Research Article Portfolio Strategy of Financial Market with ...Research Article Portfolio Strategy...
10
Research Article Portfolio Strategy of Financial Market with Regime Switching Driven by Geometric Lévy Process Liuwei Zhou and Zhijie Wang College of Information Science and Technology, Donghua University, Shanghai 201620, China Correspondence should be addressed to Liuwei Zhou; [email protected] Received 18 January 2014; Accepted 24 February 2014; Published 25 March 2014 Academic Editor: Zhengguang Wu Copyright © 2014 L. Zhou and Z. Wang. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e problem of a portfolio strategy for financial market with regime switching driven by geometric L´ evy process is investigated in this paper. e considered financial market includes one bond and multiple stocks which has few researches up to now. A new and general Black-Scholes (B-S) model is set up, in which the interest rate of the bond, the rate of return, and the volatility of the stocks vary as the market states switching and the stock prices are driven by geometric L´ evy process. For the general B-S model of the financial market, a portfolio strategy which is determined by a partial differential equation (PDE) of parabolic type is given by using Itˆ o formula. e PDE is an extension of existing result. e solvability of the PDE is researched by making use of variables transformation. An application of the solvability of the PDE on the European options with the final data is given finally. 1. Introduction To make a portfolio strategy is to search for a best allocation of wealth among different assets in markets. Taking the Euro- pean options, for instance, how to distribute the appropriate proportions of each option to maximize total returns at expire time is the core of portfolio strategy problem. ere are two points mentioned among the relevant literatures for portfolio selection problems: setting up a market model that approximates to the real financial market and the way of solving it. Portfolio strategy researches are based on portfolio selection analysis given by Markowitz [1]. Extension of Markowitz’s work to the multiperiod model has given by Li and Ng [2] which derived the analytical optimal portfolio policy. ese previous researches were assuming that the underlying market has only one state or mode. But the real market might have more than one state and could switch among them. en, portfolio policies under regime switching have been widely discussed. In a financial market model, the key process that models the evolution of stock price should be a Brownian motion. Indeed, this can be intuitively justified on the basis of the central limit theorem if one perceives the movement of stocks. e analysis of Øksendal [3] was mainly based on the generalized Black-Scholes model which has two assets () and () as () = ()() and () = ()()+()()(), where () is a Brownian motion. In that case, Øksendal formulated optimal selling decision making as an optimal stopping problem and derived a closed-form solution. e underlying problem may be treated as a free boundary value problem, which was extended to incorporate possible regime switching by Guo and Zhang [4] and Pemy et al. [5] with the switching represented by a two-state Markov chain. e rate of return (t) in the above Black-Scholes models in [4, 5] is a Markov chain which is different from the general one. As an application, Wu and Li [6, 7] have given the strategy of multiperiod mean-variance portfolio selection with regime switching and a stochastic cash flow which depends on the states of a stochastic market following a discrete-time Markov chain. Being put in the Markov jump, Black-Scholes model with regime switching is much closer to the real market. In recent years, L´ evy process as a more general process than Brownian motion has been applied in financial portfolio Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 538041, 9 pages http://dx.doi.org/10.1155/2014/538041
Transcript of Research Article Portfolio Strategy of Financial Market with ...Research Article Portfolio Strategy...
Research ArticlePortfolio Strategy of Financial Market with Regime SwitchingDriven by Geometric Leacutevy Process
Liuwei Zhou and Zhijie Wang
College of Information Science and Technology Donghua University Shanghai 201620 China
Correspondence should be addressed to Liuwei Zhou zhouliuweihotmailcom
Received 18 January 2014 Accepted 24 February 2014 Published 25 March 2014
Academic Editor Zhengguang Wu
Copyright copy 2014 L Zhou and Z Wang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The problem of a portfolio strategy for financial market with regime switching driven by geometric Levy process is investigatedin this paper The considered financial market includes one bond and multiple stocks which has few researches up to now A newand general Black-Scholes (B-S) model is set up in which the interest rate of the bond the rate of return and the volatility of thestocks vary as the market states switching and the stock prices are driven by geometric Levy process For the general B-S model ofthe financial market a portfolio strategy which is determined by a partial differential equation (PDE) of parabolic type is given byusing Ito formula The PDE is an extension of existing result The solvability of the PDE is researched by making use of variablestransformation An application of the solvability of the PDE on the European options with the final data is given finally
1 Introduction
To make a portfolio strategy is to search for a best allocationof wealth among different assets in markets Taking the Euro-pean options for instance how to distribute the appropriateproportions of each option to maximize total returns atexpire time is the core of portfolio strategy problem Thereare two points mentioned among the relevant literatures forportfolio selection problems setting up a market model thatapproximates to the real financial market and the way ofsolving it
Portfolio strategy researches are based on portfolioselection analysis given by Markowitz [1] Extension ofMarkowitzrsquos work to the multiperiod model has given by Liand Ng [2] which derived the analytical optimal portfoliopolicy These previous researches were assuming that theunderlying market has only one state or mode But the realmarket might have more than one state and could switchamong themThen portfolio policies under regime switchinghave been widely discussed In a financial market model thekey process 119878 that models the evolution of stock price shouldbe a Brownianmotion Indeed this can be intuitively justified
on the basis of the central limit theorem if one perceivesthe movement of stocks The analysis of Oslashksendal [3] wasmainly based on the generalized Black-Scholes model whichhas two assets 119861(119905) and 119878(119905) as 119889119861(119905) = 120588(119905)119861(119905)119889119905 and119889119878(119905) = 120572(119905)119878(119905)119889119905+120573(119905)119878(119905)119889119882(119905) where119882(119905) is a Brownianmotion In that case Oslashksendal formulated optimal sellingdecisionmaking as an optimal stopping problem and deriveda closed-form solution The underlying problem may betreated as a free boundary value problemwhichwas extendedto incorporate possible regime switching by Guo and Zhang[4] and Pemy et al [5] with the switching represented by atwo-state Markov chain The rate of return 120572(t) in the aboveBlack-Scholes models in [4 5] is a Markov chain which isdifferent from the general one As an application Wu and Li[6 7] have given the strategy of multiperiod mean-varianceportfolio selection with regime switching and a stochasticcash flow which depends on the states of a stochastic marketfollowing a discrete-time Markov chain Being put in theMarkov jump Black-Scholes model with regime switching ismuch closer to the real market
In recent years Levy process as a more general processthan Brownianmotion has been applied in financial portfolio
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 538041 9 pageshttpdxdoiorg1011552014538041
2 Abstract and Applied Analysis
optimization Kallsen [8] gave an optimal portfolio strat-egy of securities market under exponential Levy processMore specific than exponential Levy process a financialmarket model with stock price following the geometric Levyprocess was discussed by Applebaum [9] in which a Levyprocess 119883(119905) and geometric Levy motion 119878(119905) = 119890
119883(119905) wereintroduced Taking 119883 to be a Levy process could force ourstock prices clearly not moving continuously and a morerealistic approach is that the stock price is allowed to havesmall jumps in small time intervals Some applications offinancial market driven by Levy process are taken on lifeinsurance Vandaele and Vanmaele [10] show the real risk-minimizing hedging strategy for unit-linked life insurance infinancial market driven by a Levy process while Weng [11]has analyzed the constant proportion portfolio insurance byassuming that the risky asset price follows a regime switchingexponential Levy process and obtained the analytical formsof the shortfall probability expected shortfall and expectedgain Optimizing proportional reinsurance and investmentpolicies in a multidimensional Levy-driven insurance modelis discussed by Bauerle and Blatter [12] Moreover under agenerally method Yuen and Yin [13] have considered theoptimal dividend problem for the insurance risk process ina general Levy process which shows that if the Levy density isa completely monotone function then the optimal dividendstrategy is a barrier strategy
Among all the above literatures those portfolios arealways based on one risk-free asset and only one riskyasset which may limit the chosen stocks However in a realfinancialmarket there always existsmore than one risky assetin a portfolio That is why we are going to extend the single-stock financial marketmodel to amultistock financial marketmodel driven by geometric Levy process which ismore closerto the real market than proposed portfolios cited above Inthis paper we set up a general Black-Scholes model withgeometric Levy process For the general Black-Scholes modelof the financial market a portfolio strategy which is deter-mined by a partial differential equation (PDE) of parabolictype is given by using Ito formula The solvability of the PDEis researched by making use of variables transformation Anapplication of the solvability of the PDE on the Europeanoptions with the final data is given finally The contributionsof this paper are as follows (i) The B-S market model isextended into general form in which the interest rate of thebond the rate of return and the volatility of the stock vary asthe market states switching and the stock prices are drivenby geometric Levy process (ii) The PDE determining theportfolio strategy and its solvability are extensions of theexisting results
2 Problem Formulation
Assume that (ΩF 119875) is a complete probability space andF119905 119905 ge 0 is a nondecreasing family of 120590-algebra subfields
of F 120572(119905) 119905 ge 0 denotes a Markov chain in (ΩF 119875) asthe regime of financial market for example the bull marketor bear market of a stock market Let 119872 = 1 2 119898 be
the regime space of this Markov chain and let Γ = (120574119894119895)119898times119898
be the transition rate matrix which is satisfying
119875 120572 (119905 + Δ) = 119895 | 120572 (119905) = 119894 = 120574119894119895Δ + 119900 (Δ) 119894 = 119895
1 + 120574119894119894Δ + 119900 (Δ) 119894 = 119895
(1)
where Δ gt 0 is the increment of time 120574119894119895ge 0 (119894 = 119895) 120574
119894119894=
minussum119898
119895 = 119894119895=1120574119894119895
In this paper we consider a financial market modeldriven by geometric Levy process The market consists ofone risk-free asset denoted by 119861 and 119899 risky assets denotedby 1198781 1198782 119878
119899 The price process of these assets obeys the
following dynamic equations in which the price process ofthe risky assets follows the geometric Levy process that is
119889119861 (119905) = 119861 (119905) 119903 (119905 120572 (119905)) 119889119905 119861 (0) = 1198610
119889119878119896 (119905) = 119878
119896 (119905) [120583119896 (119905 120572 (119905)) 119889119905 + 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+int119877minus0
119911119896 (119889119905 119889119911)]
119878119896 (0) = 119878
0
119896gt 0
(2)
where 119861(119905) is the price of 119861 with the interest rate 119903(119905 120572(119905))and 119878
119896(119905) is the price of 119878
119896with the expect rate of return
120583119896(119905 120572(119905)) and the volatility 120590
119896(119905 120572(119905)) which follow the
regime switching of financialmarket 1198781(119905) 1198782(119905) 119878
119899(119905) are
independent from each other119882119896(119905) is the Brownian motion
which is independent from 120572(119905) 119905 ge 0 119896(sdot sdot) is defined as
below
119896 (119889119905 119889119911) = 119873
119896 (119889119905 119889119911) minus 120578119896 (119889119911) 119889119905 (3)
where119873119896(119889119905 119889119911) and 120578
119896(119889119911)119889119905 indicate the number of jumps
and average number of jumps within time 119889119905 and jump range119889119911 of price process 119878
119896(119905) respectively That is
120578119896 (119889119911) 119889t = E [119873
119896 (119889119905 119889119911)] (4)
where E is the expectation operator Moreover we assumethat119873
119896(119889119905 119889119911) 120572(119905) and119882
119896(119905) (119896 = 1 2 119899) are indepen-
dent of each other
Remark 1 Thefinancemarketmodel (2) is an extension of theB-S market model in which the interest rate of the bond therate of return and the volatility of the stock vary as themarketstates switching and the stock prices are driven by geometricLevy process
For finance market model (2) we introduce the conceptof self-financing portfolio as follows
Definition 2 A self-financing portfolio (120593 120595) = (120593 1205951
1205952 120595
119899) for the financial market model (2) is a series of
predictable processes
120593 (119905)119905ge0 120595
119896 (119905)119905ge0 (119896 = 1 2 119899) (5)
Abstract and Applied Analysis 3
that is for each 119879 gt 0
int
119879
0
1003816100381610038161003816120593(119904)1003816100381610038161003816
2119889119904 +
119899
sum
119896=1
int
119879
0
1003816100381610038161003816120595119896(119904)1003816100381610038161003816
2119889119904 lt infin (6)
and the corresponding wealth process 119881(119905)119905ge0
defined by
119881 (119905) = 120593 (119905) 119861 (119905) +
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 119905 ge 0 (7)
is an Ito process satisfying
119889119881 (119905) = 120593 (119905) 119889119861 (119905) +
119899
sum
119896=1
120595119896 (119905) 119889119878119896 (119905) 119905 ge 0 (8)
Problem Formulation In this note we will propose a port-folio strategy for the financial market model (2) whichis determined by a partial differential equation (PDE) ofparabolic type by using Ito formula The solvability of thePDE is researched bymaking use of variables transformationFurthermore the relationship between the solution of thePDE and the wealth process will be discussed
3 Main Results and Proofs
In this section wewill give the following fundamental resultsFor the sake of simplification wewrite 119903(119905 120572(119905)) as 119903119891(119905 119878(119905))as 119891 and so forth
To obtain the main result we give the solution of (2) andthe characteristic of the derivation (8) of the wealth process
The exact solutions of 119861(119905) in (2) can be found as follows
119861 (119905) = 119861 (0) exp(int119905
0
119903 (119904 120572 (119904)) 119889119904) (9)
To solve the second equation in (2) for 119878119896(119905) it follows
from the Ito formula that
119889 ln 119878119896 (119905) =
1
119878119896 (119905)
[119878119896 (119905) 120583119896 (119905 120572 (119905)) 119889119905
+119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)]
minus1
2
1
1198782
119896(119905)
1198782
119896(119905) 1205902
119896(119905 120572 (119905)) 119889119905
+ int119877minus0
[ln (119878119896 (119905) + 119911119878119896 (119905))
minus ln (119878119896 (119905))] 119896 (119889119905 119889119911)
+ int119877minus0
[ ln (119878119896 (119905) + 119911119878119896 (119905)) minus ln (119878
119896 (119905))
minus119911119878119896 (119905)
1
119878119896 (119905)
] 120578119896 (119889119911) 119889119905
+
119904
sum
119895=1
