119881 (119905 119878 119860) +

1

2211987821205902119881119878119878

(119905 119878 119860) + 119903119878119881119878(119905 119878 119860)

+119878 minus 119860

119905119881119860(119905 119878 119860) minus 119903119881 (119905 119878 119860) = 0

119881 (119879 119878 119860) = (119860 minus 119870)+

(77)

Equation (77) is convection dominated in the 119860 direc-tion because there is no diffusion effect in this dimensionBarraquand and Pudet [16] find that an explicit centrallyweighted scheme for (77) is unstable In particular theconvective term in the 119860 dimension becomes very large as119905 rarr 0 Barraquand and Pudet also note that implicit cen-trally weighted schemes will generally produce unsatisfactoryresults because of the numerical diffusion introduced by thisfirst order accurate in time scheme

Zvan et al [17] point out that under the conditionof convection dominated solutions generated by using a

centrally weighted scheme on the convective term for (77)cannot be ensured to be free of oscillations And solutionsgenerated by the first order upstream weighting scheme onthe convective termare no longer oscillatory but their profilesare too diffuse To produce oscillation free solutions withoutthe excessive diffusion of first order upstream weightingZvan et al [17] suggest the nonlinear van Leer flux limiter inconjunction with the Crank-Nicolson scheme Since the fluxlimiter is nonlinear the solutions should be obtained by usingfull Newton iteration at each time level

The ADI method pioneered in the United States by Dou-glas Rachford Peaceman Gunn and others has a numberof advantages First explicit difference methods are rarelyused to solve initial boundary value problems because oftheir poor stability Implicit methods have superior stabilityproperties but unfortunately they are difficult to solve in twoand more dimensions Consequently ADI methods becomean alternative because they can be programmed by solvinga simple trigonal system of equations For the convection-dominated problems we could use the exponentially fittedschemes in each leg of the approximate scheme to ensure thestability of the results [40] Since Barraquand and Pudet [16]have used the forward shooting grid method and Zvan et al[17] have used the van Leer flux limiter we will test the ADImethod in this paper by comparing our results with theirs

For convenience let us first convert (55) to be thefollowing forward equation in time by substituting 119905 with120591 = 119879 minus 119905

minus 120597120591119881 (120591 119878 119868) +

1

2211987821205902119881119878119878

(120591 119878 119868) + 119903119878119881119878(120591 119878 119868)

+ 119878119881119868(120591 119878 119868) minus 119903119881 (120591 119878 119868) = 0

119881 (0 119878 119868) = (119868

119879minus 119870)

+

119881 (120591 0 119868) = 119890minus119903120591

(119868

119879minus 119870)

+

119881 (119905 119878max 119868) = max ( 119868

119879minus 119870) 119890

minus119903120591+

119878max119903119879

(1 minus 119890minus119903120591

) 0

119881 (120591 119878 119870119879) =1

119903119879(1 minus 119890

minus119903120591) 119878

(78)

We let 120591119899 119899 = 0 1 119873 be a set of portion points in

[0 119879] satisfying 0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 and use even

partition of time that is 120591119899= 119899Δ120591 119899 = 0 1 119873 Define the

partitions of space variables as 0 = 1198780lt 119878

1lt sdot sdot sdot lt 119878

119872= 119878max

and 0 = 1198680lt 119868

1lt sdot sdot sdot lt 119868

119869= 119868max

With ADI we march from time level 119899 to time level119899 + 12 and then from time level 119899 + 12 to time level 119899 + 1In this case we use exponential fitting in all space variables

10 Mathematical Problems in Engineering

and implicit Euler in time The first leg is given by thescheme

minus

119881119899+12

119894119895minus 119881

119899

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899

119894119895+1minus 119881

119899

119894119895minus1

Δ119868119895+1

+ Δ119868119895

minus 119903119881119899+12

119894119895= 0

(79)

with

120588119894119895

=119903119878

119894Δ119878

119894+1

2coth(

119903119878119894Δ119878

119894+1

12059022119878

2

119894

) coth (119909) =1198902119909

+ 1

1198902119909 minus 1

(80)

Let

120572119894= minus

120588119894119895

Δ119878119894Δ119878

119894+1

120573119894= minus

120588119894119895

(Δ119878119894+1

)2minus

119903119878119894

Δ119878119894+1

120574119894= minus 120572

119894minus 120573

119894+ 119903

(81)

Then the scheme (79) can be written as

120572119894119881

119899+12

119894minus1119895+ (120574

119894+

2

Δ120591)119881

119899+12

119894119895+ 120573

119894119881

119899+12

119894+1119895

=2

Δ120591119881

119899

119894119895+

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895+1minus

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895minus1

(82)

We let

119881119899

119895= [119881

119899

1119895 119881

119899

2119895 119881

119899

119872minus1119895]119879

119881119899+12

119895= [119881

119899+12

1119895 119881

119899+12

2119895 119881

119899+12

119872minus1119895]119879

119891119899+12

119895= [120572

1119881

119899+12

0119895 0 0 120573

119872minus1119881

119899+12

119872119895]119879

119903119899

119895= [(

2

Δ120591119881

119899

1119895+

1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895+1

minus1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895minus1)

(2

Δ120591119881

119899

119872minus1119895+

119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895+1

minus119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895minus1)]

119879

1198601=

[[[

[

1205741

1205731

1205722

1205742

1205732

d d d120572119872minus1

120574119872minus1

]]]

](119872minus1)times(119872minus1)

(83)

Then we can write (82) as the following equivalent matrixform

1198601119881

119899+12

119895+ 119891

119899+12

119895= 119903

119899

119895 (84)

The second leg of the discretized PDE in (78) is given bythe scheme

minus

119881119899+1

119894119895minus 119881

119899+12

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899+1

119894119895+1minus 119881

119899+1

119894119895

Δ119868119895+1

minus 119903119881119899+12

119894119895= 0

(85)

Let

120575119895=

2

Δ120591+

119878119894

Δ119868119895+1

120598119895= minus

119878119894

Δ119868119895+1

(86)

Then the scheme (85) can be written as

120575119895119881

119899+1

119894119895minus 120598

119895119881

119899+1

119894119895+1= minus 120572

119894119881

119899+12

119894minus1119895minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119895

minus 120573119894119881

119899+12

119894+1119895

(87)

