Research ArticleAn Efficient Method for Solving Spread Option Pricing ProblemNumerical Analysis and Computing

R Company V N Egorova and L Joacutedar

Instituto de Matematica Multidisciplinar Universitat Politecnica de Valencia Camino de Vera sn 46022 Valencia Spain

Correspondence should be addressed to V N Egorova egorovavngmailcom

Received 6 September 2016 Accepted 10 November 2016

Academic Editor Ivan Ivanov

Copyright copy 2016 R Company et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with numerical analysis and computing of spread option pricing problem described by a two-spatial variablespartial differential equation Both European and American cases are treated Taking advantage of a cross derivative removingtechnique an explicit difference scheme is developed retaining the benefits of the one-dimensional finite difference methodpreserving positivity accuracy and computational time efficiency Numerical results illustrate the interest of the approach

1 Introduction

Multiasset option pricing problem has two main challengesarising from their high dimensions and the correlationsbetween assets This is a type of option that derives itsvalue from the difference between the prices of two or moreassets Spread options can be written on all types of financialproducts including equities bonds and currencies Spreadoptions are frequently traded in the energy market [1] Twoexamples are as follows

Crack Spreads Crack spreads are the options on the spreadbetween refined petroleum products and crude oil Thespread represents the refinement margin made by the oilrefinery by ldquocrackingrdquo the crude oil into a refined petroleumproduct

Spark Spreads Spark spreads are the options on the spreadbetween electricity and some type of fuel The spread repre-sents the margin of the power plant which takes fuel to runits generator to produce electricity

Since the closed form solution is not available for suchproblems many numerical methods are employed Probablythe most popular methods of solving multiasset optionpricing problems are those around Monte Carlo method andits modifications [2 3] However their simulation are usuallytime consuming and slow convergent [4]

Analytical-numerical methods are also existing in theliterature as it occurs in the one asset case So Kirk and Aron[5] provide an approximation method to price a bivariatespread option problem but it is inaccurate when the strikeprices are high [1 6 7] Alexander and Venkatramananimprove Kirkrsquos approach in [6] based on the hypothesis thatthe spread option price is the sum of the prices of twocompound exchange options which avoids any strike conven-tion

Fourier Transform approach and numerical integrationtechniques are used by Chiarella and Ziveyi in [8 9] fornumerical evaluation of American options written on twounderlying assets

Among the numerical methods we mention the so-calledlattice methods [10 11] Recently authors in [4] improve theresults of [10 11] and give fast and accurate results although itrequires some minor correlation restrictions

Dealing with pure finite difference methods the existenceof the cross derivative term in the partial differential equation(PDE) problem due to asset correlation introduces additionalchallenges in the numerical solution due to the existence ofnegative coefficient terms in the numerical scheme as well asinvolving expensive stencil schemes Both facts may producenumerical drawbacks coming from the dominance of theconvection versus the diffusion as well as more expensivecomputational cost [12ndash15]

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2016 Article ID 1549492 11 pageshttpdxdoiorg10115520161549492

2 Abstract and Applied Analysis

In order to overcome these difficulties the authors haverecently proposed several strategies dealing with finite dif-ference approach high order compact schemes are proposedin [16] using central difference approximations for stochasticvolatility HestonmodelThe authors in [17] use ADImethodsand clever discretizations to obtain stable numerical schemesunder minor coefficients restrictions in the PDEThe authorsof [14 18] use special finite difference approximations of thecross derivative term based on a seven-point stencil instead ofthe nine-point stencil scheme resulting from the central finitedifference approach

In this paper we continue removing cross derivativeapproach based on the transformation of the PDE probleminitiated in [12] for the case of one asset stochastic volatilityHeston model and applied to the Bates stochastic volatilityjump diffusion model in [19] Apart from avoiding negativecoefficients in the proposed scheme it has the additionalcomputational advantage that uses only a five-point stencil

Herewe consider spread option pricing problems for bothEuropean and American cases The paper is organized asfollowsThe first part of the paper deals with European spreadcall option with payoff 119892(1198781 1198782) = (1198781 minus 1198782 minus 119864)+ dependingon two assets 1198781 and 1198782 and strike price 119864 ge 0 In Section 2the PDE with the boundary conditions is established for thespread option pricing problem In Section 3 we remove thecross derivative term of the PDE using the canonical formtransformation method [20] In that section we also developthe discretization of the continuous European spread calloption problem achieving an explicit finite difference schemeSection 4 is devoted to the study of the properties of themethod like consistence and conditional stability and posi-tivity In Section 5 we extend the study of previous sectionsto the European spread put option summarizing the mainresults including the treatment of the boundary conditionsSection 6 deals withAmerican spread option pricing problemusing the LCP technique based on the proposed differencescheme Finally in Section 7 we illustrate the numericalresults computing simulating and comparing accuracy andCPU time with other well acknowledge methods

Throughout the paper we use the following notation119860infin = max |119886119894119895| 1 le 119894 le 119873 1 le 119895 le 119872 will denote asupremum norm for a given matrix 119860 = (119886119894119895) isin R119873times1198722 Spread Option Pricing Problem andBoundary Conditions

The price 119880(1198781 1198782 120591) of a spread option is the solution of thePDE

120597119880120597120591 = 121205902111987821 120597211988012059711987821 +

121205902211987822 120597211988012059711987822 + 1205881205901120590211987811198782

120597211988012059711987811205971198782+ (119903 minus 1199021) 1198781 1205971198801205971198781 + (119903 minus 1199022) 1198782

1205971198801205971198782 minus 1199031198800 lt 1198781 lt infin 0 lt 1198782 lt infin 0 lt 120591 le 119879

(1)

where 120591 = 119879minus119905 is the time to maturity 119879 119903 is the interest rate120590119894 is the volatility of the asset 119878119894 119902119894 is the dividend yield of

asset 119878119894 and 120588 is the correlation parameter see [1] Since weconsider backwards time it changes the signs of the coeffi-cients of (1) and instead of a terminal condition the followinginitial condition is considered

119880(1198781 1198782 0) = 119892 (1198781 1198782) = max (1198781 minus 1198782 minus 119864 0)= (1198781 minus 1198782 minus 119864)+ (2)

The case 119864 = 0 corresponds to exchange options that canbe reduced to 1D problem by the change of variables 1199091 =11987811198782 119888(1199091 120591) = 119880(1198781 1198782 120591)1198782

Since the numerical solution of European multiassetoptions requires the selection of a bounded numericaldomain and the translation of the boundary conditions tothe boundary of the domain it is important to pay attentionto such conditions For the initial problem (1)-(2) suitableboundary conditions areas follows

(1) For 1198781 = 0 the payoff (1198781minus1198782minus119864)+ suggests Dirichletrsquoscondition

119880 (0 1198782 120591) = 0 0 lt 1198782 lt infin 0 lt 120591 le 119879 (3)

as one-dimensional call option(2) Taking the ideas developed by some authors for

instance Duffy for the case of basket option (see [21]p 270) for 1198782 = 0we take the value of the closed formsolution of the basic Black-Scholes equation of a callfor 1198781 with given strike 119864119880 (1198781 0 120591) = 119890minus119903120591 (119865119873 (1206011) minus 119864119873 (1206012)) (4)

where

119865 = 1198781119890(119903minus1199021)1205911206011 = 11205901radic120591 [log

119865119864 + 1212059021120591] 1206012 = 1206011 minus 1205901radic120591

(5)

where119873(119909) is the standard normal cumulative distri-bution function

(3) As 1198781 is large versus 1198782 + 119864 then the behaviour ofthe solution looks like the asymptotic value of a one-dimensional vanilla call option with the strike (1198782+119864)for 1198781 ≫ 1198782 + 119864 (see [22] p 157)

119880 (1198781 1198782 120591) asymp 119890minus11990211205911198781 minus 119890minus119903120591 (1198782 + 119864) 1198781 ≫ 1198782 + 119864 (6)

(4) For large values of 1198782 we assume that the values arealmost constants when 1198782 changes therefore we canuse Neumanrsquos boundary condition

1205971198801205971198782 = 0 (7)

These boundary conditions will be validated with thenumerical examples

Abstract and Applied Analysis 3

3 Removing the Correlation Term andProblem Discretization

Webegin this section by transforming the PDE problem (1) inorder to remove the cross derivative term Firstlywe eliminatethe reaction term 119903119880 by means of the substitution

119881 = 119890119903120591119880 (8)

obtaining

120597119881120597119905 = 121205902111987821 120597211988112059711987821 +

121205902211987822 120597211988112059711987822 + 1205881205901120590211987811198782

120597211988112059711987811205971198782+ (119903 minus 1199021) 1198781 1205971198811205971198781 + (119903 minus 1199022) 1198782

1205971198811205971198782 (9)

In order to remove the cross derivative term note thatthe right-hand side of (9) is a linear differential operator oftwo variables and classical techniques to obtain the canonicalform of second-order linear PDEs can be applied see forinstance chapter 3 of [20] Under the assumption of corre-lated variables with minus1 lt 120588 lt 1 the sign of the discriminantis

119886212 minus 41198861111988622 = 12059021120590221198782111987822 (1205882 minus 1) lt 0 (10)

where

11988611 = 12120590211198782111988612 = 120588120590112059021198781119878211988622 = 121205902211987822

(11)

Therefore right-hand side of (9) becomes of elliptic typeand a convenient substitution to obtain the canonical form isgiven by solving the ordinary differential equation

11988911987821198891198781 +1205902119878212059011198781 (minus120588 plusmn 119894) = 0 (12)

where

= radic1 minus 1205882 (13)

and (1205902119878212059011198781)(minus120588 plusmn 119894) are the conjugate roots of 119886111199092 +11988612119909 + 11988622 = 0 Solving (12) one gets11205902 log 1198782 +

minus120588 plusmn 1198941205901 log 1198781 = 1198620 (14)

where one can relate the integration constant 1198620 to the newvariables by

119909 = Im (1198620) 119910 = minusRe (1198620) (15)

From (14) and (15) it follows the expression of the newvariables

119909 = 1205901 log 1198781119910 = 1205881205901 log 1198781 minus

11205902 log 1198782(16)

Note that

119910 minus 119898119909 = minus 11205902 log 1198782 119898 = 120588 (17)

By denoting

119882(119909 119910 120591) = 119881 (1198781 1198782 120591) (18)

(9) takes the following form without cross derivative term

120597119882120597120591 = 2

2 (12059721198821205971199092 + 120597

21198821205971199102 ) + 1205901 (119903 minus 1199021 minus120590212 ) 120597119882120597119909

+ ( 1205881205901 (119903 minus 1199021 minus120590212 ) minus 11205902 (119903 minus 1199022 minus

120590222 )) 120597119882120597119910 (119909 119910) isin R2 0 lt 120591 le 119879

(19)

In accordance with [23 24] and (6) we choose therectangular domain (1198781 1198782) isin [1198861 1198871] times [1198862 1198872] = Ω where119886119894 gt 0 119894 = 1 2 are small positive values 1198872 about 3119864 and 1198871about 3(1198872+119864) For the transformed problem (19) due to (16)the rectangular domainΩ becomes a rhomboid with vertices119860119861119862119863 (see Figure 1) From (16) let us denote

1198881 = 1205901 log 11988611198882 = log 11988621205902 1198891 = 1205901 log 11988711198892 = log 11988721205902

(20)

By (17) the rhomboid domain has the sides

119860119863 = (119909 119910) 119909 = 1198881 1198981198881 minus 1198892 le 119910 le 1198981198881 minus 1198882 119860119861 = (119909 119910) 1198881 le 119909 le 1198891 119910 = 119898119909 minus 1198892 119861119862 = (119909 119910) 119909 = 1198891 1198981198891 minus 1198892 le 119910 le 1198981198891 minus 1198882 119863119862 = (119909 119910) 1198881 le 119909 le 1198891 119910 = 119898119909 minus 1198882

(21)

4 Abstract and Applied Analysis

A

B C

D

x

y

Figure 1 Transformed numerical domain

Themesh points in the rhomboid domain are (119909119894 119910119895) suchthat

119909119894 = 1198881 + 119894ℎ 119909119894 isin [1198881 1198891] 0 le 119894 le 119873119909Δ 119909 = ℎ = 1198891 minus 1198881119873119909 Δ 119910 = |119898| ℎ119873119910 = 1198892 minus 1198882|119898| ℎ 119910119895 = 119898119909119894 minus 1198892 + (119895 minus 119894) |119898| ℎ 119894 le 119895 le 119873119910 + 119894

(22)

The approximations of the derivatives appearing in (19) atthe point (119909119894 119910119895 120591119899) are as follows

120597119882120597119909 sim 119908119899119894+1119895 minus 119908119899119894minus11198952ℎ 120597119882120597119910 sim 119908119899119894119895+1 minus 119908119899119894119895minus12 |119898| ℎ 12059721198821205971199092 sim

119908119899119894+1119895 minus 2119908119899119894119895 + 119908119899119894minus1119895ℎ2 12059721198821205971199102 sim

119908119899119894119895+1 minus 2119908119899119894119895 + 119908119899119894119895minus11198982ℎ2 120597119882120597120591 sim

119908119899+1119894119895 minus 119908119899119894119895119896

(23)

where 119908119899119894119895 sim 119882(119909119894 119910119895 120591119899) 120591119899 = 119899119896Substituting these finite difference approximations of the

derivatives into (19) is approximated by the explicit differencescheme with five-point stencil

119908119899+1119894119895 = 1205721119908119899119894minus1119895 + 1205722119908119899119894+1119895 + 1205723119908119899119894119895 + 1205724119908119899119894119895minus1+ 1205725119908119899119894119895+1 (24)

where

12057212 = 22 119896ℎ2 ∓ 1205901 (119903 minus 1199021 minus120590212 ) 1198962ℎ (25)

1205723 = 1 minus 2 119896ℎ2 (1 + 11198982 ) (26)

12057245= 211989621198982ℎ2∓ [ (119903 minus 1199021 minus 120590212)1205901 minus 119903 minus 1199022 minus 1205902221205902 ] 1198962 |119898| ℎ

(27)

The discretization of the spatial boundary is given by

119875 (119860119863) = (1199090 119910119895) 0 le 119895 le 119873119910 119875 (119860119861) = (119909119894 119910119894) 0 le 119894 le 119873119909 119875 (119861119862) = (119909119873119909 119910119895) 119873119909 le 119895 le 119873119909 + 119873119910 119875 (119863119862) = (119909119894 119910119873119910+119894) 0 le 119894 le 119873119909

(28)

Both initial and boundary conditions (2) (3) (4) (6) and(7) are transformed throughout (8) (16) (18) For the initialcondition we have

1199080119894119895 = (1198901205901119909119894 minus 119890minus1205902(119910119895minus119898119909119894) minus 119864)+ 0 le 119894 le 119873119909 119894 le 119895 le 119894 + 119873119910 (29)

Boundary condition for side 119860119863 is as follows

1199081198990119895 = 0 0 le 119895 le 119873119910 1 le 119899 le 119873120591 (30)

For 119863C with small value of 1198782 we have transformedBlack-Scholes solution so

119908119899119894119873119910+119894 = 119865119899119894 119873(1206011198991119894) minus 119864119873(1206011198992119894) 0 lt 119894 lt 119873119909 1 le 119899 le 119873120591

(31)

where

119865119899119894 = exp(1205901 119909119894 + (119903 minus 1199021) 120591119899) 1206011198991119894 = 1

1205901radic120591119899 [log119865119894119864 + 1212059021120591119899]

1206011198992119894 = 1206011198991119894 minus 1205901radic120591119899(32)

Side 119861119862 corresponds to large values of 1198781 so behaviour ofthe option there leads to

119908119899119873119909119895 = exp(1205901 119909119873119909 + (119903 minus 1199021) 120591119899)minus (exp (1205902 (119898119909119873119909 minus 119910119895)) + 119864)

119873119909 + 1 le 119895 le 119873119909 + 119873119910 1 le 119899 le 119873120591(33)

Abstract and Applied Analysis 5

Boundary 119860119861 corresponds with the boundary for largevalues of 1198782 Condition (7) means that component 1198782 isconstant and the null first derivative is approximated by theforward difference (119908119899119894119894+1 minus 119908119899119894119894)|119898|ℎ = 0

119908119899119894119894 = 119908119899119894119894+1 1 le 119894 le 119873119909 1 le 119899 le 119873120591 (34)

4 Positivity Stability and Consistency

In this section we show that the proposed numerical scheme(24)ndash(27) presents suitable qualitative and computationalproperties such as consistency and conditional positivity andstability

In order to guarantee positivity of the numerical solutionlet us check the positivity of the coefficients 120572119894

From (25) and (27) it is easy to check that coefficients1205721 1205722 1205724 and 1205725 of the scheme (24) are positive under thecondition

ℎ lt min 12059011003816100381610038161003816119903 minus 1199021 minus 1205902121003816100381610038161003816 21003816100381610038161003816119898 [ (119903 minus 1199021 minus 120590212) 1205901 minus (119903 minus 1199022 minus 120590222) 1205902]1003816100381610038161003816

(35)

Coefficient 1205723 is positive if119896 lt 1198982ℎ2

2 (1198982 + 1) (36)

Under conditions (35) and (36) values in interior pointsof the numerical domain 119908119899+1119894119895 1 le 119894 le 119873119909 minus 1 119894 + 1 le119895 le 119873119910 + 119894 minus 1 calculated by the scheme (24) preservenonnegativity from the previous time level 119899 at all interior andboundary points 119908119899119894119895 ge 0 0 le 119894 le 119873119909 119894 le 119895 le 119873119910 + 119894

Let us pay attention now to the positivity of the numericalsolution at the boundary of the numerical domain Onthe boundary 119860119863 from (30) we have zero values On theboundary 119863119862 formula (31) is used preserving the positivitysince it is the transformed Black-Scholes formula through-out expression (16) Along the line 119860119861 we use Neumannboundary condition (34) Therefore the positivity at theneighbour interior point guarantees the positivity at theboundary Finally on the line 119861119862 boundary conditions aredetermined by formula (33) This is transformed expressionfor the asymptotic behaviour of the call option (6) that ispositive under the condition

1198781maxgt (1198782max

+ 119864) exp ((1199021 minus 119903) 120591119899) 1 le 119899 le 119873120591 (37)

Following the ideas of Kangro and Nicolaides [24] thecomputational domain has to be large enough to translate theboundary conditions therefore

1198781max

ge max (1198782max+ 119864) exp ((1199021 minus 119903) 119879) 3 (1198782max

+ 119864) (38)

Therefore choosing 1198781maxaccording to (38) for the trans-

formed problem119908119899119873119909 119895 ge 0 for any119873119909+1 le 119895 le 119873119909+119873119910 1 le119899 le 119873120591In summary the nonnegativity of the numerical solution

follows from the nonnegativity of the initial values andpositivity preservation under the scheme and boundaryconditions

As authors manage several stability criteria in literaturewe recall the concept of stability used in this paper

Definition 1 Let 119908119899 isin R(119873119909+1)times(119873119910+1) be the matrix involvingthe numerical solution 119908119899119894119895 of PDE problem (19) computedby scheme (24) (29) (30) (31) (33) and (34)

119908119899 = (119908119899119894119895)0le119894le119873119909 119894le119895le119873119910+119894 0 le 119899 le 119873120591 (39)

The numerical scheme (24) (29) (30) (31) (33) and (34) issaid to be uniformly sdot infin-stable in the domain [1198881 1198891] times[1198882 1198892]times[0 119879] if for every partitionwith 119896ℎ as defined abovecondition 10038171003817100381710038171199081198991003817100381710038171003817infin le 119860 0 le 119899 le 119873120591 (40)

holds true where119860 is independent of ℎ 119896 When the schemeis uniformly stable under a condition imposed to the stepsize discretization then the scheme is said to be conditionallyuniformly stable

Let us find the maximum value of the 119908119899119894119895 with respect to119894 119895 for each fixed 119899 For 119899 = 0 the values119908119899119894119895 are defined by theinitial condition (29) Let us consider the following function

119891 (119909 119910) = exp(1205901 119909) minus exp (minus1205902 (119910 minus 119898119909)) minus 119864 (41)

and find the maximum value on the domain [1198881 1198891]times [1198882 1198892]The first derivatives are120597119891120597119909 = 1205901 exp(1205901 119909) + 1205902119898 exp (minus1205902 (119910 minus 119898119909)) gt 0120597119891120597119910 = 1205902exp (minus1205902 (119910 minus 119898119909)) gt 0

(42)

Analysing (42) one concludes that the function 119891(119909 119910) isincreasing with respect to 119909 and 119910 Therefore the maximumvalue is reached at the point (1198891 1198892)

max(119909119910)isin[1198881 1198891]times[1198882 1198892]

119891 (119909 119910) = 119891 (1198891 1198892)= exp(1205901 1198891) minus exp (minus1205902 (1198892 minus 1198981198891)) minus 119864 = 119862= const

(43)

For the next time levels 119899 ge 1 let us consider thenumerical scheme (24) Under conditions (35) and (36) thecoefficients 120572119894 gt 0 119894 = 1 5 Moreover it is easy to checkthat

5sum119894=1

120572119894 = 1 (44)

6 Abstract and Applied Analysis

Therefore the values in interior points at the (119899+1)th level119899 ge 0 can be bounded by the following

119908119899+1119894119895 le max 119908119899119894minus1119895 119908119899119894+1119895 119908119899119894119895 119908119899119894119895minus1 119908119899119894119895+1le max119894119895119908119899119894119895 (45)

This maximum can be reached on the boundary There-fore let us study the boundary conditions From (30) one getsthat

max119860119863

119908119899+1119894119895 = 0 (46)

The values on the boundary A119861 are equal to the interiorpoints therefore the inequality (45) can be applied

The values on the boundary 119861119862 are increasing withrespect to the index 119895 and reach the maximum at the point(119873119909 119873119909 + 119873119910 minus 1) From the other side this value can bebounded by the value at the point (119873119909 119873119909 + 119873119910) that iscalculated by formula (31) From the theory of option pricingthe price of European call is increasing with respect to theasset price 119878 andmoreover it is always greater than its asymp-totic function (see [25] p 268 formula (131)) The proposedtransformation preserves monotonicity therefore

119908119899+1119873119909119873119909+119873119910 = 119865119899+1119873119909 119873(120572119899+11119873119909) minus 119864119873(120572119899+12119873119909)ge 119890((1205901)119909119873119909+(119903minus1199021)120591119899+1)minus (119890(1205902(119898119909119873119909minus119910119873119909+119873119910minus1)) + 119864)

= 119908119899+1119873119909 119873119909+119873119910minus1(47)

In summary we can conclude that

max119894119895119908119899+1119894119895 = 119908119899+1119873119909119873119909+119873119910 (48)

This value is transformed call option price for the assetprice 1198781max

that is the solution of the 1D Black-Scholesequation at the fixed point European call option gives a rightto the option holder to purchase one share stock for a certainprice The option itself cannot be worth more than the stockIn other words

119880 (1198781 0 120591) = 119890minus11990211205911198781119873(1198891) minus 119890minus119903120591119864119873 (1198892)le 119890minus1199021205911198781119873(1198891) le 119890minus1199021205911198781

119908119899+1119873119909119873119909+119873119910 le 119890119903120591119899+1119890minus1199021120591119899+11198781maxle 119890|119903minus1199021|1198791198781max

0 le 119899 le 119873120591 minus 1

(49)

Then the following result can be stated

Theorem 2 With previous notations under conditions (35)and (36) numerical scheme (24) (29) (30) (31) (33) and(34) is conditionally uniformly sdot infin-stable with upper bound119860 = 119890|119903minus1199021|1198791198781max

Now consistency of the numerical scheme (24) with PDE(19) will be studied [26]

Let 119865(119908119899119894119895) = 0 be approximating difference equation(24) The scheme (24) is consistent with (19) if the localtruncation error 119879119899119894119895 satisfies119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) 997888rarr 0 as ℎ 997888rarr 0 119896 997888rarr 0 (50)

where119882119899119894119895 denotes the theoretical solution of (19) evaluatedat the point (119909119894 119910119895 120591119899) and 119871 is the operator119871 (119882) = 120597119882120597120591 minus

2

2 (12059721198821205971199092 + 120597

21198821205971199102 )minus 1205901 (119903 minus 1199021 minus

120590212 ) 120597119882120597119909minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 )) 120597119882120597119910

(51)

Assuming that theoretical solution119882of the PDEproblemis four times continuously differentiable with respect to 119909 and119910 and twice with respect to 120591 and using Taylor expansionabout (119909119894 119910119895 120591119899) it is easy to check that

119882119899+1119894119895 minus119882119899119894minus1119895119896 = 120597119882120597120591 (119909119894 119910119895 120591119899) + 119896119864119899119894119895 (1) (52)

where

119864119899119894119895 (1) = 12 12059721198821205971205912 (119909119894 119910119895 120575) 119899119896 lt 120575 lt (119899 + 1) 119896 (53)

10038161003816100381610038161003816119864119899119894119895 (1)10038161003816100381610038161003816 le 12 10038161003816100381610038161003816119863119899119894119895 (1)10038161003816100381610038161003816max

= 12max10038161003816100381610038161003816100381610038161003816100381612059721198821205971205912 (119909119894 119910119895 120575)

100381610038161003816100381610038161003816100381610038161003816 0 lt 120575 lt 119879 (54)

For the spatial discretization one gets

119882119899119894+1119895 minus119882119899119894minus11198952ℎ = 120597119882120597119909 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (2) (55)

where

119864119899119894119895 (2) = 16 12059731198821205971199093 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (56)

10038161003816100381610038161003816119864119899119894119895 (2)10038161003816100381610038161003816 le 16 10038161003816100381610038161003816119863119899119894119895 (2)10038161003816100381610038161003816max

= 16 max10038161003816100381610038161003816100381610038161003816100381612059731198821205971199093 (120585 119910119895 120591119899)

100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891 (57)

Analogously

119882119899119894119895+1 minus119882119899119894119895minus12 |119898| ℎ = 120597119882120597119910 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (3) (58)

Abstract and Applied Analysis 7

where

119864119899119894119895 (2) = 11989826 12059731198821205971199103 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (59)

thus10038161003816100381610038161003816119864119899119894119895 (3)10038161003816100381610038161003816le 11989826 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(60)

The discretization of the second spatial derivatives is asfollows

119882119899119894+1119895 minus 2119882119899119894119895 +119882119899119894minus1119895ℎ2= 12059721198821205971199092 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (4)

(61)

where

119864119899119894119895 (4) = 112 12059741198821205971199094 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (62)

10038161003816100381610038161003816119864119899119894119895 (4)10038161003816100381610038161003816le 112 max100381610038161003816100381610038161003816100381610038161003816

12059741198821205971199094 (120585 119910119895 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891

(63)

Analogously

119908119899119894119895+1 minus 2119908119899119894119895 + 119908119899119894119895minus11198982ℎ2= 12059721198821205971199102 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (5)

(64)

where

119864119899119894119895 (5) = 119898412 12059741198821205971199104 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (65)

10038161003816100381610038161003816119864119899119894119895 (5)10038161003816100381610038161003816le 119898412 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(66)

Then the local truncation error takes the following form

119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) = 119896119864119899119894119895 (1) minus 2

2sdot ℎ4 (119864119899119894119895 (4) + 119864119899119894119895 (5)) minus 1205901 (119903 minus 1199021 minus

120590212 )sdot ℎ2119864119899119894119895 (2)minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 ))

sdot ℎ2119864119899119894119895 (3)

(67)

From (54) (57) (60) (63) and (66) one gets1003816100381610038161003816100381611987911989911989411989510038161003816100381610038161003816 = 119874 (119896) + 119874 (ℎ2) (68)

Theorem 3 Assuming that the solution of the PDE (19) admitstwo times continuous partial derivativewith respect to time andup to order four with respect to each of space directions solutionthe numerical solution computed by scheme (24) is consistentwith (19)

5 European Spread Put Option Case

For the sake of brevity and in order to avoid repetition ofproofs we summarize in this section the results proved inprevious sections for the call option case

Let us consider a European spread put option with payoff

119880 (1198781 1198782 0) = ((1198782 + 119864) minus 1198781)+ 119864 gt 0 (69)

Equation (1) and all the transformed versions of it do notchange for the put option Like in the study of the previouscall option case due to different nature of the put option weneed to establish the boundary conditions and translate themto the boundary of the numerical domain

(1) If 1198782 = 0 the problem is reduced to the European putoption on asset 1198781 and strike 119864 Therefore the optionvalue on this boundary is

119880 (1198781 0 120591) = 119864119890minus119903120591119873(minus1206012) minus 11987811198901199021120591119873(minus1206011) (70)

where 1206011 and 1206012 are defined by (5)(2) If 1198781 = 0 we consider 119880(0 1198782 120591) as a put option price

on asset 1198781 = 0 with the strike 1198782 + 119864119880 (0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 (71)

(3) For large values of 1198782 we suppose that the slope of thecurve tends to 1 that is

1205971198801205971198782 (1198781 1198782 997888rarr infin 120591) = 119890minus1199022120591 (72)

(4) For 1198781 ≫ 1198782 + 119864 we consider 119880(1198781 1198782 120591) as a putoption price with large values of 1198781 and therefore

119880(1198781 1198782 120591) = 0 (73)

Under transformations (8) and (16) the boundary con-ditions (70)ndash(73) are translated to the rhomboid numericaldomain 119860119861119862119863 as follows

119863119862 119908119899119894119873119910+119894 = 119864119873 (minus1205722119894)+ 119890(1205901)119909119894minus1199021120591119899119873(minus1205721119894)

(74)

119860119863 1199081198990119895 = 119864 + 119890minus1205902(119910119895minus1198981199090) (75)

119860119861 119908119899119894119894 = 119908119899119894119894+1 + |119898| ℎ1205902119890minus1205902(119910119894+1minus119898119909119894)minus119903120591 (76)

119861119862 119908119899119873119909+1119873119909+119895 = 0 (77)

8 Abstract and Applied Analysis

Positivity As the numerical scheme (24) for the interiorpoints is the same and the new boundary conditions (74)ndash(77) are nonnegative then under conditions (35) and (36)the numerical solution for the European spread put optionis nonnegative

Stability Stability of the scheme is preservedwith newbound-ary conditions (74)ndash(77) because analogously to the calloption case the maximum value of the transformed function119882(119909 119910 120591) is reached at a corner (in this case (1198881 1198892) wherewe use the boundary condition (75)) At the correspondingpoint of the original domain we use boundary condition (71)In the original coordinates it can be bounded as

119880(0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 le 1198782max+ 119864 = 4119864 = const (78)

Therefore the transformed function also can be boundedas

max119894119895119908119899119894119895 = 4119864119890119903119879 (79)

6 American Spread Options

In the case of American type option the holder has the rightto exercise option at any moment before expiring It leadsto a free boundary problem [27] In order to simplify thecomputational procedure this problem can be considered asa linear complementarity problem (LCP)

(120597119880120597120591 minus 119871119880) (119880 minus 119892) = 0120597119880120597120591 minus 119871119880 ge 0

119880 minus 119892 (1198781 1198782) ge 0(80)

where 119871 represents the spatial differential operator of (1)If we rewrite (19) in the form

120597119882120597120591 minusL119882 = 0 (81)

then under transformation (16) LCP (80) has the followingform

(120597119882120597120591 minusL119882)(119882 minus 119891 (119909 119910)) = 0120597119882120597120591 minusL119882 ge 0119882 minus 119891 (119909 119910) ge 0

(82)

whereL is the right-hand side of (19) and 119891(119909 119910) is given by(41)

Numerical solution for problem (82) can be easilyobtained by a small modification of the calculating proceduredescribed above in Sections 3 and 4 On each time level

119899119894119895 = max 119908119899119894119895 119891 (119909119894 119910119895) ge 0 (83)

where 119908119899119894119895 is defined by the finite difference equation (24)

01500

500

4001000

1000

300

1500

200500100

0 0

U(S

1S

2T

)

S1 S2

Figure 2 Parameters ℎ and 119896 satisfy (85)

7 Numerical Examples

In this section we illustrate numerical results for bothEuropean and American spread options

In Example 1 we show that the stability conditions (35)and (36) for the European spread option can not be disre-garded

Example 1 We tested the algorithm for the European spreadcall option pricing problem with the parameters

119879 = 1119864 = 1001205901 = 021205902 = 01119903 = 011199021 = 0011199022 = 005120588 = 01

(84)

and taking 1198781 and 1198782 large enough in particular 1198782max= 4001198781max

= 3(1198782max+ 119864) = 1500

For parameters (84) the stability conditions (35) and (36)on space and time steps are as follows