120574119894119895ln (119878119896 (119905))
= [120583119896 (119905 120572 (119905)) minus
1
21205902
119896(119905 120572 (119905))] 119889119905
+ 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+ int119877minus0
ln (1 + 119911) 119896 (119889119905 119889119911)
+ int119877minus0
[ln (1 + 119911) minus 119911] 120578119896 (119889119911) 119889119905
(10)
Integrating both sides of the above equation from 0 to 119905 wehave
119878119896 (119905) = 119878
0
119896expint
119905
0
(120583119896 (119904 120572 (119904)) minus
1
21205902
119896(119904 120572 (119904))] 119889119904
+ int
119905
0
120590119896 (119904 120572 (119904)) 119889119882119896 (119904)
+ int
119905
0
int119877minus0
ln (1 + 119911) 119896 (119889119904 119889119911)
+ int
119905
0
int119877minus0
[ln (1 + 119911) minus 119911] 120578119896 (119889119911) 119889119904
(11)
Proposition 3 Consider the price model (2) of a financialmarket If a portfolio (120593 120595) is a self-financing strategy then thewealth process 119881(119905)
119905ge0defined by (7) satisfies
119889119881 (119905) = 119903 (119905 120572 (119905)) 119881 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) [120583119896 (119905 120572 (119905)) minus 119903 (119905 120572 (119905))
minusint119877minus0
119911120578119896 (119889119911)] 119889119905
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) int
119877minus0
119911119873119896 (119889119905 119889119911)
(12)
Conversely consider the model (2) of a financial market If apair (120593 120595) of predictable processes following the wealth process119881(119905)
119905ge0defined by formula (7) satisfies (12) then (120593 120595) is a
self-financing strategy
Proof Substituting (2) into (8) we have
119889119881 (119905) = 120593 (119905) 119889119861 (119905) +
119899
sum
119896=1
120595119896 (119905) 119889119878119896 (119905)
= 120593 (119905) 119861 (119905) 119903 (119905 120572 (119905)) 119889119905 +
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905)
4 Abstract and Applied Analysis
times [120583119896 (119905 120572 (119905)) 119889119905 + 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+ int119877minus0
119911119896 (119889119905 119889119911)]
= [119881 (119905) minus
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905)] 119903 (119905 120572 (119905))
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120583119896 (119905 120572 (119905)) 119889119905
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) int
119877minus0
119911119896 (119889119905 119889119911)
= 119903 (119905 120572 (119905)) 119881 (119905) +
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905)
times [120583119896 (119905 120572 (119905)) minus 119903 (119905 120572 (119905))
minusint119877minus0
119911120578119896 (119889119911)] 119889119905
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) int
119877minus0
119911119873119896 (119889119905 119889119911)
(13)
which is (12)Conversely from (2) and (12) we can obtain (8)This completes the proof of the above proposition
Now we give the following fundamental results
Theorem 4 Consider the model (2) of a financial marketAssume that the portfolio (120593 120595
1 1205952 120595
119899) is a self-financing
strategy and 119881(119905)119905ge0
is the wealth process defined by (7) andsum119899
119896=1120595119896119878119896int119877minus0
119911120578119896(119889119911) = sum
119899
119896=1int119877minus0
119911120595119896119878119896120578119896(119889119911) If there
exists a function 119891(119905 119878) of 11986212 class (the set of functions whichare once differentiable in 119905 and continuously twice differentiablein 119878) such that
119881 (119905) = 119891 (119905 119878 (119905)) 119905 isin [0 119879]
119878 (119905) = (1198781 (119905) 1198782 (119905) 119878119899 (119905))
(14)
which holds true then the portfolio (120593 1205951 1205952 120595
119899) satisfies
120593 (119905) =119891 minus (120597119891120597119878) 119878
119879
B (119905) 119905 ge 0 (15)
120595 (119905) = (120597119891
1205971198781
120597119891
1205971198782
120597119891
120597119878119899
) =120597119891
120597119878 119905 ge 0 (16)
and the function 119891(119905 119878) solves the following backward PDE ofparabolic type
120597119891
120597119905+ 119903
119899
sum
119896=1
120597119891
120597119878119896
119878119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895= 119903119891
119905 lt 119879 119878 gt 0
(17)
Moreover if 119881(119879) = 119892(119878(119879)) then the function 119891(119905 119878)
satisfies the following equation
119891 (119879 119878) = 119892 (119878) 119878 gt 0 (18)
For the converse part we assume that 119879 gt 0 If thereexists a function 119891(119905 119878) of 11986212 class such that (17) and (18)are satisfied then the process (120593 120595) defined by (16) and (15) isa self-financing strategy The wealth process 119881 = 119881(119905)
119905isin[0119879]
corresponding to (120593 120595) satisfies (14)
Proof We proof the direct part of Theorem 4 firstlyFor
119881 (119905) = 119891 (119905 119878 (119905)) (19)
by applying the Ito formula we can infer that
119889119881 (119905) =120597119891
120597119905(119905 119878 (119905)) 119889119905
+
119899
sum
119896=1
120597119891
120597119878119896
(119905 119878 (119905)) (119878119896120583119896119889119905 + 119878119896120590119896119889119882119896)
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
(119905 119878 (119905)) 119878119894120590119894119878119895120590119895119889119905
+
119899
sum
119896=1
int119877minus0
(119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)) 119896 (119889119905 119889119911)
+
119899
sum
119896=1
int119877minus0
[119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)
minus119911120597119891
120597119878119896
(119905 119878) 119878119896] 120578119896 (119889119911) 119889119905
+
119898
sum
119895=1
120574119894119895119891 (119905 119878 (119905))
= [
[
120597119891
120597119905+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120583119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
minus
119899
sum
119896=1
int119877minus0
119911120597119891
120597119878119896
119878119896120578119896 (119889119911)] 119889119905
+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120590119896119889119882119896
+
119899
sum
119896=1
int119877minus0
[119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)]119873 (119889119905 119889119911)
(20)
Abstract and Applied Analysis 5
On the other hand since our strategy is self-financing theformula (12) is satisfied
Thus the rate of return and the volatility in (20) and (12)should be coincided and hence
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 =
119899
sum
119896=1
120597119891
120597119878119896
(119905 119878) 119878119896120590119896
119903 (119905 120572 (119905)) 119891 (119905 119878) +
119899
sum
119896=1
120595119896119878119896(120583119896minus 119903)
=120597119891
120597119905+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120583119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
(21)
We can easily get 119878119896ge 0 from (11) which together with
the first equation of (21) and the independence of 119878119896(119896 =
1 2 119899) yields (16)From the first equation of (21) (7) and (14) we have
119903120593119861 = 119891 minus
119899
sum
119896=1
120597119891
120597119878119896
119878119896 (22)
So that
120593 =119891 minus sum
119899
119896=1(120597119891120597119878
119896) 119878119896
119861=119891 minus 119891119878119878119879
119861 (23)
Substituting (16) into the second equation of (21) we have
119903119891 minus
119899
sum
119896=1
120595119896119878119896119903 =
120597119891
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895 (24)
which is (17)Conversely assume that 119891 = 119891(119905 119878) is a 119862
12-classfunction which is a solution of the PDE (17) and that (120593 120595)is a process defined by (16) and (15)
Firstly we will show that a process 119881 = 119881(119905) 119905 isin [0 119879]defined by (7) satisfies the equation
119881 (119905) = 119891 (119905 119878 (119905)) 119905 isin [0 119879] (25)
In fact substituting formulas (16) and (15) into the right handside of (7) we have
119881 (119905) = 120593119861 +
119899
sum
119896=1
120595119896119878119896=119891 minus sum
119899
119896=1(120597119891120597119878
119896) 119878119896
119861119861
+
119899
sum
119896=1
120597119891
120597119878119896
119878119896= 119891 119905 ge 0
(26)
This proves (25)Next we will show that (120593 120595) is a self-financing strategy
that is (12) holdsBy applying the Ito formula to the process119881 and function
119891 we have that (20) is satisfied
Furthermore by (17)
120597119891
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895= 119903119891 minus 119903
119899
sum
119896=1
119878119896
120597119891
120597119878119896
120597119891
120597119905+
119899
sum
119896=1
119878119896120583119896
120597119891
120597119878119896
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
= 119903119891 +
119899
sum
119896=1
(120583119896minus 119903) 119878
119896
120597119891
120597119878119896
(27)
Then by (25) and (16) we have
119903119881 +
119899
sum
119896=1
120595119896119878119896(120583119896minus 119903) =
120597119891
120597119905+
119899
sum
119896=1
119878119896120583119896
120597119891
120597119878119896
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
(28)
119899
sum
119896=1
120595119896119878119896120590119896=
119899
sum
119896=1
120597119891
120597119878119896
119878119896120590119896 (29)
Those together with (16) yield that (20) implies (12) Theproof of Theorem 4 is completed
Remark 5 In order to determine the portfolio strategy (120601 120595)and obtain the final value 119881(119905) from Theorem 4 we shouldfind the solution of the PDF (17) with the final data (18)This is the key problem in the rest of this section Wehave the following result in terms of method of variablestransformation
Theorem 6 Let 119903(119905 120572(119905)) in (2) be a constant 119903 The function119891(119905 119878) 119905 le 119879 119878 gt 0 given by the following formula
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942119892 (0 0 119878
119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(30)
is a solution of the general Black-Scholes equation (17) with thefinal data (18)
Proof We are going to do some equivalent transformationsof general B-S equation (17) in order to get an appropriateequivalent equation with analytic solutions The procedurewill be divided into four steps
Step I Let
119891 (119905 1198781 119878
119899) = 119890119903(119905minus119879)
119902 (119905 ln 1198781minus (119903 minus
1
21205902
1) (119905 minus 119879)
ln 119878119899minus (119903 minus
1
21205902
119899) (119905 minus 119879))
(31)
6 Abstract and Applied Analysis
and denote 119910119894= ln 119878
119894minus (119903 minus (12)120590
2
119894)(119905 minus 119879) (119894 = 1 2 119899)
and then
120597119891
120597119905=
119889 (119890119903(119905minus119879)
)
119889119905119902 + 119890119903(119905minus119879)
119902119905
= 119890119903(119905minus119879)
119902 (119889119903
119889119905(119905 minus 119879) + 119903)
+ 119890119903(119905minus119879)
[119902119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
= 119903119890119903(119905minus119879)
119902 + 119890119903(119905minus119879)
119902119889119903
119889119905(119905 minus 119879)
+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
120597119891
120597119878119894
= 119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
1205972119891
120597119878119894120597119878119895
=
120597 (119890119903(119905minus119879)
(120597119902120597119910119894) (1119878
119894))
120597119878119895
=
119890119903(119905minus119879)
1205972119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119894 = 119895
119890119903(119905minus119879)
(1205972119902
120597119910119894120597119910119894
1
119878119894
1
119878119894
minus120597119902
120597119910119894
1
1198782
119894
) 119894 = 119895
(32)
Inserting the above formulas into (17) we get
119903119891 +119889119903
119889119905(119905 minus 119879) 119902119890
119903(119905minus119879)+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
+ 119903
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
119878119894+1
2
119899
sum
119894=1
119899
sum
119895=1
119890119903(119905minus119879) 120597
2119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119878119894120590119894119878119895120590119895
minus1
2
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1198782
119894
1
1198782
119894
1198782
119894= 119903119891
(33)
which can be simplified as
119889119903
119889119905(119905 minus 119879) 119902 +
120597119902
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119902
120597119910119894120597119910119895
120590119894120590119895= 0 (34)
The final data 119891(119879 119878) = 119892(119878) can be rewritten as
119902 (119879 119878) = 119892 (1198901198781 1198901198782 119890
119878119899) (35)
Step II We introduce another variable and a new function asfollows
120591 = 119879 minus 119905 gt 0 119905 = 119879 minus 120591 120591 ge 0 119905 le 119879
119902 (119905 119910) = 119906 (119879 minus 119905 119910) or 119906 (120591 119910) = 119902 (119879 minus 120591 119910)
(36)
It can be computed that
119902119905(119905 119910) = minus119906
120591(119879 minus 119905 119910)
120597119902
120597119910119894
=120597119906
120597119910119894
(119879 minus 119905 119910)
1205972119902
120597119910119894120597119910119895
=1205972119906
120597119910119894120597119910119895
(119879 minus 119905 119910)
(37)
Substituting the above formulas into (34) we get
119889119881
119889119905(119905 minus 119879) 119906 (119879 minus 119905 119910) minus 119906
120591(119879 minus 119905 119910) +
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= 0
119906 (0 119910) = 119892 (119890119910)
(38)
Since 119903(119905 120572(119905)) is assumed as a constant 119903 (38) can be changedinto
119906120591(120591 119910) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895= 0
119906 (0 119910) = 119892 (119890119910)
(39)
Step III We claim that the unique solution of (39) is
119906 (119905 1199101 1199102 119910
119899)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0) 119889119909
119894
(40)
In fact
119906120591(120591 119910) = minus
1
2radic2120587120591120591
times
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
times(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
Abstract and Applied Analysis 7
120597119906
120597119910119894
=1
radic2120587120591int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0)
times (minus119910119894minus 119909119894
1205902
119894120591
)119889119909119894
1205972119906
120597119910119894120597119910119895
=
0 119894 = 119895
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
119890minus(119910119894minus119909119894)221205902
119894120591
times((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909i 119894 = 119895
(41)
So
119906120591(120591 119892) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= minus1
2radic2120587120591120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
minus1
2
119899
sum
119894=1
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
times 119890minus(119910119894minus119909119894)221205902
119894120591((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909119894= 0
(42)
Step IV By introducing a change of variables 119911119894= 119909119894minus 119910119894
we have 119909119894= 119911119894+ 119910119894and 119889119909
119894= 119889119911119894 where 119911
119894isin (minusinfininfin) It
follows that
119906 (120591 119910)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119911119894+119910119894 0 0) 119889119911
119894
(43)
In order to get rid of the denominator 1205902119894120591 in the exponent in
the above formula we make another change of variables as
119911119894= 120590119894radic120591119909119894 (44)
So 119889119911119894= 120590119894radic120591119889119909
119894
Recalling the relationship between 119902 and 119906 described in(36) we therefore have
119902 (119905 119910)
=1
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+119910119894 0 0) 119889119909
119894
(45)
Hence by formula (31) we have
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+ln 119878119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(46)
Since 119890ln 119878 = 119878 then
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(47)
In this way we provedTheorem 6
4 A Financial Example