Mathematical Problems in Engineering 11

300

300

250

200

200

150

100

100

100

50

50

0

0 0

Call

valu

e

sI

Figure 1 Fixed strike Asian call value when = 08 120590 = 05 119903 =

010 119879 = 1 and 119870 = 100

We let

119881119899+1

119894= [119881

119899+1

1198940 119881

119899+1

1198941 119881

119899+1

119894119869minus1]119879

119881119899+12

119894= [119881

119899+12

1198940 119881

119899+12

1198941 119881

119899+12

119894119869minus1]119879

119865119894= [0 0 0 120598

119869minus1119881

119899+1

119894119869]119879

119869

119877119899+12

119894= [(minus120572

119894119881

119899+12

119894minus10minus(120574

119894minus

2

Δ120591)119881

119899+12

1198940minus120573

119894119881

119899+12

119894+10)

(minus120572119894119881

119899+12

119894minus1119869minus1minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119869minus1minus 120573

119894119881

119899+12

119894+1119869minus1)]

119879

1198602=

[[[

[

1205750

minus1205980

0 1205751

minus1205981

d d minus120598119869minus2

120575119869minus1

]]]

]119869times119869

(88)

The matrix form of (87) is

1198602119881

119899+1

119894+ 119865

119899+1

119894= 119877

119899+12

119894 (89)

Thematrix equations (84) and (89) can be solved by using LUdecomposition

The goal of the computation is to find the fair price of theAsian option at emission that is119881(119878

0 119868

0 0)Weneed tomake

sense of any value 119868 isin [0 119868max] and stock price 119878 isin [0 119878max] attime 119905 = 0 There are no problems with the stock prices butthe integral at time zero can only be 0 for continuous averagesthat is 119868

0= 0

Figure 1 is the results we obtained by using the abovediscretization scheme with = 08 120590 = 05 119903 = 010 119879 = 1and 119870 = 100 The point marked by lowast is the price 119881(0 119878 0)

of the Asian option We choose 119878max = 3119870 to ensure thedesirable accuracy

Table 1 shows the convergence of the results with therefinement of the grid From (78) the values of variable 119868

lie in the interval [0 119870119879] where 119870119879 is the boundary and0 is the point where the continuous averages Asian options

are valued so there is no point that is less important on theinterval [0 119870119879] Hence we use uniform grids in direction 119868

In our uncertain volatility model the pricing PDEs (48)are similar to the PDEs in certain volatility model withvolatility 120590 = 120590 Barraquand and Pudet [16] and Zvan etal [17] give numerical results of Asian options with certainvolatilities In the uncertain volatility model when =

08 120590 = 05 the Asian option PDE is the same as that ofthe Asian option with certain volatility 120590 = 04 When theparameters 119903 = 010 119879 = 1 119870 = 95 and grid Δ119868 =

011875 ΔS = 285 the price of the Asian call option wecalculate is 1382 The results with the same parameters ofBarraquand and Pudet [16] and Zvan et al [17] are 13825 and13721 respectively

Table 2 is the comparison of our results with those ofZvan et al [17] Z F and V in Table 2 refer to the resultof Zvan et al [17] Our results are denoted by Asian fixedstrike which calculated with Δ120591 = 001 002 004 Δ119868 =

0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Since119868max = 119870119879 the partitions in 119868 direction are related with thematurity 119879 The execution time is much less than theirs Thegrid spacing was chosen to achieve an accuracy of at least010 of 119878 Our results are close to that of Zvan et al [17] andthe difference is less than 010 of 119878

Our calculation procedure is different from Zvan et al[17] as well as Barraquand and Pudet [16] in the followingfour aspects First our pricing PDE (78) is in terms of therunning sum 119868

119905 Zvan et al [17] and Barraquand and Pudet

[16] use the pricing PDE

minus 120597120591119881 (120591 119878 119860) +

1

211987821205902119881119878119878

(120591 119878 119860) + 119903119878119881119878(120591 119878 119860)

+119878 minus 119860

119905119881119860(120591 119878 119860) minus 119903119881 (120591 119878 119860) = 0

(90)

where 119860119905is defined as 119860

119905= 119868

119905119905 But PDE (78) and (90) are

equivalent for pricing fixed strike European Asian optionsSecond since the PDEs are different PDE (78) and (90) musthave different boundary conditions We do not know theboundary conditions used by Zvan et al [17] The forwardshooting grid method used by Barraquand and Pudet [16]proceeds without any boundary conditions Third our PDE(78) is defined on the domain 119863 = (119905 119878 119868) 0 le 119905 le 119879 0 le

119878 le 119878max 0 le 119868 le 119870119879 and 119868max = 119870119879 is the boundary and119868 = 0 is the point where the options are valued So there isno point that is less important on the interval [0 119870119879] Henceour numerical schemes use uniform grid spacing while Zvanet al [17] use nonuniform grid spacing of 41 times 45 on domain[0 300] times [0 300] To achieve an accuracy of at least 01 of119878 our grids are much finer than Zvan et al [17]

32 Numerical Computation of Floating Strike Asian OptionsTo value the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69)) for 119867(119905 119877) andthe values of the options will be 119881(119905 119878 119868) = 119878119867(119905 119877) with119877 = 119868119878 Actually what we really care about is the optionvalues at time 119905 = 0 while from the definition of 119868

119905 it is

obvious that 1198680= 0 and therefore 119877

0= 0 so our goal of the

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

10 Mathematical Problems in Engineering

and implicit Euler in time The first leg is given by thescheme

minus

119881119899+12

119894119895minus 119881

119899

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899

119894119895+1minus 119881

119899

119894119895minus1

Δ119868119895+1

+ Δ119868119895

minus 119903119881119899+12

119894119895= 0

(79)

with

120588119894119895

=119903119878

119894Δ119878

119894+1

2coth(

119903119878119894Δ119878

119894+1

12059022119878

2

119894

) coth (119909) =1198902119909

+ 1

1198902119909 minus 1

(80)

Let

120572119894= minus

120588119894119895

Δ119878119894Δ119878

119894+1

120573119894= minus

120588119894119895

(Δ119878119894+1

)2minus

119903119878119894

Δ119878119894+1

120574119894= minus 120572

119894minus 120573

119894+ 119903

(81)