ℎ lt min 28428 237358 = 28428 119896 lt 00112 (85)

These conditions are crucial for the qualitative propertiessuch as positivity and stability of the scheme In Figure 2 thenumerical solution with ℎ and 119896 satisfying (85) is presentedOn the other hand in Figure 3 there is a numerical simulationwith time step 119896 = 0012 gt 00112 The solution is unstableand it has negative values

Example 2 illustrates the computational time aswell as thecomparison with analytical approximation provided by [6]

Abstract and Applied Analysis 9

1500

0500

400

1000

1000

1500

300

20002500

200500100

0 0

minus500

U(S

1S

2T

)

S1

S2

Figure 3 Stability condition is broken

Example 2 Consider the European spread option with thedata of Example 1 (84) and choosing step sizes satisfying thestability condition

The price of European spread option can be expressedas the price of a compound exchange option (see [6])The known derived analytical approximation for the spreadoption reads

119880 (1198781 1198782 119879) = 1198781119890minus1199021119879119873(1198891)minus (119864119890minus119903119879 + 1198782) 119890minus(119903minusminus2)119879119873(1198892) (86)

where

1198891 = 1120590radic119879 (ln

11987811198782 + 119864119890minus119903119879+ (119903 minus + 2 minus 1199021 + 12059022 )119879) 1198892 = 1198891 minus 120590radic119879

120590 = radic12059021 minus 212058812059011205902 11987821198782 + 119864119890minus119903119879 + (11987821198782 + 119864119890minus119903119879)

2 12059022 = 11987821198782 + 119864119890minus119903119879 1199032 = 11987821198782 + 119864119890minus119903119879 1199022

(87)

The comparison of our proposed numerical method andthe analytical approximation (86) is presented in Table 1Analytical approximation is calculated for the same arraysof 1198781 and 1198782 using MATLAB-function cdf for calculat-ing cumulative distribution function that requires a lot ofcomputational resources for each point In the proposedfinite difference method (FDM) cdf is only used for theboundary conditions Last row in the Table 1 presents theCPU time for each method for the same sizes of array Forapproximation method cdf-time is 925018 sec It is the main

Table 1 Comparison proposed method (FDM) with analyticalapproximation (86)

1198782 1198781 FDM Approximation

100

400 2105978 2097261200 242895 234543100 01150 0012350 31179119890 minus 05 minus47741119890 minus 10

90

400 2200678 2201714200 298745 298724100 01691 0037250 52150119890 minus 05 minus19574119890 minus 09

CPU time (sec) 106188 929467

00400

100

300 500

200

200

300

1000100

400

15000

500

U(S

1S

2T

)

S 1S2

Figure 4 European spread put option with parameters (84)

part of computational time In the case of FDM cdf takes lessthan half of computational time - 48867 sec

Furthermore unsuitable negative values appear in ana-lytical approximation for small 1198781 and 1198782 while the proposedmethod preserves nonnegativity of the solution

In the next example we confirm that for the spreadput option case the removing technique is fruitful and theselection of the boundary conditions at the boundary of thenumerical domain is appropriated because the behaviour ofthe solution at the boundaries fits the solution at the interiorpoints as it is shown in Figure 4

Example 3 With the parameters in (84) of Example 1 butwith payoff (1198782+119864minus1198781)+ in Figure 4we show the simulation ofthe numerical solution obtained by the proposed numericalscheme

Finally in Example 4 we compare the numerical solutionobtained with our scheme (83) using LCP approach withother tested efficient methods presented in [28]

10 Abstract and Applied Analysis

Table 2 Comparing several methods for American option pricing

119864 Analytic FFT MC FDM0 8513201 8513079 8516613 85081982 7542296 7542242 7545496 75340644 6653060 6652976 6656364 6648792

Example 4 Let us consider the American spread call optionproblem with parameters

119879 = 11205901 = 011205902 = 02119903 = 011199021 = 0051199022 = 005120588 = 05

(88)

Table 2 shows the option price at fixed asset prices 1198781 = 96and 1198782 = 100 for different values of strike price 119864 evaluatedby one-dimensional integration analytic method (Analytic)Fast Fourier Transform (FFT) Monte Carlo (MC) methodand our proposed method The accuracy is competitive withother methods such as Fourier Transform with high numberof discretization (4096) andMonte Carlo with big number oftime steps (2000)

Another set of parameters can be considered in order tocompare it with data in [9]

119879 = 05119864 = 1001205901 = 0251205902 = 03119903 = 0031199021 = 0061199022 = 002120588 = 05

(89)

In [9] three methods are considered to evaluate thespread option price for data 74 and fixed values 1198781 = 100and 1198782 = 200 a numerical integration method (NIM) amethod of lines (MOL) and a Monte Carlo approach NIMused in [9] is based in the recursive integration techniquesdeveloped byHuang et al [29] InNIM the option is treated asa Bermudan that is an option that can be exercised at discretepoints of time and Simpsons rule is used For the experimentthe authors used 1024 spatial nodes see [8 9] for moredetails The MOL is implemented by a spatial discretizationof 1438 times 200 mesh points and a three-time level scheme isused for the integration in time [9] 50 000 simulations have

Table 3 CPU time in sec (first row) and absolute difference (secondrow) for different methods depending on number of time steps forfixed number of space steps for parameters (89)

119873120591 NIM MOL Monte Carlo FDM

50 296598 5920 47 970 265470003 0001 01735 00036

32 127069 5512 19 811 1791100035 0006 01148 00122

16 33493 51718 5 149 943200012 00236 0132 00371

8 10360 4901 1 350 478700105 0074 01316 00907

4 4117 4547 35575 281500364 02512 01351 02254

been used in the Monte Carlo method described in [30] Inour method FDM the step size discretization considered isℎ = 01 To compare FDM with the three previous methodsabsolute errors are computed in Table 3 with respect to areference value obtained by the Fourier space time-steppingapproach of Surkov [31] with large number of nodes 4096spatial nodes and 300 time steps From Table 3 we can statethat the proposed method has the same order of accuracy asMOL more efficient and requires less computational timethan Monte Carlo method It is slightly less efficient than theNIM but we prove that it possesses a very highly desirablequalitative behaviour in finance

Competing Interests

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

Acknowledgments

This work has been partially supported by the EuropeanUnion in the FP7- PEOPLE-2012-ITN Program under GrantAgreement no 304617 (FP7 Marie Curie Action ProjectMulti-ITN STRIKE-Novel Methods in ComputationalFinance) and the Ministerio de Economıa y CompetitividadSpanish Grant MTM2013-41765-P

References

[1] R Carmona and V Durrleman ldquoPricing and hedging spreadoptionsrdquo SIAM Review vol 45 no 4 pp 627ndash685 2003

[2] P P Boyle ldquoOptions a Monte Carlo approachrdquo Journal ofFinancial Economics vol 4 no 3 pp 323ndash338 1977

[3] C Joy P P Boyle and K S Tan ldquoQuasi-Monte Carlo methodsin numerical financerdquo Management Science vol 42 no 6 pp926ndash938 1996

[4] W-H Kao Y-D Lyuu and K-W Wen ldquoThe hexanomiallattice for pricingmulti-asset optionsrdquoAppliedMathematics andComputation vol 233 pp 463ndash479 2014

Abstract and Applied Analysis 11

[5] E Kirk and J Aron ldquoCorrelation in the energy marketsrdquo inManaging Energy Price Risk V Kaminski Ed pp 71ndash78 RiskBooks London UK 1995

[6] C Alexander andA Venkatramanan ldquoAnalytic approximationsfor spread optionsrdquo ICMACentre Discussion Papers in FinanceDP2007-11 ICMA Centre 2007

[7] N D Pearson ldquoAn efficient approach for pricing spreadoptionsrdquoThe Journal of Derivatives vol 3 no 1 pp 76ndash91 1995

[8] C Chiarella and J Ziveyi ldquoPricingAmerican options written ontwo underlying assetsrdquo Quantitative Finance vol 14 no 3 pp409ndash426 2014

[9] J Ziveyi The evaluation of early exercise exotic options [PhDthesis] University of Technology Sydney Australia 2011

[10] P Boyle ldquoA lattice framework for option pricing with two statevariablesrdquo Journal of Financial and Quantitative Analysis vol23 no 1 pp 1ndash12 1988

[11] M Rubinstein ldquoReturn to Ozrdquo RISK vol 7 no 11 pp 67ndash711994

[12] R Company L Jodar M Fakharany and M-C CasabanldquoRemoving the correlation term in option pricing Hestonmodel numerical analysis and computingrdquo Abstract andApplied Analysis vol 2013 Article ID 246724 11 pages 2013

[13] S Salmi J Toivanen and L von Sydow ldquoIterative methodsfor pricing american options under the bates modelrdquo ProcediaComputer Science vol 18 pp 1136ndash1144 2013

[14] J Toivanen ldquoA componentwise splitting method for pricingamerican options under the bates modelrdquo in Applied andNumerical Partial Differential Equations vol 15 of Computa-tional Methods in Applied Sciences pp 213ndash227 Springer 2010

[15] R Zvan P A Forsyth and K R Vetzal ldquoNegative coefficientsin two-factor option pricing modelsrdquo Journal of ComputationalFinance vol 7 no 1 pp 37ndash73 2003

[16] B During M Fournie and C Heuer ldquoHigh-order compactfinite difference schemes for option pricing in stochastic volatil-ity models on non-uniform gridsrdquo Journal of Computationaland Applied Mathematics vol 271 pp 247ndash266 2014

[17] K J in rsquot Hout and C Mishra ldquoStability of ADI schemes formultidimensional diffusion equations with mixed derivativetermsrdquoApplied NumericalMathematics vol 74 pp 83ndash94 2013

[18] C Chiarella B Kang G H Mayer and A Ziogas ldquoTheevaluation of American option prices under stochastic volatilityand jump-diffusion dynamics using the method of linesrdquoInternational Journal of Theoretical and Applied Finance vol 12no 3 pp 393ndash425 2009

[19] M Fakharany R Company and L Jodar ldquoPositive finite dif-ference schemes for a partial integro-differential option pricingmodelrdquo Applied Mathematics and Computation vol 249 pp320ndash332 2014

[20] P R Garabedian Partial Differential Equations AMS ChelseaPublishing Providence RI USA 1998

[21] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp SonsChichester UK 2006

[22] R U Seydel Tools for Computational Finance Springer 3rdedition 2006

[23] M Ehrhardt Discrete artificial boundary conditions [PhDthesis] Technische Universitat Berlin Berlin Germany 2001

[24] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[25] J C HullOptions Futures andOther Derivatives Prentice-HallUpper Saddle River NJ USA 2000

[26] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods Clarendon Press Oxford UK3rd edition 1985

[27] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[28] M A H Dempster and S S G Hong Spread Option Valuationand the Fast Fourier Transform WP 262000 Judge Institute ofManagement Studies University of Cambridge 2000

[29] J-Z Huang M G Subrahmanyam and G G Yu ldquoPricingand hedging american options a recursive integrationmethodrdquoReview of Financial Studies vol 9 no 1 pp 277ndash300 1996

[30] A Ibanez and F Zapatero ldquoMonte Carlo valuation of Americanoptions through computation of the optimal exercise frontierrdquoJournal of Financial and Quantitative Analysis vol 39 no 2 pp253ndash275 2004

[31] V Surkov ldquoParallel option pricing with fourier space time-stepping method on graphics processingrdquo in Proceedings of the1st International Workshop on Parallel and Distributed Comput-ing in Finance (PDCoF rsquo08) Miami Fla USA 2008

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

In order to overcome these difficulties the authors haverecently proposed several strategies dealing with finite dif-ference approach high order compact schemes are proposedin [16] using central difference approximations for stochasticvolatility HestonmodelThe authors in [17] use ADImethodsand clever discretizations to obtain stable numerical schemesunder minor coefficients restrictions in the PDEThe authorsof [14 18] use special finite difference approximations of thecross derivative term based on a seven-point stencil instead ofthe nine-point stencil scheme resulting from the central finitedifference approach

In this paper we continue removing cross derivativeapproach based on the transformation of the PDE probleminitiated in [12] for the case of one asset stochastic volatilityHeston model and applied to the Bates stochastic volatilityjump diffusion model in [19] Apart from avoiding negativecoefficients in the proposed scheme it has the additionalcomputational advantage that uses only a five-point stencil

Herewe consider spread option pricing problems for bothEuropean and American cases The paper is organized asfollowsThe first part of the paper deals with European spreadcall option with payoff 119892(1198781 1198782) = (1198781 minus 1198782 minus 119864)+ dependingon two assets 1198781 and 1198782 and strike price 119864 ge 0 In Section 2the PDE with the boundary conditions is established for thespread option pricing problem In Section 3 we remove thecross derivative term of the PDE using the canonical formtransformation method [20] In that section we also developthe discretization of the continuous European spread calloption problem achieving an explicit finite difference schemeSection 4 is devoted to the study of the properties of themethod like consistence and conditional stability and posi-tivity In Section 5 we extend the study of previous sectionsto the European spread put option summarizing the mainresults including the treatment of the boundary conditionsSection 6 deals withAmerican spread option pricing problemusing the LCP technique based on the proposed differencescheme Finally in Section 7 we illustrate the numericalresults computing simulating and comparing accuracy andCPU time with other well acknowledge methods

Throughout the paper we use the following notation119860infin = max |119886119894119895| 1 le 119894 le 119873 1 le 119895 le 119872 will denote asupremum norm for a given matrix 119860 = (119886119894119895) isin R119873times1198722 Spread Option Pricing Problem andBoundary Conditions

The price 119880(1198781 1198782 120591) of a spread option is the solution of thePDE

120597119880120597120591 = 121205902111987821 120597211988012059711987821 +

121205902211987822 120597211988012059711987822 + 1205881205901120590211987811198782

120597211988012059711987811205971198782+ (119903 minus 1199021) 1198781 1205971198801205971198781 + (119903 minus 1199022) 1198782

1205971198801205971198782 minus 1199031198800 lt 1198781 lt infin 0 lt 1198782 lt infin 0 lt 120591 le 119879

(1)

where 120591 = 119879minus119905 is the time to maturity 119879 119903 is the interest rate120590119894 is the volatility of the asset 119878119894 119902119894 is the dividend yield of

asset 119878119894 and 120588 is the correlation parameter see [1] Since weconsider backwards time it changes the signs of the coeffi-cients of (1) and instead of a terminal condition the followinginitial condition is considered

119880(1198781 1198782 0) = 119892 (1198781 1198782) = max (1198781 minus 1198782 minus 119864 0)= (1198781 minus 1198782 minus 119864)+ (2)

The case 119864 = 0 corresponds to exchange options that canbe reduced to 1D problem by the change of variables 1199091 =11987811198782 119888(1199091 120591) = 119880(1198781 1198782 120591)1198782

Since the numerical solution of European multiassetoptions requires the selection of a bounded numericaldomain and the translation of the boundary conditions tothe boundary of the domain it is important to pay attentionto such conditions For the initial problem (1)-(2) suitableboundary conditions areas follows

(1) For 1198781 = 0 the payoff (1198781minus1198782minus119864)+ suggests Dirichletrsquoscondition

119880 (0 1198782 120591) = 0 0 lt 1198782 lt infin 0 lt 120591 le 119879 (3)

as one-dimensional call option(2) Taking the ideas developed by some authors for

instance Duffy for the case of basket option (see [21]p 270) for 1198782 = 0we take the value of the closed formsolution of the basic Black-Scholes equation of a callfor 1198781 with given strike 119864119880 (1198781 0 120591) = 119890minus119903120591 (119865119873 (1206011) minus 119864119873 (1206012)) (4)

where

119865 = 1198781119890(119903minus1199021)1205911206011 = 11205901radic120591 [log

119865119864 + 1212059021120591] 1206012 = 1206011 minus 1205901radic120591

(5)

where119873(119909) is the standard normal cumulative distri-bution function

(3) As 1198781 is large versus 1198782 + 119864 then the behaviour ofthe solution looks like the asymptotic value of a one-dimensional vanilla call option with the strike (1198782+119864)for 1198781 ≫ 1198782 + 119864 (see [22] p 157)

119880 (1198781 1198782 120591) asymp 119890minus11990211205911198781 minus 119890minus119903120591 (1198782 + 119864) 1198781 ≫ 1198782 + 119864 (6)

(4) For large values of 1198782 we assume that the values arealmost constants when 1198782 changes therefore we canuse Neumanrsquos boundary condition

1205971198801205971198782 = 0 (7)

These boundary conditions will be validated with thenumerical examples

Abstract and Applied Analysis 3

3 Removing the Correlation Term andProblem Discretization

Webegin this section by transforming the PDE problem (1) inorder to remove the cross derivative term Firstlywe eliminatethe reaction term 119903119880 by means of the substitution

119881 = 119890119903120591119880 (8)

obtaining

120597119881120597119905 = 121205902111987821 120597211988112059711987821 +

121205902211987822 120597211988112059711987822 + 1205881205901120590211987811198782

120597211988112059711987811205971198782+ (119903 minus 1199021) 1198781 1205971198811205971198781 + (119903 minus 1199022) 1198782

1205971198811205971198782 (9)

In order to remove the cross derivative term note thatthe right-hand side of (9) is a linear differential operator oftwo variables and classical techniques to obtain the canonicalform of second-order linear PDEs can be applied see forinstance chapter 3 of [20] Under the assumption of corre-lated variables with minus1 lt 120588 lt 1 the sign of the discriminantis

119886212 minus 41198861111988622 = 12059021120590221198782111987822 (1205882 minus 1) lt 0 (10)

where

11988611 = 12120590211198782111988612 = 120588120590112059021198781119878211988622 = 121205902211987822

(11)

Therefore right-hand side of (9) becomes of elliptic typeand a convenient substitution to obtain the canonical form isgiven by solving the ordinary differential equation

11988911987821198891198781 +1205902119878212059011198781 (minus120588 plusmn 119894) = 0 (12)

where

= radic1 minus 1205882 (13)

and (1205902119878212059011198781)(minus120588 plusmn 119894) are the conjugate roots of 119886111199092 +11988612119909 + 11988622 = 0 Solving (12) one gets11205902 log 1198782 +

minus120588 plusmn 1198941205901 log 1198781 = 1198620 (14)

where one can relate the integration constant 1198620 to the newvariables by

119909 = Im (1198620) 119910 = minusRe (1198620) (15)

From (14) and (15) it follows the expression of the newvariables

119909 = 1205901 log 1198781119910 = 1205881205901 log 1198781 minus

11205902 log 1198782(16)

Note that

119910 minus 119898119909 = minus 11205902 log 1198782 119898 = 120588 (17)

By denoting

119882(119909 119910 120591) = 119881 (1198781 1198782 120591) (18)

(9) takes the following form without cross derivative term

120597119882120597120591 = 2

2 (12059721198821205971199092 + 120597

21198821205971199102 ) + 1205901 (119903 minus 1199021 minus120590212 ) 120597119882120597119909

+ ( 1205881205901 (119903 minus 1199021 minus120590212 ) minus 11205902 (119903 minus 1199022 minus

120590222 )) 120597119882120597119910 (119909 119910) isin R2 0 lt 120591 le 119879

(19)

In accordance with [23 24] and (6) we choose therectangular domain (1198781 1198782) isin [1198861 1198871] times [1198862 1198872] = Ω where119886119894 gt 0 119894 = 1 2 are small positive values 1198872 about 3119864 and 1198871about 3(1198872+119864) For the transformed problem (19) due to (16)the rectangular domainΩ becomes a rhomboid with vertices119860119861119862119863 (see Figure 1) From (16) let us denote

1198881 = 1205901 log 11988611198882 = log 11988621205902 1198891 = 1205901 log 11988711198892 = log 11988721205902

(20)

By (17) the rhomboid domain has the sides

119860119863 = (119909 119910) 119909 = 1198881 1198981198881 minus 1198892 le 119910 le 1198981198881 minus 1198882 119860119861 = (119909 119910) 1198881 le 119909 le 1198891 119910 = 119898119909 minus 1198892 119861119862 = (119909 119910) 119909 = 1198891 1198981198891 minus 1198892 le 119910 le 1198981198891 minus 1198882 119863119862 = (119909 119910) 1198881 le 119909 le 1198891 119910 = 119898119909 minus 1198882

(21)

4 Abstract and Applied Analysis

A

B C

D

x

y

Figure 1 Transformed numerical domain

Themesh points in the rhomboid domain are (119909119894 119910119895) suchthat

119909119894 = 1198881 + 119894ℎ 119909119894 isin [1198881 1198891] 0 le 119894 le 119873119909Δ 119909 = ℎ = 1198891 minus 1198881119873119909 Δ 119910 = |119898| ℎ119873119910 = 1198892 minus 1198882|119898| ℎ 119910119895 = 119898119909119894 minus 1198892 + (119895 minus 119894) |119898| ℎ 119894 le 119895 le 119873119910 + 119894

(22)

The approximations of the derivatives appearing in (19) atthe point (119909119894 119910119895 120591119899) are as follows

120597119882120597119909 sim 119908119899119894+1119895 minus 119908119899119894minus11198952ℎ 120597119882120597119910 sim 119908119899119894119895+1 minus 119908119899119894119895minus12 |119898| ℎ 12059721198821205971199092 sim

119908119899119894+1119895 minus 2119908119899119894119895 + 119908119899119894minus1119895ℎ2 12059721198821205971199102 sim

119908119899119894119895+1 minus 2119908119899119894119895 + 119908119899119894119895minus11198982ℎ2 120597119882120597120591 sim

119908119899+1119894119895 minus 119908119899119894119895119896

(23)

where 119908119899119894119895 sim 119882(119909119894 119910119895 120591119899) 120591119899 = 119899119896Substituting these finite difference approximations of the

derivatives into (19) is approximated by the explicit differencescheme with five-point stencil

119908119899+1119894119895 = 1205721119908119899119894minus1119895 + 1205722119908119899119894+1119895 + 1205723119908119899119894119895 + 1205724119908119899119894119895minus1+ 1205725119908119899119894119895+1 (24)

where

12057212 = 22 119896ℎ2 ∓ 1205901 (119903 minus 1199021 minus120590212 ) 1198962ℎ (25)

1205723 = 1 minus 2 119896ℎ2 (1 + 11198982 ) (26)

12057245= 211989621198982ℎ2∓ [ (119903 minus 1199021 minus 120590212)1205901 minus 119903 minus 1199022 minus 1205902221205902 ] 1198962 |119898| ℎ

(27)

The discretization of the spatial boundary is given by

119875 (119860119863) = (1199090 119910119895) 0 le 119895 le 119873119910 119875 (119860119861) = (119909119894 119910119894) 0 le 119894 le 119873119909 119875 (119861119862) = (119909119873119909 119910119895) 119873119909 le 119895 le 119873119909 + 119873119910 119875 (119863119862) = (119909119894 119910119873119910+119894) 0 le 119894 le 119873119909

(28)

Both initial and boundary conditions (2) (3) (4) (6) and(7) are transformed throughout (8) (16) (18) For the initialcondition we have

1199080119894119895 = (1198901205901119909119894 minus 119890minus1205902(119910119895minus119898119909119894) minus 119864)+ 0 le 119894 le 119873119909 119894 le 119895 le 119894 + 119873119910 (29)

Boundary condition for side 119860119863 is as follows

1199081198990119895 = 0 0 le 119895 le 119873119910 1 le 119899 le 119873120591 (30)

For 119863C with small value of 1198782 we have transformedBlack-Scholes solution so

119908119899119894119873119910+119894 = 119865119899119894 119873(1206011198991119894) minus 119864119873(1206011198992119894) 0 lt 119894 lt 119873119909 1 le 119899 le 119873120591

(31)

where

119865119899119894 = exp(1205901 119909119894 + (119903 minus 1199021) 120591119899) 1206011198991119894 = 1

1205901radic120591119899 [log119865119894119864 + 1212059021120591119899]

1206011198992119894 = 1206011198991119894 minus 1205901radic120591119899(32)

Side 119861119862 corresponds to large values of 1198781 so behaviour ofthe option there leads to

119908119899119873119909119895 = exp(1205901 119909119873119909 + (119903 minus 1199021) 120591119899)minus (exp (1205902 (119898119909119873119909 minus 119910119895)) + 119864)

119873119909 + 1 le 119895 le 119873119909 + 119873119910 1 le 119899 le 119873120591(33)

Abstract and Applied Analysis 5

Boundary 119860119861 corresponds with the boundary for largevalues of 1198782 Condition (7) means that component 1198782 isconstant and the null first derivative is approximated by theforward difference (119908119899119894119894+1 minus 119908119899119894119894)|119898|ℎ = 0

119908119899119894119894 = 119908119899119894119894+1 1 le 119894 le 119873119909 1 le 119899 le 119873120591 (34)

4 Positivity Stability and Consistency

In this section we show that the proposed numerical scheme(24)ndash(27) presents suitable qualitative and computationalproperties such as consistency and conditional positivity andstability

In order to guarantee positivity of the numerical solutionlet us check the positivity of the coefficients 120572119894

From (25) and (27) it is easy to check that coefficients1205721 1205722 1205724 and 1205725 of the scheme (24) are positive under thecondition

ℎ lt min 12059011003816100381610038161003816119903 minus 1199021 minus 1205902121003816100381610038161003816 21003816100381610038161003816119898 [ (119903 minus 1199021 minus 120590212) 1205901 minus (119903 minus 1199022 minus 120590222) 1205902]1003816100381610038161003816

(35)

Coefficient 1205723 is positive if119896 lt 1198982ℎ2

2 (1198982 + 1) (36)

Under conditions (35) and (36) values in interior pointsof the numerical domain 119908119899+1119894119895 1 le 119894 le 119873119909 minus 1 119894 + 1 le119895 le 119873119910 + 119894 minus 1 calculated by the scheme (24) preservenonnegativity from the previous time level 119899 at all interior andboundary points 119908119899119894119895 ge 0 0 le 119894 le 119873119909 119894 le 119895 le 119873119910 + 119894

Let us pay attention now to the positivity of the numericalsolution at the boundary of the numerical domain Onthe boundary 119860119863 from (30) we have zero values On theboundary 119863119862 formula (31) is used preserving the positivitysince it is the transformed Black-Scholes formula through-out expression (16) Along the line 119860119861 we use Neumannboundary condition (34) Therefore the positivity at theneighbour interior point guarantees the positivity at theboundary Finally on the line 119861119862 boundary conditions aredetermined by formula (33) This is transformed expressionfor the asymptotic behaviour of the call option (6) that ispositive under the condition

1198781maxgt (1198782max

+ 119864) exp ((1199021 minus 119903) 120591119899) 1 le 119899 le 119873120591 (37)

Following the ideas of Kangro and Nicolaides [24] thecomputational domain has to be large enough to translate theboundary conditions therefore

1198781max

ge max (1198782max+ 119864) exp ((1199021 minus 119903) 119879) 3 (1198782max

+ 119864) (38)

Therefore choosing 1198781maxaccording to (38) for the trans-

formed problem119908119899119873119909 119895 ge 0 for any119873119909+1 le 119895 le 119873119909+119873119910 1 le119899 le 119873120591In summary the nonnegativity of the numerical solution

follows from the nonnegativity of the initial values andpositivity preservation under the scheme and boundaryconditions

As authors manage several stability criteria in literaturewe recall the concept of stability used in this paper

Definition 1 Let 119908119899 isin R(119873119909+1)times(119873119910+1) be the matrix involvingthe numerical solution 119908119899119894119895 of PDE problem (19) computedby scheme (24) (29) (30) (31) (33) and (34)

119908119899 = (119908119899119894119895)0le119894le119873119909 119894le119895le119873119910+119894 0 le 119899 le 119873120591 (39)

The numerical scheme (24) (29) (30) (31) (33) and (34) issaid to be uniformly sdot infin-stable in the domain [1198881 1198891] times[1198882 1198892]times[0 119879] if for every partitionwith 119896ℎ as defined abovecondition 10038171003817100381710038171199081198991003817100381710038171003817infin le 119860 0 le 119899 le 119873120591 (40)

holds true where119860 is independent of ℎ 119896 When the schemeis uniformly stable under a condition imposed to the stepsize discretization then the scheme is said to be conditionallyuniformly stable

Let us find the maximum value of the 119908119899119894119895 with respect to119894 119895 for each fixed 119899 For 119899 = 0 the values119908119899119894119895 are defined by theinitial condition (29) Let us consider the following function

119891 (119909 119910) = exp(1205901 119909) minus exp (minus1205902 (119910 minus 119898119909)) minus 119864 (41)

and find the maximum value on the domain [1198881 1198891]times [1198882 1198892]The first derivatives are120597119891120597119909 = 1205901 exp(1205901 119909) + 1205902119898 exp (minus1205902 (119910 minus 119898119909)) gt 0120597119891120597119910 = 1205902exp (minus1205902 (119910 minus 119898119909)) gt 0

(42)

Analysing (42) one concludes that the function 119891(119909 119910) isincreasing with respect to 119909 and 119910 Therefore the maximumvalue is reached at the point (1198891 1198892)

max(119909119910)isin[1198881 1198891]times[1198882 1198892]

119891 (119909 119910) = 119891 (1198891 1198892)= exp(1205901 1198891) minus exp (minus1205902 (1198892 minus 1198981198891)) minus 119864 = 119862= const

(43)

For the next time levels 119899 ge 1 let us consider thenumerical scheme (24) Under conditions (35) and (36) thecoefficients 120572119894 gt 0 119894 = 1 5 Moreover it is easy to checkthat

5sum119894=1

120572119894 = 1 (44)

6 Abstract and Applied Analysis

Therefore the values in interior points at the (119899+1)th level119899 ge 0 can be bounded by the following

119908119899+1119894119895 le max 119908119899119894minus1119895 119908119899119894+1119895 119908119899119894119895 119908119899119894119895minus1 119908119899119894119895+1le max119894119895119908119899119894119895 (45)

This maximum can be reached on the boundary There-fore let us study the boundary conditions From (30) one getsthat

max119860119863

119908119899+1119894119895 = 0 (46)

The values on the boundary A119861 are equal to the interiorpoints therefore the inequality (45) can be applied

The values on the boundary 119861119862 are increasing withrespect to the index 119895 and reach the maximum at the point(119873119909 119873119909 + 119873119910 minus 1) From the other side this value can bebounded by the value at the point (119873119909 119873119909 + 119873119910) that iscalculated by formula (31) From the theory of option pricingthe price of European call is increasing with respect to theasset price 119878 andmoreover it is always greater than its asymp-totic function (see [25] p 268 formula (131)) The proposedtransformation preserves monotonicity therefore

119908119899+1119873119909119873119909+119873119910 = 119865119899+1119873119909 119873(120572119899+11119873119909) minus 119864119873(120572119899+12119873119909)ge 119890((1205901)119909119873119909+(119903minus1199021)120591119899+1)minus (119890(1205902(119898119909119873119909minus119910119873119909+119873119910minus1)) + 119864)

= 119908119899+1119873119909 119873119909+119873119910minus1(47)

In summary we can conclude that

max119894119895119908119899+1119894119895 = 119908119899+1119873119909119873119909+119873119910 (48)

This value is transformed call option price for the assetprice 1198781max

that is the solution of the 1D Black-Scholesequation at the fixed point European call option gives a rightto the option holder to purchase one share stock for a certainprice The option itself cannot be worth more than the stockIn other words

119880 (1198781 0 120591) = 119890minus11990211205911198781119873(1198891) minus 119890minus119903120591119864119873 (1198892)le 119890minus1199021205911198781119873(1198891) le 119890minus1199021205911198781

119908119899+1119873119909119873119909+119873119910 le 119890119903120591119899+1119890minus1199021120591119899+11198781maxle 119890|119903minus1199021|1198791198781max

0 le 119899 le 119873120591 minus 1

(49)

Then the following result can be stated

Theorem 2 With previous notations under conditions (35)and (36) numerical scheme (24) (29) (30) (31) (33) and(34) is conditionally uniformly sdot infin-stable with upper bound119860 = 119890|119903minus1199021|1198791198781max

Now consistency of the numerical scheme (24) with PDE(19) will be studied [26]

Let 119865(119908119899119894119895) = 0 be approximating difference equation(24) The scheme (24) is consistent with (19) if the localtruncation error 119879119899119894119895 satisfies119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) 997888rarr 0 as ℎ 997888rarr 0 119896 997888rarr 0 (50)

where119882119899119894119895 denotes the theoretical solution of (19) evaluatedat the point (119909119894 119910119895 120591119899) and 119871 is the operator119871 (119882) = 120597119882120597120591 minus