As an application we consider the European call option InTheorem 6 we have given the solution of the general B-Sequation (17) which depends on the final data (18) that is119891(119879 119904) = 119892(119904) More specific we take the final data 119892(119904) forthe European call option as
119892 (119878) = 119892 (119878minus1198961
1 119878minus1198962
2 119878
minus119896119899
119899) =
119899
sum
119894=1
(119878119894minus 119870i)+ (48)
where 119878119894gt 0 and 119870
119894gt 0 are the strike price of 119878
119894 Then we
have the following corollary fromTheorem 6
Corollary 7 For the European call option the solution to thegeneral Black-Scholes value problem (17) with the final data(48) is given by
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(49)
8 Abstract and Applied Analysis
where
minus119860119894=
(119903 minus 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radic119879 minus 119905
= 1198892
minus119860119894+ 120590119894radic119879 minus 119905 =
(119903 + 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radicT minus 119905
= 1198891
(50)
that is
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(1198892) (51)
In particular
119891 (0 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903119879
119899
sum
119894=1
119870119894Φ(1198892) (52)
Proof For a European call option we infer that
119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
gt 119870119894 (53)
Dividing (53) by 119878119894and taking the ln we get
120590119894radic119879 minus 119905119909
119894minus (119903 minus
1205902
119894
2) (119905 minus 119879) gt ln
119870119894
119878119894
(54)
that is
119909119894gt
ln (119870119894119878119894) minus (119903 minus 120590
2
1198942) (119879 minus 119905)
120590119894radic119879 minus 119905
= 119860119894 (55)
Hence from (30) and (48) it follows that
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
119860119894
119890minus1199092
1198942119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890(119903minus1205902
1198942)(119879minus119905)
119890minus1199092
1198942+120590119894radic119879minus119905119909
119894119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890minus(1205902
1198942)(119879minus119905)
times 119890minus(12)(119909
119894minus120590119894radic119879minus119905)
2
+(12)1205902
119894(119879minus119905)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894minus120590119894radic119879minus119905
119878119894119890minus11991122119889119911
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894
1
radic2120587int
minus119860119894+120590119894radic119879minus119905
minusinfin
119890minus1199092
1198942119889119909119894
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894
1
radic2120587int
minus119860119894
minusinfin
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(56)
where Φ(119905) is the probability distribution function of astandard Gaussion random variable119873(0 1) that is
Φ (119905) =1
radic2120587int
119905
minusinfin
119890minus11990922119889119909 119905 isin 119877 (57)
In this way we have proved Corollary 7
Remark 8 Theabove result is about the European call optionA similar representation to those from the above corollaryin the European put option case can be obtained by taking119892(119878) = sum
119899
119894=1(119870119894minus 119878119894)+ 119878119894gt 0 for some fixed 119870
119894gt 0
5 Conclusion
In this paper we have considered a financial market modelwith regime switching driven by geometric Levy processThis financial market model is based on the multiple riskyassets 119878
1 1198782 119878
119899driven by Levy process Ito formula and
equivalent transformation methods have been used to solvethis complicated financial market model An example of theportfolio strategy and the final value problem to applying ourmethod to the European call option has been given in the endof this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 9
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (Grants no 61075105 and no 71371046)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] B Oslashksendal Stochastic Differential Equations Springer 6thedition 2005
[4] X Guo and Q Zhang ldquoOptimal selling rules in a regimeswitching modelrdquo IEEE Transactions on Automatic Control vol50 no 9 pp 1450ndash1455 2005
[5] M Pemy Q Zhang and G G Yin ldquoLiquidation of a large blockof stock with regime switchingrdquoMathematical Finance vol 18no 4 pp 629ndash648 2008
[6] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with Markov regime switching and uncertain time-horizonrdquo Journal of Systems Science amp Complexity vol 24 no 1pp 140ndash155 2011
[7] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with regime switching and a stochastic cash flowrdquoInsuranceMathematics amp Economics vol 50 no 3 pp 371ndash3842012
[8] J Kallsen ldquoOptimal portfolios for exponential Levy processesrdquoMathematical Methods of Operations Research vol 51 no 3 pp357ndash374 2000
[9] D Applebaum Levy Processes and Stochastic Calculus vol 93Cambridge University Press Cambridge UK 2004
[10] N Vandaele and M Vanmaele ldquoA locally risk-minimizinghedging strategy for unit-linked life insurance contracts ina Levy process financial marketrdquo Insurance Mathematics ampEconomics vol 42 no 3 pp 1128ndash1137 2008
[11] C Weng ldquoConstant proportion portfolio insurance under aregime switching exponential Levy processrdquo Insurance Math-ematics amp Economics vol 52 no 3 pp 508ndash521 2013
[12] N Bauerle and A Blatter ldquoOptimal control and dependencemodeling of insurance portfolios with Levy dynamicsrdquo Insur-ance Mathematics amp Economics vol 48 no 3 pp 398ndash4052011
[13] K C Yuen and C Yin ldquoOn optimality of the barrier strategyfor a general Levy risk processrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 1700ndash1707 2011
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
2 Abstract and Applied Analysis
optimization Kallsen [8] gave an optimal portfolio strat-egy of securities market under exponential Levy processMore specific than exponential Levy process a financialmarket model with stock price following the geometric Levyprocess was discussed by Applebaum [9] in which a Levyprocess 119883(119905) and geometric Levy motion 119878(119905) = 119890
119883(119905) wereintroduced Taking 119883 to be a Levy process could force ourstock prices clearly not moving continuously and a morerealistic approach is that the stock price is allowed to havesmall jumps in small time intervals Some applications offinancial market driven by Levy process are taken on lifeinsurance Vandaele and Vanmaele [10] show the real risk-minimizing hedging strategy for unit-linked life insurance infinancial market driven by a Levy process while Weng [11]has analyzed the constant proportion portfolio insurance byassuming that the risky asset price follows a regime switchingexponential Levy process and obtained the analytical formsof the shortfall probability expected shortfall and expectedgain Optimizing proportional reinsurance and investmentpolicies in a multidimensional Levy-driven insurance modelis discussed by Bauerle and Blatter [12] Moreover under agenerally method Yuen and Yin [13] have considered theoptimal dividend problem for the insurance risk process ina general Levy process which shows that if the Levy density isa completely monotone function then the optimal dividendstrategy is a barrier strategy
Among all the above literatures those portfolios arealways based on one risk-free asset and only one riskyasset which may limit the chosen stocks However in a realfinancialmarket there always existsmore than one risky assetin a portfolio That is why we are going to extend the single-stock financial marketmodel to amultistock financial marketmodel driven by geometric Levy process which ismore closerto the real market than proposed portfolios cited above Inthis paper we set up a general Black-Scholes model withgeometric Levy process For the general Black-Scholes modelof the financial market a portfolio strategy which is deter-mined by a partial differential equation (PDE) of parabolictype is given by using Ito formula The solvability of the PDEis researched by making use of variables transformation Anapplication of the solvability of the PDE on the Europeanoptions with the final data is given finally The contributionsof this paper are as follows (i) The B-S market model isextended into general form in which the interest rate of thebond the rate of return and the volatility of the stock vary asthe market states switching and the stock prices are drivenby geometric Levy process (ii) The PDE determining theportfolio strategy and its solvability are extensions of theexisting results
2 Problem Formulation
Assume that (ΩF 119875) is a complete probability space andF119905 119905 ge 0 is a nondecreasing family of 120590-algebra subfields
of F 120572(119905) 119905 ge 0 denotes a Markov chain in (ΩF 119875) asthe regime of financial market for example the bull marketor bear market of a stock market Let 119872 = 1 2 119898 be
the regime space of this Markov chain and let Γ = (120574119894119895)119898times119898
be the transition rate matrix which is satisfying
119875 120572 (119905 + Δ) = 119895 | 120572 (119905) = 119894 = 120574119894119895Δ + 119900 (Δ) 119894 = 119895
1 + 120574119894119894Δ + 119900 (Δ) 119894 = 119895
(1)
where Δ gt 0 is the increment of time 120574119894119895ge 0 (119894 = 119895) 120574
119894119894=
minussum119898
119895 = 119894119895=1120574119894119895
In this paper we consider a financial market modeldriven by geometric Levy process The market consists ofone risk-free asset denoted by 119861 and 119899 risky assets denotedby 1198781 1198782 119878
119899 The price process of these assets obeys the
following dynamic equations in which the price process ofthe risky assets follows the geometric Levy process that is
119889119861 (119905) = 119861 (119905) 119903 (119905 120572 (119905)) 119889119905 119861 (0) = 1198610
119889119878119896 (119905) = 119878
119896 (119905) [120583119896 (119905 120572 (119905)) 119889119905 + 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+int119877minus0
119911119896 (119889119905 119889119911)]
119878119896 (0) = 119878
0
119896gt 0
(2)
where 119861(119905) is the price of 119861 with the interest rate 119903(119905 120572(119905))and 119878
119896(119905) is the price of 119878
119896with the expect rate of return
120583119896(119905 120572(119905)) and the volatility 120590
119896(119905 120572(119905)) which follow the
regime switching of financialmarket 1198781(119905) 1198782(119905) 119878
119899(119905) are
independent from each other119882119896(119905) is the Brownian motion
which is independent from 120572(119905) 119905 ge 0 119896(sdot sdot) is defined as
below
119896 (119889119905 119889119911) = 119873
119896 (119889119905 119889119911) minus 120578119896 (119889119911) 119889119905 (3)
where119873119896(119889119905 119889119911) and 120578
119896(119889119911)119889119905 indicate the number of jumps
and average number of jumps within time 119889119905 and jump range119889119911 of price process 119878
119896(119905) respectively That is
120578119896 (119889119911) 119889t = E [119873
119896 (119889119905 119889119911)] (4)
where E is the expectation operator Moreover we assumethat119873
119896(119889119905 119889119911) 120572(119905) and119882
119896(119905) (119896 = 1 2 119899) are indepen-
dent of each other
Remark 1 Thefinancemarketmodel (2) is an extension of theB-S market model in which the interest rate of the bond therate of return and the volatility of the stock vary as themarketstates switching and the stock prices are driven by geometricLevy process
For finance market model (2) we introduce the conceptof self-financing portfolio as follows
Definition 2 A self-financing portfolio (120593 120595) = (120593 1205951
1205952 120595
119899) for the financial market model (2) is a series of
predictable processes
120593 (119905)119905ge0 120595
119896 (119905)119905ge0 (119896 = 1 2 119899) (5)
Abstract and Applied Analysis 3
that is for each 119879 gt 0
int
119879
0
1003816100381610038161003816120593(119904)1003816100381610038161003816
2119889119904 +
119899
sum
119896=1
int
119879
0
1003816100381610038161003816120595119896(119904)1003816100381610038161003816
2119889119904 lt infin (6)
and the corresponding wealth process 119881(119905)119905ge0
defined by
119881 (119905) = 120593 (119905) 119861 (119905) +
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 119905 ge 0 (7)
is an Ito process satisfying
119889119881 (119905) = 120593 (119905) 119889119861 (119905) +
119899
sum
119896=1
120595119896 (119905) 119889119878119896 (119905) 119905 ge 0 (8)
Problem Formulation In this note we will propose a port-folio strategy for the financial market model (2) whichis determined by a partial differential equation (PDE) ofparabolic type by using Ito formula The solvability of thePDE is researched bymaking use of variables transformationFurthermore the relationship between the solution of thePDE and the wealth process will be discussed
3 Main Results and Proofs
In this section wewill give the following fundamental resultsFor the sake of simplification wewrite 119903(119905 120572(119905)) as 119903119891(119905 119878(119905))as 119891 and so forth
To obtain the main result we give the solution of (2) andthe characteristic of the derivation (8) of the wealth process
The exact solutions of 119861(119905) in (2) can be found as follows
119861 (119905) = 119861 (0) exp(int119905
0
119903 (119904 120572 (119904)) 119889119904) (9)
To solve the second equation in (2) for 119878119896(119905) it follows
from the Ito formula that
119889 ln 119878119896 (119905) =
1
119878119896 (119905)
[119878119896 (119905) 120583119896 (119905 120572 (119905)) 119889119905
+119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)]
minus1
2
1
1198782
119896(119905)
1198782
119896(119905) 1205902
119896(119905 120572 (119905)) 119889119905
+ int119877minus0
[ln (119878119896 (119905) + 119911119878119896 (119905))
minus ln (119878119896 (119905))] 119896 (119889119905 119889119911)
+ int119877minus0
[ ln (119878119896 (119905) + 119911119878119896 (119905)) minus ln (119878
119896 (119905))
minus119911119878119896 (119905)
1
119878119896 (119905)
] 120578119896 (119889119911) 119889119905
+
119904
sum
119895=1
120574119894119895ln (119878119896 (119905))
= [120583119896 (119905 120572 (119905)) minus
1
21205902
119896(119905 120572 (119905))] 119889119905
+ 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+ int119877minus0
ln (1 + 119911) 119896 (119889119905 119889119911)
+ int119877minus0
[ln (1 + 119911) minus 119911] 120578119896 (119889119911) 119889119905
(10)
Integrating both sides of the above equation from 0 to 119905 wehave
119878119896 (119905) = 119878
0
119896expint
119905
0
(120583119896 (119904 120572 (119904)) minus
1
21205902
119896(119904 120572 (119904))] 119889119904
+ int
119905
0
120590119896 (119904 120572 (119904)) 119889119882119896 (119904)
+ int
119905
0
int119877minus0
ln (1 + 119911) 119896 (119889119904 119889119911)
+ int
119905
0
int119877minus0
[ln (1 + 119911) minus 119911] 120578119896 (119889119911) 119889119904
(11)
Proposition 3 Consider the price model (2) of a financialmarket If a portfolio (120593 120595) is a self-financing strategy then thewealth process 119881(119905)
119905ge0defined by (7) satisfies
119889119881 (119905) = 119903 (119905 120572 (119905)) 119881 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) [120583119896 (119905 120572 (119905)) minus 119903 (119905 120572 (119905))
minusint119877minus0
119911120578119896 (119889119911)] 119889119905
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) int
119877minus0
119911119873119896 (119889119905 119889119911)
(12)
Conversely consider the model (2) of a financial market If apair (120593 120595) of predictable processes following the wealth process119881(119905)
119905ge0defined by formula (7) satisfies (12) then (120593 120595) is a
self-financing strategy
Proof Substituting (2) into (8) we have
119889119881 (119905) = 120593 (119905) 119889119861 (119905) +
119899
sum
119896=1
120595119896 (119905) 119889119878119896 (119905)
= 120593 (119905) 119861 (119905) 119903 (119905 120572 (119905)) 119889119905 +
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905)
4 Abstract and Applied Analysis
times [120583119896 (119905 120572 (119905)) 119889119905 + 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+ int119877minus0
119911119896 (119889119905 119889119911)]
= [119881 (119905) minus
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905)] 119903 (119905 120572 (119905))