Then the scheme (79) can be written as

120572119894119881

119899+12

119894minus1119895+ (120574

119894+

2

Δ120591)119881

119899+12

119894119895+ 120573

119894119881

119899+12

119894+1119895

=2

Δ120591119881

119899

119894119895+

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895+1minus

119878119894

Δ119868119895+1

+ Δ119868119895

119881119899

119894119895minus1

(82)

We let

119881119899

119895= [119881

119899

1119895 119881

119899

2119895 119881

119899

119872minus1119895]119879

119881119899+12

119895= [119881

119899+12

1119895 119881

119899+12

2119895 119881

119899+12

119872minus1119895]119879

119891119899+12

119895= [120572

1119881

119899+12

0119895 0 0 120573

119872minus1119881

119899+12

119872119895]119879

119903119899

119895= [(

2

Δ120591119881

119899

1119895+

1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895+1

minus1198781

Δ119868119895+1

+ Δ119868119895

119881119899

1119895minus1)

(2

Δ120591119881

119899

119872minus1119895+

119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895+1

minus119878119872minus1

Δ119868119895+1

+ Δ119868119895

119881119899

119872minus1119895minus1)]

119879

1198601=

[[[

[

1205741

1205731

1205722

1205742

1205732

d d d120572119872minus1

120574119872minus1

]]]

](119872minus1)times(119872minus1)

(83)

Then we can write (82) as the following equivalent matrixform

1198601119881

119899+12

119895+ 119891

119899+12

119895= 119903

119899

119895 (84)

The second leg of the discretized PDE in (78) is given bythe scheme

minus

119881119899+1

119894119895minus 119881

119899+12

119894119895

(12) Δ120591+ 120588

119894119895[

1

Δ119878119894Δ119878

119894+1

119881119899+12

119894minus1119895

minus (1

Δ1198782

119894+1

+1

Δ119878119894Δ119878

119894+1

)119881119899+12

119894119895

+1

Δ1198782

119894+1

119881119899+12

119894+1119895]

+ 119903119878119894

119881119899+12

119894+1119895minus 119881

119899+12

119894119895

Δ119878119894+1

+ 119878119894

119881119899+1

119894119895+1minus 119881

119899+1

119894119895

Δ119868119895+1

minus 119903119881119899+12

119894119895= 0

(85)

Let

120575119895=

2

Δ120591+

119878119894

Δ119868119895+1

120598119895= minus

119878119894

Δ119868119895+1

(86)

Then the scheme (85) can be written as

120575119895119881

119899+1

119894119895minus 120598

119895119881

119899+1

119894119895+1= minus 120572

119894119881

119899+12

119894minus1119895minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119895

minus 120573119894119881

119899+12

119894+1119895

(87)

Mathematical Problems in Engineering 11

300

300

250

200

200

150

100

100

100

50

50

0

0 0

Call

valu

e

sI

Figure 1 Fixed strike Asian call value when = 08 120590 = 05 119903 =

010 119879 = 1 and 119870 = 100

We let

119881119899+1

119894= [119881

119899+1

1198940 119881

119899+1

1198941 119881

119899+1

119894119869minus1]119879

119881119899+12

119894= [119881

119899+12

1198940 119881

119899+12

1198941 119881

119899+12

119894119869minus1]119879

119865119894= [0 0 0 120598

119869minus1119881

119899+1

119894119869]119879

119869

119877119899+12

119894= [(minus120572

119894119881

119899+12

119894minus10minus(120574

119894minus

2

Δ120591)119881

119899+12

1198940minus120573

119894119881

119899+12

119894+10)

(minus120572119894119881

119899+12

119894minus1119869minus1minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119869minus1minus 120573

119894119881

119899+12

119894+1119869minus1)]

119879

1198602=

[[[

[

1205750

minus1205980

0 1205751

minus1205981

d d minus120598119869minus2

120575119869minus1

]]]

]119869times119869

(88)

The matrix form of (87) is

1198602119881

119899+1

119894+ 119865

119899+1

119894= 119877

119899+12

119894 (89)

Thematrix equations (84) and (89) can be solved by using LUdecomposition

The goal of the computation is to find the fair price of theAsian option at emission that is119881(119878

0 119868

0 0)Weneed tomake

sense of any value 119868 isin [0 119868max] and stock price 119878 isin [0 119878max] attime 119905 = 0 There are no problems with the stock prices butthe integral at time zero can only be 0 for continuous averagesthat is 119868

0= 0

Figure 1 is the results we obtained by using the abovediscretization scheme with = 08 120590 = 05 119903 = 010 119879 = 1and 119870 = 100 The point marked by lowast is the price 119881(0 119878 0)

of the Asian option We choose 119878max = 3119870 to ensure thedesirable accuracy

Table 1 shows the convergence of the results with therefinement of the grid From (78) the values of variable 119868

lie in the interval [0 119870119879] where 119870119879 is the boundary and0 is the point where the continuous averages Asian options

are valued so there is no point that is less important on theinterval [0 119870119879] Hence we use uniform grids in direction 119868

In our uncertain volatility model the pricing PDEs (48)are similar to the PDEs in certain volatility model withvolatility 120590 = 120590 Barraquand and Pudet [16] and Zvan etal [17] give numerical results of Asian options with certainvolatilities In the uncertain volatility model when =

08 120590 = 05 the Asian option PDE is the same as that ofthe Asian option with certain volatility 120590 = 04 When theparameters 119903 = 010 119879 = 1 119870 = 95 and grid Δ119868 =

011875 ΔS = 285 the price of the Asian call option wecalculate is 1382 The results with the same parameters ofBarraquand and Pudet [16] and Zvan et al [17] are 13825 and13721 respectively

Table 2 is the comparison of our results with those ofZvan et al [17] Z F and V in Table 2 refer to the resultof Zvan et al [17] Our results are denoted by Asian fixedstrike which calculated with Δ120591 = 001 002 004 Δ119868 =

0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Since119868max = 119870119879 the partitions in 119868 direction are related with thematurity 119879 The execution time is much less than theirs Thegrid spacing was chosen to achieve an accuracy of at least010 of 119878 Our results are close to that of Zvan et al [17] andthe difference is less than 010 of 119878

Our calculation procedure is different from Zvan et al[17] as well as Barraquand and Pudet [16] in the followingfour aspects First our pricing PDE (78) is in terms of therunning sum 119868