2

2 (12059721198821205971199092 + 120597

21198821205971199102 )minus 1205901 (119903 minus 1199021 minus

120590212 ) 120597119882120597119909minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 )) 120597119882120597119910

(51)

Assuming that theoretical solution119882of the PDEproblemis four times continuously differentiable with respect to 119909 and119910 and twice with respect to 120591 and using Taylor expansionabout (119909119894 119910119895 120591119899) it is easy to check that

119882119899+1119894119895 minus119882119899119894minus1119895119896 = 120597119882120597120591 (119909119894 119910119895 120591119899) + 119896119864119899119894119895 (1) (52)

where

119864119899119894119895 (1) = 12 12059721198821205971205912 (119909119894 119910119895 120575) 119899119896 lt 120575 lt (119899 + 1) 119896 (53)

10038161003816100381610038161003816119864119899119894119895 (1)10038161003816100381610038161003816 le 12 10038161003816100381610038161003816119863119899119894119895 (1)10038161003816100381610038161003816max

= 12max10038161003816100381610038161003816100381610038161003816100381612059721198821205971205912 (119909119894 119910119895 120575)

100381610038161003816100381610038161003816100381610038161003816 0 lt 120575 lt 119879 (54)

For the spatial discretization one gets

119882119899119894+1119895 minus119882119899119894minus11198952ℎ = 120597119882120597119909 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (2) (55)

where

119864119899119894119895 (2) = 16 12059731198821205971199093 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (56)

10038161003816100381610038161003816119864119899119894119895 (2)10038161003816100381610038161003816 le 16 10038161003816100381610038161003816119863119899119894119895 (2)10038161003816100381610038161003816max

= 16 max10038161003816100381610038161003816100381610038161003816100381612059731198821205971199093 (120585 119910119895 120591119899)

100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891 (57)

Analogously

119882119899119894119895+1 minus119882119899119894119895minus12 |119898| ℎ = 120597119882120597119910 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (3) (58)

Abstract and Applied Analysis 7

where

119864119899119894119895 (2) = 11989826 12059731198821205971199103 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (59)

thus10038161003816100381610038161003816119864119899119894119895 (3)10038161003816100381610038161003816le 11989826 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(60)

The discretization of the second spatial derivatives is asfollows

119882119899119894+1119895 minus 2119882119899119894119895 +119882119899119894minus1119895ℎ2= 12059721198821205971199092 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (4)

(61)

where

119864119899119894119895 (4) = 112 12059741198821205971199094 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (62)

10038161003816100381610038161003816119864119899119894119895 (4)10038161003816100381610038161003816le 112 max100381610038161003816100381610038161003816100381610038161003816

12059741198821205971199094 (120585 119910119895 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891

(63)

Analogously

119908119899119894119895+1 minus 2119908119899119894119895 + 119908119899119894119895minus11198982ℎ2= 12059721198821205971199102 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (5)

(64)

where

119864119899119894119895 (5) = 119898412 12059741198821205971199104 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (65)

10038161003816100381610038161003816119864119899119894119895 (5)10038161003816100381610038161003816le 119898412 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(66)

Then the local truncation error takes the following form

119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) = 119896119864119899119894119895 (1) minus 2

2sdot ℎ4 (119864119899119894119895 (4) + 119864119899119894119895 (5)) minus 1205901 (119903 minus 1199021 minus

120590212 )sdot ℎ2119864119899119894119895 (2)minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 ))

sdot ℎ2119864119899119894119895 (3)

(67)

From (54) (57) (60) (63) and (66) one gets1003816100381610038161003816100381611987911989911989411989510038161003816100381610038161003816 = 119874 (119896) + 119874 (ℎ2) (68)

Theorem 3 Assuming that the solution of the PDE (19) admitstwo times continuous partial derivativewith respect to time andup to order four with respect to each of space directions solutionthe numerical solution computed by scheme (24) is consistentwith (19)

5 European Spread Put Option Case

For the sake of brevity and in order to avoid repetition ofproofs we summarize in this section the results proved inprevious sections for the call option case

Let us consider a European spread put option with payoff

119880 (1198781 1198782 0) = ((1198782 + 119864) minus 1198781)+ 119864 gt 0 (69)

Equation (1) and all the transformed versions of it do notchange for the put option Like in the study of the previouscall option case due to different nature of the put option weneed to establish the boundary conditions and translate themto the boundary of the numerical domain

(1) If 1198782 = 0 the problem is reduced to the European putoption on asset 1198781 and strike 119864 Therefore the optionvalue on this boundary is

119880 (1198781 0 120591) = 119864119890minus119903120591119873(minus1206012) minus 11987811198901199021120591119873(minus1206011) (70)

where 1206011 and 1206012 are defined by (5)(2) If 1198781 = 0 we consider 119880(0 1198782 120591) as a put option price

on asset 1198781 = 0 with the strike 1198782 + 119864119880 (0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 (71)

(3) For large values of 1198782 we suppose that the slope of thecurve tends to 1 that is

1205971198801205971198782 (1198781 1198782 997888rarr infin 120591) = 119890minus1199022120591 (72)

(4) For 1198781 ≫ 1198782 + 119864 we consider 119880(1198781 1198782 120591) as a putoption price with large values of 1198781 and therefore

119880(1198781 1198782 120591) = 0 (73)

Under transformations (8) and (16) the boundary con-ditions (70)ndash(73) are translated to the rhomboid numericaldomain 119860119861119862119863 as follows

119863119862 119908119899119894119873119910+119894 = 119864119873 (minus1205722119894)+ 119890(1205901)119909119894minus1199021120591119899119873(minus1205721119894)

(74)

119860119863 1199081198990119895 = 119864 + 119890minus1205902(119910119895minus1198981199090) (75)

119860119861 119908119899119894119894 = 119908119899119894119894+1 + |119898| ℎ1205902119890minus1205902(119910119894+1minus119898119909119894)minus119903120591 (76)

119861119862 119908119899119873119909+1119873119909+119895 = 0 (77)

8 Abstract and Applied Analysis

Positivity As the numerical scheme (24) for the interiorpoints is the same and the new boundary conditions (74)ndash(77) are nonnegative then under conditions (35) and (36)the numerical solution for the European spread put optionis nonnegative

Stability Stability of the scheme is preservedwith newbound-ary conditions (74)ndash(77) because analogously to the calloption case the maximum value of the transformed function119882(119909 119910 120591) is reached at a corner (in this case (1198881 1198892) wherewe use the boundary condition (75)) At the correspondingpoint of the original domain we use boundary condition (71)In the original coordinates it can be bounded as

119880(0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 le 1198782max+ 119864 = 4119864 = const (78)

Therefore the transformed function also can be boundedas

max119894119895119908119899119894119895 = 4119864119890119903119879 (79)

6 American Spread Options

In the case of American type option the holder has the rightto exercise option at any moment before expiring It leadsto a free boundary problem [27] In order to simplify thecomputational procedure this problem can be considered asa linear complementarity problem (LCP)

(120597119880120597120591 minus 119871119880) (119880 minus 119892) = 0120597119880120597120591 minus 119871119880 ge 0

119880 minus 119892 (1198781 1198782) ge 0(80)

where 119871 represents the spatial differential operator of (1)If we rewrite (19) in the form

120597119882120597120591 minusL119882 = 0 (81)

then under transformation (16) LCP (80) has the followingform

(120597119882120597120591 minusL119882)(119882 minus 119891 (119909 119910)) = 0120597119882120597120591 minusL119882 ge 0119882 minus 119891 (119909 119910) ge 0

(82)

whereL is the right-hand side of (19) and 119891(119909 119910) is given by(41)

Numerical solution for problem (82) can be easilyobtained by a small modification of the calculating proceduredescribed above in Sections 3 and 4 On each time level

119899119894119895 = max 119908119899119894119895 119891 (119909119894 119910119895) ge 0 (83)

where 119908119899119894119895 is defined by the finite difference equation (24)

01500

500

4001000

1000

300

1500

200500100

0 0

U(S

1S

2T

)

S1 S2

Figure 2 Parameters ℎ and 119896 satisfy (85)

7 Numerical Examples

In this section we illustrate numerical results for bothEuropean and American spread options

In Example 1 we show that the stability conditions (35)and (36) for the European spread option can not be disre-garded

Example 1 We tested the algorithm for the European spreadcall option pricing problem with the parameters

119879 = 1119864 = 1001205901 = 021205902 = 01119903 = 011199021 = 0011199022 = 005120588 = 01

(84)

and taking 1198781 and 1198782 large enough in particular 1198782max= 4001198781max

= 3(1198782max+ 119864) = 1500

For parameters (84) the stability conditions (35) and (36)on space and time steps are as follows

ℎ lt min 28428 237358 = 28428 119896 lt 00112 (85)

These conditions are crucial for the qualitative propertiessuch as positivity and stability of the scheme In Figure 2 thenumerical solution with ℎ and 119896 satisfying (85) is presentedOn the other hand in Figure 3 there is a numerical simulationwith time step 119896 = 0012 gt 00112 The solution is unstableand it has negative values

Example 2 illustrates the computational time aswell as thecomparison with analytical approximation provided by [6]

Abstract and Applied Analysis 9

1500

0500

400

1000

1000

1500

300

20002500

200500100

0 0

minus500

U(S

1S

2T

)

S1

S2

Figure 3 Stability condition is broken

Example 2 Consider the European spread option with thedata of Example 1 (84) and choosing step sizes satisfying thestability condition

The price of European spread option can be expressedas the price of a compound exchange option (see [6])The known derived analytical approximation for the spreadoption reads

119880 (1198781 1198782 119879) = 1198781119890minus1199021119879119873(1198891)minus (119864119890minus119903119879 + 1198782) 119890minus(119903minusminus2)119879119873(1198892) (86)

where

1198891 = 1120590radic119879 (ln

11987811198782 + 119864119890minus119903119879+ (119903 minus + 2 minus 1199021 + 12059022 )119879) 1198892 = 1198891 minus 120590radic119879

120590 = radic12059021 minus 212058812059011205902 11987821198782 + 119864119890minus119903119879 + (11987821198782 + 119864119890minus119903119879)

2 12059022 = 11987821198782 + 119864119890minus119903119879 1199032 = 11987821198782 + 119864119890minus119903119879 1199022

(87)

The comparison of our proposed numerical method andthe analytical approximation (86) is presented in Table 1Analytical approximation is calculated for the same arraysof 1198781 and 1198782 using MATLAB-function cdf for calculat-ing cumulative distribution function that requires a lot ofcomputational resources for each point In the proposedfinite difference method (FDM) cdf is only used for theboundary conditions Last row in the Table 1 presents theCPU time for each method for the same sizes of array Forapproximation method cdf-time is 925018 sec It is the main

Table 1 Comparison proposed method (FDM) with analyticalapproximation (86)

1198782 1198781 FDM Approximation

100

400 2105978 2097261200 242895 234543100 01150 0012350 31179119890 minus 05 minus47741119890 minus 10

90

400 2200678 2201714200 298745 298724100 01691 0037250 52150119890 minus 05 minus19574119890 minus 09

CPU time (sec) 106188 929467

00400

100

300 500

200

200

300

1000100

400

15000

500

U(S

1S

2T

)

S 1S2

Figure 4 European spread put option with parameters (84)

part of computational time In the case of FDM cdf takes lessthan half of computational time - 48867 sec

Furthermore unsuitable negative values appear in ana-lytical approximation for small 1198781 and 1198782 while the proposedmethod preserves nonnegativity of the solution

In the next example we confirm that for the spreadput option case the removing technique is fruitful and theselection of the boundary conditions at the boundary of thenumerical domain is appropriated because the behaviour ofthe solution at the boundaries fits the solution at the interiorpoints as it is shown in Figure 4

Example 3 With the parameters in (84) of Example 1 butwith payoff (1198782+119864minus1198781)+ in Figure 4we show the simulation ofthe numerical solution obtained by the proposed numericalscheme

Finally in Example 4 we compare the numerical solutionobtained with our scheme (83) using LCP approach withother tested efficient methods presented in [28]

10 Abstract and Applied Analysis

Table 2 Comparing several methods for American option pricing

119864 Analytic FFT MC FDM0 8513201 8513079 8516613 85081982 7542296 7542242 7545496 75340644 6653060 6652976 6656364 6648792

Example 4 Let us consider the American spread call optionproblem with parameters

119879 = 11205901 = 011205902 = 02119903 = 011199021 = 0051199022 = 005120588 = 05

(88)

Table 2 shows the option price at fixed asset prices 1198781 = 96and 1198782 = 100 for different values of strike price 119864 evaluatedby one-dimensional integration analytic method (Analytic)Fast Fourier Transform (FFT) Monte Carlo (MC) methodand our proposed method The accuracy is competitive withother methods such as Fourier Transform with high numberof discretization (4096) andMonte Carlo with big number oftime steps (2000)

Another set of parameters can be considered in order tocompare it with data in [9]

119879 = 05119864 = 1001205901 = 0251205902 = 03119903 = 0031199021 = 0061199022 = 002120588 = 05

(89)

In [9] three methods are considered to evaluate thespread option price for data 74 and fixed values 1198781 = 100and 1198782 = 200 a numerical integration method (NIM) amethod of lines (MOL) and a Monte Carlo approach NIMused in [9] is based in the recursive integration techniquesdeveloped byHuang et al [29] InNIM the option is treated asa Bermudan that is an option that can be exercised at discretepoints of time and Simpsons rule is used For the experimentthe authors used 1024 spatial nodes see [8 9] for moredetails The MOL is implemented by a spatial discretizationof 1438 times 200 mesh points and a three-time level scheme isused for the integration in time [9] 50 000 simulations have

Table 3 CPU time in sec (first row) and absolute difference (secondrow) for different methods depending on number of time steps forfixed number of space steps for parameters (89)

119873120591 NIM MOL Monte Carlo FDM

50 296598 5920 47 970 265470003 0001 01735 00036

32 127069 5512 19 811 1791100035 0006 01148 00122

16 33493 51718 5 149 943200012 00236 0132 00371

8 10360 4901 1 350 478700105 0074 01316 00907

4 4117 4547 35575 281500364 02512 01351 02254

been used in the Monte Carlo method described in [30] Inour method FDM the step size discretization considered isℎ = 01 To compare FDM with the three previous methodsabsolute errors are computed in Table 3 with respect to areference value obtained by the Fourier space time-steppingapproach of Surkov [31] with large number of nodes 4096spatial nodes and 300 time steps From Table 3 we can statethat the proposed method has the same order of accuracy asMOL more efficient and requires less computational timethan Monte Carlo method It is slightly less efficient than theNIM but we prove that it possesses a very highly desirablequalitative behaviour in finance

Competing Interests

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

Acknowledgments

This work has been partially supported by the EuropeanUnion in the FP7- PEOPLE-2012-ITN Program under GrantAgreement no 304617 (FP7 Marie Curie Action ProjectMulti-ITN STRIKE-Novel Methods in ComputationalFinance) and the Ministerio de Economıa y CompetitividadSpanish Grant MTM2013-41765-P

References

[1] R Carmona and V Durrleman ldquoPricing and hedging spreadoptionsrdquo SIAM Review vol 45 no 4 pp 627ndash685 2003

[2] P P Boyle ldquoOptions a Monte Carlo approachrdquo Journal ofFinancial Economics vol 4 no 3 pp 323ndash338 1977

[3] C Joy P P Boyle and K S Tan ldquoQuasi-Monte Carlo methodsin numerical financerdquo Management Science vol 42 no 6 pp926ndash938 1996

[4] W-H Kao Y-D Lyuu and K-W Wen ldquoThe hexanomiallattice for pricingmulti-asset optionsrdquoAppliedMathematics andComputation vol 233 pp 463ndash479 2014

Abstract and Applied Analysis 11

[5] E Kirk and J Aron ldquoCorrelation in the energy marketsrdquo inManaging Energy Price Risk V Kaminski Ed pp 71ndash78 RiskBooks London UK 1995

[6] C Alexander andA Venkatramanan ldquoAnalytic approximationsfor spread optionsrdquo ICMACentre Discussion Papers in FinanceDP2007-11 ICMA Centre 2007

[7] N D Pearson ldquoAn efficient approach for pricing spreadoptionsrdquoThe Journal of Derivatives vol 3 no 1 pp 76ndash91 1995

[8] C Chiarella and J Ziveyi ldquoPricingAmerican options written ontwo underlying assetsrdquo Quantitative Finance vol 14 no 3 pp409ndash426 2014

[9] J Ziveyi The evaluation of early exercise exotic options [PhDthesis] University of Technology Sydney Australia 2011

[10] P Boyle ldquoA lattice framework for option pricing with two statevariablesrdquo Journal of Financial and Quantitative Analysis vol23 no 1 pp 1ndash12 1988

[11] M Rubinstein ldquoReturn to Ozrdquo RISK vol 7 no 11 pp 67ndash711994

[12] R Company L Jodar M Fakharany and M-C CasabanldquoRemoving the correlation term in option pricing Hestonmodel numerical analysis and computingrdquo Abstract andApplied Analysis vol 2013 Article ID 246724 11 pages 2013

[13] S Salmi J Toivanen and L von Sydow ldquoIterative methodsfor pricing american options under the bates modelrdquo ProcediaComputer Science vol 18 pp 1136ndash1144 2013

[14] J Toivanen ldquoA componentwise splitting method for pricingamerican options under the bates modelrdquo in Applied andNumerical Partial Differential Equations vol 15 of Computa-tional Methods in Applied Sciences pp 213ndash227 Springer 2010

[15] R Zvan P A Forsyth and K R Vetzal ldquoNegative coefficientsin two-factor option pricing modelsrdquo Journal of ComputationalFinance vol 7 no 1 pp 37ndash73 2003

[16] B During M Fournie and C Heuer ldquoHigh-order compactfinite difference schemes for option pricing in stochastic volatil-ity models on non-uniform gridsrdquo Journal of Computationaland Applied Mathematics vol 271 pp 247ndash266 2014

[17] K J in rsquot Hout and C Mishra ldquoStability of ADI schemes formultidimensional diffusion equations with mixed derivativetermsrdquoApplied NumericalMathematics vol 74 pp 83ndash94 2013

[18] C Chiarella B Kang G H Mayer and A Ziogas ldquoTheevaluation of American option prices under stochastic volatilityand jump-diffusion dynamics using the method of linesrdquoInternational Journal of Theoretical and Applied Finance vol 12no 3 pp 393ndash425 2009

[19] M Fakharany R Company and L Jodar ldquoPositive finite dif-ference schemes for a partial integro-differential option pricingmodelrdquo Applied Mathematics and Computation vol 249 pp320ndash332 2014

[20] P R Garabedian Partial Differential Equations AMS ChelseaPublishing Providence RI USA 1998

[21] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp SonsChichester UK 2006

[22] R U Seydel Tools for Computational Finance Springer 3rdedition 2006

[23] M Ehrhardt Discrete artificial boundary conditions [PhDthesis] Technische Universitat Berlin Berlin Germany 2001

[24] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[25] J C HullOptions Futures andOther Derivatives Prentice-HallUpper Saddle River NJ USA 2000

[26] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods Clarendon Press Oxford UK3rd edition 1985

[27] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[28] M A H Dempster and S S G Hong Spread Option Valuationand the Fast Fourier Transform WP 262000 Judge Institute ofManagement Studies University of Cambridge 2000

[29] J-Z Huang M G Subrahmanyam and G G Yu ldquoPricingand hedging american options a recursive integrationmethodrdquoReview of Financial Studies vol 9 no 1 pp 277ndash300 1996

[30] A Ibanez and F Zapatero ldquoMonte Carlo valuation of Americanoptions through computation of the optimal exercise frontierrdquoJournal of Financial and Quantitative Analysis vol 39 no 2 pp253ndash275 2004

[31] V Surkov ldquoParallel option pricing with fourier space time-stepping method on graphics processingrdquo in Proceedings of the1st International Workshop on Parallel and Distributed Comput-ing in Finance (PDCoF rsquo08) Miami Fla USA 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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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

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

3 Removing the Correlation Term andProblem Discretization

Webegin this section by transforming the PDE problem (1) inorder to remove the cross derivative term Firstlywe eliminatethe reaction term 119903119880 by means of the substitution

119881 = 119890119903120591119880 (8)

obtaining

120597119881120597119905 = 121205902111987821 120597211988112059711987821 +

121205902211987822 120597211988112059711987822 + 1205881205901120590211987811198782

120597211988112059711987811205971198782+ (119903 minus 1199021) 1198781 1205971198811205971198781 + (119903 minus 1199022) 1198782

1205971198811205971198782 (9)

In order to remove the cross derivative term note thatthe right-hand side of (9) is a linear differential operator oftwo variables and classical techniques to obtain the canonicalform of second-order linear PDEs can be applied see forinstance chapter 3 of [20] Under the assumption of corre-lated variables with minus1 lt 120588 lt 1 the sign of the discriminantis

119886212 minus 41198861111988622 = 12059021120590221198782111987822 (1205882 minus 1) lt 0 (10)

where

11988611 = 12120590211198782111988612 = 120588120590112059021198781119878211988622 = 121205902211987822

(11)

Therefore right-hand side of (9) becomes of elliptic typeand a convenient substitution to obtain the canonical form isgiven by solving the ordinary differential equation

11988911987821198891198781 +1205902119878212059011198781 (minus120588 plusmn 119894) = 0 (12)

where

= radic1 minus 1205882 (13)

and (1205902119878212059011198781)(minus120588 plusmn 119894) are the conjugate roots of 119886111199092 +11988612119909 + 11988622 = 0 Solving (12) one gets11205902 log 1198782 +

minus120588 plusmn 1198941205901 log 1198781 = 1198620 (14)

where one can relate the integration constant 1198620 to the newvariables by

119909 = Im (1198620) 119910 = minusRe (1198620) (15)

From (14) and (15) it follows the expression of the newvariables

119909 = 1205901 log 1198781119910 = 1205881205901 log 1198781 minus

11205902 log 1198782(16)

Note that

119910 minus 119898119909 = minus 11205902 log 1198782 119898 = 120588 (17)

By denoting

119882(119909 119910 120591) = 119881 (1198781 1198782 120591) (18)

(9) takes the following form without cross derivative term

120597119882120597120591 = 2

2 (12059721198821205971199092 + 120597

21198821205971199102 ) + 1205901 (119903 minus 1199021 minus120590212 ) 120597119882120597119909

+ ( 1205881205901 (119903 minus 1199021 minus120590212 ) minus 11205902 (119903 minus 1199022 minus

120590222 )) 120597119882120597119910 (119909 119910) isin R2 0 lt 120591 le 119879

(19)

In accordance with [23 24] and (6) we choose therectangular domain (1198781 1198782) isin [1198861 1198871] times [1198862 1198872] = Ω where119886119894 gt 0 119894 = 1 2 are small positive values 1198872 about 3119864 and 1198871about 3(1198872+119864) For the transformed problem (19) due to (16)the rectangular domainΩ becomes a rhomboid with vertices119860119861119862119863 (see Figure 1) From (16) let us denote

1198881 = 1205901 log 11988611198882 = log 11988621205902 1198891 = 1205901 log 11988711198892 = log 11988721205902

(20)

By (17) the rhomboid domain has the sides

119860119863 = (119909 119910) 119909 = 1198881 1198981198881 minus 1198892 le 119910 le 1198981198881 minus 1198882 119860119861 = (119909 119910) 1198881 le 119909 le 1198891 119910 = 119898119909 minus 1198892 119861119862 = (119909 119910) 119909 = 1198891 1198981198891 minus 1198892 le 119910 le 1198981198891 minus 1198882 119863119862 = (119909 119910) 1198881 le 119909 le 1198891 119910 = 119898119909 minus 1198882

(21)

4 Abstract and Applied Analysis

A

B C

D

x

y

Figure 1 Transformed numerical domain

Themesh points in the rhomboid domain are (119909119894 119910119895) suchthat

119909119894 = 1198881 + 119894ℎ 119909119894 isin [1198881 1198891] 0 le 119894 le 119873119909Δ 119909 = ℎ = 1198891 minus 1198881119873119909 Δ 119910 = |119898| ℎ119873119910 = 1198892 minus 1198882|119898| ℎ 119910119895 = 119898119909119894 minus 1198892 + (119895 minus 119894) |119898| ℎ 119894 le 119895 le 119873119910 + 119894

(22)

The approximations of the derivatives appearing in (19) atthe point (119909119894 119910119895 120591119899) are as follows

120597119882120597119909 sim 119908119899119894+1119895 minus 119908119899119894minus11198952ℎ 120597119882120597119910 sim 119908119899119894119895+1 minus 119908119899119894119895minus12 |119898| ℎ 12059721198821205971199092 sim

119908119899119894+1119895 minus 2119908119899119894119895 + 119908119899119894minus1119895ℎ2 12059721198821205971199102 sim

119908119899119894119895+1 minus 2119908119899119894119895 + 119908119899119894119895minus11198982ℎ2 120597119882120597120591 sim

119908119899+1119894119895 minus 119908119899119894119895119896

(23)

where 119908119899119894119895 sim 119882(119909119894 119910119895 120591119899) 120591119899 = 119899119896Substituting these finite difference approximations of the

derivatives into (19) is approximated by the explicit differencescheme with five-point stencil

119908119899+1119894119895 = 1205721119908119899119894minus1119895 + 1205722119908119899119894+1119895 + 1205723119908119899119894119895 + 1205724119908119899119894119895minus1+ 1205725119908119899119894119895+1 (24)

where

12057212 = 22 119896ℎ2 ∓ 1205901 (119903 minus 1199021 minus120590212 ) 1198962ℎ (25)

1205723 = 1 minus 2 119896ℎ2 (1 + 11198982 ) (26)

12057245= 211989621198982ℎ2∓ [ (119903 minus 1199021 minus 120590212)1205901 minus 119903 minus 1199022 minus 1205902221205902 ] 1198962 |119898| ℎ

(27)

The discretization of the spatial boundary is given by

119875 (119860119863) = (1199090 119910119895) 0 le 119895 le 119873119910 119875 (119860119861) = (119909119894 119910119894) 0 le 119894 le 119873119909 119875 (119861119862) = (119909119873119909 119910119895) 119873119909 le 119895 le 119873119909 + 119873119910 119875 (119863119862) = (119909119894 119910119873119910+119894) 0 le 119894 le 119873119909

(28)

Both initial and boundary conditions (2) (3) (4) (6) and(7) are transformed throughout (8) (16) (18) For the initialcondition we have

1199080119894119895 = (1198901205901119909119894 minus 119890minus1205902(119910119895minus119898119909119894) minus 119864)+ 0 le 119894 le 119873119909 119894 le 119895 le 119894 + 119873119910 (29)

Boundary condition for side 119860119863 is as follows

1199081198990119895 = 0 0 le 119895 le 119873119910 1 le 119899 le 119873120591 (30)

For 119863C with small value of 1198782 we have transformedBlack-Scholes solution so

119908119899119894119873119910+119894 = 119865119899119894 119873(1206011198991119894) minus 119864119873(1206011198992119894) 0 lt 119894 lt 119873119909 1 le 119899 le 119873120591

(31)

where

119865119899119894 = exp(1205901 119909119894 + (119903 minus 1199021) 120591119899) 1206011198991119894 = 1

1205901radic120591119899 [log119865119894119864 + 1212059021120591119899]

1206011198992119894 = 1206011198991119894 minus 1205901radic120591119899(32)

Side 119861119862 corresponds to large values of 1198781 so behaviour ofthe option there leads to

119908119899119873119909119895 = exp(1205901 119909119873119909 + (119903 minus 1199021) 120591119899)minus (exp (1205902 (119898119909119873119909 minus 119910119895)) + 119864)

119873119909 + 1 le 119895 le 119873119909 + 119873119910 1 le 119899 le 119873120591(33)

Abstract and Applied Analysis 5

Boundary 119860119861 corresponds with the boundary for largevalues of 1198782 Condition (7) means that component 1198782 isconstant and the null first derivative is approximated by theforward difference (119908119899119894119894+1 minus 119908119899119894119894)|119898|ℎ = 0

119908119899119894119894 = 119908119899119894119894+1 1 le 119894 le 119873119909 1 le 119899 le 119873120591 (34)

4 Positivity Stability and Consistency

In this section we show that the proposed numerical scheme(24)ndash(27) presents suitable qualitative and computationalproperties such as consistency and conditional positivity andstability

In order to guarantee positivity of the numerical solutionlet us check the positivity of the coefficients 120572119894

From (25) and (27) it is easy to check that coefficients1205721 1205722 1205724 and 1205725 of the scheme (24) are positive under thecondition

ℎ lt min 12059011003816100381610038161003816119903 minus 1199021 minus 1205902121003816100381610038161003816 21003816100381610038161003816119898 [ (119903 minus 1199021 minus 120590212) 1205901 minus (119903 minus 1199022 minus 120590222) 1205902]1003816100381610038161003816

(35)

Coefficient 1205723 is positive if119896 lt 1198982ℎ2

2 (1198982 + 1) (36)

Under conditions (35) and (36) values in interior pointsof the numerical domain 119908119899+1119894119895 1 le 119894 le 119873119909 minus 1 119894 + 1 le119895 le 119873119910 + 119894 minus 1 calculated by the scheme (24) preservenonnegativity from the previous time level 119899 at all interior andboundary points 119908119899119894119895 ge 0 0 le 119894 le 119873119909 119894 le 119895 le 119873119910 + 119894

Let us pay attention now to the positivity of the numericalsolution at the boundary of the numerical domain Onthe boundary 119860119863 from (30) we have zero values On theboundary 119863119862 formula (31) is used preserving the positivitysince it is the transformed Black-Scholes formula through-out expression (16) Along the line 119860119861 we use Neumannboundary condition (34) Therefore the positivity at theneighbour interior point guarantees the positivity at theboundary Finally on the line 119861119862 boundary conditions aredetermined by formula (33) This is transformed expressionfor the asymptotic behaviour of the call option (6) that ispositive under the condition

1198781maxgt (1198782max

+ 119864) exp ((1199021 minus 119903) 120591119899) 1 le 119899 le 119873120591 (37)

Following the ideas of Kangro and Nicolaides [24] thecomputational domain has to be large enough to translate theboundary conditions therefore

1198781max

ge max (1198782max+ 119864) exp ((1199021 minus 119903) 119879) 3 (1198782max

+ 119864) (38)

Therefore choosing 1198781maxaccording to (38) for the trans-

formed problem119908119899119873119909 119895 ge 0 for any119873119909+1 le 119895 le 119873119909+119873119910 1 le119899 le 119873120591In summary the nonnegativity of the numerical solution

follows from the nonnegativity of the initial values andpositivity preservation under the scheme and boundaryconditions

As authors manage several stability criteria in literaturewe recall the concept of stability used in this paper

Definition 1 Let 119908119899 isin R(119873119909+1)times(119873119910+1) be the matrix involvingthe numerical solution 119908119899119894119895 of PDE problem (19) computedby scheme (24) (29) (30) (31) (33) and (34)

119908119899 = (119908119899119894119895)0le119894le119873119909 119894le119895le119873119910+119894 0 le 119899 le 119873120591 (39)

The numerical scheme (24) (29) (30) (31) (33) and (34) issaid to be uniformly sdot infin-stable in the domain [1198881 1198891] times[1198882 1198892]times[0 119879] if for every partitionwith 119896ℎ as defined abovecondition 10038171003817100381710038171199081198991003817100381710038171003817infin le 119860 0 le 119899 le 119873120591 (40)

holds true where119860 is independent of ℎ 119896 When the schemeis uniformly stable under a condition imposed to the stepsize discretization then the scheme is said to be conditionallyuniformly stable

Let us find the maximum value of the 119908119899119894119895 with respect to119894 119895 for each fixed 119899 For 119899 = 0 the values119908119899119894119895 are defined by theinitial condition (29) Let us consider the following function

119891 (119909 119910) = exp(1205901 119909) minus exp (minus1205902 (119910 minus 119898119909)) minus 119864 (41)

and find the maximum value on the domain [1198881 1198891]times [1198882 1198892]The first derivatives are120597119891120597119909 = 1205901 exp(1205901 119909) + 1205902119898 exp (minus1205902 (119910 minus 119898119909)) gt 0120597119891120597119910 = 1205902exp (minus1205902 (119910 minus 119898119909)) gt 0

(42)

Analysing (42) one concludes that the function 119891(119909 119910) isincreasing with respect to 119909 and 119910 Therefore the maximumvalue is reached at the point (1198891 1198892)

max(119909119910)isin[1198881 1198891]times[1198882 1198892]