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120583119896 (119905 120572 (119905)) 119889119905
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) int
119877minus0
119911119896 (119889119905 119889119911)
= 119903 (119905 120572 (119905)) 119881 (119905) +
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905)
times [120583119896 (119905 120572 (119905)) minus 119903 (119905 120572 (119905))
minusint119877minus0
119911120578119896 (119889119911)] 119889119905
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) int
119877minus0
119911119873119896 (119889119905 119889119911)
(13)
which is (12)Conversely from (2) and (12) we can obtain (8)This completes the proof of the above proposition
Now we give the following fundamental results
Theorem 4 Consider the model (2) of a financial marketAssume that the portfolio (120593 120595
1 1205952 120595
119899) is a self-financing
strategy and 119881(119905)119905ge0
is the wealth process defined by (7) andsum119899
119896=1120595119896119878119896int119877minus0
119911120578119896(119889119911) = sum
119899
119896=1int119877minus0
119911120595119896119878119896120578119896(119889119911) If there
exists a function 119891(119905 119878) of 11986212 class (the set of functions whichare once differentiable in 119905 and continuously twice differentiablein 119878) such that
119881 (119905) = 119891 (119905 119878 (119905)) 119905 isin [0 119879]
119878 (119905) = (1198781 (119905) 1198782 (119905) 119878119899 (119905))
(14)
which holds true then the portfolio (120593 1205951 1205952 120595
119899) satisfies
120593 (119905) =119891 minus (120597119891120597119878) 119878
119879
B (119905) 119905 ge 0 (15)
120595 (119905) = (120597119891
1205971198781
120597119891
1205971198782
120597119891
120597119878119899
) =120597119891
120597119878 119905 ge 0 (16)
and the function 119891(119905 119878) solves the following backward PDE ofparabolic type
120597119891
120597119905+ 119903
119899
sum
119896=1
120597119891
120597119878119896
119878119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895= 119903119891
119905 lt 119879 119878 gt 0
(17)
Moreover if 119881(119879) = 119892(119878(119879)) then the function 119891(119905 119878)
satisfies the following equation
119891 (119879 119878) = 119892 (119878) 119878 gt 0 (18)
For the converse part we assume that 119879 gt 0 If thereexists a function 119891(119905 119878) of 11986212 class such that (17) and (18)are satisfied then the process (120593 120595) defined by (16) and (15) isa self-financing strategy The wealth process 119881 = 119881(119905)
119905isin[0119879]
corresponding to (120593 120595) satisfies (14)
Proof We proof the direct part of Theorem 4 firstlyFor
119881 (119905) = 119891 (119905 119878 (119905)) (19)
by applying the Ito formula we can infer that
119889119881 (119905) =120597119891
120597119905(119905 119878 (119905)) 119889119905
+
119899
sum
119896=1
120597119891
120597119878119896
(119905 119878 (119905)) (119878119896120583119896119889119905 + 119878119896120590119896119889119882119896)
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
(119905 119878 (119905)) 119878119894120590119894119878119895120590119895119889119905
+
119899
sum
119896=1
int119877minus0
(119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)) 119896 (119889119905 119889119911)
+
119899
sum
119896=1
int119877minus0
[119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)
minus119911120597119891
120597119878119896
(119905 119878) 119878119896] 120578119896 (119889119911) 119889119905
+
119898
sum
119895=1
120574119894119895119891 (119905 119878 (119905))
= [
[
120597119891
120597119905+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120583119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
minus
119899
sum
119896=1
int119877minus0
119911120597119891
120597119878119896
119878119896120578119896 (119889119911)] 119889119905
+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120590119896119889119882119896
+
119899
sum
119896=1
int119877minus0
[119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)]119873 (119889119905 119889119911)
(20)
Abstract and Applied Analysis 5
On the other hand since our strategy is self-financing theformula (12) is satisfied
Thus the rate of return and the volatility in (20) and (12)should be coincided and hence
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 =
119899
sum
119896=1
120597119891
120597119878119896
(119905 119878) 119878119896120590119896
119903 (119905 120572 (119905)) 119891 (119905 119878) +
119899
sum
119896=1
120595119896119878119896(120583119896minus 119903)
=120597119891
120597119905+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120583119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
(21)
We can easily get 119878119896ge 0 from (11) which together with
the first equation of (21) and the independence of 119878119896(119896 =
1 2 119899) yields (16)From the first equation of (21) (7) and (14) we have
119903120593119861 = 119891 minus
119899
sum
119896=1
120597119891
120597119878119896
119878119896 (22)
So that
120593 =119891 minus sum
119899
119896=1(120597119891120597119878
119896) 119878119896
119861=119891 minus 119891119878119878119879
119861 (23)
Substituting (16) into the second equation of (21) we have
119903119891 minus
119899
sum
119896=1
120595119896119878119896119903 =
120597119891
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895 (24)
which is (17)Conversely assume that 119891 = 119891(119905 119878) is a 119862
12-classfunction which is a solution of the PDE (17) and that (120593 120595)is a process defined by (16) and (15)
Firstly we will show that a process 119881 = 119881(119905) 119905 isin [0 119879]defined by (7) satisfies the equation
119881 (119905) = 119891 (119905 119878 (119905)) 119905 isin [0 119879] (25)
In fact substituting formulas (16) and (15) into the right handside of (7) we have
119881 (119905) = 120593119861 +
119899
sum
119896=1
120595119896119878119896=119891 minus sum
119899
119896=1(120597119891120597119878
119896) 119878119896
119861119861
+
119899
sum
119896=1
120597119891
120597119878119896
119878119896= 119891 119905 ge 0
(26)
This proves (25)Next we will show that (120593 120595) is a self-financing strategy
that is (12) holdsBy applying the Ito formula to the process119881 and function
119891 we have that (20) is satisfied
Furthermore by (17)
120597119891
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895= 119903119891 minus 119903
119899
sum
119896=1
119878119896
120597119891
120597119878119896
120597119891
120597119905+
119899
sum
119896=1
119878119896120583119896
120597119891
120597119878119896
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
= 119903119891 +
119899
sum
119896=1
(120583119896minus 119903) 119878
119896
120597119891
120597119878119896
(27)
Then by (25) and (16) we have
119903119881 +
119899
sum
119896=1
120595119896119878119896(120583119896minus 119903) =
120597119891
120597119905+
119899
sum
119896=1
119878119896120583119896
120597119891
120597119878119896
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
(28)
119899
sum
119896=1
120595119896119878119896120590119896=
119899
sum
119896=1
120597119891
120597119878119896
119878119896120590119896 (29)
Those together with (16) yield that (20) implies (12) Theproof of Theorem 4 is completed
Remark 5 In order to determine the portfolio strategy (120601 120595)and obtain the final value 119881(119905) from Theorem 4 we shouldfind the solution of the PDF (17) with the final data (18)This is the key problem in the rest of this section Wehave the following result in terms of method of variablestransformation
Theorem 6 Let 119903(119905 120572(119905)) in (2) be a constant 119903 The function119891(119905 119878) 119905 le 119879 119878 gt 0 given by the following formula
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942119892 (0 0 119878
119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(30)
is a solution of the general Black-Scholes equation (17) with thefinal data (18)
Proof We are going to do some equivalent transformationsof general B-S equation (17) in order to get an appropriateequivalent equation with analytic solutions The procedurewill be divided into four steps
Step I Let
119891 (119905 1198781 119878
119899) = 119890119903(119905minus119879)
119902 (119905 ln 1198781minus (119903 minus
1
21205902
1) (119905 minus 119879)
ln 119878119899minus (119903 minus
1
21205902
119899) (119905 minus 119879))
(31)
6 Abstract and Applied Analysis
and denote 119910119894= ln 119878
119894minus (119903 minus (12)120590
2
119894)(119905 minus 119879) (119894 = 1 2 119899)
and then
120597119891
120597119905=
119889 (119890119903(119905minus119879)
)
119889119905119902 + 119890119903(119905minus119879)
119902119905
= 119890119903(119905minus119879)
119902 (119889119903
119889119905(119905 minus 119879) + 119903)
+ 119890119903(119905minus119879)
[119902119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
= 119903119890119903(119905minus119879)
119902 + 119890119903(119905minus119879)
119902119889119903
119889119905(119905 minus 119879)
+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
120597119891
120597119878119894
= 119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
1205972119891
120597119878119894120597119878119895
=
120597 (119890119903(119905minus119879)
(120597119902120597119910119894) (1119878
119894))
120597119878119895
=
119890119903(119905minus119879)
1205972119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119894 = 119895
119890119903(119905minus119879)
(1205972119902
120597119910119894120597119910119894
1
119878119894
1
119878119894
minus120597119902
120597119910119894
1
1198782
119894
) 119894 = 119895
(32)
Inserting the above formulas into (17) we get
119903119891 +119889119903
119889119905(119905 minus 119879) 119902119890
119903(119905minus119879)+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
+ 119903
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
119878119894+1
2
119899
sum
119894=1
119899
sum
119895=1
119890119903(119905minus119879) 120597
2119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119878119894120590119894119878119895120590119895
minus1
2
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1198782
119894
1
1198782
119894
1198782
119894= 119903119891
(33)
which can be simplified as
119889119903
119889119905(119905 minus 119879) 119902 +
120597119902
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119902
120597119910119894120597119910119895
120590119894120590119895= 0 (34)
The final data 119891(119879 119878) = 119892(119878) can be rewritten as
119902 (119879 119878) = 119892 (1198901198781 1198901198782 119890
119878119899) (35)
Step II We introduce another variable and a new function asfollows
120591 = 119879 minus 119905 gt 0 119905 = 119879 minus 120591 120591 ge 0 119905 le 119879
119902 (119905 119910) = 119906 (119879 minus 119905 119910) or 119906 (120591 119910) = 119902 (119879 minus 120591 119910)
(36)
It can be computed that
119902119905(119905 119910) = minus119906
120591(119879 minus 119905 119910)
120597119902
120597119910119894
=120597119906
120597119910119894
(119879 minus 119905 119910)
1205972119902
120597119910119894120597119910119895
=1205972119906
120597119910119894120597119910119895
(119879 minus 119905 119910)
(37)
Substituting the above formulas into (34) we get
119889119881
119889119905(119905 minus 119879) 119906 (119879 minus 119905 119910) minus 119906
120591(119879 minus 119905 119910) +
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= 0
119906 (0 119910) = 119892 (119890119910)
(38)
Since 119903(119905 120572(119905)) is assumed as a constant 119903 (38) can be changedinto
119906120591(120591 119910) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895= 0
119906 (0 119910) = 119892 (119890119910)
(39)
Step III We claim that the unique solution of (39) is
119906 (119905 1199101 1199102 119910
119899)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0) 119889119909
119894
(40)
In fact
119906120591(120591 119910) = minus
1
2radic2120587120591120591
times
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
times(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
Abstract and Applied Analysis 7
120597119906
120597119910119894
=1
radic2120587120591int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0)
times (minus119910119894minus 119909119894
1205902
119894120591
)119889119909119894
1205972119906
120597119910119894120597119910119895
=
0 119894 = 119895
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
119890minus(119910119894minus119909119894)221205902
119894120591
times((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909i 119894 = 119895
(41)
So
119906120591(120591 119892) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= minus1
2radic2120587120591120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
minus1
2
119899
sum
119894=1
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
times 119890minus(119910119894minus119909119894)221205902
119894120591((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909119894= 0
(42)
Step IV By introducing a change of variables 119911119894= 119909119894minus 119910119894
we have 119909119894= 119911119894+ 119910119894and 119889119909
119894= 119889119911119894 where 119911
119894isin (minusinfininfin) It
follows that
119906 (120591 119910)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119911119894+119910119894 0 0) 119889119911
119894
(43)
In order to get rid of the denominator 1205902119894120591 in the exponent in
the above formula we make another change of variables as
119911119894= 120590119894radic120591119909119894 (44)
So 119889119911119894= 120590119894radic120591119889119909
119894
Recalling the relationship between 119902 and 119906 described in(36) we therefore have
119902 (119905 119910)
=1
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+119910119894 0 0) 119889119909
119894
(45)
Hence by formula (31) we have
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+ln 119878119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(46)
Since 119890ln 119878 = 119878 then
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(47)
In this way we provedTheorem 6
4 A Financial Example
As an application we consider the European call option InTheorem 6 we have given the solution of the general B-Sequation (17) which depends on the final data (18) that is119891(119879 119904) = 119892(119904) More specific we take the final data 119892(119904) forthe European call option as
119892 (119878) = 119892 (119878minus1198961
1 119878minus1198962
2 119878
minus119896119899
119899) =
119899
sum
119894=1
(119878119894minus 119870i)+ (48)
where 119878119894gt 0 and 119870
119894gt 0 are the strike price of 119878
119894 Then we
have the following corollary fromTheorem 6
Corollary 7 For the European call option the solution to thegeneral Black-Scholes value problem (17) with the final data(48) is given by
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(49)
8 Abstract and Applied Analysis
where
minus119860119894=
(119903 minus 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radic119879 minus 119905
= 1198892
minus119860119894+ 120590119894radic119879 minus 119905 =
(119903 + 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radicT minus 119905
= 1198891
(50)
that is
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(1198892) (51)
In particular
119891 (0 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903119879
119899
sum
119894=1
119870119894Φ(1198892) (52)
Proof For a European call option we infer that
119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
gt 119870119894 (53)
Dividing (53) by 119878119894and taking the ln we get
120590119894radic119879 minus 119905119909
119894minus (119903 minus
1205902
119894
2) (119905 minus 119879) gt ln
119870119894
119878119894
(54)
that is
119909119894gt
ln (119870119894119878119894) minus (119903 minus 120590
2
1198942) (119879 minus 119905)
120590119894radic119879 minus 119905
= 119860119894 (55)
Hence from (30) and (48) it follows that
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
119860119894
119890minus1199092
1198942119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890(119903minus1205902
1198942)(119879minus119905)
119890minus1199092
1198942+120590119894radic119879minus119905119909
119894119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890minus(1205902
1198942)(119879minus119905)
times 119890minus(12)(119909
119894minus120590119894radic119879minus119905)
2
+(12)1205902
119894(119879minus119905)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894minus120590119894radic119879minus119905