119905 Zvan et al [17] and Barraquand and Pudet

[16] use the pricing PDE

minus 120597120591119881 (120591 119878 119860) +

1

211987821205902119881119878119878

(120591 119878 119860) + 119903119878119881119878(120591 119878 119860)

+119878 minus 119860

119905119881119860(120591 119878 119860) minus 119903119881 (120591 119878 119860) = 0

(90)

where 119860119905is defined as 119860

119905= 119868

119905119905 But PDE (78) and (90) are

equivalent for pricing fixed strike European Asian optionsSecond since the PDEs are different PDE (78) and (90) musthave different boundary conditions We do not know theboundary conditions used by Zvan et al [17] The forwardshooting grid method used by Barraquand and Pudet [16]proceeds without any boundary conditions Third our PDE(78) is defined on the domain 119863 = (119905 119878 119868) 0 le 119905 le 119879 0 le

119878 le 119878max 0 le 119868 le 119870119879 and 119868max = 119870119879 is the boundary and119868 = 0 is the point where the options are valued So there isno point that is less important on the interval [0 119870119879] Henceour numerical schemes use uniform grid spacing while Zvanet al [17] use nonuniform grid spacing of 41 times 45 on domain[0 300] times [0 300] To achieve an accuracy of at least 01 of119878 our grids are much finer than Zvan et al [17]

32 Numerical Computation of Floating Strike Asian OptionsTo value the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69)) for 119867(119905 119877) andthe values of the options will be 119881(119905 119878 119868) = 119878119867(119905 119877) with119877 = 119868119878 Actually what we really care about is the optionvalues at time 119905 = 0 while from the definition of 119868

119905 it is

obvious that 1198680= 0 and therefore 119877

0= 0 so our goal of the

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

Mathematical Problems in Engineering 11

300

300

250

200

200

150

100

100

100

50

50

0

0 0

Call

valu

e

sI

Figure 1 Fixed strike Asian call value when = 08 120590 = 05 119903 =

010 119879 = 1 and 119870 = 100

We let

119881119899+1

119894= [119881

119899+1

1198940 119881

119899+1

1198941 119881

119899+1

119894119869minus1]119879

119881119899+12

119894= [119881

119899+12

1198940 119881

119899+12

1198941 119881

119899+12

119894119869minus1]119879

119865119894= [0 0 0 120598

119869minus1119881

119899+1

119894119869]119879

119869

119877119899+12

119894= [(minus120572

119894119881

119899+12

119894minus10minus(120574

119894minus

2

Δ120591)119881

119899+12

1198940minus120573

119894119881

119899+12

119894+10)

(minus120572119894119881

119899+12

119894minus1119869minus1minus (120574

119894minus

2

Δ120591)119881

119899+12

119894119869minus1minus 120573

119894119881

119899+12

119894+1119869minus1)]

119879

1198602=

[[[

[

1205750

minus1205980

0 1205751

minus1205981

d d minus120598119869minus2

120575119869minus1

]]]

]119869times119869

(88)

The matrix form of (87) is

1198602119881

119899+1

119894+ 119865

119899+1

119894= 119877

119899+12

119894 (89)

Thematrix equations (84) and (89) can be solved by using LUdecomposition

The goal of the computation is to find the fair price of theAsian option at emission that is119881(119878

0 119868

0 0)Weneed tomake

sense of any value 119868 isin [0 119868max] and stock price 119878 isin [0 119878max] attime 119905 = 0 There are no problems with the stock prices butthe integral at time zero can only be 0 for continuous averagesthat is 119868

0= 0

Figure 1 is the results we obtained by using the abovediscretization scheme with = 08 120590 = 05 119903 = 010 119879 = 1and 119870 = 100 The point marked by lowast is the price 119881(0 119878 0)

of the Asian option We choose 119878max = 3119870 to ensure thedesirable accuracy

Table 1 shows the convergence of the results with therefinement of the grid From (78) the values of variable 119868

lie in the interval [0 119870119879] where 119870119879 is the boundary and0 is the point where the continuous averages Asian options

are valued so there is no point that is less important on theinterval [0 119870119879] Hence we use uniform grids in direction 119868

In our uncertain volatility model the pricing PDEs (48)are similar to the PDEs in certain volatility model withvolatility 120590 = 120590 Barraquand and Pudet [16] and Zvan etal [17] give numerical results of Asian options with certainvolatilities In the uncertain volatility model when =

08 120590 = 05 the Asian option PDE is the same as that ofthe Asian option with certain volatility 120590 = 04 When theparameters 119903 = 010 119879 = 1 119870 = 95 and grid Δ119868 =

011875 ΔS = 285 the price of the Asian call option wecalculate is 1382 The results with the same parameters ofBarraquand and Pudet [16] and Zvan et al [17] are 13825 and13721 respectively

Table 2 is the comparison of our results with those ofZvan et al [17] Z F and V in Table 2 refer to the resultof Zvan et al [17] Our results are denoted by Asian fixedstrike which calculated with Δ120591 = 001 002 004 Δ119868 =

0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Since119868max = 119870119879 the partitions in 119868 direction are related with thematurity 119879 The execution time is much less than theirs Thegrid spacing was chosen to achieve an accuracy of at least010 of 119878 Our results are close to that of Zvan et al [17] andthe difference is less than 010 of 119878

Our calculation procedure is different from Zvan et al[17] as well as Barraquand and Pudet [16] in the followingfour aspects First our pricing PDE (78) is in terms of therunning sum 119868

119905 Zvan et al [17] and Barraquand and Pudet

[16] use the pricing PDE

minus 120597120591119881 (120591 119878 119860) +

1

211987821205902119881119878119878

(120591 119878 119860) + 119903119878119881119878(120591 119878 119860)

+119878 minus 119860

119905119881119860(120591 119878 119860) minus 119903119881 (120591 119878 119860) = 0

(90)

where 119860119905is defined as 119860

119905= 119868

119905119905 But PDE (78) and (90) are

equivalent for pricing fixed strike European Asian optionsSecond since the PDEs are different PDE (78) and (90) musthave different boundary conditions We do not know theboundary conditions used by Zvan et al [17] The forwardshooting grid method used by Barraquand and Pudet [16]proceeds without any boundary conditions Third our PDE(78) is defined on the domain 119863 = (119905 119878 119868) 0 le 119905 le 119879 0 le