119891 (119909 119910) = 119891 (1198891 1198892)= exp(1205901 1198891) minus exp (minus1205902 (1198892 minus 1198981198891)) minus 119864 = 119862= const

(43)

For the next time levels 119899 ge 1 let us consider thenumerical scheme (24) Under conditions (35) and (36) thecoefficients 120572119894 gt 0 119894 = 1 5 Moreover it is easy to checkthat

5sum119894=1

120572119894 = 1 (44)

6 Abstract and Applied Analysis

Therefore the values in interior points at the (119899+1)th level119899 ge 0 can be bounded by the following

119908119899+1119894119895 le max 119908119899119894minus1119895 119908119899119894+1119895 119908119899119894119895 119908119899119894119895minus1 119908119899119894119895+1le max119894119895119908119899119894119895 (45)

This maximum can be reached on the boundary There-fore let us study the boundary conditions From (30) one getsthat

max119860119863

119908119899+1119894119895 = 0 (46)

The values on the boundary A119861 are equal to the interiorpoints therefore the inequality (45) can be applied

The values on the boundary 119861119862 are increasing withrespect to the index 119895 and reach the maximum at the point(119873119909 119873119909 + 119873119910 minus 1) From the other side this value can bebounded by the value at the point (119873119909 119873119909 + 119873119910) that iscalculated by formula (31) From the theory of option pricingthe price of European call is increasing with respect to theasset price 119878 andmoreover it is always greater than its asymp-totic function (see [25] p 268 formula (131)) The proposedtransformation preserves monotonicity therefore

119908119899+1119873119909119873119909+119873119910 = 119865119899+1119873119909 119873(120572119899+11119873119909) minus 119864119873(120572119899+12119873119909)ge 119890((1205901)119909119873119909+(119903minus1199021)120591119899+1)minus (119890(1205902(119898119909119873119909minus119910119873119909+119873119910minus1)) + 119864)

= 119908119899+1119873119909 119873119909+119873119910minus1(47)

In summary we can conclude that

max119894119895119908119899+1119894119895 = 119908119899+1119873119909119873119909+119873119910 (48)

This value is transformed call option price for the assetprice 1198781max

that is the solution of the 1D Black-Scholesequation at the fixed point European call option gives a rightto the option holder to purchase one share stock for a certainprice The option itself cannot be worth more than the stockIn other words

119880 (1198781 0 120591) = 119890minus11990211205911198781119873(1198891) minus 119890minus119903120591119864119873 (1198892)le 119890minus1199021205911198781119873(1198891) le 119890minus1199021205911198781

119908119899+1119873119909119873119909+119873119910 le 119890119903120591119899+1119890minus1199021120591119899+11198781maxle 119890|119903minus1199021|1198791198781max

0 le 119899 le 119873120591 minus 1

(49)

Then the following result can be stated

Theorem 2 With previous notations under conditions (35)and (36) numerical scheme (24) (29) (30) (31) (33) and(34) is conditionally uniformly sdot infin-stable with upper bound119860 = 119890|119903minus1199021|1198791198781max

Now consistency of the numerical scheme (24) with PDE(19) will be studied [26]

Let 119865(119908119899119894119895) = 0 be approximating difference equation(24) The scheme (24) is consistent with (19) if the localtruncation error 119879119899119894119895 satisfies119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) 997888rarr 0 as ℎ 997888rarr 0 119896 997888rarr 0 (50)

where119882119899119894119895 denotes the theoretical solution of (19) evaluatedat the point (119909119894 119910119895 120591119899) and 119871 is the operator119871 (119882) = 120597119882120597120591 minus

2

2 (12059721198821205971199092 + 120597

21198821205971199102 )minus 1205901 (119903 minus 1199021 minus

120590212 ) 120597119882120597119909minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 )) 120597119882120597119910

(51)

Assuming that theoretical solution119882of the PDEproblemis four times continuously differentiable with respect to 119909 and119910 and twice with respect to 120591 and using Taylor expansionabout (119909119894 119910119895 120591119899) it is easy to check that

119882119899+1119894119895 minus119882119899119894minus1119895119896 = 120597119882120597120591 (119909119894 119910119895 120591119899) + 119896119864119899119894119895 (1) (52)

where

119864119899119894119895 (1) = 12 12059721198821205971205912 (119909119894 119910119895 120575) 119899119896 lt 120575 lt (119899 + 1) 119896 (53)

10038161003816100381610038161003816119864119899119894119895 (1)10038161003816100381610038161003816 le 12 10038161003816100381610038161003816119863119899119894119895 (1)10038161003816100381610038161003816max

= 12max10038161003816100381610038161003816100381610038161003816100381612059721198821205971205912 (119909119894 119910119895 120575)

100381610038161003816100381610038161003816100381610038161003816 0 lt 120575 lt 119879 (54)

For the spatial discretization one gets

119882119899119894+1119895 minus119882119899119894minus11198952ℎ = 120597119882120597119909 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (2) (55)

where

119864119899119894119895 (2) = 16 12059731198821205971199093 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (56)

10038161003816100381610038161003816119864119899119894119895 (2)10038161003816100381610038161003816 le 16 10038161003816100381610038161003816119863119899119894119895 (2)10038161003816100381610038161003816max

= 16 max10038161003816100381610038161003816100381610038161003816100381612059731198821205971199093 (120585 119910119895 120591119899)

100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891 (57)

Analogously

119882119899119894119895+1 minus119882119899119894119895minus12 |119898| ℎ = 120597119882120597119910 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (3) (58)

Abstract and Applied Analysis 7

where

119864119899119894119895 (2) = 11989826 12059731198821205971199103 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (59)

thus10038161003816100381610038161003816119864119899119894119895 (3)10038161003816100381610038161003816le 11989826 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(60)

The discretization of the second spatial derivatives is asfollows

119882119899119894+1119895 minus 2119882119899119894119895 +119882119899119894minus1119895ℎ2= 12059721198821205971199092 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (4)

(61)

where

119864119899119894119895 (4) = 112 12059741198821205971199094 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (62)

10038161003816100381610038161003816119864119899119894119895 (4)10038161003816100381610038161003816le 112 max100381610038161003816100381610038161003816100381610038161003816

12059741198821205971199094 (120585 119910119895 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891

(63)

Analogously

119908119899119894119895+1 minus 2119908119899119894119895 + 119908119899119894119895minus11198982ℎ2= 12059721198821205971199102 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (5)

(64)

where

119864119899119894119895 (5) = 119898412 12059741198821205971199104 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (65)

10038161003816100381610038161003816119864119899119894119895 (5)10038161003816100381610038161003816le 119898412 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(66)

Then the local truncation error takes the following form

119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) = 119896119864119899119894119895 (1) minus 2

2sdot ℎ4 (119864119899119894119895 (4) + 119864119899119894119895 (5)) minus 1205901 (119903 minus 1199021 minus

120590212 )sdot ℎ2119864119899119894119895 (2)minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 ))

sdot ℎ2119864119899119894119895 (3)

(67)

From (54) (57) (60) (63) and (66) one gets1003816100381610038161003816100381611987911989911989411989510038161003816100381610038161003816 = 119874 (119896) + 119874 (ℎ2) (68)

Theorem 3 Assuming that the solution of the PDE (19) admitstwo times continuous partial derivativewith respect to time andup to order four with respect to each of space directions solutionthe numerical solution computed by scheme (24) is consistentwith (19)

5 European Spread Put Option Case

For the sake of brevity and in order to avoid repetition ofproofs we summarize in this section the results proved inprevious sections for the call option case

Let us consider a European spread put option with payoff

119880 (1198781 1198782 0) = ((1198782 + 119864) minus 1198781)+ 119864 gt 0 (69)

Equation (1) and all the transformed versions of it do notchange for the put option Like in the study of the previouscall option case due to different nature of the put option weneed to establish the boundary conditions and translate themto the boundary of the numerical domain

(1) If 1198782 = 0 the problem is reduced to the European putoption on asset 1198781 and strike 119864 Therefore the optionvalue on this boundary is

119880 (1198781 0 120591) = 119864119890minus119903120591119873(minus1206012) minus 11987811198901199021120591119873(minus1206011) (70)

where 1206011 and 1206012 are defined by (5)(2) If 1198781 = 0 we consider 119880(0 1198782 120591) as a put option price

on asset 1198781 = 0 with the strike 1198782 + 119864119880 (0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 (71)

(3) For large values of 1198782 we suppose that the slope of thecurve tends to 1 that is

1205971198801205971198782 (1198781 1198782 997888rarr infin 120591) = 119890minus1199022120591 (72)

(4) For 1198781 ≫ 1198782 + 119864 we consider 119880(1198781 1198782 120591) as a putoption price with large values of 1198781 and therefore

119880(1198781 1198782 120591) = 0 (73)

Under transformations (8) and (16) the boundary con-ditions (70)ndash(73) are translated to the rhomboid numericaldomain 119860119861119862119863 as follows

119863119862 119908119899119894119873119910+119894 = 119864119873 (minus1205722119894)+ 119890(1205901)119909119894minus1199021120591119899119873(minus1205721119894)

(74)

119860119863 1199081198990119895 = 119864 + 119890minus1205902(119910119895minus1198981199090) (75)

119860119861 119908119899119894119894 = 119908119899119894119894+1 + |119898| ℎ1205902119890minus1205902(119910119894+1minus119898119909119894)minus119903120591 (76)

119861119862 119908119899119873119909+1119873119909+119895 = 0 (77)

8 Abstract and Applied Analysis

Positivity As the numerical scheme (24) for the interiorpoints is the same and the new boundary conditions (74)ndash(77) are nonnegative then under conditions (35) and (36)the numerical solution for the European spread put optionis nonnegative

Stability Stability of the scheme is preservedwith newbound-ary conditions (74)ndash(77) because analogously to the calloption case the maximum value of the transformed function119882(119909 119910 120591) is reached at a corner (in this case (1198881 1198892) wherewe use the boundary condition (75)) At the correspondingpoint of the original domain we use boundary condition (71)In the original coordinates it can be bounded as

119880(0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 le 1198782max+ 119864 = 4119864 = const (78)

Therefore the transformed function also can be boundedas

max119894119895119908119899119894119895 = 4119864119890119903119879 (79)

6 American Spread Options

In the case of American type option the holder has the rightto exercise option at any moment before expiring It leadsto a free boundary problem [27] In order to simplify thecomputational procedure this problem can be considered asa linear complementarity problem (LCP)

(120597119880120597120591 minus 119871119880) (119880 minus 119892) = 0120597119880120597120591 minus 119871119880 ge 0

119880 minus 119892 (1198781 1198782) ge 0(80)

where 119871 represents the spatial differential operator of (1)If we rewrite (19) in the form

120597119882120597120591 minusL119882 = 0 (81)

then under transformation (16) LCP (80) has the followingform

(120597119882120597120591 minusL119882)(119882 minus 119891 (119909 119910)) = 0120597119882120597120591 minusL119882 ge 0119882 minus 119891 (119909 119910) ge 0

(82)

whereL is the right-hand side of (19) and 119891(119909 119910) is given by(41)

Numerical solution for problem (82) can be easilyobtained by a small modification of the calculating proceduredescribed above in Sections 3 and 4 On each time level

119899119894119895 = max 119908119899119894119895 119891 (119909119894 119910119895) ge 0 (83)

where 119908119899119894119895 is defined by the finite difference equation (24)

01500

500

4001000

1000

300

1500

200500100

0 0

U(S

1S

2T

)

S1 S2

Figure 2 Parameters ℎ and 119896 satisfy (85)

7 Numerical Examples

In this section we illustrate numerical results for bothEuropean and American spread options

In Example 1 we show that the stability conditions (35)and (36) for the European spread option can not be disre-garded

Example 1 We tested the algorithm for the European spreadcall option pricing problem with the parameters

119879 = 1119864 = 1001205901 = 021205902 = 01119903 = 011199021 = 0011199022 = 005120588 = 01

(84)

and taking 1198781 and 1198782 large enough in particular 1198782max= 4001198781max

= 3(1198782max+ 119864) = 1500

For parameters (84) the stability conditions (35) and (36)on space and time steps are as follows

ℎ lt min 28428 237358 = 28428 119896 lt 00112 (85)

These conditions are crucial for the qualitative propertiessuch as positivity and stability of the scheme In Figure 2 thenumerical solution with ℎ and 119896 satisfying (85) is presentedOn the other hand in Figure 3 there is a numerical simulationwith time step 119896 = 0012 gt 00112 The solution is unstableand it has negative values

Example 2 illustrates the computational time aswell as thecomparison with analytical approximation provided by [6]

Abstract and Applied Analysis 9

1500

0500

400

1000

1000

1500

300

20002500

200500100

0 0

minus500

U(S

1S

2T

)

S1

S2

Figure 3 Stability condition is broken

Example 2 Consider the European spread option with thedata of Example 1 (84) and choosing step sizes satisfying thestability condition

The price of European spread option can be expressedas the price of a compound exchange option (see [6])The known derived analytical approximation for the spreadoption reads

119880 (1198781 1198782 119879) = 1198781119890minus1199021119879119873(1198891)minus (119864119890minus119903119879 + 1198782) 119890minus(119903minusminus2)119879119873(1198892) (86)

where

1198891 = 1120590radic119879 (ln

11987811198782 + 119864119890minus119903119879+ (119903 minus + 2 minus 1199021 + 12059022 )119879) 1198892 = 1198891 minus 120590radic119879

120590 = radic12059021 minus 212058812059011205902 11987821198782 + 119864119890minus119903119879 + (11987821198782 + 119864119890minus119903119879)

2 12059022 = 11987821198782 + 119864119890minus119903119879 1199032 = 11987821198782 + 119864119890minus119903119879 1199022

(87)

The comparison of our proposed numerical method andthe analytical approximation (86) is presented in Table 1Analytical approximation is calculated for the same arraysof 1198781 and 1198782 using MATLAB-function cdf for calculat-ing cumulative distribution function that requires a lot ofcomputational resources for each point In the proposedfinite difference method (FDM) cdf is only used for theboundary conditions Last row in the Table 1 presents theCPU time for each method for the same sizes of array Forapproximation method cdf-time is 925018 sec It is the main

Table 1 Comparison proposed method (FDM) with analyticalapproximation (86)

1198782 1198781 FDM Approximation

100

400 2105978 2097261200 242895 234543100 01150 0012350 31179119890 minus 05 minus47741119890 minus 10

90

400 2200678 2201714200 298745 298724100 01691 0037250 52150119890 minus 05 minus19574119890 minus 09

CPU time (sec) 106188 929467

00400

100

300 500

200

200

300

1000100

400

15000

500

U(S

1S

2T

)

S 1S2

Figure 4 European spread put option with parameters (84)

part of computational time In the case of FDM cdf takes lessthan half of computational time - 48867 sec

Furthermore unsuitable negative values appear in ana-lytical approximation for small 1198781 and 1198782 while the proposedmethod preserves nonnegativity of the solution

In the next example we confirm that for the spreadput option case the removing technique is fruitful and theselection of the boundary conditions at the boundary of thenumerical domain is appropriated because the behaviour ofthe solution at the boundaries fits the solution at the interiorpoints as it is shown in Figure 4

Example 3 With the parameters in (84) of Example 1 butwith payoff (1198782+119864minus1198781)+ in Figure 4we show the simulation ofthe numerical solution obtained by the proposed numericalscheme

Finally in Example 4 we compare the numerical solutionobtained with our scheme (83) using LCP approach withother tested efficient methods presented in [28]

10 Abstract and Applied Analysis

Table 2 Comparing several methods for American option pricing

119864 Analytic FFT MC FDM0 8513201 8513079 8516613 85081982 7542296 7542242 7545496 75340644 6653060 6652976 6656364 6648792

Example 4 Let us consider the American spread call optionproblem with parameters

119879 = 11205901 = 011205902 = 02119903 = 011199021 = 0051199022 = 005120588 = 05

(88)

Table 2 shows the option price at fixed asset prices 1198781 = 96and 1198782 = 100 for different values of strike price 119864 evaluatedby one-dimensional integration analytic method (Analytic)Fast Fourier Transform (FFT) Monte Carlo (MC) methodand our proposed method The accuracy is competitive withother methods such as Fourier Transform with high numberof discretization (4096) andMonte Carlo with big number oftime steps (2000)

Another set of parameters can be considered in order tocompare it with data in [9]

119879 = 05119864 = 1001205901 = 0251205902 = 03119903 = 0031199021 = 0061199022 = 002120588 = 05

(89)

In [9] three methods are considered to evaluate thespread option price for data 74 and fixed values 1198781 = 100and 1198782 = 200 a numerical integration method (NIM) amethod of lines (MOL) and a Monte Carlo approach NIMused in [9] is based in the recursive integration techniquesdeveloped byHuang et al [29] InNIM the option is treated asa Bermudan that is an option that can be exercised at discretepoints of time and Simpsons rule is used For the experimentthe authors used 1024 spatial nodes see [8 9] for moredetails The MOL is implemented by a spatial discretizationof 1438 times 200 mesh points and a three-time level scheme isused for the integration in time [9] 50 000 simulations have

Table 3 CPU time in sec (first row) and absolute difference (secondrow) for different methods depending on number of time steps forfixed number of space steps for parameters (89)

119873120591 NIM MOL Monte Carlo FDM

50 296598 5920 47 970 265470003 0001 01735 00036

32 127069 5512 19 811 1791100035 0006 01148 00122

16 33493 51718 5 149 943200012 00236 0132 00371

8 10360 4901 1 350 478700105 0074 01316 00907

4 4117 4547 35575 281500364 02512 01351 02254

been used in the Monte Carlo method described in [30] Inour method FDM the step size discretization considered isℎ = 01 To compare FDM with the three previous methodsabsolute errors are computed in Table 3 with respect to areference value obtained by the Fourier space time-steppingapproach of Surkov [31] with large number of nodes 4096spatial nodes and 300 time steps From Table 3 we can statethat the proposed method has the same order of accuracy asMOL more efficient and requires less computational timethan Monte Carlo method It is slightly less efficient than theNIM but we prove that it possesses a very highly desirablequalitative behaviour in finance

Competing Interests

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

Acknowledgments

This work has been partially supported by the EuropeanUnion in the FP7- PEOPLE-2012-ITN Program under GrantAgreement no 304617 (FP7 Marie Curie Action ProjectMulti-ITN STRIKE-Novel Methods in ComputationalFinance) and the Ministerio de Economıa y CompetitividadSpanish Grant MTM2013-41765-P

References

[1] R Carmona and V Durrleman ldquoPricing and hedging spreadoptionsrdquo SIAM Review vol 45 no 4 pp 627ndash685 2003

[2] P P Boyle ldquoOptions a Monte Carlo approachrdquo Journal ofFinancial Economics vol 4 no 3 pp 323ndash338 1977

[3] C Joy P P Boyle and K S Tan ldquoQuasi-Monte Carlo methodsin numerical financerdquo Management Science vol 42 no 6 pp926ndash938 1996

[4] W-H Kao Y-D Lyuu and K-W Wen ldquoThe hexanomiallattice for pricingmulti-asset optionsrdquoAppliedMathematics andComputation vol 233 pp 463ndash479 2014

Abstract and Applied Analysis 11

[5] E Kirk and J Aron ldquoCorrelation in the energy marketsrdquo inManaging Energy Price Risk V Kaminski Ed pp 71ndash78 RiskBooks London UK 1995

[6] C Alexander andA Venkatramanan ldquoAnalytic approximationsfor spread optionsrdquo ICMACentre Discussion Papers in FinanceDP2007-11 ICMA Centre 2007

[7] N D Pearson ldquoAn efficient approach for pricing spreadoptionsrdquoThe Journal of Derivatives vol 3 no 1 pp 76ndash91 1995

[8] C Chiarella and J Ziveyi ldquoPricingAmerican options written ontwo underlying assetsrdquo Quantitative Finance vol 14 no 3 pp409ndash426 2014

[9] J Ziveyi The evaluation of early exercise exotic options [PhDthesis] University of Technology Sydney Australia 2011

[10] P Boyle ldquoA lattice framework for option pricing with two statevariablesrdquo Journal of Financial and Quantitative Analysis vol23 no 1 pp 1ndash12 1988

[11] M Rubinstein ldquoReturn to Ozrdquo RISK vol 7 no 11 pp 67ndash711994

[12] R Company L Jodar M Fakharany and M-C CasabanldquoRemoving the correlation term in option pricing Hestonmodel numerical analysis and computingrdquo Abstract andApplied Analysis vol 2013 Article ID 246724 11 pages 2013

[13] S Salmi J Toivanen and L von Sydow ldquoIterative methodsfor pricing american options under the bates modelrdquo ProcediaComputer Science vol 18 pp 1136ndash1144 2013

[14] J Toivanen ldquoA componentwise splitting method for pricingamerican options under the bates modelrdquo in Applied andNumerical Partial Differential Equations vol 15 of Computa-tional Methods in Applied Sciences pp 213ndash227 Springer 2010

[15] R Zvan P A Forsyth and K R Vetzal ldquoNegative coefficientsin two-factor option pricing modelsrdquo Journal of ComputationalFinance vol 7 no 1 pp 37ndash73 2003

[16] B During M Fournie and C Heuer ldquoHigh-order compactfinite difference schemes for option pricing in stochastic volatil-ity models on non-uniform gridsrdquo Journal of Computationaland Applied Mathematics vol 271 pp 247ndash266 2014

[17] K J in rsquot Hout and C Mishra ldquoStability of ADI schemes formultidimensional diffusion equations with mixed derivativetermsrdquoApplied NumericalMathematics vol 74 pp 83ndash94 2013

[18] C Chiarella B Kang G H Mayer and A Ziogas ldquoTheevaluation of American option prices under stochastic volatilityand jump-diffusion dynamics using the method of linesrdquoInternational Journal of Theoretical and Applied Finance vol 12no 3 pp 393ndash425 2009

[19] M Fakharany R Company and L Jodar ldquoPositive finite dif-ference schemes for a partial integro-differential option pricingmodelrdquo Applied Mathematics and Computation vol 249 pp320ndash332 2014

[20] P R Garabedian Partial Differential Equations AMS ChelseaPublishing Providence RI USA 1998

[21] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp SonsChichester UK 2006

[22] R U Seydel Tools for Computational Finance Springer 3rdedition 2006

[23] M Ehrhardt Discrete artificial boundary conditions [PhDthesis] Technische Universitat Berlin Berlin Germany 2001

[24] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[25] J C HullOptions Futures andOther Derivatives Prentice-HallUpper Saddle River NJ USA 2000

[26] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods Clarendon Press Oxford UK3rd edition 1985

[27] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[28] M A H Dempster and S S G Hong Spread Option Valuationand the Fast Fourier Transform WP 262000 Judge Institute ofManagement Studies University of Cambridge 2000

[29] J-Z Huang M G Subrahmanyam and G G Yu ldquoPricingand hedging american options a recursive integrationmethodrdquoReview of Financial Studies vol 9 no 1 pp 277ndash300 1996

[30] A Ibanez and F Zapatero ldquoMonte Carlo valuation of Americanoptions through computation of the optimal exercise frontierrdquoJournal of Financial and Quantitative Analysis vol 39 no 2 pp253ndash275 2004

[31] V Surkov ldquoParallel option pricing with fourier space time-stepping method on graphics processingrdquo in Proceedings of the1st International Workshop on Parallel and Distributed Comput-ing in Finance (PDCoF rsquo08) Miami Fla USA 2008

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

A

B C

D

x

y

Figure 1 Transformed numerical domain

Themesh points in the rhomboid domain are (119909119894 119910119895) suchthat

119909119894 = 1198881 + 119894ℎ 119909119894 isin [1198881 1198891] 0 le 119894 le 119873119909Δ 119909 = ℎ = 1198891 minus 1198881119873119909 Δ 119910 = |119898| ℎ119873119910 = 1198892 minus 1198882|119898| ℎ 119910119895 = 119898119909119894 minus 1198892 + (119895 minus 119894) |119898| ℎ 119894 le 119895 le 119873119910 + 119894

(22)

The approximations of the derivatives appearing in (19) atthe point (119909119894 119910119895 120591119899) are as follows

120597119882120597119909 sim 119908119899119894+1119895 minus 119908119899119894minus11198952ℎ 120597119882120597119910 sim 119908119899119894119895+1 minus 119908119899119894119895minus12 |119898| ℎ 12059721198821205971199092 sim

119908119899119894+1119895 minus 2119908119899119894119895 + 119908119899119894minus1119895ℎ2 12059721198821205971199102 sim

119908119899119894119895+1 minus 2119908119899119894119895 + 119908119899119894119895minus11198982ℎ2 120597119882120597120591 sim

119908119899+1119894119895 minus 119908119899119894119895119896

(23)

where 119908119899119894119895 sim 119882(119909119894 119910119895 120591119899) 120591119899 = 119899119896Substituting these finite difference approximations of the

derivatives into (19) is approximated by the explicit differencescheme with five-point stencil

119908119899+1119894119895 = 1205721119908119899119894minus1119895 + 1205722119908119899119894+1119895 + 1205723119908119899119894119895 + 1205724119908119899119894119895minus1+ 1205725119908119899119894119895+1 (24)

where

12057212 = 22 119896ℎ2 ∓ 1205901 (119903 minus 1199021 minus120590212 ) 1198962ℎ (25)

1205723 = 1 minus 2 119896ℎ2 (1 + 11198982 ) (26)

12057245= 211989621198982ℎ2∓ [ (119903 minus 1199021 minus 120590212)1205901 minus 119903 minus 1199022 minus 1205902221205902 ] 1198962 |119898| ℎ

(27)

The discretization of the spatial boundary is given by

119875 (119860119863) = (1199090 119910119895) 0 le 119895 le 119873119910 119875 (119860119861) = (119909119894 119910119894) 0 le 119894 le 119873119909 119875 (119861119862) = (119909119873119909 119910119895) 119873119909 le 119895 le 119873119909 + 119873119910 119875 (119863119862) = (119909119894 119910119873119910+119894) 0 le 119894 le 119873119909

(28)

Both initial and boundary conditions (2) (3) (4) (6) and(7) are transformed throughout (8) (16) (18) For the initialcondition we have

1199080119894119895 = (1198901205901119909119894 minus 119890minus1205902(119910119895minus119898119909119894) minus 119864)+ 0 le 119894 le 119873119909 119894 le 119895 le 119894 + 119873119910 (29)

Boundary condition for side 119860119863 is as follows

1199081198990119895 = 0 0 le 119895 le 119873119910 1 le 119899 le 119873120591 (30)

For 119863C with small value of 1198782 we have transformedBlack-Scholes solution so

119908119899119894119873119910+119894 = 119865119899119894 119873(1206011198991119894) minus 119864119873(1206011198992119894) 0 lt 119894 lt 119873119909 1 le 119899 le 119873120591

(31)

where

119865119899119894 = exp(1205901 119909119894 + (119903 minus 1199021) 120591119899) 1206011198991119894 = 1

1205901radic120591119899 [log119865119894119864 + 1212059021120591119899]

1206011198992119894 = 1206011198991119894 minus 1205901radic120591119899(32)

Side 119861119862 corresponds to large values of 1198781 so behaviour ofthe option there leads to

119908119899119873119909119895 = exp(1205901 119909119873119909 + (119903 minus 1199021) 120591119899)minus (exp (1205902 (119898119909119873119909 minus 119910119895)) + 119864)

119873119909 + 1 le 119895 le 119873119909 + 119873119910 1 le 119899 le 119873120591(33)

Abstract and Applied Analysis 5

Boundary 119860119861 corresponds with the boundary for largevalues of 1198782 Condition (7) means that component 1198782 isconstant and the null first derivative is approximated by theforward difference (119908119899119894119894+1 minus 119908119899119894119894)|119898|ℎ = 0

119908119899119894119894 = 119908119899119894119894+1 1 le 119894 le 119873119909 1 le 119899 le 119873120591 (34)

4 Positivity Stability and Consistency

In this section we show that the proposed numerical scheme(24)ndash(27) presents suitable qualitative and computationalproperties such as consistency and conditional positivity andstability

In order to guarantee positivity of the numerical solutionlet us check the positivity of the coefficients 120572119894

From (25) and (27) it is easy to check that coefficients1205721 1205722 1205724 and 1205725 of the scheme (24) are positive under thecondition

ℎ lt min 12059011003816100381610038161003816119903 minus 1199021 minus 1205902121003816100381610038161003816 21003816100381610038161003816119898 [ (119903 minus 1199021 minus 120590212) 1205901 minus (119903 minus 1199022 minus 120590222) 1205902]1003816100381610038161003816

(35)

Coefficient 1205723 is positive if119896 lt 1198982ℎ2

2 (1198982 + 1) (36)

Under conditions (35) and (36) values in interior pointsof the numerical domain 119908119899+1119894119895 1 le 119894 le 119873119909 minus 1 119894 + 1 le119895 le 119873119910 + 119894 minus 1 calculated by the scheme (24) preservenonnegativity from the previous time level 119899 at all interior andboundary points 119908119899119894119895 ge 0 0 le 119894 le 119873119909 119894 le 119895 le 119873119910 + 119894

Let us pay attention now to the positivity of the numericalsolution at the boundary of the numerical domain Onthe boundary 119860119863 from (30) we have zero values On theboundary 119863119862 formula (31) is used preserving the positivitysince it is the transformed Black-Scholes formula through-out expression (16) Along the line 119860119861 we use Neumannboundary condition (34) Therefore the positivity at theneighbour interior point guarantees the positivity at theboundary Finally on the line 119861119862 boundary conditions aredetermined by formula (33) This is transformed expressionfor the asymptotic behaviour of the call option (6) that ispositive under the condition

1198781maxgt (1198782max

+ 119864) exp ((1199021 minus 119903) 120591119899) 1 le 119899 le 119873120591 (37)

Following the ideas of Kangro and Nicolaides [24] thecomputational domain has to be large enough to translate theboundary conditions therefore

1198781max

ge max (1198782max+ 119864) exp ((1199021 minus 119903) 119879) 3 (1198782max

+ 119864) (38)

Therefore choosing 1198781maxaccording to (38) for the trans-

formed problem119908119899119873119909 119895 ge 0 for any119873119909+1 le 119895 le 119873119909+119873119910 1 le119899 le 119873120591In summary the nonnegativity of the numerical solution

follows from the nonnegativity of the initial values andpositivity preservation under the scheme and boundaryconditions

As authors manage several stability criteria in literaturewe recall the concept of stability used in this paper

Definition 1 Let 119908119899 isin R(119873119909+1)times(119873119910+1) be the matrix involvingthe numerical solution 119908119899119894119895 of PDE problem (19) computedby scheme (24) (29) (30) (31) (33) and (34)

119908119899 = (119908119899119894119895)0le119894le119873119909 119894le119895le119873119910+119894 0 le 119899 le 119873120591 (39)

The numerical scheme (24) (29) (30) (31) (33) and (34) issaid to be uniformly sdot infin-stable in the domain [1198881 1198891] times[1198882 1198892]times[0 119879] if for every partitionwith 119896ℎ as defined abovecondition 10038171003817100381710038171199081198991003817100381710038171003817infin le 119860 0 le 119899 le 119873120591 (40)

holds true where119860 is independent of ℎ 119896 When the schemeis uniformly stable under a condition imposed to the stepsize discretization then the scheme is said to be conditionallyuniformly stable

Let us find the maximum value of the 119908119899119894119895 with respect to119894 119895 for each fixed 119899 For 119899 = 0 the values119908119899119894119895 are defined by theinitial condition (29) Let us consider the following function

119891 (119909 119910) = exp(1205901 119909) minus exp (minus1205902 (119910 minus 119898119909)) minus 119864 (41)

and find the maximum value on the domain [1198881 1198891]times [1198882 1198892]The first derivatives are120597119891120597119909 = 1205901 exp(1205901 119909) + 1205902119898 exp (minus1205902 (119910 minus 119898119909)) gt 0120597119891120597119910 = 1205902exp (minus1205902 (119910 minus 119898119909)) gt 0

(42)

Analysing (42) one concludes that the function 119891(119909 119910) isincreasing with respect to 119909 and 119910 Therefore the maximumvalue is reached at the point (1198891 1198892)

max(119909119910)isin[1198881 1198891]times[1198882 1198892]

119891 (119909 119910) = 119891 (1198891 1198892)= exp(1205901 1198891) minus exp (minus1205902 (1198892 minus 1198981198891)) minus 119864 = 119862= const

(43)

For the next time levels 119899 ge 1 let us consider thenumerical scheme (24) Under conditions (35) and (36) thecoefficients 120572119894 gt 0 119894 = 1 5 Moreover it is easy to checkthat