119878119894119890minus11991122119889119911
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894
1
radic2120587int
minus119860119894+120590119894radic119879minus119905
minusinfin
119890minus1199092
1198942119889119909119894
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894
1
radic2120587int
minus119860119894
minusinfin
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(56)
where Φ(119905) is the probability distribution function of astandard Gaussion random variable119873(0 1) that is
Φ (119905) =1
radic2120587int
119905
minusinfin
119890minus11990922119889119909 119905 isin 119877 (57)
In this way we have proved Corollary 7
Remark 8 Theabove result is about the European call optionA similar representation to those from the above corollaryin the European put option case can be obtained by taking119892(119878) = sum
119899
119894=1(119870119894minus 119878119894)+ 119878119894gt 0 for some fixed 119870
119894gt 0
5 Conclusion
In this paper we have considered a financial market modelwith regime switching driven by geometric Levy processThis financial market model is based on the multiple riskyassets 119878
1 1198782 119878
119899driven by Levy process Ito formula and
equivalent transformation methods have been used to solvethis complicated financial market model An example of theportfolio strategy and the final value problem to applying ourmethod to the European call option has been given in the endof this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 9
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (Grants no 61075105 and no 71371046)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] B Oslashksendal Stochastic Differential Equations Springer 6thedition 2005
[4] X Guo and Q Zhang ldquoOptimal selling rules in a regimeswitching modelrdquo IEEE Transactions on Automatic Control vol50 no 9 pp 1450ndash1455 2005
[5] M Pemy Q Zhang and G G Yin ldquoLiquidation of a large blockof stock with regime switchingrdquoMathematical Finance vol 18no 4 pp 629ndash648 2008
[6] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with Markov regime switching and uncertain time-horizonrdquo Journal of Systems Science amp Complexity vol 24 no 1pp 140ndash155 2011
[7] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with regime switching and a stochastic cash flowrdquoInsuranceMathematics amp Economics vol 50 no 3 pp 371ndash3842012
[8] J Kallsen ldquoOptimal portfolios for exponential Levy processesrdquoMathematical Methods of Operations Research vol 51 no 3 pp357ndash374 2000
[9] D Applebaum Levy Processes and Stochastic Calculus vol 93Cambridge University Press Cambridge UK 2004
[10] N Vandaele and M Vanmaele ldquoA locally risk-minimizinghedging strategy for unit-linked life insurance contracts ina Levy process financial marketrdquo Insurance Mathematics ampEconomics vol 42 no 3 pp 1128ndash1137 2008
[11] C Weng ldquoConstant proportion portfolio insurance under aregime switching exponential Levy processrdquo Insurance Math-ematics amp Economics vol 52 no 3 pp 508ndash521 2013
[12] N Bauerle and A Blatter ldquoOptimal control and dependencemodeling of insurance portfolios with Levy dynamicsrdquo Insur-ance Mathematics amp Economics vol 48 no 3 pp 398ndash4052011
[13] K C Yuen and C Yin ldquoOn optimality of the barrier strategyfor a general Levy risk processrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 1700ndash1707 2011
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 3
that is for each 119879 gt 0
int
119879
0
1003816100381610038161003816120593(119904)1003816100381610038161003816
2119889119904 +
119899
sum
119896=1
int
119879
0
1003816100381610038161003816120595119896(119904)1003816100381610038161003816
2119889119904 lt infin (6)
and the corresponding wealth process 119881(119905)119905ge0
defined by
119881 (119905) = 120593 (119905) 119861 (119905) +
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 119905 ge 0 (7)
is an Ito process satisfying
119889119881 (119905) = 120593 (119905) 119889119861 (119905) +
119899
sum
119896=1
120595119896 (119905) 119889119878119896 (119905) 119905 ge 0 (8)
Problem Formulation In this note we will propose a port-folio strategy for the financial market model (2) whichis determined by a partial differential equation (PDE) ofparabolic type by using Ito formula The solvability of thePDE is researched bymaking use of variables transformationFurthermore the relationship between the solution of thePDE and the wealth process will be discussed
3 Main Results and Proofs
In this section wewill give the following fundamental resultsFor the sake of simplification wewrite 119903(119905 120572(119905)) as 119903119891(119905 119878(119905))as 119891 and so forth
To obtain the main result we give the solution of (2) andthe characteristic of the derivation (8) of the wealth process
The exact solutions of 119861(119905) in (2) can be found as follows
119861 (119905) = 119861 (0) exp(int119905
0
119903 (119904 120572 (119904)) 119889119904) (9)
To solve the second equation in (2) for 119878119896(119905) it follows
from the Ito formula that
119889 ln 119878119896 (119905) =
1
119878119896 (119905)
[119878119896 (119905) 120583119896 (119905 120572 (119905)) 119889119905
+119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)]
minus1
2
1
1198782
119896(119905)
1198782
119896(119905) 1205902
119896(119905 120572 (119905)) 119889119905
+ int119877minus0
[ln (119878119896 (119905) + 119911119878119896 (119905))
minus ln (119878119896 (119905))] 119896 (119889119905 119889119911)
+ int119877minus0
[ ln (119878119896 (119905) + 119911119878119896 (119905)) minus ln (119878
119896 (119905))
minus119911119878119896 (119905)
1
119878119896 (119905)
] 120578119896 (119889119911) 119889119905
+
119904
sum
119895=1
120574119894119895ln (119878119896 (119905))
= [120583119896 (119905 120572 (119905)) minus
1
21205902
119896(119905 120572 (119905))] 119889119905
+ 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+ int119877minus0
ln (1 + 119911) 119896 (119889119905 119889119911)
+ int119877minus0
[ln (1 + 119911) minus 119911] 120578119896 (119889119911) 119889119905
(10)
Integrating both sides of the above equation from 0 to 119905 wehave
119878119896 (119905) = 119878
0
119896expint
119905
0
(120583119896 (119904 120572 (119904)) minus
1
21205902
119896(119904 120572 (119904))] 119889119904
+ int
119905
0
120590119896 (119904 120572 (119904)) 119889119882119896 (119904)
+ int
119905
0
int119877minus0
ln (1 + 119911) 119896 (119889119904 119889119911)
+ int
119905
0
int119877minus0
[ln (1 + 119911) minus 119911] 120578119896 (119889119911) 119889119904
(11)
Proposition 3 Consider the price model (2) of a financialmarket If a portfolio (120593 120595) is a self-financing strategy then thewealth process 119881(119905)
119905ge0defined by (7) satisfies
119889119881 (119905) = 119903 (119905 120572 (119905)) 119881 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) [120583119896 (119905 120572 (119905)) minus 119903 (119905 120572 (119905))
minusint119877minus0
119911120578119896 (119889119911)] 119889119905
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) int
119877minus0
119911119873119896 (119889119905 119889119911)
(12)
Conversely consider the model (2) of a financial market If apair (120593 120595) of predictable processes following the wealth process119881(119905)
119905ge0defined by formula (7) satisfies (12) then (120593 120595) is a
self-financing strategy
Proof Substituting (2) into (8) we have
119889119881 (119905) = 120593 (119905) 119889119861 (119905) +
119899
sum
119896=1
120595119896 (119905) 119889119878119896 (119905)
= 120593 (119905) 119861 (119905) 119903 (119905 120572 (119905)) 119889119905 +
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905)
4 Abstract and Applied Analysis
times [120583119896 (119905 120572 (119905)) 119889119905 + 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+ int119877minus0
119911119896 (119889119905 119889119911)]
= [119881 (119905) minus
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905)] 119903 (119905 120572 (119905))
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120583119896 (119905 120572 (119905)) 119889119905
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) int
119877minus0
119911119896 (119889119905 119889119911)
= 119903 (119905 120572 (119905)) 119881 (119905) +
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905)
times [120583119896 (119905 120572 (119905)) minus 119903 (119905 120572 (119905))
minusint119877minus0
119911120578119896 (119889119911)] 119889119905
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) int
119877minus0
119911119873119896 (119889119905 119889119911)
(13)
which is (12)Conversely from (2) and (12) we can obtain (8)This completes the proof of the above proposition
Now we give the following fundamental results
Theorem 4 Consider the model (2) of a financial marketAssume that the portfolio (120593 120595
1 1205952 120595
119899) is a self-financing
strategy and 119881(119905)119905ge0
is the wealth process defined by (7) andsum119899
119896=1120595119896119878119896int119877minus0
119911120578119896(119889119911) = sum
119899
119896=1int119877minus0
119911120595119896119878119896120578119896(119889119911) If there
exists a function 119891(119905 119878) of 11986212 class (the set of functions whichare once differentiable in 119905 and continuously twice differentiablein 119878) such that
119881 (119905) = 119891 (119905 119878 (119905)) 119905 isin [0 119879]
119878 (119905) = (1198781 (119905) 1198782 (119905) 119878119899 (119905))
(14)
which holds true then the portfolio (120593 1205951 1205952 120595
119899) satisfies
120593 (119905) =119891 minus (120597119891120597119878) 119878
119879
B (119905) 119905 ge 0 (15)
120595 (119905) = (120597119891
1205971198781
120597119891
1205971198782
120597119891
120597119878119899
) =120597119891
120597119878 119905 ge 0 (16)
and the function 119891(119905 119878) solves the following backward PDE ofparabolic type
120597119891
120597119905+ 119903
119899
sum
119896=1
120597119891
120597119878119896
119878119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895= 119903119891
119905 lt 119879 119878 gt 0
(17)
Moreover if 119881(119879) = 119892(119878(119879)) then the function 119891(119905 119878)
satisfies the following equation
119891 (119879 119878) = 119892 (119878) 119878 gt 0 (18)
For the converse part we assume that 119879 gt 0 If thereexists a function 119891(119905 119878) of 11986212 class such that (17) and (18)are satisfied then the process (120593 120595) defined by (16) and (15) isa self-financing strategy The wealth process 119881 = 119881(119905)
119905isin[0119879]
corresponding to (120593 120595) satisfies (14)
Proof We proof the direct part of Theorem 4 firstlyFor
119881 (119905) = 119891 (119905 119878 (119905)) (19)
by applying the Ito formula we can infer that
119889119881 (119905) =120597119891
120597119905(119905 119878 (119905)) 119889119905
+
119899
sum
119896=1
120597119891
120597119878119896
(119905 119878 (119905)) (119878119896120583119896119889119905 + 119878119896120590119896119889119882119896)
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
(119905 119878 (119905)) 119878119894120590119894119878119895120590119895119889119905
+
119899
sum
119896=1
int119877minus0
(119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)) 119896 (119889119905 119889119911)
+
119899
sum
119896=1
int119877minus0
[119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)
minus119911120597119891
120597119878119896
(119905 119878) 119878119896] 120578119896 (119889119911) 119889119905
+
119898
sum
119895=1
120574119894119895119891 (119905 119878 (119905))
= [
[
120597119891
120597119905+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120583119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
minus
119899
sum
119896=1
int119877minus0
119911120597119891
120597119878119896
119878119896120578119896 (119889119911)] 119889119905
+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120590119896119889119882119896
+
119899
sum
119896=1
int119877minus0
[119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)]119873 (119889119905 119889119911)
(20)
Abstract and Applied Analysis 5
On the other hand since our strategy is self-financing theformula (12) is satisfied
Thus the rate of return and the volatility in (20) and (12)should be coincided and hence
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 =
119899
sum
119896=1
120597119891
120597119878119896
(119905 119878) 119878119896120590119896
119903 (119905 120572 (119905)) 119891 (119905 119878) +
119899
sum
119896=1
120595119896119878119896(120583119896minus 119903)
=120597119891
120597119905+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120583119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
(21)
We can easily get 119878119896ge 0 from (11) which together with
the first equation of (21) and the independence of 119878119896(119896 =
1 2 119899) yields (16)From the first equation of (21) (7) and (14) we have
119903120593119861 = 119891 minus
119899
sum
119896=1
120597119891
120597119878119896
119878119896 (22)
So that
120593 =119891 minus sum
119899
119896=1(120597119891120597119878
119896) 119878119896
119861=119891 minus 119891119878119878119879
119861 (23)
Substituting (16) into the second equation of (21) we have
119903119891 minus
119899
sum
119896=1
120595119896119878119896119903 =
120597119891
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895 (24)
which is (17)Conversely assume that 119891 = 119891(119905 119878) is a 119862
12-classfunction which is a solution of the PDE (17) and that (120593 120595)is a process defined by (16) and (15)
Firstly we will show that a process 119881 = 119881(119905) 119905 isin [0 119879]defined by (7) satisfies the equation
119881 (119905) = 119891 (119905 119878 (119905)) 119905 isin [0 119879] (25)
In fact substituting formulas (16) and (15) into the right handside of (7) we have
119881 (119905) = 120593119861 +
119899
sum
119896=1
120595119896119878119896=119891 minus sum
119899
119896=1(120597119891120597119878
119896) 119878119896
119861119861
+
119899
sum
119896=1
120597119891
120597119878119896
119878119896= 119891 119905 ge 0
(26)
This proves (25)Next we will show that (120593 120595) is a self-financing strategy
that is (12) holdsBy applying the Ito formula to the process119881 and function
119891 we have that (20) is satisfied
Furthermore by (17)
120597119891
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895= 119903119891 minus 119903
119899
sum
119896=1
119878119896
120597119891
120597119878119896
120597119891
120597119905+
119899
sum
119896=1
119878119896120583119896
120597119891
120597119878119896
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
= 119903119891 +
119899
sum
119896=1
(120583119896minus 119903) 119878
119896
120597119891
120597119878119896
(27)
Then by (25) and (16) we have
119903119881 +
119899
sum
119896=1
120595119896119878119896(120583119896minus 119903) =
120597119891
120597119905+
119899
sum
119896=1
119878119896120583119896
120597119891
120597119878119896
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
(28)
119899
sum
119896=1
120595119896119878119896120590119896=
119899
sum
119896=1
120597119891
120597119878119896
119878119896120590119896 (29)
Those together with (16) yield that (20) implies (12) Theproof of Theorem 4 is completed
Remark 5 In order to determine the portfolio strategy (120601 120595)and obtain the final value 119881(119905) from Theorem 4 we shouldfind the solution of the PDF (17) with the final data (18)This is the key problem in the rest of this section Wehave the following result in terms of method of variablestransformation
Theorem 6 Let 119903(119905 120572(119905)) in (2) be a constant 119903 The function119891(119905 119878) 119905 le 119879 119878 gt 0 given by the following formula
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942119892 (0 0 119878
119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(30)
is a solution of the general Black-Scholes equation (17) with thefinal data (18)
Proof We are going to do some equivalent transformationsof general B-S equation (17) in order to get an appropriateequivalent equation with analytic solutions The procedurewill be divided into four steps
Step I Let
119891 (119905 1198781 119878
119899) = 119890119903(119905minus119879)
119902 (119905 ln 1198781minus (119903 minus
1
21205902
1) (119905 minus 119879)
ln 119878119899minus (119903 minus
1
21205902
119899) (119905 minus 119879))
(31)
6 Abstract and Applied Analysis