119878 le 119878max 0 le 119868 le 119870119879 and 119868max = 119870119879 is the boundary and119868 = 0 is the point where the options are valued So there isno point that is less important on the interval [0 119870119879] Henceour numerical schemes use uniform grid spacing while Zvanet al [17] use nonuniform grid spacing of 41 times 45 on domain[0 300] times [0 300] To achieve an accuracy of at least 01 of119878 our grids are much finer than Zvan et al [17]

32 Numerical Computation of Floating Strike Asian OptionsTo value the arithmetic average floating strike Asian call (put)options we need to solve (67) (see (69)) for 119867(119905 119877) andthe values of the options will be 119881(119905 119878 119868) = 119878119867(119905 119877) with119877 = 119868119878 Actually what we really care about is the optionvalues at time 119905 = 0 while from the definition of 119868

119905 it is

obvious that 1198680= 0 and therefore 119877

0= 0 so our goal of the

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

12 Mathematical Problems in Engineering

Table 1 Successive grid refinements demonstrating convergence of the results with the refinements when 119903 = 010 119879 = 1 = 08 120590 = 05and 119870 = 95 The values are the option prices at 119878 = 100 Δ119878 and Δ119868 denote the grid spacing in directions 119878 and 119868 respectively Δ119905 = 004Execution times (in seconds) are in parentheses

Δ119868 = 095 Δ119868 = 0475 Δ119868 = 02375 Δ119868 = 011875 Δ119868 = 0059375

Δ119878 = 57014532 1419 14021 1394 1389(101) (210) (7081) (27225) (90)

Δ119878 = 2851442 1408 13914 1382 13781(308) (713) (23012) (7651) (292)

Δ119878 = 14251439 14053 13879 13790 1374(1022) (2471) (7601) (238) (940)

Table 2 Fixed strike Asian call values when 119903 = 010 and 119870 = 95The values are the option prices at 119878 = 100 Our results are denotedby Asian fixed strike which calculated with Δ120591 = 001 002 004Δ119868 = 0059375 011875 02375 and Δ119878 = 285 for maturities ofone quarter half a year and one year respectively Z F and V referto the result of Zvan et al [17] Execution times (in seconds) are inparentheses

120590 119879 Asian fixed strike Z F and V

02

05025 6190 (230) 613305 7199 (240) 72441 9204 (236) 9316

1025 656 (230) 650105 799 (241) 79211 1044 (250) 10309

2025 830 (241) 812305 1055 (244) 103571 1391 (252) 13721

computation is to find the prices of the Asian options at 119905 = 0that is 119881(0 119878 0) = 119878119867(119905 0)

To solve (67) we use the fully implicit positive coefficientdiscretization which will ensure convergence to the viscositysolution

Let 120591 = 119879 minus 119905 and then the PDE in (67) can be changedinto the following forward equation in time

minus 120597120591119867(120591 119877) + sup

1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877) = 0

(91)

Now let us consider the discretization of (91) Define agrid 0 = 119877

0lt 119877

1lt sdot sdot sdot lt 119877

119872= 119877max and a set of timesteps

0 = 1205910lt 120591

1lt sdot sdot sdot lt 120591

119873= 119879 Let the discretization parameters

be given by

Δ119877119894= 119877

119894+1minus 119877

119894 119894 = 0 119872 (92)

we use even discretization in time that is 120591119899+1 minus 120591119899

= Δ120591119899 = 0 119873 minus 1

Let119867119899

119894be the approximate values of the solution at 120591119899 119877

119894

that is 119867119899

119894≃ 119867(120591

119899 119877

119894) If 1 minus 119903119877

119894ge 0 we use forward

difference to term 119867119877(120591 119877) Otherwise we use backward

difference to term119867119877(120591 119877)

The general form of the discretized PDE (91) is

120572119899+1

119894119867

119899+1

119894minus1+ 120574

119899+1

119894119867

119899+1

119894+ 120573

119899+1

119894119867

119899+1

119894+1=

1

Δ120591119867

119899

119894 (93)

with

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

minus1 minus 119903119877

119894

Δ119877119894+1

if 1 minus 119903119877119894ge 0

120572119899+1

119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ119877119894Δ119877

119894+1

minus119903119877

119894minus 1

Δ119877119894

120573119894= minus

2119877

2

119894(

119899+1

119894)2

2Δ1198772

119894+1

if 1 minus 119903119877119894lt 0

120574119899+1

119894= minus 120572

119899+1

119894minus 120573

119899+1

119894+

1

Δ120591

(94)

where

119899+1

119894

= argmax120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

]

(95)

Of course we have120572119899+1

119894le 0 120573

119899+1

119894le 0 and 120574

119899+1

119894ge 0 So the

discretization equation (93) satisfies the positive coefficientcondition of Forsyth and Labahn [41]

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

Mathematical Problems in Engineering 13

Let

119867119899= [119867

119899

1 119867

119899

119872minus1]119879

120590119899+1

= [120590119899+1

1 120590

119899+1

119872minus1]119879

119891119899+1

= [120572119899+1

1119867

119899+1

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

119894)

=

[[[[[[[[

[

120574119899+1

1120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]

](119872minus1)times(119872minus1)

(96)

Then (93) can be written as the following matrix form

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(97)

By the boundary condition

120597119905119867(119905 119877) + 120597

119877119867(119905 119877) = 0 119877 = 0 (98)

we replace 119867119899+1

0in 119891

119899+1 with (Δ120591(Δ1198771

+ Δ120591))119867119899+1

1+

(Δ1198771(Δ119877

1+ Δ120591))119867

119899

0 Then (97) can be written as

119860119899+1

119867119899+1

=1

Δ120591119867

119899minus 119891

119899+1

119899+1

= argmax120590le120590119899+1

le120590

[119860119899+1

119867119899+1

+ 119891119899+1

]

(99)

with

119891119899+1

= [120572119899+1

1Δ119877

1

Δ1198771+ Δ120591

119867119899

0 0 0 120573

119899+1

119872minus1119867

119899+1

119872]

119879

119872minus1

119860119899+1

= 119860119899+1

(120590119899+1

)