5sum119894=1

120572119894 = 1 (44)

6 Abstract and Applied Analysis

Therefore the values in interior points at the (119899+1)th level119899 ge 0 can be bounded by the following

119908119899+1119894119895 le max 119908119899119894minus1119895 119908119899119894+1119895 119908119899119894119895 119908119899119894119895minus1 119908119899119894119895+1le max119894119895119908119899119894119895 (45)

This maximum can be reached on the boundary There-fore let us study the boundary conditions From (30) one getsthat

max119860119863

119908119899+1119894119895 = 0 (46)

The values on the boundary A119861 are equal to the interiorpoints therefore the inequality (45) can be applied

The values on the boundary 119861119862 are increasing withrespect to the index 119895 and reach the maximum at the point(119873119909 119873119909 + 119873119910 minus 1) From the other side this value can bebounded by the value at the point (119873119909 119873119909 + 119873119910) that iscalculated by formula (31) From the theory of option pricingthe price of European call is increasing with respect to theasset price 119878 andmoreover it is always greater than its asymp-totic function (see [25] p 268 formula (131)) The proposedtransformation preserves monotonicity therefore

119908119899+1119873119909119873119909+119873119910 = 119865119899+1119873119909 119873(120572119899+11119873119909) minus 119864119873(120572119899+12119873119909)ge 119890((1205901)119909119873119909+(119903minus1199021)120591119899+1)minus (119890(1205902(119898119909119873119909minus119910119873119909+119873119910minus1)) + 119864)

= 119908119899+1119873119909 119873119909+119873119910minus1(47)

In summary we can conclude that

max119894119895119908119899+1119894119895 = 119908119899+1119873119909119873119909+119873119910 (48)

This value is transformed call option price for the assetprice 1198781max

that is the solution of the 1D Black-Scholesequation at the fixed point European call option gives a rightto the option holder to purchase one share stock for a certainprice The option itself cannot be worth more than the stockIn other words

119880 (1198781 0 120591) = 119890minus11990211205911198781119873(1198891) minus 119890minus119903120591119864119873 (1198892)le 119890minus1199021205911198781119873(1198891) le 119890minus1199021205911198781

119908119899+1119873119909119873119909+119873119910 le 119890119903120591119899+1119890minus1199021120591119899+11198781maxle 119890|119903minus1199021|1198791198781max

0 le 119899 le 119873120591 minus 1

(49)

Then the following result can be stated

Theorem 2 With previous notations under conditions (35)and (36) numerical scheme (24) (29) (30) (31) (33) and(34) is conditionally uniformly sdot infin-stable with upper bound119860 = 119890|119903minus1199021|1198791198781max

Now consistency of the numerical scheme (24) with PDE(19) will be studied [26]

Let 119865(119908119899119894119895) = 0 be approximating difference equation(24) The scheme (24) is consistent with (19) if the localtruncation error 119879119899119894119895 satisfies119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) 997888rarr 0 as ℎ 997888rarr 0 119896 997888rarr 0 (50)

where119882119899119894119895 denotes the theoretical solution of (19) evaluatedat the point (119909119894 119910119895 120591119899) and 119871 is the operator119871 (119882) = 120597119882120597120591 minus

2

2 (12059721198821205971199092 + 120597

21198821205971199102 )minus 1205901 (119903 minus 1199021 minus

120590212 ) 120597119882120597119909minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 )) 120597119882120597119910

(51)

Assuming that theoretical solution119882of the PDEproblemis four times continuously differentiable with respect to 119909 and119910 and twice with respect to 120591 and using Taylor expansionabout (119909119894 119910119895 120591119899) it is easy to check that

119882119899+1119894119895 minus119882119899119894minus1119895119896 = 120597119882120597120591 (119909119894 119910119895 120591119899) + 119896119864119899119894119895 (1) (52)

where

119864119899119894119895 (1) = 12 12059721198821205971205912 (119909119894 119910119895 120575) 119899119896 lt 120575 lt (119899 + 1) 119896 (53)

10038161003816100381610038161003816119864119899119894119895 (1)10038161003816100381610038161003816 le 12 10038161003816100381610038161003816119863119899119894119895 (1)10038161003816100381610038161003816max

= 12max10038161003816100381610038161003816100381610038161003816100381612059721198821205971205912 (119909119894 119910119895 120575)

100381610038161003816100381610038161003816100381610038161003816 0 lt 120575 lt 119879 (54)

For the spatial discretization one gets

119882119899119894+1119895 minus119882119899119894minus11198952ℎ = 120597119882120597119909 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (2) (55)

where

119864119899119894119895 (2) = 16 12059731198821205971199093 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (56)

10038161003816100381610038161003816119864119899119894119895 (2)10038161003816100381610038161003816 le 16 10038161003816100381610038161003816119863119899119894119895 (2)10038161003816100381610038161003816max

= 16 max10038161003816100381610038161003816100381610038161003816100381612059731198821205971199093 (120585 119910119895 120591119899)

100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891 (57)

Analogously

119882119899119894119895+1 minus119882119899119894119895minus12 |119898| ℎ = 120597119882120597119910 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (3) (58)

Abstract and Applied Analysis 7

where

119864119899119894119895 (2) = 11989826 12059731198821205971199103 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (59)

thus10038161003816100381610038161003816119864119899119894119895 (3)10038161003816100381610038161003816le 11989826 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(60)

The discretization of the second spatial derivatives is asfollows

119882119899119894+1119895 minus 2119882119899119894119895 +119882119899119894minus1119895ℎ2= 12059721198821205971199092 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (4)

(61)

where

119864119899119894119895 (4) = 112 12059741198821205971199094 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (62)

10038161003816100381610038161003816119864119899119894119895 (4)10038161003816100381610038161003816le 112 max100381610038161003816100381610038161003816100381610038161003816

12059741198821205971199094 (120585 119910119895 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891

(63)

Analogously

119908119899119894119895+1 minus 2119908119899119894119895 + 119908119899119894119895minus11198982ℎ2= 12059721198821205971199102 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (5)

(64)

where

119864119899119894119895 (5) = 119898412 12059741198821205971199104 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (65)

10038161003816100381610038161003816119864119899119894119895 (5)10038161003816100381610038161003816le 119898412 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(66)

Then the local truncation error takes the following form

119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) = 119896119864119899119894119895 (1) minus 2

2sdot ℎ4 (119864119899119894119895 (4) + 119864119899119894119895 (5)) minus 1205901 (119903 minus 1199021 minus

120590212 )sdot ℎ2119864119899119894119895 (2)minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 ))

sdot ℎ2119864119899119894119895 (3)

(67)

From (54) (57) (60) (63) and (66) one gets1003816100381610038161003816100381611987911989911989411989510038161003816100381610038161003816 = 119874 (119896) + 119874 (ℎ2) (68)

Theorem 3 Assuming that the solution of the PDE (19) admitstwo times continuous partial derivativewith respect to time andup to order four with respect to each of space directions solutionthe numerical solution computed by scheme (24) is consistentwith (19)

5 European Spread Put Option Case

For the sake of brevity and in order to avoid repetition ofproofs we summarize in this section the results proved inprevious sections for the call option case

Let us consider a European spread put option with payoff

119880 (1198781 1198782 0) = ((1198782 + 119864) minus 1198781)+ 119864 gt 0 (69)

Equation (1) and all the transformed versions of it do notchange for the put option Like in the study of the previouscall option case due to different nature of the put option weneed to establish the boundary conditions and translate themto the boundary of the numerical domain

(1) If 1198782 = 0 the problem is reduced to the European putoption on asset 1198781 and strike 119864 Therefore the optionvalue on this boundary is

119880 (1198781 0 120591) = 119864119890minus119903120591119873(minus1206012) minus 11987811198901199021120591119873(minus1206011) (70)

where 1206011 and 1206012 are defined by (5)(2) If 1198781 = 0 we consider 119880(0 1198782 120591) as a put option price

on asset 1198781 = 0 with the strike 1198782 + 119864119880 (0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 (71)

(3) For large values of 1198782 we suppose that the slope of thecurve tends to 1 that is

1205971198801205971198782 (1198781 1198782 997888rarr infin 120591) = 119890minus1199022120591 (72)

(4) For 1198781 ≫ 1198782 + 119864 we consider 119880(1198781 1198782 120591) as a putoption price with large values of 1198781 and therefore

119880(1198781 1198782 120591) = 0 (73)

Under transformations (8) and (16) the boundary con-ditions (70)ndash(73) are translated to the rhomboid numericaldomain 119860119861119862119863 as follows

119863119862 119908119899119894119873119910+119894 = 119864119873 (minus1205722119894)+ 119890(1205901)119909119894minus1199021120591119899119873(minus1205721119894)

(74)

119860119863 1199081198990119895 = 119864 + 119890minus1205902(119910119895minus1198981199090) (75)

119860119861 119908119899119894119894 = 119908119899119894119894+1 + |119898| ℎ1205902119890minus1205902(119910119894+1minus119898119909119894)minus119903120591 (76)

119861119862 119908119899119873119909+1119873119909+119895 = 0 (77)

8 Abstract and Applied Analysis

Positivity As the numerical scheme (24) for the interiorpoints is the same and the new boundary conditions (74)ndash(77) are nonnegative then under conditions (35) and (36)the numerical solution for the European spread put optionis nonnegative

Stability Stability of the scheme is preservedwith newbound-ary conditions (74)ndash(77) because analogously to the calloption case the maximum value of the transformed function119882(119909 119910 120591) is reached at a corner (in this case (1198881 1198892) wherewe use the boundary condition (75)) At the correspondingpoint of the original domain we use boundary condition (71)In the original coordinates it can be bounded as

119880(0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 le 1198782max+ 119864 = 4119864 = const (78)

Therefore the transformed function also can be boundedas

max119894119895119908119899119894119895 = 4119864119890119903119879 (79)

6 American Spread Options

In the case of American type option the holder has the rightto exercise option at any moment before expiring It leadsto a free boundary problem [27] In order to simplify thecomputational procedure this problem can be considered asa linear complementarity problem (LCP)

(120597119880120597120591 minus 119871119880) (119880 minus 119892) = 0120597119880120597120591 minus 119871119880 ge 0

119880 minus 119892 (1198781 1198782) ge 0(80)

where 119871 represents the spatial differential operator of (1)If we rewrite (19) in the form

120597119882120597120591 minusL119882 = 0 (81)

then under transformation (16) LCP (80) has the followingform

(120597119882120597120591 minusL119882)(119882 minus 119891 (119909 119910)) = 0120597119882120597120591 minusL119882 ge 0119882 minus 119891 (119909 119910) ge 0

(82)

whereL is the right-hand side of (19) and 119891(119909 119910) is given by(41)

Numerical solution for problem (82) can be easilyobtained by a small modification of the calculating proceduredescribed above in Sections 3 and 4 On each time level

119899119894119895 = max 119908119899119894119895 119891 (119909119894 119910119895) ge 0 (83)

where 119908119899119894119895 is defined by the finite difference equation (24)

01500

500

4001000

1000

300

1500

200500100

0 0

U(S

1S

2T

)

S1 S2

Figure 2 Parameters ℎ and 119896 satisfy (85)

7 Numerical Examples

In this section we illustrate numerical results for bothEuropean and American spread options

In Example 1 we show that the stability conditions (35)and (36) for the European spread option can not be disre-garded

Example 1 We tested the algorithm for the European spreadcall option pricing problem with the parameters

119879 = 1119864 = 1001205901 = 021205902 = 01119903 = 011199021 = 0011199022 = 005120588 = 01

(84)

and taking 1198781 and 1198782 large enough in particular 1198782max= 4001198781max

= 3(1198782max+ 119864) = 1500

For parameters (84) the stability conditions (35) and (36)on space and time steps are as follows

ℎ lt min 28428 237358 = 28428 119896 lt 00112 (85)

These conditions are crucial for the qualitative propertiessuch as positivity and stability of the scheme In Figure 2 thenumerical solution with ℎ and 119896 satisfying (85) is presentedOn the other hand in Figure 3 there is a numerical simulationwith time step 119896 = 0012 gt 00112 The solution is unstableand it has negative values

Example 2 illustrates the computational time aswell as thecomparison with analytical approximation provided by [6]

Abstract and Applied Analysis 9

1500

0500

400

1000

1000

1500

300

20002500

200500100

0 0

minus500

U(S

1S

2T

)

S1

S2

Figure 3 Stability condition is broken

Example 2 Consider the European spread option with thedata of Example 1 (84) and choosing step sizes satisfying thestability condition

The price of European spread option can be expressedas the price of a compound exchange option (see [6])The known derived analytical approximation for the spreadoption reads

119880 (1198781 1198782 119879) = 1198781119890minus1199021119879119873(1198891)minus (119864119890minus119903119879 + 1198782) 119890minus(119903minusminus2)119879119873(1198892) (86)

where

1198891 = 1120590radic119879 (ln

11987811198782 + 119864119890minus119903119879+ (119903 minus + 2 minus 1199021 + 12059022 )119879) 1198892 = 1198891 minus 120590radic119879

120590 = radic12059021 minus 212058812059011205902 11987821198782 + 119864119890minus119903119879 + (11987821198782 + 119864119890minus119903119879)

2 12059022 = 11987821198782 + 119864119890minus119903119879 1199032 = 11987821198782 + 119864119890minus119903119879 1199022

(87)

The comparison of our proposed numerical method andthe analytical approximation (86) is presented in Table 1Analytical approximation is calculated for the same arraysof 1198781 and 1198782 using MATLAB-function cdf for calculat-ing cumulative distribution function that requires a lot ofcomputational resources for each point In the proposedfinite difference method (FDM) cdf is only used for theboundary conditions Last row in the Table 1 presents theCPU time for each method for the same sizes of array Forapproximation method cdf-time is 925018 sec It is the main

Table 1 Comparison proposed method (FDM) with analyticalapproximation (86)

1198782 1198781 FDM Approximation

100

400 2105978 2097261200 242895 234543100 01150 0012350 31179119890 minus 05 minus47741119890 minus 10

90

400 2200678 2201714200 298745 298724100 01691 0037250 52150119890 minus 05 minus19574119890 minus 09

CPU time (sec) 106188 929467

00400

100

300 500

200

200

300

1000100

400

15000

500

U(S

1S

2T

)

S 1S2

Figure 4 European spread put option with parameters (84)

part of computational time In the case of FDM cdf takes lessthan half of computational time - 48867 sec

Furthermore unsuitable negative values appear in ana-lytical approximation for small 1198781 and 1198782 while the proposedmethod preserves nonnegativity of the solution

In the next example we confirm that for the spreadput option case the removing technique is fruitful and theselection of the boundary conditions at the boundary of thenumerical domain is appropriated because the behaviour ofthe solution at the boundaries fits the solution at the interiorpoints as it is shown in Figure 4

Example 3 With the parameters in (84) of Example 1 butwith payoff (1198782+119864minus1198781)+ in Figure 4we show the simulation ofthe numerical solution obtained by the proposed numericalscheme

Finally in Example 4 we compare the numerical solutionobtained with our scheme (83) using LCP approach withother tested efficient methods presented in [28]

10 Abstract and Applied Analysis

Table 2 Comparing several methods for American option pricing

119864 Analytic FFT MC FDM0 8513201 8513079 8516613 85081982 7542296 7542242 7545496 75340644 6653060 6652976 6656364 6648792

Example 4 Let us consider the American spread call optionproblem with parameters

119879 = 11205901 = 011205902 = 02119903 = 011199021 = 0051199022 = 005120588 = 05

(88)

Table 2 shows the option price at fixed asset prices 1198781 = 96and 1198782 = 100 for different values of strike price 119864 evaluatedby one-dimensional integration analytic method (Analytic)Fast Fourier Transform (FFT) Monte Carlo (MC) methodand our proposed method The accuracy is competitive withother methods such as Fourier Transform with high numberof discretization (4096) andMonte Carlo with big number oftime steps (2000)

Another set of parameters can be considered in order tocompare it with data in [9]

119879 = 05119864 = 1001205901 = 0251205902 = 03119903 = 0031199021 = 0061199022 = 002120588 = 05

(89)

In [9] three methods are considered to evaluate thespread option price for data 74 and fixed values 1198781 = 100and 1198782 = 200 a numerical integration method (NIM) amethod of lines (MOL) and a Monte Carlo approach NIMused in [9] is based in the recursive integration techniquesdeveloped byHuang et al [29] InNIM the option is treated asa Bermudan that is an option that can be exercised at discretepoints of time and Simpsons rule is used For the experimentthe authors used 1024 spatial nodes see [8 9] for moredetails The MOL is implemented by a spatial discretizationof 1438 times 200 mesh points and a three-time level scheme isused for the integration in time [9] 50 000 simulations have

Table 3 CPU time in sec (first row) and absolute difference (secondrow) for different methods depending on number of time steps forfixed number of space steps for parameters (89)

119873120591 NIM MOL Monte Carlo FDM

50 296598 5920 47 970 265470003 0001 01735 00036

32 127069 5512 19 811 1791100035 0006 01148 00122

16 33493 51718 5 149 943200012 00236 0132 00371

8 10360 4901 1 350 478700105 0074 01316 00907

4 4117 4547 35575 281500364 02512 01351 02254

been used in the Monte Carlo method described in [30] Inour method FDM the step size discretization considered isℎ = 01 To compare FDM with the three previous methodsabsolute errors are computed in Table 3 with respect to areference value obtained by the Fourier space time-steppingapproach of Surkov [31] with large number of nodes 4096spatial nodes and 300 time steps From Table 3 we can statethat the proposed method has the same order of accuracy asMOL more efficient and requires less computational timethan Monte Carlo method It is slightly less efficient than theNIM but we prove that it possesses a very highly desirablequalitative behaviour in finance

Competing Interests

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

Acknowledgments

This work has been partially supported by the EuropeanUnion in the FP7- PEOPLE-2012-ITN Program under GrantAgreement no 304617 (FP7 Marie Curie Action ProjectMulti-ITN STRIKE-Novel Methods in ComputationalFinance) and the Ministerio de Economıa y CompetitividadSpanish Grant MTM2013-41765-P

References

[1] R Carmona and V Durrleman ldquoPricing and hedging spreadoptionsrdquo SIAM Review vol 45 no 4 pp 627ndash685 2003

[2] P P Boyle ldquoOptions a Monte Carlo approachrdquo Journal ofFinancial Economics vol 4 no 3 pp 323ndash338 1977

[3] C Joy P P Boyle and K S Tan ldquoQuasi-Monte Carlo methodsin numerical financerdquo Management Science vol 42 no 6 pp926ndash938 1996

[4] W-H Kao Y-D Lyuu and K-W Wen ldquoThe hexanomiallattice for pricingmulti-asset optionsrdquoAppliedMathematics andComputation vol 233 pp 463ndash479 2014

Abstract and Applied Analysis 11

[5] E Kirk and J Aron ldquoCorrelation in the energy marketsrdquo inManaging Energy Price Risk V Kaminski Ed pp 71ndash78 RiskBooks London UK 1995

[6] C Alexander andA Venkatramanan ldquoAnalytic approximationsfor spread optionsrdquo ICMACentre Discussion Papers in FinanceDP2007-11 ICMA Centre 2007

[7] N D Pearson ldquoAn efficient approach for pricing spreadoptionsrdquoThe Journal of Derivatives vol 3 no 1 pp 76ndash91 1995

[8] C Chiarella and J Ziveyi ldquoPricingAmerican options written ontwo underlying assetsrdquo Quantitative Finance vol 14 no 3 pp409ndash426 2014

[9] J Ziveyi The evaluation of early exercise exotic options [PhDthesis] University of Technology Sydney Australia 2011

[10] P Boyle ldquoA lattice framework for option pricing with two statevariablesrdquo Journal of Financial and Quantitative Analysis vol23 no 1 pp 1ndash12 1988

[11] M Rubinstein ldquoReturn to Ozrdquo RISK vol 7 no 11 pp 67ndash711994

[12] R Company L Jodar M Fakharany and M-C CasabanldquoRemoving the correlation term in option pricing Hestonmodel numerical analysis and computingrdquo Abstract andApplied Analysis vol 2013 Article ID 246724 11 pages 2013

[13] S Salmi J Toivanen and L von Sydow ldquoIterative methodsfor pricing american options under the bates modelrdquo ProcediaComputer Science vol 18 pp 1136ndash1144 2013

[14] J Toivanen ldquoA componentwise splitting method for pricingamerican options under the bates modelrdquo in Applied andNumerical Partial Differential Equations vol 15 of Computa-tional Methods in Applied Sciences pp 213ndash227 Springer 2010

[15] R Zvan P A Forsyth and K R Vetzal ldquoNegative coefficientsin two-factor option pricing modelsrdquo Journal of ComputationalFinance vol 7 no 1 pp 37ndash73 2003

[16] B During M Fournie and C Heuer ldquoHigh-order compactfinite difference schemes for option pricing in stochastic volatil-ity models on non-uniform gridsrdquo Journal of Computationaland Applied Mathematics vol 271 pp 247ndash266 2014

[17] K J in rsquot Hout and C Mishra ldquoStability of ADI schemes formultidimensional diffusion equations with mixed derivativetermsrdquoApplied NumericalMathematics vol 74 pp 83ndash94 2013

[18] C Chiarella B Kang G H Mayer and A Ziogas ldquoTheevaluation of American option prices under stochastic volatilityand jump-diffusion dynamics using the method of linesrdquoInternational Journal of Theoretical and Applied Finance vol 12no 3 pp 393ndash425 2009

[19] M Fakharany R Company and L Jodar ldquoPositive finite dif-ference schemes for a partial integro-differential option pricingmodelrdquo Applied Mathematics and Computation vol 249 pp320ndash332 2014

[20] P R Garabedian Partial Differential Equations AMS ChelseaPublishing Providence RI USA 1998

[21] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp SonsChichester UK 2006

[22] R U Seydel Tools for Computational Finance Springer 3rdedition 2006

[23] M Ehrhardt Discrete artificial boundary conditions [PhDthesis] Technische Universitat Berlin Berlin Germany 2001

[24] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[25] J C HullOptions Futures andOther Derivatives Prentice-HallUpper Saddle River NJ USA 2000

[26] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods Clarendon Press Oxford UK3rd edition 1985

[27] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[28] M A H Dempster and S S G Hong Spread Option Valuationand the Fast Fourier Transform WP 262000 Judge Institute ofManagement Studies University of Cambridge 2000

[29] J-Z Huang M G Subrahmanyam and G G Yu ldquoPricingand hedging american options a recursive integrationmethodrdquoReview of Financial Studies vol 9 no 1 pp 277ndash300 1996

[30] A Ibanez and F Zapatero ldquoMonte Carlo valuation of Americanoptions through computation of the optimal exercise frontierrdquoJournal of Financial and Quantitative Analysis vol 39 no 2 pp253ndash275 2004

[31] V Surkov ldquoParallel option pricing with fourier space time-stepping method on graphics processingrdquo in Proceedings of the1st International Workshop on Parallel and Distributed Comput-ing in Finance (PDCoF rsquo08) Miami Fla USA 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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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

Abstract and Applied Analysis 5

Boundary 119860119861 corresponds with the boundary for largevalues of 1198782 Condition (7) means that component 1198782 isconstant and the null first derivative is approximated by theforward difference (119908119899119894119894+1 minus 119908119899119894119894)|119898|ℎ = 0

119908119899119894119894 = 119908119899119894119894+1 1 le 119894 le 119873119909 1 le 119899 le 119873120591 (34)

4 Positivity Stability and Consistency

In this section we show that the proposed numerical scheme(24)ndash(27) presents suitable qualitative and computationalproperties such as consistency and conditional positivity andstability

In order to guarantee positivity of the numerical solutionlet us check the positivity of the coefficients 120572119894

From (25) and (27) it is easy to check that coefficients1205721 1205722 1205724 and 1205725 of the scheme (24) are positive under thecondition

ℎ lt min 12059011003816100381610038161003816119903 minus 1199021 minus 1205902121003816100381610038161003816 21003816100381610038161003816119898 [ (119903 minus 1199021 minus 120590212) 1205901 minus (119903 minus 1199022 minus 120590222) 1205902]1003816100381610038161003816

(35)

Coefficient 1205723 is positive if119896 lt 1198982ℎ2

2 (1198982 + 1) (36)

Under conditions (35) and (36) values in interior pointsof the numerical domain 119908119899+1119894119895 1 le 119894 le 119873119909 minus 1 119894 + 1 le119895 le 119873119910 + 119894 minus 1 calculated by the scheme (24) preservenonnegativity from the previous time level 119899 at all interior andboundary points 119908119899119894119895 ge 0 0 le 119894 le 119873119909 119894 le 119895 le 119873119910 + 119894

Let us pay attention now to the positivity of the numericalsolution at the boundary of the numerical domain Onthe boundary 119860119863 from (30) we have zero values On theboundary 119863119862 formula (31) is used preserving the positivitysince it is the transformed Black-Scholes formula through-out expression (16) Along the line 119860119861 we use Neumannboundary condition (34) Therefore the positivity at theneighbour interior point guarantees the positivity at theboundary Finally on the line 119861119862 boundary conditions aredetermined by formula (33) This is transformed expressionfor the asymptotic behaviour of the call option (6) that ispositive under the condition

1198781maxgt (1198782max

+ 119864) exp ((1199021 minus 119903) 120591119899) 1 le 119899 le 119873120591 (37)

Following the ideas of Kangro and Nicolaides [24] thecomputational domain has to be large enough to translate theboundary conditions therefore

1198781max

ge max (1198782max+ 119864) exp ((1199021 minus 119903) 119879) 3 (1198782max

+ 119864) (38)

Therefore choosing 1198781maxaccording to (38) for the trans-

formed problem119908119899119873119909 119895 ge 0 for any119873119909+1 le 119895 le 119873119909+119873119910 1 le119899 le 119873120591In summary the nonnegativity of the numerical solution

follows from the nonnegativity of the initial values andpositivity preservation under the scheme and boundaryconditions

As authors manage several stability criteria in literaturewe recall the concept of stability used in this paper

Definition 1 Let 119908119899 isin R(119873119909+1)times(119873119910+1) be the matrix involvingthe numerical solution 119908119899119894119895 of PDE problem (19) computedby scheme (24) (29) (30) (31) (33) and (34)

119908119899 = (119908119899119894119895)0le119894le119873119909 119894le119895le119873119910+119894 0 le 119899 le 119873120591 (39)

The numerical scheme (24) (29) (30) (31) (33) and (34) issaid to be uniformly sdot infin-stable in the domain [1198881 1198891] times[1198882 1198892]times[0 119879] if for every partitionwith 119896ℎ as defined abovecondition 10038171003817100381710038171199081198991003817100381710038171003817infin le 119860 0 le 119899 le 119873120591 (40)

holds true where119860 is independent of ℎ 119896 When the schemeis uniformly stable under a condition imposed to the stepsize discretization then the scheme is said to be conditionallyuniformly stable

Let us find the maximum value of the 119908119899119894119895 with respect to119894 119895 for each fixed 119899 For 119899 = 0 the values119908119899119894119895 are defined by theinitial condition (29) Let us consider the following function

119891 (119909 119910) = exp(1205901 119909) minus exp (minus1205902 (119910 minus 119898119909)) minus 119864 (41)

and find the maximum value on the domain [1198881 1198891]times [1198882 1198892]The first derivatives are120597119891120597119909 = 1205901 exp(1205901 119909) + 1205902119898 exp (minus1205902 (119910 minus 119898119909)) gt 0120597119891120597119910 = 1205902exp (minus1205902 (119910 minus 119898119909)) gt 0

(42)

Analysing (42) one concludes that the function 119891(119909 119910) isincreasing with respect to 119909 and 119910 Therefore the maximumvalue is reached at the point (1198891 1198892)

max(119909119910)isin[1198881 1198891]times[1198882 1198892]

119891 (119909 119910) = 119891 (1198891 1198892)= exp(1205901 1198891) minus exp (minus1205902 (1198892 minus 1198981198891)) minus 119864 = 119862= const

(43)

For the next time levels 119899 ge 1 let us consider thenumerical scheme (24) Under conditions (35) and (36) thecoefficients 120572119894 gt 0 119894 = 1 5 Moreover it is easy to checkthat

5sum119894=1

120572119894 = 1 (44)

6 Abstract and Applied Analysis

Therefore the values in interior points at the (119899+1)th level119899 ge 0 can be bounded by the following

119908119899+1119894119895 le max 119908119899119894minus1119895 119908119899119894+1119895 119908119899119894119895 119908119899119894119895minus1 119908119899119894119895+1le max119894119895119908119899119894119895 (45)

This maximum can be reached on the boundary There-fore let us study the boundary conditions From (30) one getsthat

max119860119863

119908119899+1119894119895 = 0 (46)

The values on the boundary A119861 are equal to the interiorpoints therefore the inequality (45) can be applied

The values on the boundary 119861119862 are increasing withrespect to the index 119895 and reach the maximum at the point(119873119909 119873119909 + 119873119910 minus 1) From the other side this value can bebounded by the value at the point (119873119909 119873119909 + 119873119910) that iscalculated by formula (31) From the theory of option pricingthe price of European call is increasing with respect to theasset price 119878 andmoreover it is always greater than its asymp-totic function (see [25] p 268 formula (131)) The proposedtransformation preserves monotonicity therefore

119908119899+1119873119909119873119909+119873119910 = 119865119899+1119873119909 119873(120572119899+11119873119909) minus 119864119873(120572119899+12119873119909)ge 119890((1205901)119909119873119909+(119903minus1199021)120591119899+1)minus (119890(1205902(119898119909119873119909minus119910119873119909+119873119910minus1)) + 119864)

= 119908119899+1119873119909 119873119909+119873119910minus1(47)

In summary we can conclude that

max119894119895119908119899+1119894119895 = 119908119899+1119873119909119873119909+119873119910 (48)

This value is transformed call option price for the assetprice 1198781max

that is the solution of the 1D Black-Scholesequation at the fixed point European call option gives a rightto the option holder to purchase one share stock for a certainprice The option itself cannot be worth more than the stockIn other words

119880 (1198781 0 120591) = 119890minus11990211205911198781119873(1198891) minus 119890minus119903120591119864119873 (1198892)le 119890minus1199021205911198781119873(1198891) le 119890minus1199021205911198781

119908119899+1119873119909119873119909+119873119910 le 119890119903120591119899+1119890minus1199021120591119899+11198781maxle 119890|119903minus1199021|1198791198781max

0 le 119899 le 119873120591 minus 1

(49)

Then the following result can be stated

Theorem 2 With previous notations under conditions (35)and (36) numerical scheme (24) (29) (30) (31) (33) and(34) is conditionally uniformly sdot infin-stable with upper bound119860 = 119890|119903minus1199021|1198791198781max

Now consistency of the numerical scheme (24) with PDE(19) will be studied [26]

Let 119865(119908119899119894119895) = 0 be approximating difference equation(24) The scheme (24) is consistent with (19) if the localtruncation error 119879119899119894119895 satisfies119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) 997888rarr 0 as ℎ 997888rarr 0 119896 997888rarr 0 (50)

where119882119899119894119895 denotes the theoretical solution of (19) evaluatedat the point (119909119894 119910119895 120591119899) and 119871 is the operator119871 (119882) = 120597119882120597120591 minus

2

2 (12059721198821205971199092 + 120597

21198821205971199102 )minus 1205901 (119903 minus 1199021 minus

120590212 ) 120597119882120597119909minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 )) 120597119882120597119910

(51)

Assuming that theoretical solution119882of the PDEproblemis four times continuously differentiable with respect to 119909 and119910 and twice with respect to 120591 and using Taylor expansionabout (119909119894 119910119895 120591119899) it is easy to check that

119882119899+1119894119895 minus119882119899119894minus1119895119896 = 120597119882120597120591 (119909119894 119910119895 120591119899) + 119896119864119899119894119895 (1) (52)

where

119864119899119894119895 (1) = 12 12059721198821205971205912 (119909119894 119910119895 120575) 119899119896 lt 120575 lt (119899 + 1) 119896 (53)

10038161003816100381610038161003816119864119899119894119895 (1)10038161003816100381610038161003816 le 12 10038161003816100381610038161003816119863119899119894119895 (1)10038161003816100381610038161003816max

= 12max10038161003816100381610038161003816100381610038161003816100381612059721198821205971205912 (119909119894 119910119895 120575)

100381610038161003816100381610038161003816100381610038161003816 0 lt 120575 lt 119879 (54)

For the spatial discretization one gets

119882119899119894+1119895 minus119882119899119894minus11198952ℎ = 120597119882120597119909 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (2) (55)

where

119864119899119894119895 (2) = 16 12059731198821205971199093 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (56)