and denote 119910119894= ln 119878
119894minus (119903 minus (12)120590
2
119894)(119905 minus 119879) (119894 = 1 2 119899)
and then
120597119891
120597119905=
119889 (119890119903(119905minus119879)
)
119889119905119902 + 119890119903(119905minus119879)
119902119905
= 119890119903(119905minus119879)
119902 (119889119903
119889119905(119905 minus 119879) + 119903)
+ 119890119903(119905minus119879)
[119902119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
= 119903119890119903(119905minus119879)
119902 + 119890119903(119905minus119879)
119902119889119903
119889119905(119905 minus 119879)
+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
120597119891
120597119878119894
= 119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
1205972119891
120597119878119894120597119878119895
=
120597 (119890119903(119905minus119879)
(120597119902120597119910119894) (1119878
119894))
120597119878119895
=
119890119903(119905minus119879)
1205972119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119894 = 119895
119890119903(119905minus119879)
(1205972119902
120597119910119894120597119910119894
1
119878119894
1
119878119894
minus120597119902
120597119910119894
1
1198782
119894
) 119894 = 119895
(32)
Inserting the above formulas into (17) we get
119903119891 +119889119903
119889119905(119905 minus 119879) 119902119890
119903(119905minus119879)+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
+ 119903
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
119878119894+1
2
119899
sum
119894=1
119899
sum
119895=1
119890119903(119905minus119879) 120597
2119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119878119894120590119894119878119895120590119895
minus1
2
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1198782
119894
1
1198782
119894
1198782
119894= 119903119891
(33)
which can be simplified as
119889119903
119889119905(119905 minus 119879) 119902 +
120597119902
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119902
120597119910119894120597119910119895
120590119894120590119895= 0 (34)
The final data 119891(119879 119878) = 119892(119878) can be rewritten as
119902 (119879 119878) = 119892 (1198901198781 1198901198782 119890
119878119899) (35)
Step II We introduce another variable and a new function asfollows
120591 = 119879 minus 119905 gt 0 119905 = 119879 minus 120591 120591 ge 0 119905 le 119879
119902 (119905 119910) = 119906 (119879 minus 119905 119910) or 119906 (120591 119910) = 119902 (119879 minus 120591 119910)
(36)
It can be computed that
119902119905(119905 119910) = minus119906
120591(119879 minus 119905 119910)
120597119902
120597119910119894
=120597119906
120597119910119894
(119879 minus 119905 119910)
1205972119902
120597119910119894120597119910119895
=1205972119906
120597119910119894120597119910119895
(119879 minus 119905 119910)
(37)
Substituting the above formulas into (34) we get
119889119881
119889119905(119905 minus 119879) 119906 (119879 minus 119905 119910) minus 119906
120591(119879 minus 119905 119910) +
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= 0
119906 (0 119910) = 119892 (119890119910)
(38)
Since 119903(119905 120572(119905)) is assumed as a constant 119903 (38) can be changedinto
119906120591(120591 119910) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895= 0
119906 (0 119910) = 119892 (119890119910)
(39)
Step III We claim that the unique solution of (39) is
119906 (119905 1199101 1199102 119910
119899)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0) 119889119909
119894
(40)
In fact
119906120591(120591 119910) = minus
1
2radic2120587120591120591
times
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
times(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
Abstract and Applied Analysis 7
120597119906
120597119910119894
=1
radic2120587120591int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0)
times (minus119910119894minus 119909119894
1205902
119894120591
)119889119909119894
1205972119906
120597119910119894120597119910119895
=
0 119894 = 119895
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
119890minus(119910119894minus119909119894)221205902
119894120591
times((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909i 119894 = 119895
(41)
So
119906120591(120591 119892) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= minus1
2radic2120587120591120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
minus1
2
119899
sum
119894=1
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
times 119890minus(119910119894minus119909119894)221205902
119894120591((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909119894= 0
(42)
Step IV By introducing a change of variables 119911119894= 119909119894minus 119910119894
we have 119909119894= 119911119894+ 119910119894and 119889119909
119894= 119889119911119894 where 119911
119894isin (minusinfininfin) It
follows that
119906 (120591 119910)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119911119894+119910119894 0 0) 119889119911
119894
(43)
In order to get rid of the denominator 1205902119894120591 in the exponent in
the above formula we make another change of variables as
119911119894= 120590119894radic120591119909119894 (44)
So 119889119911119894= 120590119894radic120591119889119909
119894
Recalling the relationship between 119902 and 119906 described in(36) we therefore have
119902 (119905 119910)
=1
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+119910119894 0 0) 119889119909
119894
(45)
Hence by formula (31) we have
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+ln 119878119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(46)
Since 119890ln 119878 = 119878 then
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(47)
In this way we provedTheorem 6
4 A Financial Example
As an application we consider the European call option InTheorem 6 we have given the solution of the general B-Sequation (17) which depends on the final data (18) that is119891(119879 119904) = 119892(119904) More specific we take the final data 119892(119904) forthe European call option as
119892 (119878) = 119892 (119878minus1198961
1 119878minus1198962
2 119878
minus119896119899
119899) =
119899
sum
119894=1
(119878119894minus 119870i)+ (48)
where 119878119894gt 0 and 119870
119894gt 0 are the strike price of 119878
119894 Then we
have the following corollary fromTheorem 6
Corollary 7 For the European call option the solution to thegeneral Black-Scholes value problem (17) with the final data(48) is given by
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(49)
8 Abstract and Applied Analysis
where
minus119860119894=
(119903 minus 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radic119879 minus 119905
= 1198892
minus119860119894+ 120590119894radic119879 minus 119905 =
(119903 + 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radicT minus 119905
= 1198891
(50)
that is
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(1198892) (51)
In particular
119891 (0 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903119879
119899
sum
119894=1
119870119894Φ(1198892) (52)
Proof For a European call option we infer that
119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
gt 119870119894 (53)
Dividing (53) by 119878119894and taking the ln we get
120590119894radic119879 minus 119905119909
119894minus (119903 minus
1205902
119894
2) (119905 minus 119879) gt ln
119870119894
119878119894
(54)
that is
119909119894gt
ln (119870119894119878119894) minus (119903 minus 120590
2
1198942) (119879 minus 119905)
120590119894radic119879 minus 119905
= 119860119894 (55)
Hence from (30) and (48) it follows that
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
119860119894
119890minus1199092
1198942119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890(119903minus1205902
1198942)(119879minus119905)
119890minus1199092
1198942+120590119894radic119879minus119905119909
119894119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890minus(1205902
1198942)(119879minus119905)
times 119890minus(12)(119909
119894minus120590119894radic119879minus119905)
2
+(12)1205902
119894(119879minus119905)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894minus120590119894radic119879minus119905
119878119894119890minus11991122119889119911
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894
1
radic2120587int
minus119860119894+120590119894radic119879minus119905
minusinfin
119890minus1199092
1198942119889119909119894
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894
1
radic2120587int
minus119860119894
minusinfin
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(56)
where Φ(119905) is the probability distribution function of astandard Gaussion random variable119873(0 1) that is
Φ (119905) =1
radic2120587int
119905
minusinfin
119890minus11990922119889119909 119905 isin 119877 (57)
In this way we have proved Corollary 7
Remark 8 Theabove result is about the European call optionA similar representation to those from the above corollaryin the European put option case can be obtained by taking119892(119878) = sum
119899
119894=1(119870119894minus 119878119894)+ 119878119894gt 0 for some fixed 119870
119894gt 0
5 Conclusion
In this paper we have considered a financial market modelwith regime switching driven by geometric Levy processThis financial market model is based on the multiple riskyassets 119878
1 1198782 119878
119899driven by Levy process Ito formula and
equivalent transformation methods have been used to solvethis complicated financial market model An example of theportfolio strategy and the final value problem to applying ourmethod to the European call option has been given in the endof this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 9
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (Grants no 61075105 and no 71371046)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] B Oslashksendal Stochastic Differential Equations Springer 6thedition 2005
[4] X Guo and Q Zhang ldquoOptimal selling rules in a regimeswitching modelrdquo IEEE Transactions on Automatic Control vol50 no 9 pp 1450ndash1455 2005
[5] M Pemy Q Zhang and G G Yin ldquoLiquidation of a large blockof stock with regime switchingrdquoMathematical Finance vol 18no 4 pp 629ndash648 2008
[6] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with Markov regime switching and uncertain time-horizonrdquo Journal of Systems Science amp Complexity vol 24 no 1pp 140ndash155 2011
[7] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with regime switching and a stochastic cash flowrdquoInsuranceMathematics amp Economics vol 50 no 3 pp 371ndash3842012
[8] J Kallsen ldquoOptimal portfolios for exponential Levy processesrdquoMathematical Methods of Operations Research vol 51 no 3 pp357ndash374 2000
[9] D Applebaum Levy Processes and Stochastic Calculus vol 93Cambridge University Press Cambridge UK 2004
[10] N Vandaele and M Vanmaele ldquoA locally risk-minimizinghedging strategy for unit-linked life insurance contracts ina Levy process financial marketrdquo Insurance Mathematics ampEconomics vol 42 no 3 pp 1128ndash1137 2008
[11] C Weng ldquoConstant proportion portfolio insurance under aregime switching exponential Levy processrdquo Insurance Math-ematics amp Economics vol 52 no 3 pp 508ndash521 2013
[12] N Bauerle and A Blatter ldquoOptimal control and dependencemodeling of insurance portfolios with Levy dynamicsrdquo Insur-ance Mathematics amp Economics vol 48 no 3 pp 398ndash4052011
[13] K C Yuen and C Yin ldquoOn optimality of the barrier strategyfor a general Levy risk processrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 1700ndash1707 2011
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
4 Abstract and Applied Analysis
times [120583119896 (119905 120572 (119905)) 119889119905 + 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+ int119877minus0
119911119896 (119889119905 119889119911)]
= [119881 (119905) minus
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905)] 119903 (119905 120572 (119905))
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120583119896 (119905 120572 (119905)) 119889119905
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) int
119877minus0
119911119896 (119889119905 119889119911)
= 119903 (119905 120572 (119905)) 119881 (119905) +
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905)
times [120583119896 (119905 120572 (119905)) minus 119903 (119905 120572 (119905))
minusint119877minus0
119911120578119896 (119889119911)] 119889119905
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 (119905 120572 (119905)) 119889119882119896 (119905)
+
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) int
119877minus0
119911119873119896 (119889119905 119889119911)
(13)
which is (12)Conversely from (2) and (12) we can obtain (8)This completes the proof of the above proposition
Now we give the following fundamental results
Theorem 4 Consider the model (2) of a financial marketAssume that the portfolio (120593 120595
1 1205952 120595
119899) is a self-financing
strategy and 119881(119905)119905ge0
is the wealth process defined by (7) andsum119899
119896=1120595119896119878119896int119877minus0
119911120578119896(119889119911) = sum
119899
119896=1int119877minus0
119911120595119896119878119896120578119896(119889119911) If there
exists a function 119891(119905 119878) of 11986212 class (the set of functions whichare once differentiable in 119905 and continuously twice differentiablein 119878) such that
119881 (119905) = 119891 (119905 119878 (119905)) 119905 isin [0 119879]
119878 (119905) = (1198781 (119905) 1198782 (119905) 119878119899 (119905))
(14)
which holds true then the portfolio (120593 1205951 1205952 120595
119899) satisfies
120593 (119905) =119891 minus (120597119891120597119878) 119878
119879
B (119905) 119905 ge 0 (15)
120595 (119905) = (120597119891
1205971198781
120597119891
1205971198782
120597119891
120597119878119899
) =120597119891
120597119878 119905 ge 0 (16)
and the function 119891(119905 119878) solves the following backward PDE ofparabolic type
120597119891
120597119905+ 119903
119899
sum
119896=1
120597119891
120597119878119896
119878119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895= 119903119891
119905 lt 119879 119878 gt 0
(17)
Moreover if 119881(119879) = 119892(119878(119879)) then the function 119891(119905 119878)
satisfies the following equation
119891 (119879 119878) = 119892 (119878) 119878 gt 0 (18)
For the converse part we assume that 119879 gt 0 If thereexists a function 119891(119905 119878) of 11986212 class such that (17) and (18)are satisfied then the process (120593 120595) defined by (16) and (15) isa self-financing strategy The wealth process 119881 = 119881(119905)
119905isin[0119879]
corresponding to (120593 120595) satisfies (14)
Proof We proof the direct part of Theorem 4 firstlyFor
119881 (119905) = 119891 (119905 119878 (119905)) (19)
by applying the Ito formula we can infer that
119889119881 (119905) =120597119891
120597119905(119905 119878 (119905)) 119889119905
+
119899
sum
119896=1
120597119891
120597119878119896
(119905 119878 (119905)) (119878119896120583119896119889119905 + 119878119896120590119896119889119882119896)
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
(119905 119878 (119905)) 119878119894120590119894119878119895120590119895119889119905
+
119899
sum
119896=1
int119877minus0
(119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)) 119896 (119889119905 119889119911)
+
119899
sum
119896=1
int119877minus0
[119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)
minus119911120597119891
120597119878119896
(119905 119878) 119878119896] 120578119896 (119889119911) 119889119905
+
119898
sum
119895=1
120574119894119895119891 (119905 119878 (119905))
= [
[
120597119891
120597119905+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120583119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
minus
119899
sum
119896=1
int119877minus0
119911120597119891
120597119878119896
119878119896120578119896 (119889119911)] 119889119905
+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120590119896119889119882119896
+
119899
sum
119896=1
int119877minus0
[119891 (119905 119878 + 119911119878) minus 119891 (119905 119878)]119873 (119889119905 119889119911)
(20)
Abstract and Applied Analysis 5
On the other hand since our strategy is self-financing theformula (12) is satisfied
Thus the rate of return and the volatility in (20) and (12)should be coincided and hence
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 =
119899
sum
119896=1
120597119891
120597119878119896
(119905 119878) 119878119896120590119896
119903 (119905 120572 (119905)) 119891 (119905 119878) +
119899
sum
119896=1