=

[[[[[[[[[

[

120574119899+1

1+

120572119899+1

1Δ120591

Δ1198771+ Δ120591

120573119899+1

1

120572119899+1

2120574119899+1

2120573119899+1

2

d d d

120572119899+1

119872minus1120574119899+1

119872minus1

]]]]]]]]]

](119872minus1)times(119872minus1)

(100)

It is clear that119860119899+1 is119872-matrix From the property of119872-matrix the solution of (99) exists and is unique

It is important to ensure that we generate a numericalsolution which converges to the viscosity solution It has been

shown that (67) satisfies the strong comparison property [42ndash44]Then from Barles and Souganidis [45] and Barles [46] anumerical scheme converges to the viscosity solution if themethod is consistent stable and monotone Thus we willshow that our numerical scheme satisfies these conditions

Lemma 6 (Stability) The discretization (93) is stable

Proof It follows from (93) that

120574119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894

10038161003816100381610038161003816le

1

Δ120591

1003816100381610038161003816119867119899

119894

1003816100381610038161003816 minus 120572119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894minus1

10038161003816100381610038161003816minus 120573

119899+1

119894

10038161003816100381610038161003816119867

119899+1

119894+1

10038161003816100381610038161003816

le1

Δ120591

10038171003817100381710038171198671198991003817100381710038171003817infin minus (120572

119899+1

119894+ 120573

119899+1

119894)10038171003817100381710038171003817119867

119899+110038171003817100381710038171003817infin

(101)

since 120572119899+1

119894le 0 120573

119899+1

119894le 0

Denote

119867119899+1

= [119867119899+1

0 119867

119899+1

119872]119879

(102)

If 119867119899+1

infin

= |119867119899+1

119895| 0 lt 119895 lt 119872 then we have

100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle

(1Δ120591)1003817100381710038171003817119867

1198991003817100381710038171003817infin

120574119899+1

119894+ 120572

119899+1

119894+ 120573

119899+1

119894

=1003817100381710038171003817119867

1198991003817100381710038171003817infin (103)

Otherwise 119867119899+1

infin

= |119867119899+1

0| or 119867

119899+1

infin

= |119867119899+1

119872|

Therefore100381710038171003817100381710038171003817119867

119899+1100381710038171003817100381710038171003817infinle max (10038171003817100381710038171003817119867

010038171003817100381710038171003817infin1003816100381610038161003816119867

119899

0

1003816100381610038161003816 1003816100381610038161003816119867

119899

119873

1003816100381610038161003816) (104)

with119867119899

0 119867119899

119873being the given boundary conditions

Lemma 7 (Monotonicity) The discretization (93) is mono-tonic

Proof The lemma is trivially true on the boundary so let usjust see the cases 0 lt 119894 lt 119872 and 0 lt 119899 le 119873 Denote

119866119899+1

119894=

119867119899+1

119894minus 119867

119899

119894

Δ120591

+ sup120590le120590119899+1

119894le120590

120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894

(105)For any 120598 gt 0

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1+ 120598119867

119899+1

119894minus1 119867

119899

119894)

minus 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

= sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894) (119867

119899+1

119894+1+ 120598)

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

minus sup120590le120590119899+1

119894le120590

(120572119899+1

119894(120590

119899+1

119894)119867

119899+1

119894minus1+ 120573

119899+1

119894(120590

119899+1

119894)119867

119899+1

119894+1

+ 120574119899+1

119894(120590

119899+1

119894)119867

119899+1

119894)

le sup120590le120590119899+1

119894le120590

(120573119899+1

119894(120590

119899+1

119894) 120598) le 0

(106)

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 14: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

14 Mathematical Problems in Engineering

since 120573119899+1

119894(120590

119899+1

119894) le 0 and sup

120590le120590le120590119866

1(120590) minus sup

120590le120590le120590119866

2(120590) le

sup120590le120590le120590

(1198661(120590) minus 119866

2(120590))

Similarly

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1+ 120598119867

119899

119894)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894+ 120598)

le 119866119899+1

119894(Δ119877119867

119899+1

119894 119867

119899+1

119894+1 119867

119899+1

119894minus1 119867

119899

119894)

(107)

Lemma 8 (Consistency) The discretization (93) is consistent

Proof Suppose 120601(120591 119877) is a smooth test function withbounded derivatives of all orders with respect to (120591 119877) Let

L119867(120591 119877) = sup1205902le1205902le1205902

1

22119877

21205902119867

119877119877(120591 119877)

+ (1 minus 119903119877)119867119877(120591 119877)

(108)

Then10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816

119867119899+1

119894minus 119867

119899

119894

Δ120591minus 119867

120591(120591 119877)

100381610038161003816100381610038161003816100381610038161003816

+10038161003816100381610038161003816L119867(120591 119877) minusL

119899+1

119894119867(120591 119877)

10038161003816100381610038161003816

(109)

where

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894+1minus 119867

119899+1

119894

Δ119877(1 minus 119903119877

119894)

(110)

if 1 minus 119903119877119894ge 0 otherwise

L119899+1

119894119867(120591 119877)

=1

2sup

120590le120590119899+1

119894le120590

2119877

2

119894(120590

119899+1

119894)2

times [119867

119899+1

119894minus1

Δ119877119894Δ119877

119894+1

minus (119867

119899+1

119894

Δ1198772

119894+1

+119867

119899+1

119894

Δ119877119894Δ119877

119894+1

)

+119867

119899

119894+1

Δ1198772

119894+1

] +119867

119899+1

119894minus 119867

119899+1

119894minus1

Δ119877(1 minus 119903119877

119894)

(111)

Using Taylor series expansions to (109) we have10038161003816100381610038161003816119866

119899+1

119894minus 119867

120591(120591 119877) +L119867(120591 119877)

10038161003816100381610038161003816le 119874 (Δ120591) + 119874 (Δ119877) (112)

that is (93) is consistent

Table 3 Successive grid refinements demonstrating convergence ofthe results with the refinements when 119903 = 010 120590 = 05 120590 = 1 = 04 119879 = 1 and tolerance = 10

minus4 The values are the option priceat 119878 = 100 Δ119877 denotes the grid spacing and Δ120591 denotes the timestepping Execution times (in seconds) are in parentheses

Δ119877 = 0008 Δ119877 = 0004 Δ119877 = 0002 Δ119877 = 0001 Δ119877 = 00005

Δ120591 = 0008 Δ120591 = 0004 Δ119905 = 0002 Δ119905 = 0001 Δ119905 = 00005

11988812698 12129 11828 11673 11592(100) (219) (908) (3566) (14138)