10038161003816100381610038161003816119864119899119894119895 (2)10038161003816100381610038161003816 le 16 10038161003816100381610038161003816119863119899119894119895 (2)10038161003816100381610038161003816max

= 16 max10038161003816100381610038161003816100381610038161003816100381612059731198821205971199093 (120585 119910119895 120591119899)

100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891 (57)

Analogously

119882119899119894119895+1 minus119882119899119894119895minus12 |119898| ℎ = 120597119882120597119910 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (3) (58)

Abstract and Applied Analysis 7

where

119864119899119894119895 (2) = 11989826 12059731198821205971199103 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (59)

thus10038161003816100381610038161003816119864119899119894119895 (3)10038161003816100381610038161003816le 11989826 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(60)

The discretization of the second spatial derivatives is asfollows

119882119899119894+1119895 minus 2119882119899119894119895 +119882119899119894minus1119895ℎ2= 12059721198821205971199092 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (4)

(61)

where

119864119899119894119895 (4) = 112 12059741198821205971199094 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (62)

10038161003816100381610038161003816119864119899119894119895 (4)10038161003816100381610038161003816le 112 max100381610038161003816100381610038161003816100381610038161003816

12059741198821205971199094 (120585 119910119895 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891

(63)

Analogously

119908119899119894119895+1 minus 2119908119899119894119895 + 119908119899119894119895minus11198982ℎ2= 12059721198821205971199102 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (5)

(64)

where

119864119899119894119895 (5) = 119898412 12059741198821205971199104 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (65)

10038161003816100381610038161003816119864119899119894119895 (5)10038161003816100381610038161003816le 119898412 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(66)

Then the local truncation error takes the following form

119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) = 119896119864119899119894119895 (1) minus 2

2sdot ℎ4 (119864119899119894119895 (4) + 119864119899119894119895 (5)) minus 1205901 (119903 minus 1199021 minus

120590212 )sdot ℎ2119864119899119894119895 (2)minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 ))

sdot ℎ2119864119899119894119895 (3)

(67)

From (54) (57) (60) (63) and (66) one gets1003816100381610038161003816100381611987911989911989411989510038161003816100381610038161003816 = 119874 (119896) + 119874 (ℎ2) (68)

Theorem 3 Assuming that the solution of the PDE (19) admitstwo times continuous partial derivativewith respect to time andup to order four with respect to each of space directions solutionthe numerical solution computed by scheme (24) is consistentwith (19)

5 European Spread Put Option Case

For the sake of brevity and in order to avoid repetition ofproofs we summarize in this section the results proved inprevious sections for the call option case

Let us consider a European spread put option with payoff

119880 (1198781 1198782 0) = ((1198782 + 119864) minus 1198781)+ 119864 gt 0 (69)

Equation (1) and all the transformed versions of it do notchange for the put option Like in the study of the previouscall option case due to different nature of the put option weneed to establish the boundary conditions and translate themto the boundary of the numerical domain

(1) If 1198782 = 0 the problem is reduced to the European putoption on asset 1198781 and strike 119864 Therefore the optionvalue on this boundary is

119880 (1198781 0 120591) = 119864119890minus119903120591119873(minus1206012) minus 11987811198901199021120591119873(minus1206011) (70)

where 1206011 and 1206012 are defined by (5)(2) If 1198781 = 0 we consider 119880(0 1198782 120591) as a put option price

on asset 1198781 = 0 with the strike 1198782 + 119864119880 (0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 (71)

(3) For large values of 1198782 we suppose that the slope of thecurve tends to 1 that is

1205971198801205971198782 (1198781 1198782 997888rarr infin 120591) = 119890minus1199022120591 (72)

(4) For 1198781 ≫ 1198782 + 119864 we consider 119880(1198781 1198782 120591) as a putoption price with large values of 1198781 and therefore

119880(1198781 1198782 120591) = 0 (73)

Under transformations (8) and (16) the boundary con-ditions (70)ndash(73) are translated to the rhomboid numericaldomain 119860119861119862119863 as follows

119863119862 119908119899119894119873119910+119894 = 119864119873 (minus1205722119894)+ 119890(1205901)119909119894minus1199021120591119899119873(minus1205721119894)

(74)

119860119863 1199081198990119895 = 119864 + 119890minus1205902(119910119895minus1198981199090) (75)

119860119861 119908119899119894119894 = 119908119899119894119894+1 + |119898| ℎ1205902119890minus1205902(119910119894+1minus119898119909119894)minus119903120591 (76)

119861119862 119908119899119873119909+1119873119909+119895 = 0 (77)

8 Abstract and Applied Analysis

Positivity As the numerical scheme (24) for the interiorpoints is the same and the new boundary conditions (74)ndash(77) are nonnegative then under conditions (35) and (36)the numerical solution for the European spread put optionis nonnegative

Stability Stability of the scheme is preservedwith newbound-ary conditions (74)ndash(77) because analogously to the calloption case the maximum value of the transformed function119882(119909 119910 120591) is reached at a corner (in this case (1198881 1198892) wherewe use the boundary condition (75)) At the correspondingpoint of the original domain we use boundary condition (71)In the original coordinates it can be bounded as

119880(0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 le 1198782max+ 119864 = 4119864 = const (78)

Therefore the transformed function also can be boundedas

max119894119895119908119899119894119895 = 4119864119890119903119879 (79)

6 American Spread Options

In the case of American type option the holder has the rightto exercise option at any moment before expiring It leadsto a free boundary problem [27] In order to simplify thecomputational procedure this problem can be considered asa linear complementarity problem (LCP)

(120597119880120597120591 minus 119871119880) (119880 minus 119892) = 0120597119880120597120591 minus 119871119880 ge 0

119880 minus 119892 (1198781 1198782) ge 0(80)

where 119871 represents the spatial differential operator of (1)If we rewrite (19) in the form

120597119882120597120591 minusL119882 = 0 (81)

then under transformation (16) LCP (80) has the followingform

(120597119882120597120591 minusL119882)(119882 minus 119891 (119909 119910)) = 0120597119882120597120591 minusL119882 ge 0119882 minus 119891 (119909 119910) ge 0

(82)

whereL is the right-hand side of (19) and 119891(119909 119910) is given by(41)

Numerical solution for problem (82) can be easilyobtained by a small modification of the calculating proceduredescribed above in Sections 3 and 4 On each time level

119899119894119895 = max 119908119899119894119895 119891 (119909119894 119910119895) ge 0 (83)

where 119908119899119894119895 is defined by the finite difference equation (24)

01500

500

4001000

1000

300

1500

200500100

0 0

U(S

1S

2T

)

S1 S2

Figure 2 Parameters ℎ and 119896 satisfy (85)

7 Numerical Examples

In this section we illustrate numerical results for bothEuropean and American spread options

In Example 1 we show that the stability conditions (35)and (36) for the European spread option can not be disre-garded

Example 1 We tested the algorithm for the European spreadcall option pricing problem with the parameters

119879 = 1119864 = 1001205901 = 021205902 = 01119903 = 011199021 = 0011199022 = 005120588 = 01

(84)

and taking 1198781 and 1198782 large enough in particular 1198782max= 4001198781max

= 3(1198782max+ 119864) = 1500

For parameters (84) the stability conditions (35) and (36)on space and time steps are as follows

ℎ lt min 28428 237358 = 28428 119896 lt 00112 (85)

These conditions are crucial for the qualitative propertiessuch as positivity and stability of the scheme In Figure 2 thenumerical solution with ℎ and 119896 satisfying (85) is presentedOn the other hand in Figure 3 there is a numerical simulationwith time step 119896 = 0012 gt 00112 The solution is unstableand it has negative values

Example 2 illustrates the computational time aswell as thecomparison with analytical approximation provided by [6]

Abstract and Applied Analysis 9

1500

0500

400

1000

1000

1500

300

20002500

200500100

0 0

minus500

U(S

1S

2T

)

S1

S2

Figure 3 Stability condition is broken

Example 2 Consider the European spread option with thedata of Example 1 (84) and choosing step sizes satisfying thestability condition

The price of European spread option can be expressedas the price of a compound exchange option (see [6])The known derived analytical approximation for the spreadoption reads

119880 (1198781 1198782 119879) = 1198781119890minus1199021119879119873(1198891)minus (119864119890minus119903119879 + 1198782) 119890minus(119903minusminus2)119879119873(1198892) (86)

where

1198891 = 1120590radic119879 (ln

11987811198782 + 119864119890minus119903119879+ (119903 minus + 2 minus 1199021 + 12059022 )119879) 1198892 = 1198891 minus 120590radic119879

120590 = radic12059021 minus 212058812059011205902 11987821198782 + 119864119890minus119903119879 + (11987821198782 + 119864119890minus119903119879)

2 12059022 = 11987821198782 + 119864119890minus119903119879 1199032 = 11987821198782 + 119864119890minus119903119879 1199022

(87)

The comparison of our proposed numerical method andthe analytical approximation (86) is presented in Table 1Analytical approximation is calculated for the same arraysof 1198781 and 1198782 using MATLAB-function cdf for calculat-ing cumulative distribution function that requires a lot ofcomputational resources for each point In the proposedfinite difference method (FDM) cdf is only used for theboundary conditions Last row in the Table 1 presents theCPU time for each method for the same sizes of array Forapproximation method cdf-time is 925018 sec It is the main

Table 1 Comparison proposed method (FDM) with analyticalapproximation (86)

1198782 1198781 FDM Approximation

100

400 2105978 2097261200 242895 234543100 01150 0012350 31179119890 minus 05 minus47741119890 minus 10

90

400 2200678 2201714200 298745 298724100 01691 0037250 52150119890 minus 05 minus19574119890 minus 09

CPU time (sec) 106188 929467

00400

100

300 500

200

200

300

1000100

400

15000

500

U(S

1S

2T

)

S 1S2

Figure 4 European spread put option with parameters (84)

part of computational time In the case of FDM cdf takes lessthan half of computational time - 48867 sec

Furthermore unsuitable negative values appear in ana-lytical approximation for small 1198781 and 1198782 while the proposedmethod preserves nonnegativity of the solution

In the next example we confirm that for the spreadput option case the removing technique is fruitful and theselection of the boundary conditions at the boundary of thenumerical domain is appropriated because the behaviour ofthe solution at the boundaries fits the solution at the interiorpoints as it is shown in Figure 4

Example 3 With the parameters in (84) of Example 1 butwith payoff (1198782+119864minus1198781)+ in Figure 4we show the simulation ofthe numerical solution obtained by the proposed numericalscheme

Finally in Example 4 we compare the numerical solutionobtained with our scheme (83) using LCP approach withother tested efficient methods presented in [28]

10 Abstract and Applied Analysis

Table 2 Comparing several methods for American option pricing

119864 Analytic FFT MC FDM0 8513201 8513079 8516613 85081982 7542296 7542242 7545496 75340644 6653060 6652976 6656364 6648792

Example 4 Let us consider the American spread call optionproblem with parameters

119879 = 11205901 = 011205902 = 02119903 = 011199021 = 0051199022 = 005120588 = 05

(88)

Table 2 shows the option price at fixed asset prices 1198781 = 96and 1198782 = 100 for different values of strike price 119864 evaluatedby one-dimensional integration analytic method (Analytic)Fast Fourier Transform (FFT) Monte Carlo (MC) methodand our proposed method The accuracy is competitive withother methods such as Fourier Transform with high numberof discretization (4096) andMonte Carlo with big number oftime steps (2000)

Another set of parameters can be considered in order tocompare it with data in [9]

119879 = 05119864 = 1001205901 = 0251205902 = 03119903 = 0031199021 = 0061199022 = 002120588 = 05

(89)

In [9] three methods are considered to evaluate thespread option price for data 74 and fixed values 1198781 = 100and 1198782 = 200 a numerical integration method (NIM) amethod of lines (MOL) and a Monte Carlo approach NIMused in [9] is based in the recursive integration techniquesdeveloped byHuang et al [29] InNIM the option is treated asa Bermudan that is an option that can be exercised at discretepoints of time and Simpsons rule is used For the experimentthe authors used 1024 spatial nodes see [8 9] for moredetails The MOL is implemented by a spatial discretizationof 1438 times 200 mesh points and a three-time level scheme isused for the integration in time [9] 50 000 simulations have

Table 3 CPU time in sec (first row) and absolute difference (secondrow) for different methods depending on number of time steps forfixed number of space steps for parameters (89)

119873120591 NIM MOL Monte Carlo FDM

50 296598 5920 47 970 265470003 0001 01735 00036

32 127069 5512 19 811 1791100035 0006 01148 00122

16 33493 51718 5 149 943200012 00236 0132 00371

8 10360 4901 1 350 478700105 0074 01316 00907

4 4117 4547 35575 281500364 02512 01351 02254

been used in the Monte Carlo method described in [30] Inour method FDM the step size discretization considered isℎ = 01 To compare FDM with the three previous methodsabsolute errors are computed in Table 3 with respect to areference value obtained by the Fourier space time-steppingapproach of Surkov [31] with large number of nodes 4096spatial nodes and 300 time steps From Table 3 we can statethat the proposed method has the same order of accuracy asMOL more efficient and requires less computational timethan Monte Carlo method It is slightly less efficient than theNIM but we prove that it possesses a very highly desirablequalitative behaviour in finance

Competing Interests

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

Acknowledgments

This work has been partially supported by the EuropeanUnion in the FP7- PEOPLE-2012-ITN Program under GrantAgreement no 304617 (FP7 Marie Curie Action ProjectMulti-ITN STRIKE-Novel Methods in ComputationalFinance) and the Ministerio de Economıa y CompetitividadSpanish Grant MTM2013-41765-P

References

[1] R Carmona and V Durrleman ldquoPricing and hedging spreadoptionsrdquo SIAM Review vol 45 no 4 pp 627ndash685 2003

[2] P P Boyle ldquoOptions a Monte Carlo approachrdquo Journal ofFinancial Economics vol 4 no 3 pp 323ndash338 1977

[3] C Joy P P Boyle and K S Tan ldquoQuasi-Monte Carlo methodsin numerical financerdquo Management Science vol 42 no 6 pp926ndash938 1996

[4] W-H Kao Y-D Lyuu and K-W Wen ldquoThe hexanomiallattice for pricingmulti-asset optionsrdquoAppliedMathematics andComputation vol 233 pp 463ndash479 2014

Abstract and Applied Analysis 11

[5] E Kirk and J Aron ldquoCorrelation in the energy marketsrdquo inManaging Energy Price Risk V Kaminski Ed pp 71ndash78 RiskBooks London UK 1995

[6] C Alexander andA Venkatramanan ldquoAnalytic approximationsfor spread optionsrdquo ICMACentre Discussion Papers in FinanceDP2007-11 ICMA Centre 2007

[7] N D Pearson ldquoAn efficient approach for pricing spreadoptionsrdquoThe Journal of Derivatives vol 3 no 1 pp 76ndash91 1995

[8] C Chiarella and J Ziveyi ldquoPricingAmerican options written ontwo underlying assetsrdquo Quantitative Finance vol 14 no 3 pp409ndash426 2014

[9] J Ziveyi The evaluation of early exercise exotic options [PhDthesis] University of Technology Sydney Australia 2011

[10] P Boyle ldquoA lattice framework for option pricing with two statevariablesrdquo Journal of Financial and Quantitative Analysis vol23 no 1 pp 1ndash12 1988

[11] M Rubinstein ldquoReturn to Ozrdquo RISK vol 7 no 11 pp 67ndash711994

[12] R Company L Jodar M Fakharany and M-C CasabanldquoRemoving the correlation term in option pricing Hestonmodel numerical analysis and computingrdquo Abstract andApplied Analysis vol 2013 Article ID 246724 11 pages 2013

[13] S Salmi J Toivanen and L von Sydow ldquoIterative methodsfor pricing american options under the bates modelrdquo ProcediaComputer Science vol 18 pp 1136ndash1144 2013

[14] J Toivanen ldquoA componentwise splitting method for pricingamerican options under the bates modelrdquo in Applied andNumerical Partial Differential Equations vol 15 of Computa-tional Methods in Applied Sciences pp 213ndash227 Springer 2010

[15] R Zvan P A Forsyth and K R Vetzal ldquoNegative coefficientsin two-factor option pricing modelsrdquo Journal of ComputationalFinance vol 7 no 1 pp 37ndash73 2003

[16] B During M Fournie and C Heuer ldquoHigh-order compactfinite difference schemes for option pricing in stochastic volatil-ity models on non-uniform gridsrdquo Journal of Computationaland Applied Mathematics vol 271 pp 247ndash266 2014

[17] K J in rsquot Hout and C Mishra ldquoStability of ADI schemes formultidimensional diffusion equations with mixed derivativetermsrdquoApplied NumericalMathematics vol 74 pp 83ndash94 2013

[18] C Chiarella B Kang G H Mayer and A Ziogas ldquoTheevaluation of American option prices under stochastic volatilityand jump-diffusion dynamics using the method of linesrdquoInternational Journal of Theoretical and Applied Finance vol 12no 3 pp 393ndash425 2009

[19] M Fakharany R Company and L Jodar ldquoPositive finite dif-ference schemes for a partial integro-differential option pricingmodelrdquo Applied Mathematics and Computation vol 249 pp320ndash332 2014

[20] P R Garabedian Partial Differential Equations AMS ChelseaPublishing Providence RI USA 1998

[21] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp SonsChichester UK 2006

[22] R U Seydel Tools for Computational Finance Springer 3rdedition 2006

[23] M Ehrhardt Discrete artificial boundary conditions [PhDthesis] Technische Universitat Berlin Berlin Germany 2001

[24] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[25] J C HullOptions Futures andOther Derivatives Prentice-HallUpper Saddle River NJ USA 2000

[26] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods Clarendon Press Oxford UK3rd edition 1985

[27] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[28] M A H Dempster and S S G Hong Spread Option Valuationand the Fast Fourier Transform WP 262000 Judge Institute ofManagement Studies University of Cambridge 2000

[29] J-Z Huang M G Subrahmanyam and G G Yu ldquoPricingand hedging american options a recursive integrationmethodrdquoReview of Financial Studies vol 9 no 1 pp 277ndash300 1996

[30] A Ibanez and F Zapatero ldquoMonte Carlo valuation of Americanoptions through computation of the optimal exercise frontierrdquoJournal of Financial and Quantitative Analysis vol 39 no 2 pp253ndash275 2004

[31] V Surkov ldquoParallel option pricing with fourier space time-stepping method on graphics processingrdquo in Proceedings of the1st International Workshop on Parallel and Distributed Comput-ing in Finance (PDCoF rsquo08) Miami Fla USA 2008

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Therefore the values in interior points at the (119899+1)th level119899 ge 0 can be bounded by the following

119908119899+1119894119895 le max 119908119899119894minus1119895 119908119899119894+1119895 119908119899119894119895 119908119899119894119895minus1 119908119899119894119895+1le max119894119895119908119899119894119895 (45)

This maximum can be reached on the boundary There-fore let us study the boundary conditions From (30) one getsthat

max119860119863

119908119899+1119894119895 = 0 (46)

The values on the boundary A119861 are equal to the interiorpoints therefore the inequality (45) can be applied

The values on the boundary 119861119862 are increasing withrespect to the index 119895 and reach the maximum at the point(119873119909 119873119909 + 119873119910 minus 1) From the other side this value can bebounded by the value at the point (119873119909 119873119909 + 119873119910) that iscalculated by formula (31) From the theory of option pricingthe price of European call is increasing with respect to theasset price 119878 andmoreover it is always greater than its asymp-totic function (see [25] p 268 formula (131)) The proposedtransformation preserves monotonicity therefore

119908119899+1119873119909119873119909+119873119910 = 119865119899+1119873119909 119873(120572119899+11119873119909) minus 119864119873(120572119899+12119873119909)ge 119890((1205901)119909119873119909+(119903minus1199021)120591119899+1)minus (119890(1205902(119898119909119873119909minus119910119873119909+119873119910minus1)) + 119864)

= 119908119899+1119873119909 119873119909+119873119910minus1(47)

In summary we can conclude that

max119894119895119908119899+1119894119895 = 119908119899+1119873119909119873119909+119873119910 (48)

This value is transformed call option price for the assetprice 1198781max

that is the solution of the 1D Black-Scholesequation at the fixed point European call option gives a rightto the option holder to purchase one share stock for a certainprice The option itself cannot be worth more than the stockIn other words

119880 (1198781 0 120591) = 119890minus11990211205911198781119873(1198891) minus 119890minus119903120591119864119873 (1198892)le 119890minus1199021205911198781119873(1198891) le 119890minus1199021205911198781

119908119899+1119873119909119873119909+119873119910 le 119890119903120591119899+1119890minus1199021120591119899+11198781maxle 119890|119903minus1199021|1198791198781max

0 le 119899 le 119873120591 minus 1

(49)

Then the following result can be stated

Theorem 2 With previous notations under conditions (35)and (36) numerical scheme (24) (29) (30) (31) (33) and(34) is conditionally uniformly sdot infin-stable with upper bound119860 = 119890|119903minus1199021|1198791198781max

Now consistency of the numerical scheme (24) with PDE(19) will be studied [26]

Let 119865(119908119899119894119895) = 0 be approximating difference equation(24) The scheme (24) is consistent with (19) if the localtruncation error 119879119899119894119895 satisfies119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) 997888rarr 0 as ℎ 997888rarr 0 119896 997888rarr 0 (50)

where119882119899119894119895 denotes the theoretical solution of (19) evaluatedat the point (119909119894 119910119895 120591119899) and 119871 is the operator119871 (119882) = 120597119882120597120591 minus

2

2 (12059721198821205971199092 + 120597

21198821205971199102 )minus 1205901 (119903 minus 1199021 minus

120590212 ) 120597119882120597119909minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 )) 120597119882120597119910

(51)

Assuming that theoretical solution119882of the PDEproblemis four times continuously differentiable with respect to 119909 and119910 and twice with respect to 120591 and using Taylor expansionabout (119909119894 119910119895 120591119899) it is easy to check that

119882119899+1119894119895 minus119882119899119894minus1119895119896 = 120597119882120597120591 (119909119894 119910119895 120591119899) + 119896119864119899119894119895 (1) (52)

where

119864119899119894119895 (1) = 12 12059721198821205971205912 (119909119894 119910119895 120575) 119899119896 lt 120575 lt (119899 + 1) 119896 (53)

10038161003816100381610038161003816119864119899119894119895 (1)10038161003816100381610038161003816 le 12 10038161003816100381610038161003816119863119899119894119895 (1)10038161003816100381610038161003816max

= 12max10038161003816100381610038161003816100381610038161003816100381612059721198821205971205912 (119909119894 119910119895 120575)

100381610038161003816100381610038161003816100381610038161003816 0 lt 120575 lt 119879 (54)

For the spatial discretization one gets

119882119899119894+1119895 minus119882119899119894minus11198952ℎ = 120597119882120597119909 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (2) (55)

where

119864119899119894119895 (2) = 16 12059731198821205971199093 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (56)

10038161003816100381610038161003816119864119899119894119895 (2)10038161003816100381610038161003816 le 16 10038161003816100381610038161003816119863119899119894119895 (2)10038161003816100381610038161003816max

= 16 max10038161003816100381610038161003816100381610038161003816100381612059731198821205971199093 (120585 119910119895 120591119899)

100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891 (57)

Analogously

119882119899119894119895+1 minus119882119899119894119895minus12 |119898| ℎ = 120597119882120597119910 (119909119894 119910119895 120591119899) + ℎ2119864119899119894119895 (3) (58)

Abstract and Applied Analysis 7

where

119864119899119894119895 (2) = 11989826 12059731198821205971199103 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (59)

thus10038161003816100381610038161003816119864119899119894119895 (3)10038161003816100381610038161003816le 11989826 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(60)

The discretization of the second spatial derivatives is asfollows

119882119899119894+1119895 minus 2119882119899119894119895 +119882119899119894minus1119895ℎ2= 12059721198821205971199092 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (4)

(61)

where

119864119899119894119895 (4) = 112 12059741198821205971199094 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (62)

10038161003816100381610038161003816119864119899119894119895 (4)10038161003816100381610038161003816le 112 max100381610038161003816100381610038161003816100381610038161003816

12059741198821205971199094 (120585 119910119895 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891

(63)

Analogously

119908119899119894119895+1 minus 2119908119899119894119895 + 119908119899119894119895minus11198982ℎ2= 12059721198821205971199102 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (5)

(64)

where

119864119899119894119895 (5) = 119898412 12059741198821205971199104 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (65)

10038161003816100381610038161003816119864119899119894119895 (5)10038161003816100381610038161003816le 119898412 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(66)

Then the local truncation error takes the following form

119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) = 119896119864119899119894119895 (1) minus 2

2sdot ℎ4 (119864119899119894119895 (4) + 119864119899119894119895 (5)) minus 1205901 (119903 minus 1199021 minus

120590212 )sdot ℎ2119864119899119894119895 (2)minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 ))

sdot ℎ2119864119899119894119895 (3)

(67)

From (54) (57) (60) (63) and (66) one gets1003816100381610038161003816100381611987911989911989411989510038161003816100381610038161003816 = 119874 (119896) + 119874 (ℎ2) (68)

Theorem 3 Assuming that the solution of the PDE (19) admitstwo times continuous partial derivativewith respect to time andup to order four with respect to each of space directions solutionthe numerical solution computed by scheme (24) is consistentwith (19)

5 European Spread Put Option Case

For the sake of brevity and in order to avoid repetition ofproofs we summarize in this section the results proved inprevious sections for the call option case

Let us consider a European spread put option with payoff

119880 (1198781 1198782 0) = ((1198782 + 119864) minus 1198781)+ 119864 gt 0 (69)

Equation (1) and all the transformed versions of it do notchange for the put option Like in the study of the previouscall option case due to different nature of the put option weneed to establish the boundary conditions and translate themto the boundary of the numerical domain

(1) If 1198782 = 0 the problem is reduced to the European putoption on asset 1198781 and strike 119864 Therefore the optionvalue on this boundary is

119880 (1198781 0 120591) = 119864119890minus119903120591119873(minus1206012) minus 11987811198901199021120591119873(minus1206011) (70)

where 1206011 and 1206012 are defined by (5)(2) If 1198781 = 0 we consider 119880(0 1198782 120591) as a put option price

on asset 1198781 = 0 with the strike 1198782 + 119864119880 (0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 (71)

(3) For large values of 1198782 we suppose that the slope of thecurve tends to 1 that is

1205971198801205971198782 (1198781 1198782 997888rarr infin 120591) = 119890minus1199022120591 (72)

(4) For 1198781 ≫ 1198782 + 119864 we consider 119880(1198781 1198782 120591) as a putoption price with large values of 1198781 and therefore

119880(1198781 1198782 120591) = 0 (73)

Under transformations (8) and (16) the boundary con-ditions (70)ndash(73) are translated to the rhomboid numericaldomain 119860119861119862119863 as follows

119863119862 119908119899119894119873119910+119894 = 119864119873 (minus1205722119894)+ 119890(1205901)119909119894minus1199021120591119899119873(minus1205721119894)

(74)

119860119863 1199081198990119895 = 119864 + 119890minus1205902(119910119895minus1198981199090) (75)

119860119861 119908119899119894119894 = 119908119899119894119894+1 + |119898| ℎ1205902119890minus1205902(119910119894+1minus119898119909119894)minus119903120591 (76)

119861119862 119908119899119873119909+1119873119909+119895 = 0 (77)

8 Abstract and Applied Analysis

Positivity As the numerical scheme (24) for the interiorpoints is the same and the new boundary conditions (74)ndash(77) are nonnegative then under conditions (35) and (36)the numerical solution for the European spread put optionis nonnegative

Stability Stability of the scheme is preservedwith newbound-ary conditions (74)ndash(77) because analogously to the calloption case the maximum value of the transformed function119882(119909 119910 120591) is reached at a corner (in this case (1198881 1198892) wherewe use the boundary condition (75)) At the correspondingpoint of the original domain we use boundary condition (71)In the original coordinates it can be bounded as

119880(0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 le 1198782max+ 119864 = 4119864 = const (78)

Therefore the transformed function also can be boundedas

max119894119895119908119899119894119895 = 4119864119890119903119879 (79)

6 American Spread Options

In the case of American type option the holder has the rightto exercise option at any moment before expiring It leadsto a free boundary problem [27] In order to simplify thecomputational procedure this problem can be considered asa linear complementarity problem (LCP)

(120597119880120597120591 minus 119871119880) (119880 minus 119892) = 0120597119880120597120591 minus 119871119880 ge 0

119880 minus 119892 (1198781 1198782) ge 0(80)

where 119871 represents the spatial differential operator of (1)If we rewrite (19) in the form

120597119882120597120591 minusL119882 = 0 (81)

then under transformation (16) LCP (80) has the followingform

(120597119882120597120591 minusL119882)(119882 minus 119891 (119909 119910)) = 0120597119882120597120591 minusL119882 ge 0119882 minus 119891 (119909 119910) ge 0

(82)

whereL is the right-hand side of (19) and 119891(119909 119910) is given by(41)

Numerical solution for problem (82) can be easilyobtained by a small modification of the calculating proceduredescribed above in Sections 3 and 4 On each time level

119899119894119895 = max 119908119899119894119895 119891 (119909119894 119910119895) ge 0 (83)

where 119908119899119894119895 is defined by the finite difference equation (24)

01500

500

4001000

1000

300

1500

200500100

0 0

U(S

1S

2T

)

S1 S2

Figure 2 Parameters ℎ and 119896 satisfy (85)

7 Numerical Examples

In this section we illustrate numerical results for bothEuropean and American spread options

In Example 1 we show that the stability conditions (35)and (36) for the European spread option can not be disre-garded

Example 1 We tested the algorithm for the European spreadcall option pricing problem with the parameters

119879 = 1119864 = 1001205901 = 021205902 = 01119903 = 011199021 = 0011199022 = 005120588 = 01

(84)

and taking 1198781 and 1198782 large enough in particular 1198782max= 4001198781max

= 3(1198782max+ 119864) = 1500

For parameters (84) the stability conditions (35) and (36)on space and time steps are as follows

ℎ lt min 28428 237358 = 28428 119896 lt 00112 (85)

These conditions are crucial for the qualitative propertiessuch as positivity and stability of the scheme In Figure 2 thenumerical solution with ℎ and 119896 satisfying (85) is presentedOn the other hand in Figure 3 there is a numerical simulationwith time step 119896 = 0012 gt 00112 The solution is unstableand it has negative values

Example 2 illustrates the computational time aswell as thecomparison with analytical approximation provided by [6]

Abstract and Applied Analysis 9

1500

0500

400

1000

1000

1500

300

20002500

200500100

0 0

minus500

U(S

1S

2T

)

S1

S2

Figure 3 Stability condition is broken

Example 2 Consider the European spread option with thedata of Example 1 (84) and choosing step sizes satisfying thestability condition

The price of European spread option can be expressedas the price of a compound exchange option (see [6])The known derived analytical approximation for the spreadoption reads

119880 (1198781 1198782 119879) = 1198781119890minus1199021119879119873(1198891)minus (119864119890minus119903119879 + 1198782) 119890minus(119903minusminus2)119879119873(1198892) (86)

where

1198891 = 1120590radic119879 (ln

11987811198782 + 119864119890minus119903119879+ (119903 minus + 2 minus 1199021 + 12059022 )119879) 1198892 = 1198891 minus 120590radic119879

120590 = radic12059021 minus 212058812059011205902 11987821198782 + 119864119890minus119903119879 + (11987821198782 + 119864119890minus119903119879)

2 12059022 = 11987821198782 + 119864119890minus119903119879 1199032 = 11987821198782 + 119864119890minus119903119879 1199022

(87)

The comparison of our proposed numerical method andthe analytical approximation (86) is presented in Table 1Analytical approximation is calculated for the same arraysof 1198781 and 1198782 using MATLAB-function cdf for calculat-ing cumulative distribution function that requires a lot ofcomputational resources for each point In the proposedfinite difference method (FDM) cdf is only used for theboundary conditions Last row in the Table 1 presents theCPU time for each method for the same sizes of array Forapproximation method cdf-time is 925018 sec It is the main

Table 1 Comparison proposed method (FDM) with analyticalapproximation (86)

1198782 1198781 FDM Approximation

100

400 2105978 2097261200 242895 234543100 01150 0012350 31179119890 minus 05 minus47741119890 minus 10

90

400 2200678 2201714200 298745 298724100 01691 0037250 52150119890 minus 05 minus19574119890 minus 09