120595119896119878119896(120583119896minus 119903)
=120597119891
120597119905+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120583119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
(21)
We can easily get 119878119896ge 0 from (11) which together with
the first equation of (21) and the independence of 119878119896(119896 =
1 2 119899) yields (16)From the first equation of (21) (7) and (14) we have
119903120593119861 = 119891 minus
119899
sum
119896=1
120597119891
120597119878119896
119878119896 (22)
So that
120593 =119891 minus sum
119899
119896=1(120597119891120597119878
119896) 119878119896
119861=119891 minus 119891119878119878119879
119861 (23)
Substituting (16) into the second equation of (21) we have
119903119891 minus
119899
sum
119896=1
120595119896119878119896119903 =
120597119891
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895 (24)
which is (17)Conversely assume that 119891 = 119891(119905 119878) is a 119862
12-classfunction which is a solution of the PDE (17) and that (120593 120595)is a process defined by (16) and (15)
Firstly we will show that a process 119881 = 119881(119905) 119905 isin [0 119879]defined by (7) satisfies the equation
119881 (119905) = 119891 (119905 119878 (119905)) 119905 isin [0 119879] (25)
In fact substituting formulas (16) and (15) into the right handside of (7) we have
119881 (119905) = 120593119861 +
119899
sum
119896=1
120595119896119878119896=119891 minus sum
119899
119896=1(120597119891120597119878
119896) 119878119896
119861119861
+
119899
sum
119896=1
120597119891
120597119878119896
119878119896= 119891 119905 ge 0
(26)
This proves (25)Next we will show that (120593 120595) is a self-financing strategy
that is (12) holdsBy applying the Ito formula to the process119881 and function
119891 we have that (20) is satisfied
Furthermore by (17)
120597119891
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895= 119903119891 minus 119903
119899
sum
119896=1
119878119896
120597119891
120597119878119896
120597119891
120597119905+
119899
sum
119896=1
119878119896120583119896
120597119891
120597119878119896
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
= 119903119891 +
119899
sum
119896=1
(120583119896minus 119903) 119878
119896
120597119891
120597119878119896
(27)
Then by (25) and (16) we have
119903119881 +
119899
sum
119896=1
120595119896119878119896(120583119896minus 119903) =
120597119891
120597119905+
119899
sum
119896=1
119878119896120583119896
120597119891
120597119878119896
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
(28)
119899
sum
119896=1
120595119896119878119896120590119896=
119899
sum
119896=1
120597119891
120597119878119896
119878119896120590119896 (29)
Those together with (16) yield that (20) implies (12) Theproof of Theorem 4 is completed
Remark 5 In order to determine the portfolio strategy (120601 120595)and obtain the final value 119881(119905) from Theorem 4 we shouldfind the solution of the PDF (17) with the final data (18)This is the key problem in the rest of this section Wehave the following result in terms of method of variablestransformation
Theorem 6 Let 119903(119905 120572(119905)) in (2) be a constant 119903 The function119891(119905 119878) 119905 le 119879 119878 gt 0 given by the following formula
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942119892 (0 0 119878
119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(30)
is a solution of the general Black-Scholes equation (17) with thefinal data (18)
Proof We are going to do some equivalent transformationsof general B-S equation (17) in order to get an appropriateequivalent equation with analytic solutions The procedurewill be divided into four steps
Step I Let
119891 (119905 1198781 119878
119899) = 119890119903(119905minus119879)
119902 (119905 ln 1198781minus (119903 minus
1
21205902
1) (119905 minus 119879)
ln 119878119899minus (119903 minus
1
21205902
119899) (119905 minus 119879))
(31)
6 Abstract and Applied Analysis
and denote 119910119894= ln 119878
119894minus (119903 minus (12)120590
2
119894)(119905 minus 119879) (119894 = 1 2 119899)
and then
120597119891
120597119905=
119889 (119890119903(119905minus119879)
)
119889119905119902 + 119890119903(119905minus119879)
119902119905
= 119890119903(119905minus119879)
119902 (119889119903
119889119905(119905 minus 119879) + 119903)
+ 119890119903(119905minus119879)
[119902119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
= 119903119890119903(119905minus119879)
119902 + 119890119903(119905minus119879)
119902119889119903
119889119905(119905 minus 119879)
+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
120597119891
120597119878119894
= 119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
1205972119891
120597119878119894120597119878119895
=
120597 (119890119903(119905minus119879)
(120597119902120597119910119894) (1119878
119894))
120597119878119895
=
119890119903(119905minus119879)
1205972119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119894 = 119895
119890119903(119905minus119879)
(1205972119902
120597119910119894120597119910119894
1
119878119894
1
119878119894
minus120597119902
120597119910119894
1
1198782
119894
) 119894 = 119895
(32)
Inserting the above formulas into (17) we get
119903119891 +119889119903
119889119905(119905 minus 119879) 119902119890
119903(119905minus119879)+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
+ 119903
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
119878119894+1
2
119899
sum
119894=1
119899
sum
119895=1
119890119903(119905minus119879) 120597
2119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119878119894120590119894119878119895120590119895
minus1
2
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1198782
119894
1
1198782
119894
1198782
119894= 119903119891
(33)
which can be simplified as
119889119903
119889119905(119905 minus 119879) 119902 +
120597119902
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119902
120597119910119894120597119910119895
120590119894120590119895= 0 (34)
The final data 119891(119879 119878) = 119892(119878) can be rewritten as
119902 (119879 119878) = 119892 (1198901198781 1198901198782 119890
119878119899) (35)
Step II We introduce another variable and a new function asfollows
120591 = 119879 minus 119905 gt 0 119905 = 119879 minus 120591 120591 ge 0 119905 le 119879
119902 (119905 119910) = 119906 (119879 minus 119905 119910) or 119906 (120591 119910) = 119902 (119879 minus 120591 119910)
(36)
It can be computed that
119902119905(119905 119910) = minus119906
120591(119879 minus 119905 119910)
120597119902
120597119910119894
=120597119906
120597119910119894
(119879 minus 119905 119910)
1205972119902
120597119910119894120597119910119895
=1205972119906
120597119910119894120597119910119895
(119879 minus 119905 119910)
(37)
Substituting the above formulas into (34) we get
119889119881
119889119905(119905 minus 119879) 119906 (119879 minus 119905 119910) minus 119906
120591(119879 minus 119905 119910) +
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= 0
119906 (0 119910) = 119892 (119890119910)
(38)
Since 119903(119905 120572(119905)) is assumed as a constant 119903 (38) can be changedinto
119906120591(120591 119910) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895= 0
119906 (0 119910) = 119892 (119890119910)
(39)
Step III We claim that the unique solution of (39) is
119906 (119905 1199101 1199102 119910
119899)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0) 119889119909
119894
(40)
In fact
119906120591(120591 119910) = minus
1
2radic2120587120591120591
times
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
times(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
Abstract and Applied Analysis 7
120597119906
120597119910119894
=1
radic2120587120591int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0)
times (minus119910119894minus 119909119894
1205902
119894120591
)119889119909119894
1205972119906
120597119910119894120597119910119895
=
0 119894 = 119895
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
119890minus(119910119894minus119909119894)221205902
119894120591
times((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909i 119894 = 119895
(41)
So
119906120591(120591 119892) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= minus1
2radic2120587120591120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
minus1
2
119899
sum
119894=1
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
times 119890minus(119910119894minus119909119894)221205902
119894120591((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909119894= 0
(42)
Step IV By introducing a change of variables 119911119894= 119909119894minus 119910119894
we have 119909119894= 119911119894+ 119910119894and 119889119909
119894= 119889119911119894 where 119911
119894isin (minusinfininfin) It
follows that
119906 (120591 119910)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119911119894+119910119894 0 0) 119889119911
119894
(43)
In order to get rid of the denominator 1205902119894120591 in the exponent in
the above formula we make another change of variables as
119911119894= 120590119894radic120591119909119894 (44)
So 119889119911119894= 120590119894radic120591119889119909
119894
Recalling the relationship between 119902 and 119906 described in(36) we therefore have
119902 (119905 119910)
=1
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+119910119894 0 0) 119889119909
119894
(45)
Hence by formula (31) we have
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+ln 119878119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(46)
Since 119890ln 119878 = 119878 then
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(47)
In this way we provedTheorem 6
4 A Financial Example
As an application we consider the European call option InTheorem 6 we have given the solution of the general B-Sequation (17) which depends on the final data (18) that is119891(119879 119904) = 119892(119904) More specific we take the final data 119892(119904) forthe European call option as
119892 (119878) = 119892 (119878minus1198961
1 119878minus1198962
2 119878
minus119896119899
119899) =
119899
sum
119894=1
(119878119894minus 119870i)+ (48)
where 119878119894gt 0 and 119870
119894gt 0 are the strike price of 119878
119894 Then we
have the following corollary fromTheorem 6
Corollary 7 For the European call option the solution to thegeneral Black-Scholes value problem (17) with the final data(48) is given by
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(49)
8 Abstract and Applied Analysis
where
minus119860119894=
(119903 minus 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radic119879 minus 119905
= 1198892
minus119860119894+ 120590119894radic119879 minus 119905 =
(119903 + 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radicT minus 119905
= 1198891
(50)
that is
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(1198892) (51)
In particular
119891 (0 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903119879
119899
sum
119894=1
119870119894Φ(1198892) (52)
Proof For a European call option we infer that
119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
gt 119870119894 (53)
Dividing (53) by 119878119894and taking the ln we get
120590119894radic119879 minus 119905119909
119894minus (119903 minus
1205902
119894
2) (119905 minus 119879) gt ln
119870119894
119878119894
(54)
that is
119909119894gt
ln (119870119894119878119894) minus (119903 minus 120590
2
1198942) (119879 minus 119905)
120590119894radic119879 minus 119905
= 119860119894 (55)
Hence from (30) and (48) it follows that
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
119860119894
119890minus1199092
1198942119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890(119903minus1205902
1198942)(119879minus119905)
119890minus1199092
1198942+120590119894radic119879minus119905119909
119894119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890minus(1205902
1198942)(119879minus119905)
times 119890minus(12)(119909
119894minus120590119894radic119879minus119905)
2
+(12)1205902
119894(119879minus119905)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894minus120590119894radic119879minus119905
119878119894119890minus11991122119889119911
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894
1
radic2120587int
minus119860119894+120590119894radic119879minus119905
minusinfin
119890minus1199092
1198942119889119909119894
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894
1
radic2120587int
minus119860119894
minusinfin
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(56)
where Φ(119905) is the probability distribution function of astandard Gaussion random variable119873(0 1) that is
Φ (119905) =1
radic2120587int
119905
minusinfin
119890minus11990922119889119909 119905 isin 119877 (57)
In this way we have proved Corollary 7
Remark 8 Theabove result is about the European call optionA similar representation to those from the above corollaryin the European put option case can be obtained by taking119892(119878) = sum
119899
119894=1(119870119894minus 119878119894)+ 119878119894gt 0 for some fixed 119870
119894gt 0
5 Conclusion
In this paper we have considered a financial market modelwith regime switching driven by geometric Levy processThis financial market model is based on the multiple riskyassets 119878
1 1198782 119878
119899driven by Levy process Ito formula and
equivalent transformation methods have been used to solvethis complicated financial market model An example of theportfolio strategy and the final value problem to applying ourmethod to the European call option has been given in the endof this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 9
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (Grants no 61075105 and no 71371046)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] B Oslashksendal Stochastic Differential Equations Springer 6thedition 2005
[4] X Guo and Q Zhang ldquoOptimal selling rules in a regimeswitching modelrdquo IEEE Transactions on Automatic Control vol50 no 9 pp 1450ndash1455 2005
[5] M Pemy Q Zhang and G G Yin ldquoLiquidation of a large blockof stock with regime switchingrdquoMathematical Finance vol 18no 4 pp 629ndash648 2008
[6] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with Markov regime switching and uncertain time-horizonrdquo Journal of Systems Science amp Complexity vol 24 no 1pp 140ndash155 2011
[7] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with regime switching and a stochastic cash flowrdquoInsuranceMathematics amp Economics vol 50 no 3 pp 371ndash3842012
[8] J Kallsen ldquoOptimal portfolios for exponential Levy processesrdquoMathematical Methods of Operations Research vol 51 no 3 pp357ndash374 2000
[9] D Applebaum Levy Processes and Stochastic Calculus vol 93Cambridge University Press Cambridge UK 2004
[10] N Vandaele and M Vanmaele ldquoA locally risk-minimizinghedging strategy for unit-linked life insurance contracts ina Levy process financial marketrdquo Insurance Mathematics ampEconomics vol 42 no 3 pp 1128ndash1137 2008
[11] C Weng ldquoConstant proportion portfolio insurance under aregime switching exponential Levy processrdquo Insurance Math-ematics amp Economics vol 52 no 3 pp 508ndash521 2013
[12] N Bauerle and A Blatter ldquoOptimal control and dependencemodeling of insurance portfolios with Levy dynamicsrdquo Insur-ance Mathematics amp Economics vol 48 no 3 pp 398ndash4052011
[13] K C Yuen and C Yin ldquoOn optimality of the barrier strategyfor a general Levy risk processrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 1700ndash1707 2011
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 5
On the other hand since our strategy is self-financing theformula (12) is satisfied
Thus the rate of return and the volatility in (20) and (12)should be coincided and hence
119899
sum
119896=1
120595119896 (119905) 119878119896 (119905) 120590119896 =
119899
sum
119896=1
120597119891
120597119878119896
(119905 119878) 119878119896120590119896
119903 (119905 120572 (119905)) 119891 (119905 119878) +
119899
sum
119896=1
120595119896119878119896(120583119896minus 119903)
=120597119891
120597119905+
119899
sum
119896=1
120597119891
120597119878119896
119878119896120583119896+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
(21)
We can easily get 119878119896ge 0 from (11) which together with
the first equation of (21) and the independence of 119878119896(119896 =
1 2 119899) yields (16)From the first equation of (21) (7) and (14) we have
119903120593119861 = 119891 minus
119899
sum
119896=1
120597119891
120597119878119896
119878119896 (22)
So that
120593 =119891 minus sum
119899
119896=1(120597119891120597119878
119896) 119878119896
119861=119891 minus 119891119878119878119879