1199017831 7274 6981 6828 6753(101) (206) (905) (3410) (14312)

0 05 1 15 20

002

004

006

008

01

012

R

H

Figure 2 The 119867(0 119877) function of the arithmetic average floatingstrike Asian call option when 119903 = 010 119879 = 1 120590 = 05 120590 = 1 =

04 and tolerance = 10minus4

Therefore we have the following

Theorem 9 The solution of the discretization scheme (93)converges uniformly to the unique viscosity solution of PDE(91)

Table 3 shows the convergence of the values of thearithmetic average floating strike Asian call and put optionswith the refinement of the grid We use uniform grids Toensure the desirable accuracy we choose 119877max = 2 Theexecution times are less than 40 seconds

Figure 2 is the values of function 119867(0 119877) for the arith-metic average floating strike Asian call option with the givenparameters specified under the figure The correspondingoption values are 119878119867(0 0)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 15: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

Mathematical Problems in Engineering 15

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (11301011) and Beijing Natural ScienceFoundation (1112009) The authors would like to thank theanonymous referees for the comments

References

[1] Z Chen and L Epstein ldquoAmbiguity risk and asset returns incontinuous timerdquo Econometrica vol 70 no 4 pp 1403ndash14432002

[2] L G Epstein and T Wang ldquoUncertainty risk-neutral measuresand security price booms and crashesrdquo Journal of EconomicTheory vol 67 no 1 pp 40ndash82 1995

[3] B R Routledge and S Zin ldquoModel uncertainty and liquidityrdquoNBERWorking Paper 8683 2001

[4] L P Hansen T J Sargent and T D Tallarini Jr ldquoRobustpermanent income and pricingrdquo Review of Economic Studiesvol 66 no 4 pp 873ndash907 1999

[5] L Hansen T H Sargent G A Turluhambetova and NWilliams ldquoRobustness and uncertainty aversionrdquo WorkingPaper 2002

[6] L Denis and C Martini ldquoA theoretical framework for the pric-ing of contingent claims in the presence of model uncertaintyrdquoThe Annals of Applied Probability vol 16 no 2 pp 827ndash8522006

[7] M Avellaneda A Levy and A Paras ldquoPricing and hedgingderivative securities in markets with uncertain volatilitiesrdquoApplied Mathematical Finance vol 2 pp 73ndash88 1995

[8] T J Lyons ldquoUncertain volatility and the risk free synthesis ofderivativesrdquo Applied Mathematical Finance vol 2 pp 117ndash1331995

[9] F Delbaen ldquoCoherent risk measures on general probabilityspacesrdquo in Advances in Finance and Stochastics K Sandmannand P J Schonbucher Eds pp 1ndash37 Springer Berlin Germany2002

[10] H Geman and M Yor ldquoBessel processes Asian options andperpetuitiesrdquoMathmatical Finance vol 3 pp 349ndash375 1993

[11] MYorExponential Functionals of BrownianMotion andRelatedProcesses Springer Finance Springer Berlin Germany 2001

[12] AG Z Kemna andAC FVorst ldquoApricingmethod for optionsbased on average asset valuesrdquo Journal of Banking and Financevol 14 no 1 pp 113ndash129 1990

[13] J E Ingersoll Theory of Financial Decision Making BlackwellOxford UK 1987

[14] PWilmott J Dewynne and SHowisonOption Pricing OxfordFinancial Press Oxford UK 1993

[15] L C G Rogers and Z Shi ldquoThe value of an Asian optionrdquoJournal of Applied Probability vol 32 no 4 pp 1077ndash1088 1995

[16] J Barraquand and T Pudet ldquoPricing of American path-dependent contingent claimsrdquoMathematical Finance vol 6 no1 pp 17ndash51 1996

[17] R Zvan K R Vetzal and P A Forsyth ldquoPDE methods forpricing barrier optionsrdquo Journal of Economic Dynamics andControl vol 24 no 11-12 pp 1563ndash1590 2000

[18] R Zvan P A Forsyth and K R Vetzal ldquoA finite volumeapproach for contingent claims valuationrdquo IMA Journal ofNumerical Analysis vol 21 no 3 pp 703ndash731 2001

[19] S Simon M J Goovaerts and J Dhaene ldquoAn easy computableupper bound for the price of an arithmetic Asian optionrdquoInsurance Mathematics amp Economics vol 26 no 2-3 pp 175ndash183 2000

[20] R Kaas J Dhaene and M J Goovaerts ldquoUpper and lowerbounds for sums of random variablesrdquo Insurance Mathematicsamp Economics vol 27 no 2 pp 151ndash168 2000

[21] J Dhaene M Denuit M J Goovaerts R Kaas and DVyncke ldquoThe concept of comonotonicity in actuarial scienceand finance theoryrdquo Insurance Mathematics amp Economics vol31 no 1 pp 3ndash33 2002

[22] J Vecer ldquoA new PDE approach for pricing arithmetic averageAsian optionsrdquo Journal of Computational Finance vol 4 pp105ndash113 2001

[23] J E Zhang ldquoA semi-analytical method for pricing and hedgingcontinuously sampled arithmetic average rate optionsrdquo Journalof Computational Finance vol 5 pp 1ndash20 2001

[24] J E Zhang ldquoPricing continuously sampled Asian options withperturbation methodrdquo Journal of Futures Markets vol 23 no 6pp 535ndash560 2003

[25] M D Marcozzi ldquoOn the valuation of Asian options by varia-tional methodsrdquo SIAM Journal on Scientific Computing vol 24no 4 pp 1124ndash1140 2003

[26] CWOosterlee J C Frisch and F J Gaspar ldquoTVDWENOandblended BDF discretizations for Asian optionsrdquo Computing andVisualization in Science vol 6 no 2-3 pp 131ndash138 2004

[27] V Linetsky ldquoSpectral expansions for Asian (average price)optionsrdquo Operations Research vol 52 no 6 pp 856ndash867 2004

[28] G Fusai ldquoPricing Asian option via Fourier and Laplace trans-formsrdquo Journal of Computational Finance vol 7 pp 87ndash1062004