CPU time (sec) 106188 929467

00400

100

300 500

200

200

300

1000100

400

15000

500

U(S

1S

2T

)

S 1S2

Figure 4 European spread put option with parameters (84)

part of computational time In the case of FDM cdf takes lessthan half of computational time - 48867 sec

Furthermore unsuitable negative values appear in ana-lytical approximation for small 1198781 and 1198782 while the proposedmethod preserves nonnegativity of the solution

In the next example we confirm that for the spreadput option case the removing technique is fruitful and theselection of the boundary conditions at the boundary of thenumerical domain is appropriated because the behaviour ofthe solution at the boundaries fits the solution at the interiorpoints as it is shown in Figure 4

Example 3 With the parameters in (84) of Example 1 butwith payoff (1198782+119864minus1198781)+ in Figure 4we show the simulation ofthe numerical solution obtained by the proposed numericalscheme

Finally in Example 4 we compare the numerical solutionobtained with our scheme (83) using LCP approach withother tested efficient methods presented in [28]

10 Abstract and Applied Analysis

Table 2 Comparing several methods for American option pricing

119864 Analytic FFT MC FDM0 8513201 8513079 8516613 85081982 7542296 7542242 7545496 75340644 6653060 6652976 6656364 6648792

Example 4 Let us consider the American spread call optionproblem with parameters

119879 = 11205901 = 011205902 = 02119903 = 011199021 = 0051199022 = 005120588 = 05

(88)

Table 2 shows the option price at fixed asset prices 1198781 = 96and 1198782 = 100 for different values of strike price 119864 evaluatedby one-dimensional integration analytic method (Analytic)Fast Fourier Transform (FFT) Monte Carlo (MC) methodand our proposed method The accuracy is competitive withother methods such as Fourier Transform with high numberof discretization (4096) andMonte Carlo with big number oftime steps (2000)

Another set of parameters can be considered in order tocompare it with data in [9]

119879 = 05119864 = 1001205901 = 0251205902 = 03119903 = 0031199021 = 0061199022 = 002120588 = 05

(89)

In [9] three methods are considered to evaluate thespread option price for data 74 and fixed values 1198781 = 100and 1198782 = 200 a numerical integration method (NIM) amethod of lines (MOL) and a Monte Carlo approach NIMused in [9] is based in the recursive integration techniquesdeveloped byHuang et al [29] InNIM the option is treated asa Bermudan that is an option that can be exercised at discretepoints of time and Simpsons rule is used For the experimentthe authors used 1024 spatial nodes see [8 9] for moredetails The MOL is implemented by a spatial discretizationof 1438 times 200 mesh points and a three-time level scheme isused for the integration in time [9] 50 000 simulations have

Table 3 CPU time in sec (first row) and absolute difference (secondrow) for different methods depending on number of time steps forfixed number of space steps for parameters (89)

119873120591 NIM MOL Monte Carlo FDM

50 296598 5920 47 970 265470003 0001 01735 00036

32 127069 5512 19 811 1791100035 0006 01148 00122

16 33493 51718 5 149 943200012 00236 0132 00371

8 10360 4901 1 350 478700105 0074 01316 00907

4 4117 4547 35575 281500364 02512 01351 02254

been used in the Monte Carlo method described in [30] Inour method FDM the step size discretization considered isℎ = 01 To compare FDM with the three previous methodsabsolute errors are computed in Table 3 with respect to areference value obtained by the Fourier space time-steppingapproach of Surkov [31] with large number of nodes 4096spatial nodes and 300 time steps From Table 3 we can statethat the proposed method has the same order of accuracy asMOL more efficient and requires less computational timethan Monte Carlo method It is slightly less efficient than theNIM but we prove that it possesses a very highly desirablequalitative behaviour in finance

Competing Interests

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

Acknowledgments

This work has been partially supported by the EuropeanUnion in the FP7- PEOPLE-2012-ITN Program under GrantAgreement no 304617 (FP7 Marie Curie Action ProjectMulti-ITN STRIKE-Novel Methods in ComputationalFinance) and the Ministerio de Economıa y CompetitividadSpanish Grant MTM2013-41765-P

References

[1] R Carmona and V Durrleman ldquoPricing and hedging spreadoptionsrdquo SIAM Review vol 45 no 4 pp 627ndash685 2003

[2] P P Boyle ldquoOptions a Monte Carlo approachrdquo Journal ofFinancial Economics vol 4 no 3 pp 323ndash338 1977

[3] C Joy P P Boyle and K S Tan ldquoQuasi-Monte Carlo methodsin numerical financerdquo Management Science vol 42 no 6 pp926ndash938 1996

[4] W-H Kao Y-D Lyuu and K-W Wen ldquoThe hexanomiallattice for pricingmulti-asset optionsrdquoAppliedMathematics andComputation vol 233 pp 463ndash479 2014

Abstract and Applied Analysis 11

[5] E Kirk and J Aron ldquoCorrelation in the energy marketsrdquo inManaging Energy Price Risk V Kaminski Ed pp 71ndash78 RiskBooks London UK 1995

[6] C Alexander andA Venkatramanan ldquoAnalytic approximationsfor spread optionsrdquo ICMACentre Discussion Papers in FinanceDP2007-11 ICMA Centre 2007

[7] N D Pearson ldquoAn efficient approach for pricing spreadoptionsrdquoThe Journal of Derivatives vol 3 no 1 pp 76ndash91 1995

[8] C Chiarella and J Ziveyi ldquoPricingAmerican options written ontwo underlying assetsrdquo Quantitative Finance vol 14 no 3 pp409ndash426 2014

[9] J Ziveyi The evaluation of early exercise exotic options [PhDthesis] University of Technology Sydney Australia 2011

[10] P Boyle ldquoA lattice framework for option pricing with two statevariablesrdquo Journal of Financial and Quantitative Analysis vol23 no 1 pp 1ndash12 1988

[11] M Rubinstein ldquoReturn to Ozrdquo RISK vol 7 no 11 pp 67ndash711994

[12] R Company L Jodar M Fakharany and M-C CasabanldquoRemoving the correlation term in option pricing Hestonmodel numerical analysis and computingrdquo Abstract andApplied Analysis vol 2013 Article ID 246724 11 pages 2013

[13] S Salmi J Toivanen and L von Sydow ldquoIterative methodsfor pricing american options under the bates modelrdquo ProcediaComputer Science vol 18 pp 1136ndash1144 2013

[14] J Toivanen ldquoA componentwise splitting method for pricingamerican options under the bates modelrdquo in Applied andNumerical Partial Differential Equations vol 15 of Computa-tional Methods in Applied Sciences pp 213ndash227 Springer 2010

[15] R Zvan P A Forsyth and K R Vetzal ldquoNegative coefficientsin two-factor option pricing modelsrdquo Journal of ComputationalFinance vol 7 no 1 pp 37ndash73 2003

[16] B During M Fournie and C Heuer ldquoHigh-order compactfinite difference schemes for option pricing in stochastic volatil-ity models on non-uniform gridsrdquo Journal of Computationaland Applied Mathematics vol 271 pp 247ndash266 2014

[17] K J in rsquot Hout and C Mishra ldquoStability of ADI schemes formultidimensional diffusion equations with mixed derivativetermsrdquoApplied NumericalMathematics vol 74 pp 83ndash94 2013

[18] C Chiarella B Kang G H Mayer and A Ziogas ldquoTheevaluation of American option prices under stochastic volatilityand jump-diffusion dynamics using the method of linesrdquoInternational Journal of Theoretical and Applied Finance vol 12no 3 pp 393ndash425 2009

[19] M Fakharany R Company and L Jodar ldquoPositive finite dif-ference schemes for a partial integro-differential option pricingmodelrdquo Applied Mathematics and Computation vol 249 pp320ndash332 2014

[20] P R Garabedian Partial Differential Equations AMS ChelseaPublishing Providence RI USA 1998

[21] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp SonsChichester UK 2006

[22] R U Seydel Tools for Computational Finance Springer 3rdedition 2006

[23] M Ehrhardt Discrete artificial boundary conditions [PhDthesis] Technische Universitat Berlin Berlin Germany 2001

[24] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[25] J C HullOptions Futures andOther Derivatives Prentice-HallUpper Saddle River NJ USA 2000

[26] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods Clarendon Press Oxford UK3rd edition 1985

[27] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[28] M A H Dempster and S S G Hong Spread Option Valuationand the Fast Fourier Transform WP 262000 Judge Institute ofManagement Studies University of Cambridge 2000

[29] J-Z Huang M G Subrahmanyam and G G Yu ldquoPricingand hedging american options a recursive integrationmethodrdquoReview of Financial Studies vol 9 no 1 pp 277ndash300 1996

[30] A Ibanez and F Zapatero ldquoMonte Carlo valuation of Americanoptions through computation of the optimal exercise frontierrdquoJournal of Financial and Quantitative Analysis vol 39 no 2 pp253ndash275 2004

[31] V Surkov ldquoParallel option pricing with fourier space time-stepping method on graphics processingrdquo in Proceedings of the1st International Workshop on Parallel and Distributed Comput-ing in Finance (PDCoF rsquo08) Miami Fla USA 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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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

Abstract and Applied Analysis 7

where

119864119899119894119895 (2) = 11989826 12059731198821205971199103 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (59)

thus10038161003816100381610038161003816119864119899119894119895 (3)10038161003816100381610038161003816le 11989826 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(60)

The discretization of the second spatial derivatives is asfollows

119882119899119894+1119895 minus 2119882119899119894119895 +119882119899119894minus1119895ℎ2= 12059721198821205971199092 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (4)

(61)

where

119864119899119894119895 (4) = 112 12059741198821205971199094 (120585 119910119895 120591119899) 119909119894minus1 lt 120585 lt 119909119894+1 (62)

10038161003816100381610038161003816119864119899119894119895 (4)10038161003816100381610038161003816le 112 max100381610038161003816100381610038161003816100381610038161003816

12059741198821205971199094 (120585 119910119895 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198881 lt 120585 lt 1198891

(63)

Analogously

119908119899119894119895+1 minus 2119908119899119894119895 + 119908119899119894119895minus11198982ℎ2= 12059721198821205971199102 (119909119894 119910119895 120591119899) + ℎ4119864119899119894119895 (5)

(64)

where

119864119899119894119895 (5) = 119898412 12059741198821205971199104 (119909119894 120578 120591119899) 119910119895minus1 lt 120578 lt 119910119895+1 (65)

10038161003816100381610038161003816119864119899119894119895 (5)10038161003816100381610038161003816le 119898412 max100381610038161003816100381610038161003816100381610038161003816

12059731198821205971199103 (119909119894 120578 120591119899)100381610038161003816100381610038161003816100381610038161003816 1198882 lt 120578 lt 1198892

(66)

Then the local truncation error takes the following form

119879119899119894119895 = 119865 (119882119899119894119895) minus 119871 (119882119899119894119895) = 119896119864119899119894119895 (1) minus 2

2sdot ℎ4 (119864119899119894119895 (4) + 119864119899119894119895 (5)) minus 1205901 (119903 minus 1199021 minus

120590212 )sdot ℎ2119864119899119894119895 (2)minus ( 1205881205901 (119903 minus 1199021 minus

120590212 ) minus 11205902 (119903 minus 1199022 minus120590222 ))

sdot ℎ2119864119899119894119895 (3)

(67)

From (54) (57) (60) (63) and (66) one gets1003816100381610038161003816100381611987911989911989411989510038161003816100381610038161003816 = 119874 (119896) + 119874 (ℎ2) (68)

Theorem 3 Assuming that the solution of the PDE (19) admitstwo times continuous partial derivativewith respect to time andup to order four with respect to each of space directions solutionthe numerical solution computed by scheme (24) is consistentwith (19)

5 European Spread Put Option Case

For the sake of brevity and in order to avoid repetition ofproofs we summarize in this section the results proved inprevious sections for the call option case

Let us consider a European spread put option with payoff

119880 (1198781 1198782 0) = ((1198782 + 119864) minus 1198781)+ 119864 gt 0 (69)

Equation (1) and all the transformed versions of it do notchange for the put option Like in the study of the previouscall option case due to different nature of the put option weneed to establish the boundary conditions and translate themto the boundary of the numerical domain

(1) If 1198782 = 0 the problem is reduced to the European putoption on asset 1198781 and strike 119864 Therefore the optionvalue on this boundary is

119880 (1198781 0 120591) = 119864119890minus119903120591119873(minus1206012) minus 11987811198901199021120591119873(minus1206011) (70)

where 1206011 and 1206012 are defined by (5)(2) If 1198781 = 0 we consider 119880(0 1198782 120591) as a put option price

on asset 1198781 = 0 with the strike 1198782 + 119864119880 (0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 (71)

(3) For large values of 1198782 we suppose that the slope of thecurve tends to 1 that is

1205971198801205971198782 (1198781 1198782 997888rarr infin 120591) = 119890minus1199022120591 (72)

(4) For 1198781 ≫ 1198782 + 119864 we consider 119880(1198781 1198782 120591) as a putoption price with large values of 1198781 and therefore

119880(1198781 1198782 120591) = 0 (73)

Under transformations (8) and (16) the boundary con-ditions (70)ndash(73) are translated to the rhomboid numericaldomain 119860119861119862119863 as follows

119863119862 119908119899119894119873119910+119894 = 119864119873 (minus1205722119894)+ 119890(1205901)119909119894minus1199021120591119899119873(minus1205721119894)

(74)

119860119863 1199081198990119895 = 119864 + 119890minus1205902(119910119895minus1198981199090) (75)

119860119861 119908119899119894119894 = 119908119899119894119894+1 + |119898| ℎ1205902119890minus1205902(119910119894+1minus119898119909119894)minus119903120591 (76)

119861119862 119908119899119873119909+1119873119909+119895 = 0 (77)

8 Abstract and Applied Analysis

Positivity As the numerical scheme (24) for the interiorpoints is the same and the new boundary conditions (74)ndash(77) are nonnegative then under conditions (35) and (36)the numerical solution for the European spread put optionis nonnegative

Stability Stability of the scheme is preservedwith newbound-ary conditions (74)ndash(77) because analogously to the calloption case the maximum value of the transformed function119882(119909 119910 120591) is reached at a corner (in this case (1198881 1198892) wherewe use the boundary condition (75)) At the correspondingpoint of the original domain we use boundary condition (71)In the original coordinates it can be bounded as

119880(0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 le 1198782max+ 119864 = 4119864 = const (78)

Therefore the transformed function also can be boundedas

max119894119895119908119899119894119895 = 4119864119890119903119879 (79)

6 American Spread Options

In the case of American type option the holder has the rightto exercise option at any moment before expiring It leadsto a free boundary problem [27] In order to simplify thecomputational procedure this problem can be considered asa linear complementarity problem (LCP)

(120597119880120597120591 minus 119871119880) (119880 minus 119892) = 0120597119880120597120591 minus 119871119880 ge 0

119880 minus 119892 (1198781 1198782) ge 0(80)

where 119871 represents the spatial differential operator of (1)If we rewrite (19) in the form

120597119882120597120591 minusL119882 = 0 (81)

then under transformation (16) LCP (80) has the followingform

(120597119882120597120591 minusL119882)(119882 minus 119891 (119909 119910)) = 0120597119882120597120591 minusL119882 ge 0119882 minus 119891 (119909 119910) ge 0

(82)

whereL is the right-hand side of (19) and 119891(119909 119910) is given by(41)

Numerical solution for problem (82) can be easilyobtained by a small modification of the calculating proceduredescribed above in Sections 3 and 4 On each time level

119899119894119895 = max 119908119899119894119895 119891 (119909119894 119910119895) ge 0 (83)

where 119908119899119894119895 is defined by the finite difference equation (24)

01500

500

4001000

1000

300

1500

200500100

0 0

U(S

1S

2T

)

S1 S2

Figure 2 Parameters ℎ and 119896 satisfy (85)

7 Numerical Examples

In this section we illustrate numerical results for bothEuropean and American spread options

In Example 1 we show that the stability conditions (35)and (36) for the European spread option can not be disre-garded

Example 1 We tested the algorithm for the European spreadcall option pricing problem with the parameters

119879 = 1119864 = 1001205901 = 021205902 = 01119903 = 011199021 = 0011199022 = 005120588 = 01

(84)

and taking 1198781 and 1198782 large enough in particular 1198782max= 4001198781max

= 3(1198782max+ 119864) = 1500

For parameters (84) the stability conditions (35) and (36)on space and time steps are as follows

ℎ lt min 28428 237358 = 28428 119896 lt 00112 (85)

These conditions are crucial for the qualitative propertiessuch as positivity and stability of the scheme In Figure 2 thenumerical solution with ℎ and 119896 satisfying (85) is presentedOn the other hand in Figure 3 there is a numerical simulationwith time step 119896 = 0012 gt 00112 The solution is unstableand it has negative values

Example 2 illustrates the computational time aswell as thecomparison with analytical approximation provided by [6]

Abstract and Applied Analysis 9

1500

0500

400

1000

1000

1500

300

20002500

200500100

0 0

minus500

U(S

1S

2T

)

S1

S2

Figure 3 Stability condition is broken

Example 2 Consider the European spread option with thedata of Example 1 (84) and choosing step sizes satisfying thestability condition

The price of European spread option can be expressedas the price of a compound exchange option (see [6])The known derived analytical approximation for the spreadoption reads

119880 (1198781 1198782 119879) = 1198781119890minus1199021119879119873(1198891)minus (119864119890minus119903119879 + 1198782) 119890minus(119903minusminus2)119879119873(1198892) (86)

where

1198891 = 1120590radic119879 (ln

11987811198782 + 119864119890minus119903119879+ (119903 minus + 2 minus 1199021 + 12059022 )119879) 1198892 = 1198891 minus 120590radic119879

120590 = radic12059021 minus 212058812059011205902 11987821198782 + 119864119890minus119903119879 + (11987821198782 + 119864119890minus119903119879)

2 12059022 = 11987821198782 + 119864119890minus119903119879 1199032 = 11987821198782 + 119864119890minus119903119879 1199022

(87)

The comparison of our proposed numerical method andthe analytical approximation (86) is presented in Table 1Analytical approximation is calculated for the same arraysof 1198781 and 1198782 using MATLAB-function cdf for calculat-ing cumulative distribution function that requires a lot ofcomputational resources for each point In the proposedfinite difference method (FDM) cdf is only used for theboundary conditions Last row in the Table 1 presents theCPU time for each method for the same sizes of array Forapproximation method cdf-time is 925018 sec It is the main

Table 1 Comparison proposed method (FDM) with analyticalapproximation (86)

1198782 1198781 FDM Approximation

100

400 2105978 2097261200 242895 234543100 01150 0012350 31179119890 minus 05 minus47741119890 minus 10

90

400 2200678 2201714200 298745 298724100 01691 0037250 52150119890 minus 05 minus19574119890 minus 09

CPU time (sec) 106188 929467

00400

100

300 500

200

200

300

1000100

400

15000

500

U(S

1S

2T

)

S 1S2

Figure 4 European spread put option with parameters (84)

part of computational time In the case of FDM cdf takes lessthan half of computational time - 48867 sec

Furthermore unsuitable negative values appear in ana-lytical approximation for small 1198781 and 1198782 while the proposedmethod preserves nonnegativity of the solution

In the next example we confirm that for the spreadput option case the removing technique is fruitful and theselection of the boundary conditions at the boundary of thenumerical domain is appropriated because the behaviour ofthe solution at the boundaries fits the solution at the interiorpoints as it is shown in Figure 4

Example 3 With the parameters in (84) of Example 1 butwith payoff (1198782+119864minus1198781)+ in Figure 4we show the simulation ofthe numerical solution obtained by the proposed numericalscheme

Finally in Example 4 we compare the numerical solutionobtained with our scheme (83) using LCP approach withother tested efficient methods presented in [28]

10 Abstract and Applied Analysis

Table 2 Comparing several methods for American option pricing

119864 Analytic FFT MC FDM0 8513201 8513079 8516613 85081982 7542296 7542242 7545496 75340644 6653060 6652976 6656364 6648792

Example 4 Let us consider the American spread call optionproblem with parameters

119879 = 11205901 = 011205902 = 02119903 = 011199021 = 0051199022 = 005120588 = 05

(88)

Table 2 shows the option price at fixed asset prices 1198781 = 96and 1198782 = 100 for different values of strike price 119864 evaluatedby one-dimensional integration analytic method (Analytic)Fast Fourier Transform (FFT) Monte Carlo (MC) methodand our proposed method The accuracy is competitive withother methods such as Fourier Transform with high numberof discretization (4096) andMonte Carlo with big number oftime steps (2000)

Another set of parameters can be considered in order tocompare it with data in [9]

119879 = 05119864 = 1001205901 = 0251205902 = 03119903 = 0031199021 = 0061199022 = 002120588 = 05

(89)

In [9] three methods are considered to evaluate thespread option price for data 74 and fixed values 1198781 = 100and 1198782 = 200 a numerical integration method (NIM) amethod of lines (MOL) and a Monte Carlo approach NIMused in [9] is based in the recursive integration techniquesdeveloped byHuang et al [29] InNIM the option is treated asa Bermudan that is an option that can be exercised at discretepoints of time and Simpsons rule is used For the experimentthe authors used 1024 spatial nodes see [8 9] for moredetails The MOL is implemented by a spatial discretizationof 1438 times 200 mesh points and a three-time level scheme isused for the integration in time [9] 50 000 simulations have

Table 3 CPU time in sec (first row) and absolute difference (secondrow) for different methods depending on number of time steps forfixed number of space steps for parameters (89)

119873120591 NIM MOL Monte Carlo FDM

50 296598 5920 47 970 265470003 0001 01735 00036

32 127069 5512 19 811 1791100035 0006 01148 00122

16 33493 51718 5 149 943200012 00236 0132 00371

8 10360 4901 1 350 478700105 0074 01316 00907

4 4117 4547 35575 281500364 02512 01351 02254

been used in the Monte Carlo method described in [30] Inour method FDM the step size discretization considered isℎ = 01 To compare FDM with the three previous methodsabsolute errors are computed in Table 3 with respect to areference value obtained by the Fourier space time-steppingapproach of Surkov [31] with large number of nodes 4096spatial nodes and 300 time steps From Table 3 we can statethat the proposed method has the same order of accuracy asMOL more efficient and requires less computational timethan Monte Carlo method It is slightly less efficient than theNIM but we prove that it possesses a very highly desirablequalitative behaviour in finance

Competing Interests

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

Acknowledgments

This work has been partially supported by the EuropeanUnion in the FP7- PEOPLE-2012-ITN Program under GrantAgreement no 304617 (FP7 Marie Curie Action ProjectMulti-ITN STRIKE-Novel Methods in ComputationalFinance) and the Ministerio de Economıa y CompetitividadSpanish Grant MTM2013-41765-P

References

[1] R Carmona and V Durrleman ldquoPricing and hedging spreadoptionsrdquo SIAM Review vol 45 no 4 pp 627ndash685 2003

[2] P P Boyle ldquoOptions a Monte Carlo approachrdquo Journal ofFinancial Economics vol 4 no 3 pp 323ndash338 1977

[3] C Joy P P Boyle and K S Tan ldquoQuasi-Monte Carlo methodsin numerical financerdquo Management Science vol 42 no 6 pp926ndash938 1996

[4] W-H Kao Y-D Lyuu and K-W Wen ldquoThe hexanomiallattice for pricingmulti-asset optionsrdquoAppliedMathematics andComputation vol 233 pp 463ndash479 2014

Abstract and Applied Analysis 11

[5] E Kirk and J Aron ldquoCorrelation in the energy marketsrdquo inManaging Energy Price Risk V Kaminski Ed pp 71ndash78 RiskBooks London UK 1995

[6] C Alexander andA Venkatramanan ldquoAnalytic approximationsfor spread optionsrdquo ICMACentre Discussion Papers in FinanceDP2007-11 ICMA Centre 2007

[7] N D Pearson ldquoAn efficient approach for pricing spreadoptionsrdquoThe Journal of Derivatives vol 3 no 1 pp 76ndash91 1995

[8] C Chiarella and J Ziveyi ldquoPricingAmerican options written ontwo underlying assetsrdquo Quantitative Finance vol 14 no 3 pp409ndash426 2014

[9] J Ziveyi The evaluation of early exercise exotic options [PhDthesis] University of Technology Sydney Australia 2011

[10] P Boyle ldquoA lattice framework for option pricing with two statevariablesrdquo Journal of Financial and Quantitative Analysis vol23 no 1 pp 1ndash12 1988

[11] M Rubinstein ldquoReturn to Ozrdquo RISK vol 7 no 11 pp 67ndash711994

[12] R Company L Jodar M Fakharany and M-C CasabanldquoRemoving the correlation term in option pricing Hestonmodel numerical analysis and computingrdquo Abstract andApplied Analysis vol 2013 Article ID 246724 11 pages 2013

[13] S Salmi J Toivanen and L von Sydow ldquoIterative methodsfor pricing american options under the bates modelrdquo ProcediaComputer Science vol 18 pp 1136ndash1144 2013

[14] J Toivanen ldquoA componentwise splitting method for pricingamerican options under the bates modelrdquo in Applied andNumerical Partial Differential Equations vol 15 of Computa-tional Methods in Applied Sciences pp 213ndash227 Springer 2010

[15] R Zvan P A Forsyth and K R Vetzal ldquoNegative coefficientsin two-factor option pricing modelsrdquo Journal of ComputationalFinance vol 7 no 1 pp 37ndash73 2003

[16] B During M Fournie and C Heuer ldquoHigh-order compactfinite difference schemes for option pricing in stochastic volatil-ity models on non-uniform gridsrdquo Journal of Computationaland Applied Mathematics vol 271 pp 247ndash266 2014

[17] K J in rsquot Hout and C Mishra ldquoStability of ADI schemes formultidimensional diffusion equations with mixed derivativetermsrdquoApplied NumericalMathematics vol 74 pp 83ndash94 2013

[18] C Chiarella B Kang G H Mayer and A Ziogas ldquoTheevaluation of American option prices under stochastic volatilityand jump-diffusion dynamics using the method of linesrdquoInternational Journal of Theoretical and Applied Finance vol 12no 3 pp 393ndash425 2009

[19] M Fakharany R Company and L Jodar ldquoPositive finite dif-ference schemes for a partial integro-differential option pricingmodelrdquo Applied Mathematics and Computation vol 249 pp320ndash332 2014

[20] P R Garabedian Partial Differential Equations AMS ChelseaPublishing Providence RI USA 1998

[21] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp SonsChichester UK 2006

[22] R U Seydel Tools for Computational Finance Springer 3rdedition 2006

[23] M Ehrhardt Discrete artificial boundary conditions [PhDthesis] Technische Universitat Berlin Berlin Germany 2001

[24] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[25] J C HullOptions Futures andOther Derivatives Prentice-HallUpper Saddle River NJ USA 2000

[26] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods Clarendon Press Oxford UK3rd edition 1985

[27] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[28] M A H Dempster and S S G Hong Spread Option Valuationand the Fast Fourier Transform WP 262000 Judge Institute ofManagement Studies University of Cambridge 2000

[29] J-Z Huang M G Subrahmanyam and G G Yu ldquoPricingand hedging american options a recursive integrationmethodrdquoReview of Financial Studies vol 9 no 1 pp 277ndash300 1996

[30] A Ibanez and F Zapatero ldquoMonte Carlo valuation of Americanoptions through computation of the optimal exercise frontierrdquoJournal of Financial and Quantitative Analysis vol 39 no 2 pp253ndash275 2004

[31] V Surkov ldquoParallel option pricing with fourier space time-stepping method on graphics processingrdquo in Proceedings of the1st International Workshop on Parallel and Distributed Comput-ing in Finance (PDCoF rsquo08) Miami Fla USA 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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

8 Abstract and Applied Analysis

Positivity As the numerical scheme (24) for the interiorpoints is the same and the new boundary conditions (74)ndash(77) are nonnegative then under conditions (35) and (36)the numerical solution for the European spread put optionis nonnegative

Stability Stability of the scheme is preservedwith newbound-ary conditions (74)ndash(77) because analogously to the calloption case the maximum value of the transformed function119882(119909 119910 120591) is reached at a corner (in this case (1198881 1198892) wherewe use the boundary condition (75)) At the correspondingpoint of the original domain we use boundary condition (71)In the original coordinates it can be bounded as

119880(0 1198782 120591) = (1198782 + 119864) 119890minus119903120591 le 1198782max+ 119864 = 4119864 = const (78)

Therefore the transformed function also can be boundedas

max119894119895119908119899119894119895 = 4119864119890119903119879 (79)

6 American Spread Options

In the case of American type option the holder has the rightto exercise option at any moment before expiring It leadsto a free boundary problem [27] In order to simplify thecomputational procedure this problem can be considered asa linear complementarity problem (LCP)

(120597119880120597120591 minus 119871119880) (119880 minus 119892) = 0120597119880120597120591 minus 119871119880 ge 0

119880 minus 119892 (1198781 1198782) ge 0(80)

where 119871 represents the spatial differential operator of (1)If we rewrite (19) in the form

120597119882120597120591 minusL119882 = 0 (81)

then under transformation (16) LCP (80) has the followingform

(120597119882120597120591 minusL119882)(119882 minus 119891 (119909 119910)) = 0120597119882120597120591 minusL119882 ge 0119882 minus 119891 (119909 119910) ge 0

(82)

whereL is the right-hand side of (19) and 119891(119909 119910) is given by(41)

Numerical solution for problem (82) can be easilyobtained by a small modification of the calculating proceduredescribed above in Sections 3 and 4 On each time level

119899119894119895 = max 119908119899119894119895 119891 (119909119894 119910119895) ge 0 (83)

where 119908119899119894119895 is defined by the finite difference equation (24)

01500

500

4001000

1000

300

1500

200500100

0 0

U(S

1S

2T

)

S1 S2

Figure 2 Parameters ℎ and 119896 satisfy (85)

7 Numerical Examples

In this section we illustrate numerical results for bothEuropean and American spread options

In Example 1 we show that the stability conditions (35)and (36) for the European spread option can not be disre-garded

Example 1 We tested the algorithm for the European spreadcall option pricing problem with the parameters

119879 = 1119864 = 1001205901 = 021205902 = 01119903 = 011199021 = 0011199022 = 005120588 = 01

(84)

and taking 1198781 and 1198782 large enough in particular 1198782max= 4001198781max

= 3(1198782max+ 119864) = 1500

For parameters (84) the stability conditions (35) and (36)on space and time steps are as follows

ℎ lt min 28428 237358 = 28428 119896 lt 00112 (85)

These conditions are crucial for the qualitative propertiessuch as positivity and stability of the scheme In Figure 2 thenumerical solution with ℎ and 119896 satisfying (85) is presentedOn the other hand in Figure 3 there is a numerical simulationwith time step 119896 = 0012 gt 00112 The solution is unstableand it has negative values

Example 2 illustrates the computational time aswell as thecomparison with analytical approximation provided by [6]

Abstract and Applied Analysis 9

1500

0500

400

1000

1000

1500

300

20002500

200500100

0 0

minus500

U(S

1S

2T

)

S1

S2

Figure 3 Stability condition is broken

Example 2 Consider the European spread option with thedata of Example 1 (84) and choosing step sizes satisfying thestability condition

The price of European spread option can be expressedas the price of a compound exchange option (see [6])The known derived analytical approximation for the spreadoption reads

119880 (1198781 1198782 119879) = 1198781119890minus1199021119879119873(1198891)minus (119864119890minus119903119879 + 1198782) 119890minus(119903minusminus2)119879119873(1198892) (86)

where

1198891 = 1120590radic119879 (ln

11987811198782 + 119864119890minus119903119879+ (119903 minus + 2 minus 1199021 + 12059022 )119879) 1198892 = 1198891 minus 120590radic119879

120590 = radic12059021 minus 212058812059011205902 11987821198782 + 119864119890minus119903119879 + (11987821198782 + 119864119890minus119903119879)

2 12059022 = 11987821198782 + 119864119890minus119903119879 1199032 = 11987821198782 + 119864119890minus119903119879 1199022

(87)

The comparison of our proposed numerical method andthe analytical approximation (86) is presented in Table 1Analytical approximation is calculated for the same arraysof 1198781 and 1198782 using MATLAB-function cdf for calculat-ing cumulative distribution function that requires a lot ofcomputational resources for each point In the proposedfinite difference method (FDM) cdf is only used for theboundary conditions Last row in the Table 1 presents theCPU time for each method for the same sizes of array Forapproximation method cdf-time is 925018 sec It is the main

Table 1 Comparison proposed method (FDM) with analyticalapproximation (86)