[29] Y DrsquoHalluin P A Forsyth and G Labahn ldquoA semi-Lagrangianapproach for American Asian options under jump diffusionrdquoSIAM Journal on Scientific Computing vol 27 no 1 pp 315ndash3452005

[30] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[31] E Bayraktar and H Xing ldquoPricing Asian options for jumpdiffusionrdquoMathematical Finance vol 21 no 1 pp 117ndash143 2011

[32] S Peng ldquo119866-expectation 119866-Brownian motion and relatedstochastic calculus of Ito typerdquo in Stochastic Analysis andApplications F Benth G Di Nunno T Lindstroslash m B Oslashksendal and T Zhang Eds vol 2 of Abel Symp pp 541ndash567Springer Berlin Germany 2007

[33] S Peng ldquoG-Brownianmotion and dynamic risk measure undervolatility uncertaintyrdquo httparxivorgabs07112834

[34] S Peng ldquoMulti-dimensional 119866-Brownian motion and relatedstochastic calculus under 119866-expectationrdquo Stochastic Processesand Their Applications vol 118 no 12 pp 2223ndash2253 2008

[35] S Peng ldquoFiltration consistent nonlinear expectations and eval-uations of contingent claimsrdquo Acta Mathematicae ApplicataeSinica vol 20 no 2 pp 191ndash214 2004

[36] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo httparxivorgabs10024546

[37] J Hugger ldquoWellposedness of the boundary value formulationof a fixed strike Asian optionrdquo Journal of Computational andApplied Mathematics vol 185 no 2 pp 460ndash481 2006

[38] P Sun Numerical methods for pricing American and Asianoptions [Doctoral Dissertation] Shandong University 2007

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 16: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

16 Mathematical Problems in Engineering

[39] D Liang and W Zhao ldquoA high-order upwind method for theconvection-diffusion problemrdquo Computer Methods in AppliedMechanics and Engineering vol 147 no 1-2 pp 105ndash115 1997

[40] D J Duffy Finite Difference Methods in Financial EngineeringWiley Finance Series JohnWiley amp Sons Chichester UK 2006

[41] A Forsyth and G Labahn ldquoNumerical methods for controlledHamilton-Jacobi-Bellman PDEs in financerdquo Journal of Compu-tational Finance vol 11 pp 1ndash44 2007

[42] G Barles and J Burdeau ldquoThe Dirichlet problem for semilinearsecond-order degenerate elliptic equations and applicationsto stochastic exit time control problemsrdquo Communications inPartial Differential Equations vol 20 no 1-2 pp 129ndash178 1995

[43] G Barles and E Rouy ldquoA strong comparison result for the Bell-man equation arising in stochastic exit time control problemsand its applicationsrdquo Communications in Partial DifferentialEquations vol 23 no 11-12 pp 1995ndash2033 1998

[44] S Chaumont ldquoA strong comparison result for viscosity solu-tions to Hamilton-Jacobi-Bellman equations with Dirichletconditions on a non-smooth boundaryrdquo Working Paper 2004

[45] G Barles and P E Souganidis ldquoConvergence of approximationschemes for fully nonlinear secondorder equationsrdquoAsymptoticAnalysis vol 4 no 3 pp 271ndash283 1991

[46] G Barles ldquoConvergence of numerical schemes for degenerateparabolic equations arising in finance theoryrdquo in NumericalMethods in Finance L C G Rogers and D Talay Eds PublNewton Inst pp 1ndash21 CambridgeUniversity Press CambridgeUK 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 17: Research Article The Pricing of Asian Options in Uncertain ...downloads.hindawi.com/journals/mpe/2014/786391.pdf · 2. The Pricing of Asian Options in Uncertain Volatility Model..

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


South Asian Countries Power Pricing Mechanism ......2018/03/22  · South Asian Countries Power Pricing Mechanism Recommendation for CBET The Ministry of Power, India has come out

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The US drug pricing debate 2017 - EYFile/ey-the-us-drug-pricing-debate-2017.pdf · The US drug pricing debate 2017 Managing risks to results through uncertain times

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Pricing and Hedging Asian Options Using Monte Carlo and ...important class of such options is the class of Asian options. Asian options have a wide variety of application in commodities,

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Pricing Asian Options using Monte Carlo Methods

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video Asian Option 20 periods’ pay-off stock’s up/down determined by mean and volatility u – up price jump d - down price jump Asian Option Pricing.

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Coopetition Strategy and Pricing Timing in an Outsourcing Supply … · 2019-01-10 · Coopetition Strategy and Pricing Timing in an Outsourcing Supply Chain with Uncertain Operation

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Uncertain Times, Uncertain Roles - BPE Global€¦ · Uncertain Times, Uncertain Roles Export Compliance Benchmark Study ... administration may bring a dismantling of existing trade

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Challenges to JCC Pricing in Asian LNG Markets · PDF fileChallenges to JCC Pricing in Asian ... JCC Pricing in Asian LNG Markets 1 ... to this form of pricing prior to the increase

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Pricing Asian Options under a General Jump Di ffusion Model · Pricing Asian Options under a General Jump Di ffusion Model Ning Cai1 andS.G.Kou2 1Room 5521, Department of IELM,

Pricing Asian Options under a General Jump Di ffusion Model · Pricing Asian Options under a General Jump Di ffusion Model Ning Cai1 andS.G.Kou2 1Room 5521, Department of IELM,

Robustness and Pricing with Uncertain Growthpages.stern.nyu.edu/~dbackus/Exotic/1Robustness... ·  · 2004-04-12Robustness and Pricing with Uncertain Growth Marco Cagetti, Lars Peter

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Global Medium-Term Note Program: Pricing …...BNP PARIBAS Credit Suisse HSBC Nomura Standard Chartered Bank PRICING SUPPLEMENT ADB ASIAN DEVELOPMENT BANK GLOBAL MEDIUM-TERM NOTE PROGRAM

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Pricing Asian Options Using Monte Carlo - GitHub Pages · Pricing Asian Options Using Monte Carlo Joy Tolia 1. Contents 1 Introduction 2 ... The full set of results for each option

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Comparing auction designs where suppliers have uncertain ... · –nd that, for buyers, uniform-pricing is preferable to discriminatory pricing when producers™private signals are

Comparing auction designs where suppliers have uncertain ... · –nd that, for buyers, uniform-pricing is preferable to discriminatory pricing when producers™private signals are