1198782 1198781 FDM Approximation

100

400 2105978 2097261200 242895 234543100 01150 0012350 31179119890 minus 05 minus47741119890 minus 10

90

400 2200678 2201714200 298745 298724100 01691 0037250 52150119890 minus 05 minus19574119890 minus 09

CPU time (sec) 106188 929467

00400

100

300 500

200

200

300

1000100

400

15000

500

U(S

1S

2T

)

S 1S2

Figure 4 European spread put option with parameters (84)

part of computational time In the case of FDM cdf takes lessthan half of computational time - 48867 sec

Furthermore unsuitable negative values appear in ana-lytical approximation for small 1198781 and 1198782 while the proposedmethod preserves nonnegativity of the solution

In the next example we confirm that for the spreadput option case the removing technique is fruitful and theselection of the boundary conditions at the boundary of thenumerical domain is appropriated because the behaviour ofthe solution at the boundaries fits the solution at the interiorpoints as it is shown in Figure 4

Example 3 With the parameters in (84) of Example 1 butwith payoff (1198782+119864minus1198781)+ in Figure 4we show the simulation ofthe numerical solution obtained by the proposed numericalscheme

Finally in Example 4 we compare the numerical solutionobtained with our scheme (83) using LCP approach withother tested efficient methods presented in [28]

10 Abstract and Applied Analysis

Table 2 Comparing several methods for American option pricing

119864 Analytic FFT MC FDM0 8513201 8513079 8516613 85081982 7542296 7542242 7545496 75340644 6653060 6652976 6656364 6648792

Example 4 Let us consider the American spread call optionproblem with parameters

119879 = 11205901 = 011205902 = 02119903 = 011199021 = 0051199022 = 005120588 = 05

(88)

Table 2 shows the option price at fixed asset prices 1198781 = 96and 1198782 = 100 for different values of strike price 119864 evaluatedby one-dimensional integration analytic method (Analytic)Fast Fourier Transform (FFT) Monte Carlo (MC) methodand our proposed method The accuracy is competitive withother methods such as Fourier Transform with high numberof discretization (4096) andMonte Carlo with big number oftime steps (2000)

Another set of parameters can be considered in order tocompare it with data in [9]

119879 = 05119864 = 1001205901 = 0251205902 = 03119903 = 0031199021 = 0061199022 = 002120588 = 05

(89)

In [9] three methods are considered to evaluate thespread option price for data 74 and fixed values 1198781 = 100and 1198782 = 200 a numerical integration method (NIM) amethod of lines (MOL) and a Monte Carlo approach NIMused in [9] is based in the recursive integration techniquesdeveloped byHuang et al [29] InNIM the option is treated asa Bermudan that is an option that can be exercised at discretepoints of time and Simpsons rule is used For the experimentthe authors used 1024 spatial nodes see [8 9] for moredetails The MOL is implemented by a spatial discretizationof 1438 times 200 mesh points and a three-time level scheme isused for the integration in time [9] 50 000 simulations have

Table 3 CPU time in sec (first row) and absolute difference (secondrow) for different methods depending on number of time steps forfixed number of space steps for parameters (89)

119873120591 NIM MOL Monte Carlo FDM

50 296598 5920 47 970 265470003 0001 01735 00036

32 127069 5512 19 811 1791100035 0006 01148 00122

16 33493 51718 5 149 943200012 00236 0132 00371

8 10360 4901 1 350 478700105 0074 01316 00907

4 4117 4547 35575 281500364 02512 01351 02254

been used in the Monte Carlo method described in [30] Inour method FDM the step size discretization considered isℎ = 01 To compare FDM with the three previous methodsabsolute errors are computed in Table 3 with respect to areference value obtained by the Fourier space time-steppingapproach of Surkov [31] with large number of nodes 4096spatial nodes and 300 time steps From Table 3 we can statethat the proposed method has the same order of accuracy asMOL more efficient and requires less computational timethan Monte Carlo method It is slightly less efficient than theNIM but we prove that it possesses a very highly desirablequalitative behaviour in finance

Competing Interests

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

Acknowledgments

This work has been partially supported by the EuropeanUnion in the FP7- PEOPLE-2012-ITN Program under GrantAgreement no 304617 (FP7 Marie Curie Action ProjectMulti-ITN STRIKE-Novel Methods in ComputationalFinance) and the Ministerio de Economıa y CompetitividadSpanish Grant MTM2013-41765-P

References

[1] R Carmona and V Durrleman ldquoPricing and hedging spreadoptionsrdquo SIAM Review vol 45 no 4 pp 627ndash685 2003

[2] P P Boyle ldquoOptions a Monte Carlo approachrdquo Journal ofFinancial Economics vol 4 no 3 pp 323ndash338 1977

[3] C Joy P P Boyle and K S Tan ldquoQuasi-Monte Carlo methodsin numerical financerdquo Management Science vol 42 no 6 pp926ndash938 1996

[4] W-H Kao Y-D Lyuu and K-W Wen ldquoThe hexanomiallattice for pricingmulti-asset optionsrdquoAppliedMathematics andComputation vol 233 pp 463ndash479 2014

Abstract and Applied Analysis 11

[5] E Kirk and J Aron ldquoCorrelation in the energy marketsrdquo inManaging Energy Price Risk V Kaminski Ed pp 71ndash78 RiskBooks London UK 1995

[6] C Alexander andA Venkatramanan ldquoAnalytic approximationsfor spread optionsrdquo ICMACentre Discussion Papers in FinanceDP2007-11 ICMA Centre 2007

[7] N D Pearson ldquoAn efficient approach for pricing spreadoptionsrdquoThe Journal of Derivatives vol 3 no 1 pp 76ndash91 1995

[8] C Chiarella and J Ziveyi ldquoPricingAmerican options written ontwo underlying assetsrdquo Quantitative Finance vol 14 no 3 pp409ndash426 2014

[9] J Ziveyi The evaluation of early exercise exotic options [PhDthesis] University of Technology Sydney Australia 2011

[10] P Boyle ldquoA lattice framework for option pricing with two statevariablesrdquo Journal of Financial and Quantitative Analysis vol23 no 1 pp 1ndash12 1988

[11] M Rubinstein ldquoReturn to Ozrdquo RISK vol 7 no 11 pp 67ndash711994

[12] R Company L Jodar M Fakharany and M-C CasabanldquoRemoving the correlation term in option pricing Hestonmodel numerical analysis and computingrdquo Abstract andApplied Analysis vol 2013 Article ID 246724 11 pages 2013

[13] S Salmi J Toivanen and L von Sydow ldquoIterative methodsfor pricing american options under the bates modelrdquo ProcediaComputer Science vol 18 pp 1136ndash1144 2013

[14] J Toivanen ldquoA componentwise splitting method for pricingamerican options under the bates modelrdquo in Applied andNumerical Partial Differential Equations vol 15 of Computa-tional Methods in Applied Sciences pp 213ndash227 Springer 2010

[15] R Zvan P A Forsyth and K R Vetzal ldquoNegative coefficientsin two-factor option pricing modelsrdquo Journal of ComputationalFinance vol 7 no 1 pp 37ndash73 2003

[16] B During M Fournie and C Heuer ldquoHigh-order compactfinite difference schemes for option pricing in stochastic volatil-ity models on non-uniform gridsrdquo Journal of Computationaland Applied Mathematics vol 271 pp 247ndash266 2014

[17] K J in rsquot Hout and C Mishra ldquoStability of ADI schemes formultidimensional diffusion equations with mixed derivativetermsrdquoApplied NumericalMathematics vol 74 pp 83ndash94 2013

[18] C Chiarella B Kang G H Mayer and A Ziogas ldquoTheevaluation of American option prices under stochastic volatilityand jump-diffusion dynamics using the method of linesrdquoInternational Journal of Theoretical and Applied Finance vol 12no 3 pp 393ndash425 2009

[19] M Fakharany R Company and L Jodar ldquoPositive finite dif-ference schemes for a partial integro-differential option pricingmodelrdquo Applied Mathematics and Computation vol 249 pp320ndash332 2014

[20] P R Garabedian Partial Differential Equations AMS ChelseaPublishing Providence RI USA 1998

[21] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp SonsChichester UK 2006

[22] R U Seydel Tools for Computational Finance Springer 3rdedition 2006

[23] M Ehrhardt Discrete artificial boundary conditions [PhDthesis] Technische Universitat Berlin Berlin Germany 2001

[24] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[25] J C HullOptions Futures andOther Derivatives Prentice-HallUpper Saddle River NJ USA 2000

[26] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods Clarendon Press Oxford UK3rd edition 1985

[27] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[28] M A H Dempster and S S G Hong Spread Option Valuationand the Fast Fourier Transform WP 262000 Judge Institute ofManagement Studies University of Cambridge 2000

[29] J-Z Huang M G Subrahmanyam and G G Yu ldquoPricingand hedging american options a recursive integrationmethodrdquoReview of Financial Studies vol 9 no 1 pp 277ndash300 1996

[30] A Ibanez and F Zapatero ldquoMonte Carlo valuation of Americanoptions through computation of the optimal exercise frontierrdquoJournal of Financial and Quantitative Analysis vol 39 no 2 pp253ndash275 2004

[31] V Surkov ldquoParallel option pricing with fourier space time-stepping method on graphics processingrdquo in Proceedings of the1st International Workshop on Parallel and Distributed Comput-ing in Finance (PDCoF rsquo08) Miami Fla USA 2008

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

Abstract and Applied Analysis 9

1500

0500

400

1000

1000

1500

300

20002500

200500100

0 0

minus500

U(S

1S

2T

)

S1

S2

Figure 3 Stability condition is broken

Example 2 Consider the European spread option with thedata of Example 1 (84) and choosing step sizes satisfying thestability condition

The price of European spread option can be expressedas the price of a compound exchange option (see [6])The known derived analytical approximation for the spreadoption reads

119880 (1198781 1198782 119879) = 1198781119890minus1199021119879119873(1198891)minus (119864119890minus119903119879 + 1198782) 119890minus(119903minusminus2)119879119873(1198892) (86)

where

1198891 = 1120590radic119879 (ln

11987811198782 + 119864119890minus119903119879+ (119903 minus + 2 minus 1199021 + 12059022 )119879) 1198892 = 1198891 minus 120590radic119879

120590 = radic12059021 minus 212058812059011205902 11987821198782 + 119864119890minus119903119879 + (11987821198782 + 119864119890minus119903119879)

2 12059022 = 11987821198782 + 119864119890minus119903119879 1199032 = 11987821198782 + 119864119890minus119903119879 1199022

(87)

The comparison of our proposed numerical method andthe analytical approximation (86) is presented in Table 1Analytical approximation is calculated for the same arraysof 1198781 and 1198782 using MATLAB-function cdf for calculat-ing cumulative distribution function that requires a lot ofcomputational resources for each point In the proposedfinite difference method (FDM) cdf is only used for theboundary conditions Last row in the Table 1 presents theCPU time for each method for the same sizes of array Forapproximation method cdf-time is 925018 sec It is the main

Table 1 Comparison proposed method (FDM) with analyticalapproximation (86)

1198782 1198781 FDM Approximation

100

400 2105978 2097261200 242895 234543100 01150 0012350 31179119890 minus 05 minus47741119890 minus 10

90

400 2200678 2201714200 298745 298724100 01691 0037250 52150119890 minus 05 minus19574119890 minus 09

CPU time (sec) 106188 929467

00400

100

300 500

200

200

300

1000100

400

15000

500

U(S

1S

2T

)

S 1S2

Figure 4 European spread put option with parameters (84)

part of computational time In the case of FDM cdf takes lessthan half of computational time - 48867 sec

Furthermore unsuitable negative values appear in ana-lytical approximation for small 1198781 and 1198782 while the proposedmethod preserves nonnegativity of the solution

In the next example we confirm that for the spreadput option case the removing technique is fruitful and theselection of the boundary conditions at the boundary of thenumerical domain is appropriated because the behaviour ofthe solution at the boundaries fits the solution at the interiorpoints as it is shown in Figure 4

Example 3 With the parameters in (84) of Example 1 butwith payoff (1198782+119864minus1198781)+ in Figure 4we show the simulation ofthe numerical solution obtained by the proposed numericalscheme

Finally in Example 4 we compare the numerical solutionobtained with our scheme (83) using LCP approach withother tested efficient methods presented in [28]

10 Abstract and Applied Analysis

Table 2 Comparing several methods for American option pricing

119864 Analytic FFT MC FDM0 8513201 8513079 8516613 85081982 7542296 7542242 7545496 75340644 6653060 6652976 6656364 6648792

Example 4 Let us consider the American spread call optionproblem with parameters

119879 = 11205901 = 011205902 = 02119903 = 011199021 = 0051199022 = 005120588 = 05

(88)

Table 2 shows the option price at fixed asset prices 1198781 = 96and 1198782 = 100 for different values of strike price 119864 evaluatedby one-dimensional integration analytic method (Analytic)Fast Fourier Transform (FFT) Monte Carlo (MC) methodand our proposed method The accuracy is competitive withother methods such as Fourier Transform with high numberof discretization (4096) andMonte Carlo with big number oftime steps (2000)

Another set of parameters can be considered in order tocompare it with data in [9]

119879 = 05119864 = 1001205901 = 0251205902 = 03119903 = 0031199021 = 0061199022 = 002120588 = 05

(89)

In [9] three methods are considered to evaluate thespread option price for data 74 and fixed values 1198781 = 100and 1198782 = 200 a numerical integration method (NIM) amethod of lines (MOL) and a Monte Carlo approach NIMused in [9] is based in the recursive integration techniquesdeveloped byHuang et al [29] InNIM the option is treated asa Bermudan that is an option that can be exercised at discretepoints of time and Simpsons rule is used For the experimentthe authors used 1024 spatial nodes see [8 9] for moredetails The MOL is implemented by a spatial discretizationof 1438 times 200 mesh points and a three-time level scheme isused for the integration in time [9] 50 000 simulations have

Table 3 CPU time in sec (first row) and absolute difference (secondrow) for different methods depending on number of time steps forfixed number of space steps for parameters (89)

119873120591 NIM MOL Monte Carlo FDM

50 296598 5920 47 970 265470003 0001 01735 00036

32 127069 5512 19 811 1791100035 0006 01148 00122

16 33493 51718 5 149 943200012 00236 0132 00371

8 10360 4901 1 350 478700105 0074 01316 00907

4 4117 4547 35575 281500364 02512 01351 02254

been used in the Monte Carlo method described in [30] Inour method FDM the step size discretization considered isℎ = 01 To compare FDM with the three previous methodsabsolute errors are computed in Table 3 with respect to areference value obtained by the Fourier space time-steppingapproach of Surkov [31] with large number of nodes 4096spatial nodes and 300 time steps From Table 3 we can statethat the proposed method has the same order of accuracy asMOL more efficient and requires less computational timethan Monte Carlo method It is slightly less efficient than theNIM but we prove that it possesses a very highly desirablequalitative behaviour in finance

Competing Interests

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

Acknowledgments

This work has been partially supported by the EuropeanUnion in the FP7- PEOPLE-2012-ITN Program under GrantAgreement no 304617 (FP7 Marie Curie Action ProjectMulti-ITN STRIKE-Novel Methods in ComputationalFinance) and the Ministerio de Economıa y CompetitividadSpanish Grant MTM2013-41765-P

References

[1] R Carmona and V Durrleman ldquoPricing and hedging spreadoptionsrdquo SIAM Review vol 45 no 4 pp 627ndash685 2003

[2] P P Boyle ldquoOptions a Monte Carlo approachrdquo Journal ofFinancial Economics vol 4 no 3 pp 323ndash338 1977

[3] C Joy P P Boyle and K S Tan ldquoQuasi-Monte Carlo methodsin numerical financerdquo Management Science vol 42 no 6 pp926ndash938 1996

[4] W-H Kao Y-D Lyuu and K-W Wen ldquoThe hexanomiallattice for pricingmulti-asset optionsrdquoAppliedMathematics andComputation vol 233 pp 463ndash479 2014

Abstract and Applied Analysis 11

[5] E Kirk and J Aron ldquoCorrelation in the energy marketsrdquo inManaging Energy Price Risk V Kaminski Ed pp 71ndash78 RiskBooks London UK 1995

[6] C Alexander andA Venkatramanan ldquoAnalytic approximationsfor spread optionsrdquo ICMACentre Discussion Papers in FinanceDP2007-11 ICMA Centre 2007

[7] N D Pearson ldquoAn efficient approach for pricing spreadoptionsrdquoThe Journal of Derivatives vol 3 no 1 pp 76ndash91 1995

[8] C Chiarella and J Ziveyi ldquoPricingAmerican options written ontwo underlying assetsrdquo Quantitative Finance vol 14 no 3 pp409ndash426 2014

[9] J Ziveyi The evaluation of early exercise exotic options [PhDthesis] University of Technology Sydney Australia 2011

[10] P Boyle ldquoA lattice framework for option pricing with two statevariablesrdquo Journal of Financial and Quantitative Analysis vol23 no 1 pp 1ndash12 1988

[11] M Rubinstein ldquoReturn to Ozrdquo RISK vol 7 no 11 pp 67ndash711994

[12] R Company L Jodar M Fakharany and M-C CasabanldquoRemoving the correlation term in option pricing Hestonmodel numerical analysis and computingrdquo Abstract andApplied Analysis vol 2013 Article ID 246724 11 pages 2013

[13] S Salmi J Toivanen and L von Sydow ldquoIterative methodsfor pricing american options under the bates modelrdquo ProcediaComputer Science vol 18 pp 1136ndash1144 2013

[14] J Toivanen ldquoA componentwise splitting method for pricingamerican options under the bates modelrdquo in Applied andNumerical Partial Differential Equations vol 15 of Computa-tional Methods in Applied Sciences pp 213ndash227 Springer 2010

[15] R Zvan P A Forsyth and K R Vetzal ldquoNegative coefficientsin two-factor option pricing modelsrdquo Journal of ComputationalFinance vol 7 no 1 pp 37ndash73 2003

[16] B During M Fournie and C Heuer ldquoHigh-order compactfinite difference schemes for option pricing in stochastic volatil-ity models on non-uniform gridsrdquo Journal of Computationaland Applied Mathematics vol 271 pp 247ndash266 2014

[17] K J in rsquot Hout and C Mishra ldquoStability of ADI schemes formultidimensional diffusion equations with mixed derivativetermsrdquoApplied NumericalMathematics vol 74 pp 83ndash94 2013

[18] C Chiarella B Kang G H Mayer and A Ziogas ldquoTheevaluation of American option prices under stochastic volatilityand jump-diffusion dynamics using the method of linesrdquoInternational Journal of Theoretical and Applied Finance vol 12no 3 pp 393ndash425 2009

[19] M Fakharany R Company and L Jodar ldquoPositive finite dif-ference schemes for a partial integro-differential option pricingmodelrdquo Applied Mathematics and Computation vol 249 pp320ndash332 2014

[20] P R Garabedian Partial Differential Equations AMS ChelseaPublishing Providence RI USA 1998

[21] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp SonsChichester UK 2006

[22] R U Seydel Tools for Computational Finance Springer 3rdedition 2006

[23] M Ehrhardt Discrete artificial boundary conditions [PhDthesis] Technische Universitat Berlin Berlin Germany 2001

[24] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[25] J C HullOptions Futures andOther Derivatives Prentice-HallUpper Saddle River NJ USA 2000

[26] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods Clarendon Press Oxford UK3rd edition 1985

[27] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[28] M A H Dempster and S S G Hong Spread Option Valuationand the Fast Fourier Transform WP 262000 Judge Institute ofManagement Studies University of Cambridge 2000

[29] J-Z Huang M G Subrahmanyam and G G Yu ldquoPricingand hedging american options a recursive integrationmethodrdquoReview of Financial Studies vol 9 no 1 pp 277ndash300 1996

[30] A Ibanez and F Zapatero ldquoMonte Carlo valuation of Americanoptions through computation of the optimal exercise frontierrdquoJournal of Financial and Quantitative Analysis vol 39 no 2 pp253ndash275 2004

[31] V Surkov ldquoParallel option pricing with fourier space time-stepping method on graphics processingrdquo in Proceedings of the1st International Workshop on Parallel and Distributed Comput-ing in Finance (PDCoF rsquo08) Miami Fla USA 2008

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

10 Abstract and Applied Analysis

Table 2 Comparing several methods for American option pricing

119864 Analytic FFT MC FDM0 8513201 8513079 8516613 85081982 7542296 7542242 7545496 75340644 6653060 6652976 6656364 6648792

Example 4 Let us consider the American spread call optionproblem with parameters

119879 = 11205901 = 011205902 = 02119903 = 011199021 = 0051199022 = 005120588 = 05

(88)

Table 2 shows the option price at fixed asset prices 1198781 = 96and 1198782 = 100 for different values of strike price 119864 evaluatedby one-dimensional integration analytic method (Analytic)Fast Fourier Transform (FFT) Monte Carlo (MC) methodand our proposed method The accuracy is competitive withother methods such as Fourier Transform with high numberof discretization (4096) andMonte Carlo with big number oftime steps (2000)

Another set of parameters can be considered in order tocompare it with data in [9]

119879 = 05119864 = 1001205901 = 0251205902 = 03119903 = 0031199021 = 0061199022 = 002120588 = 05

(89)

In [9] three methods are considered to evaluate thespread option price for data 74 and fixed values 1198781 = 100and 1198782 = 200 a numerical integration method (NIM) amethod of lines (MOL) and a Monte Carlo approach NIMused in [9] is based in the recursive integration techniquesdeveloped byHuang et al [29] InNIM the option is treated asa Bermudan that is an option that can be exercised at discretepoints of time and Simpsons rule is used For the experimentthe authors used 1024 spatial nodes see [8 9] for moredetails The MOL is implemented by a spatial discretizationof 1438 times 200 mesh points and a three-time level scheme isused for the integration in time [9] 50 000 simulations have

Table 3 CPU time in sec (first row) and absolute difference (secondrow) for different methods depending on number of time steps forfixed number of space steps for parameters (89)

119873120591 NIM MOL Monte Carlo FDM

50 296598 5920 47 970 265470003 0001 01735 00036

32 127069 5512 19 811 1791100035 0006 01148 00122

16 33493 51718 5 149 943200012 00236 0132 00371

8 10360 4901 1 350 478700105 0074 01316 00907

4 4117 4547 35575 281500364 02512 01351 02254

been used in the Monte Carlo method described in [30] Inour method FDM the step size discretization considered isℎ = 01 To compare FDM with the three previous methodsabsolute errors are computed in Table 3 with respect to areference value obtained by the Fourier space time-steppingapproach of Surkov [31] with large number of nodes 4096spatial nodes and 300 time steps From Table 3 we can statethat the proposed method has the same order of accuracy asMOL more efficient and requires less computational timethan Monte Carlo method It is slightly less efficient than theNIM but we prove that it possesses a very highly desirablequalitative behaviour in finance

Competing Interests

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

Acknowledgments

This work has been partially supported by the EuropeanUnion in the FP7- PEOPLE-2012-ITN Program under GrantAgreement no 304617 (FP7 Marie Curie Action ProjectMulti-ITN STRIKE-Novel Methods in ComputationalFinance) and the Ministerio de Economıa y CompetitividadSpanish Grant MTM2013-41765-P

References

[1] R Carmona and V Durrleman ldquoPricing and hedging spreadoptionsrdquo SIAM Review vol 45 no 4 pp 627ndash685 2003

[2] P P Boyle ldquoOptions a Monte Carlo approachrdquo Journal ofFinancial Economics vol 4 no 3 pp 323ndash338 1977

[3] C Joy P P Boyle and K S Tan ldquoQuasi-Monte Carlo methodsin numerical financerdquo Management Science vol 42 no 6 pp926ndash938 1996

[4] W-H Kao Y-D Lyuu and K-W Wen ldquoThe hexanomiallattice for pricingmulti-asset optionsrdquoAppliedMathematics andComputation vol 233 pp 463ndash479 2014

Abstract and Applied Analysis 11

[5] E Kirk and J Aron ldquoCorrelation in the energy marketsrdquo inManaging Energy Price Risk V Kaminski Ed pp 71ndash78 RiskBooks London UK 1995

[6] C Alexander andA Venkatramanan ldquoAnalytic approximationsfor spread optionsrdquo ICMACentre Discussion Papers in FinanceDP2007-11 ICMA Centre 2007

[7] N D Pearson ldquoAn efficient approach for pricing spreadoptionsrdquoThe Journal of Derivatives vol 3 no 1 pp 76ndash91 1995

[8] C Chiarella and J Ziveyi ldquoPricingAmerican options written ontwo underlying assetsrdquo Quantitative Finance vol 14 no 3 pp409ndash426 2014

[9] J Ziveyi The evaluation of early exercise exotic options [PhDthesis] University of Technology Sydney Australia 2011

[10] P Boyle ldquoA lattice framework for option pricing with two statevariablesrdquo Journal of Financial and Quantitative Analysis vol23 no 1 pp 1ndash12 1988

[11] M Rubinstein ldquoReturn to Ozrdquo RISK vol 7 no 11 pp 67ndash711994

[12] R Company L Jodar M Fakharany and M-C CasabanldquoRemoving the correlation term in option pricing Hestonmodel numerical analysis and computingrdquo Abstract andApplied Analysis vol 2013 Article ID 246724 11 pages 2013

[13] S Salmi J Toivanen and L von Sydow ldquoIterative methodsfor pricing american options under the bates modelrdquo ProcediaComputer Science vol 18 pp 1136ndash1144 2013

[14] J Toivanen ldquoA componentwise splitting method for pricingamerican options under the bates modelrdquo in Applied andNumerical Partial Differential Equations vol 15 of Computa-tional Methods in Applied Sciences pp 213ndash227 Springer 2010

[15] R Zvan P A Forsyth and K R Vetzal ldquoNegative coefficientsin two-factor option pricing modelsrdquo Journal of ComputationalFinance vol 7 no 1 pp 37ndash73 2003

[16] B During M Fournie and C Heuer ldquoHigh-order compactfinite difference schemes for option pricing in stochastic volatil-ity models on non-uniform gridsrdquo Journal of Computationaland Applied Mathematics vol 271 pp 247ndash266 2014

[17] K J in rsquot Hout and C Mishra ldquoStability of ADI schemes formultidimensional diffusion equations with mixed derivativetermsrdquoApplied NumericalMathematics vol 74 pp 83ndash94 2013

[18] C Chiarella B Kang G H Mayer and A Ziogas ldquoTheevaluation of American option prices under stochastic volatilityand jump-diffusion dynamics using the method of linesrdquoInternational Journal of Theoretical and Applied Finance vol 12no 3 pp 393ndash425 2009

[19] M Fakharany R Company and L Jodar ldquoPositive finite dif-ference schemes for a partial integro-differential option pricingmodelrdquo Applied Mathematics and Computation vol 249 pp320ndash332 2014

[20] P R Garabedian Partial Differential Equations AMS ChelseaPublishing Providence RI USA 1998

[21] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp SonsChichester UK 2006

[22] R U Seydel Tools for Computational Finance Springer 3rdedition 2006

[23] M Ehrhardt Discrete artificial boundary conditions [PhDthesis] Technische Universitat Berlin Berlin Germany 2001

[24] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[25] J C HullOptions Futures andOther Derivatives Prentice-HallUpper Saddle River NJ USA 2000

[26] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods Clarendon Press Oxford UK3rd edition 1985

[27] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[28] M A H Dempster and S S G Hong Spread Option Valuationand the Fast Fourier Transform WP 262000 Judge Institute ofManagement Studies University of Cambridge 2000

[29] J-Z Huang M G Subrahmanyam and G G Yu ldquoPricingand hedging american options a recursive integrationmethodrdquoReview of Financial Studies vol 9 no 1 pp 277ndash300 1996

[30] A Ibanez and F Zapatero ldquoMonte Carlo valuation of Americanoptions through computation of the optimal exercise frontierrdquoJournal of Financial and Quantitative Analysis vol 39 no 2 pp253ndash275 2004

[31] V Surkov ldquoParallel option pricing with fourier space time-stepping method on graphics processingrdquo in Proceedings of the1st International Workshop on Parallel and Distributed Comput-ing in Finance (PDCoF rsquo08) Miami Fla USA 2008

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

Abstract and Applied Analysis 11

[5] E Kirk and J Aron ldquoCorrelation in the energy marketsrdquo inManaging Energy Price Risk V Kaminski Ed pp 71ndash78 RiskBooks London UK 1995

[6] C Alexander andA Venkatramanan ldquoAnalytic approximationsfor spread optionsrdquo ICMACentre Discussion Papers in FinanceDP2007-11 ICMA Centre 2007

[7] N D Pearson ldquoAn efficient approach for pricing spreadoptionsrdquoThe Journal of Derivatives vol 3 no 1 pp 76ndash91 1995

[8] C Chiarella and J Ziveyi ldquoPricingAmerican options written ontwo underlying assetsrdquo Quantitative Finance vol 14 no 3 pp409ndash426 2014

[9] J Ziveyi The evaluation of early exercise exotic options [PhDthesis] University of Technology Sydney Australia 2011

[10] P Boyle ldquoA lattice framework for option pricing with two statevariablesrdquo Journal of Financial and Quantitative Analysis vol23 no 1 pp 1ndash12 1988

[11] M Rubinstein ldquoReturn to Ozrdquo RISK vol 7 no 11 pp 67ndash711994

[12] R Company L Jodar M Fakharany and M-C CasabanldquoRemoving the correlation term in option pricing Hestonmodel numerical analysis and computingrdquo Abstract andApplied Analysis vol 2013 Article ID 246724 11 pages 2013

[13] S Salmi J Toivanen and L von Sydow ldquoIterative methodsfor pricing american options under the bates modelrdquo ProcediaComputer Science vol 18 pp 1136ndash1144 2013

[14] J Toivanen ldquoA componentwise splitting method for pricingamerican options under the bates modelrdquo in Applied andNumerical Partial Differential Equations vol 15 of Computa-tional Methods in Applied Sciences pp 213ndash227 Springer 2010

[15] R Zvan P A Forsyth and K R Vetzal ldquoNegative coefficientsin two-factor option pricing modelsrdquo Journal of ComputationalFinance vol 7 no 1 pp 37ndash73 2003

[16] B During M Fournie and C Heuer ldquoHigh-order compactfinite difference schemes for option pricing in stochastic volatil-ity models on non-uniform gridsrdquo Journal of Computationaland Applied Mathematics vol 271 pp 247ndash266 2014

[17] K J in rsquot Hout and C Mishra ldquoStability of ADI schemes formultidimensional diffusion equations with mixed derivativetermsrdquoApplied NumericalMathematics vol 74 pp 83ndash94 2013

[18] C Chiarella B Kang G H Mayer and A Ziogas ldquoTheevaluation of American option prices under stochastic volatilityand jump-diffusion dynamics using the method of linesrdquoInternational Journal of Theoretical and Applied Finance vol 12no 3 pp 393ndash425 2009

[19] M Fakharany R Company and L Jodar ldquoPositive finite dif-ference schemes for a partial integro-differential option pricingmodelrdquo Applied Mathematics and Computation vol 249 pp320ndash332 2014

[20] P R Garabedian Partial Differential Equations AMS ChelseaPublishing Providence RI USA 1998

[21] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp SonsChichester UK 2006

[22] R U Seydel Tools for Computational Finance Springer 3rdedition 2006

[23] M Ehrhardt Discrete artificial boundary conditions [PhDthesis] Technische Universitat Berlin Berlin Germany 2001

[24] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[25] J C HullOptions Futures andOther Derivatives Prentice-HallUpper Saddle River NJ USA 2000

[26] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods Clarendon Press Oxford UK3rd edition 1985

[27] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[28] M A H Dempster and S S G Hong Spread Option Valuationand the Fast Fourier Transform WP 262000 Judge Institute ofManagement Studies University of Cambridge 2000

[29] J-Z Huang M G Subrahmanyam and G G Yu ldquoPricingand hedging american options a recursive integrationmethodrdquoReview of Financial Studies vol 9 no 1 pp 277ndash300 1996

[30] A Ibanez and F Zapatero ldquoMonte Carlo valuation of Americanoptions through computation of the optimal exercise frontierrdquoJournal of Financial and Quantitative Analysis vol 39 no 2 pp253ndash275 2004

[31] V Surkov ldquoParallel option pricing with fourier space time-stepping method on graphics processingrdquo in Proceedings of the1st International Workshop on Parallel and Distributed Comput-ing in Finance (PDCoF rsquo08) Miami Fla USA 2008

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

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