119861 (23)
Substituting (16) into the second equation of (21) we have
119903119891 minus
119899
sum
119896=1
120595119896119878119896119903 =
120597119891
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895 (24)
which is (17)Conversely assume that 119891 = 119891(119905 119878) is a 119862
12-classfunction which is a solution of the PDE (17) and that (120593 120595)is a process defined by (16) and (15)
Firstly we will show that a process 119881 = 119881(119905) 119905 isin [0 119879]defined by (7) satisfies the equation
119881 (119905) = 119891 (119905 119878 (119905)) 119905 isin [0 119879] (25)
In fact substituting formulas (16) and (15) into the right handside of (7) we have
119881 (119905) = 120593119861 +
119899
sum
119896=1
120595119896119878119896=119891 minus sum
119899
119896=1(120597119891120597119878
119896) 119878119896
119861119861
+
119899
sum
119896=1
120597119891
120597119878119896
119878119896= 119891 119905 ge 0
(26)
This proves (25)Next we will show that (120593 120595) is a self-financing strategy
that is (12) holdsBy applying the Ito formula to the process119881 and function
119891 we have that (20) is satisfied
Furthermore by (17)
120597119891
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895= 119903119891 minus 119903
119899
sum
119896=1
119878119896
120597119891
120597119878119896
120597119891
120597119905+
119899
sum
119896=1
119878119896120583119896
120597119891
120597119878119896
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
= 119903119891 +
119899
sum
119896=1
(120583119896minus 119903) 119878
119896
120597119891
120597119878119896
(27)
Then by (25) and (16) we have
119903119881 +
119899
sum
119896=1
120595119896119878119896(120583119896minus 119903) =
120597119891
120597119905+
119899
sum
119896=1
119878119896120583119896
120597119891
120597119878119896
+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119891
120597119878119894120597119878119895
119878119894120590119894119878119895120590119895
(28)
119899
sum
119896=1
120595119896119878119896120590119896=
119899
sum
119896=1
120597119891
120597119878119896
119878119896120590119896 (29)
Those together with (16) yield that (20) implies (12) Theproof of Theorem 4 is completed
Remark 5 In order to determine the portfolio strategy (120601 120595)and obtain the final value 119881(119905) from Theorem 4 we shouldfind the solution of the PDF (17) with the final data (18)This is the key problem in the rest of this section Wehave the following result in terms of method of variablestransformation
Theorem 6 Let 119903(119905 120572(119905)) in (2) be a constant 119903 The function119891(119905 119878) 119905 le 119879 119878 gt 0 given by the following formula
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942119892 (0 0 119878
119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(30)
is a solution of the general Black-Scholes equation (17) with thefinal data (18)
Proof We are going to do some equivalent transformationsof general B-S equation (17) in order to get an appropriateequivalent equation with analytic solutions The procedurewill be divided into four steps
Step I Let
119891 (119905 1198781 119878
119899) = 119890119903(119905minus119879)
119902 (119905 ln 1198781minus (119903 minus
1
21205902
1) (119905 minus 119879)
ln 119878119899minus (119903 minus
1
21205902
119899) (119905 minus 119879))
(31)
6 Abstract and Applied Analysis
and denote 119910119894= ln 119878
119894minus (119903 minus (12)120590
2
119894)(119905 minus 119879) (119894 = 1 2 119899)
and then
120597119891
120597119905=
119889 (119890119903(119905minus119879)
)
119889119905119902 + 119890119903(119905minus119879)
119902119905
= 119890119903(119905minus119879)
119902 (119889119903
119889119905(119905 minus 119879) + 119903)
+ 119890119903(119905minus119879)
[119902119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
= 119903119890119903(119905minus119879)
119902 + 119890119903(119905minus119879)
119902119889119903
119889119905(119905 minus 119879)
+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
120597119891
120597119878119894
= 119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
1205972119891
120597119878119894120597119878119895
=
120597 (119890119903(119905minus119879)
(120597119902120597119910119894) (1119878
119894))
120597119878119895
=
119890119903(119905minus119879)
1205972119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119894 = 119895
119890119903(119905minus119879)
(1205972119902
120597119910119894120597119910119894
1
119878119894
1
119878119894
minus120597119902
120597119910119894
1
1198782
119894
) 119894 = 119895
(32)
Inserting the above formulas into (17) we get
119903119891 +119889119903
119889119905(119905 minus 119879) 119902119890
119903(119905minus119879)+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
+ 119903
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
119878119894+1
2
119899
sum
119894=1
119899
sum
119895=1
119890119903(119905minus119879) 120597
2119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119878119894120590119894119878119895120590119895
minus1
2
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1198782
119894
1
1198782
119894
1198782
119894= 119903119891
(33)
which can be simplified as
119889119903
119889119905(119905 minus 119879) 119902 +
120597119902
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119902
120597119910119894120597119910119895
120590119894120590119895= 0 (34)
The final data 119891(119879 119878) = 119892(119878) can be rewritten as
119902 (119879 119878) = 119892 (1198901198781 1198901198782 119890
119878119899) (35)
Step II We introduce another variable and a new function asfollows
120591 = 119879 minus 119905 gt 0 119905 = 119879 minus 120591 120591 ge 0 119905 le 119879
119902 (119905 119910) = 119906 (119879 minus 119905 119910) or 119906 (120591 119910) = 119902 (119879 minus 120591 119910)
(36)
It can be computed that
119902119905(119905 119910) = minus119906
120591(119879 minus 119905 119910)
120597119902
120597119910119894
=120597119906
120597119910119894
(119879 minus 119905 119910)
1205972119902
120597119910119894120597119910119895
=1205972119906
120597119910119894120597119910119895
(119879 minus 119905 119910)
(37)
Substituting the above formulas into (34) we get
119889119881
119889119905(119905 minus 119879) 119906 (119879 minus 119905 119910) minus 119906
120591(119879 minus 119905 119910) +
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= 0
119906 (0 119910) = 119892 (119890119910)
(38)
Since 119903(119905 120572(119905)) is assumed as a constant 119903 (38) can be changedinto
119906120591(120591 119910) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895= 0
119906 (0 119910) = 119892 (119890119910)
(39)
Step III We claim that the unique solution of (39) is
119906 (119905 1199101 1199102 119910
119899)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0) 119889119909
119894
(40)
In fact
119906120591(120591 119910) = minus
1
2radic2120587120591120591
times
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
times(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
Abstract and Applied Analysis 7
120597119906
120597119910119894
=1
radic2120587120591int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0)
times (minus119910119894minus 119909119894
1205902
119894120591
)119889119909119894
1205972119906
120597119910119894120597119910119895
=
0 119894 = 119895
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
119890minus(119910119894minus119909119894)221205902
119894120591
times((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909i 119894 = 119895
(41)
So
119906120591(120591 119892) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= minus1
2radic2120587120591120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
minus1
2
119899
sum
119894=1
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
times 119890minus(119910119894minus119909119894)221205902
119894120591((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909119894= 0
(42)
Step IV By introducing a change of variables 119911119894= 119909119894minus 119910119894
we have 119909119894= 119911119894+ 119910119894and 119889119909
119894= 119889119911119894 where 119911
119894isin (minusinfininfin) It
follows that
119906 (120591 119910)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119911119894+119910119894 0 0) 119889119911
119894
(43)
In order to get rid of the denominator 1205902119894120591 in the exponent in
the above formula we make another change of variables as
119911119894= 120590119894radic120591119909119894 (44)
So 119889119911119894= 120590119894radic120591119889119909
119894
Recalling the relationship between 119902 and 119906 described in(36) we therefore have
119902 (119905 119910)
=1
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+119910119894 0 0) 119889119909
119894
(45)
Hence by formula (31) we have
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+ln 119878119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(46)
Since 119890ln 119878 = 119878 then
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(47)
In this way we provedTheorem 6
4 A Financial Example
As an application we consider the European call option InTheorem 6 we have given the solution of the general B-Sequation (17) which depends on the final data (18) that is119891(119879 119904) = 119892(119904) More specific we take the final data 119892(119904) forthe European call option as
119892 (119878) = 119892 (119878minus1198961
1 119878minus1198962
2 119878
minus119896119899
119899) =
119899
sum
119894=1
(119878119894minus 119870i)+ (48)
where 119878119894gt 0 and 119870
119894gt 0 are the strike price of 119878
119894 Then we
have the following corollary fromTheorem 6
Corollary 7 For the European call option the solution to thegeneral Black-Scholes value problem (17) with the final data(48) is given by
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(49)
8 Abstract and Applied Analysis
where
minus119860119894=
(119903 minus 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radic119879 minus 119905
= 1198892
minus119860119894+ 120590119894radic119879 minus 119905 =
(119903 + 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radicT minus 119905
= 1198891
(50)
that is
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(1198892) (51)
In particular
119891 (0 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903119879
119899
sum
119894=1
119870119894Φ(1198892) (52)
Proof For a European call option we infer that
119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
gt 119870119894 (53)
Dividing (53) by 119878119894and taking the ln we get
120590119894radic119879 minus 119905119909
119894minus (119903 minus
1205902
119894
2) (119905 minus 119879) gt ln
119870119894
119878119894
(54)
that is
119909119894gt
ln (119870119894119878119894) minus (119903 minus 120590
2
1198942) (119879 minus 119905)
120590119894radic119879 minus 119905
= 119860119894 (55)
Hence from (30) and (48) it follows that
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
119860119894
119890minus1199092
1198942119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890(119903minus1205902
1198942)(119879minus119905)
119890minus1199092
1198942+120590119894radic119879minus119905119909
119894119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890minus(1205902
1198942)(119879minus119905)
times 119890minus(12)(119909
119894minus120590119894radic119879minus119905)
2
+(12)1205902
119894(119879minus119905)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894minus120590119894radic119879minus119905
119878119894119890minus11991122119889119911
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894
1
radic2120587int
minus119860119894+120590119894radic119879minus119905
minusinfin
119890minus1199092
1198942119889119909119894
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894
1
radic2120587int
minus119860119894
minusinfin
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(56)
where Φ(119905) is the probability distribution function of astandard Gaussion random variable119873(0 1) that is
Φ (119905) =1
radic2120587int
119905
minusinfin
119890minus11990922119889119909 119905 isin 119877 (57)
In this way we have proved Corollary 7
Remark 8 Theabove result is about the European call optionA similar representation to those from the above corollaryin the European put option case can be obtained by taking119892(119878) = sum
119899
119894=1(119870119894minus 119878119894)+ 119878119894gt 0 for some fixed 119870
119894gt 0
5 Conclusion
In this paper we have considered a financial market modelwith regime switching driven by geometric Levy processThis financial market model is based on the multiple riskyassets 119878
1 1198782 119878
119899driven by Levy process Ito formula and
equivalent transformation methods have been used to solvethis complicated financial market model An example of theportfolio strategy and the final value problem to applying ourmethod to the European call option has been given in the endof this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 9
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (Grants no 61075105 and no 71371046)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] B Oslashksendal Stochastic Differential Equations Springer 6thedition 2005
[4] X Guo and Q Zhang ldquoOptimal selling rules in a regimeswitching modelrdquo IEEE Transactions on Automatic Control vol50 no 9 pp 1450ndash1455 2005
[5] M Pemy Q Zhang and G G Yin ldquoLiquidation of a large blockof stock with regime switchingrdquoMathematical Finance vol 18no 4 pp 629ndash648 2008
[6] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with Markov regime switching and uncertain time-horizonrdquo Journal of Systems Science amp Complexity vol 24 no 1pp 140ndash155 2011
[7] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with regime switching and a stochastic cash flowrdquoInsuranceMathematics amp Economics vol 50 no 3 pp 371ndash3842012
[8] J Kallsen ldquoOptimal portfolios for exponential Levy processesrdquoMathematical Methods of Operations Research vol 51 no 3 pp357ndash374 2000
[9] D Applebaum Levy Processes and Stochastic Calculus vol 93Cambridge University Press Cambridge UK 2004
[10] N Vandaele and M Vanmaele ldquoA locally risk-minimizinghedging strategy for unit-linked life insurance contracts ina Levy process financial marketrdquo Insurance Mathematics ampEconomics vol 42 no 3 pp 1128ndash1137 2008
[11] C Weng ldquoConstant proportion portfolio insurance under aregime switching exponential Levy processrdquo Insurance Math-ematics amp Economics vol 52 no 3 pp 508ndash521 2013
[12] N Bauerle and A Blatter ldquoOptimal control and dependencemodeling of insurance portfolios with Levy dynamicsrdquo Insur-ance Mathematics amp Economics vol 48 no 3 pp 398ndash4052011
[13] K C Yuen and C Yin ldquoOn optimality of the barrier strategyfor a general Levy risk processrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 1700ndash1707 2011
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
6 Abstract and Applied Analysis
and denote 119910119894= ln 119878
119894minus (119903 minus (12)120590
2
119894)(119905 minus 119879) (119894 = 1 2 119899)
and then
120597119891
120597119905=
119889 (119890119903(119905minus119879)
)
119889119905119902 + 119890119903(119905minus119879)
119902119905
= 119890119903(119905minus119879)
119902 (119889119903
119889119905(119905 minus 119879) + 119903)
+ 119890119903(119905minus119879)
[119902119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
= 119903119890119903(119905minus119879)
119902 + 119890119903(119905minus119879)
119902119889119903
119889119905(119905 minus 119879)
+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
120597119891
120597119878119894
= 119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
1205972119891
120597119878119894120597119878119895
=
120597 (119890119903(119905minus119879)
(120597119902120597119910119894) (1119878
119894))
120597119878119895
=
119890119903(119905minus119879)
1205972119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119894 = 119895
119890119903(119905minus119879)
(1205972119902
120597119910119894120597119910119894
1
119878119894
1
119878119894
minus120597119902
120597119910119894
1
1198782
119894
) 119894 = 119895
(32)
Inserting the above formulas into (17) we get
119903119891 +119889119903
119889119905(119905 minus 119879) 119902119890
119903(119905minus119879)+ 119890119903(119905minus119879)
[120597119902
120597119905minus
119899
sum
119894=1
120597119902
120597119910119894
(119903 minus1
21205902
119894)]
+ 119903
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1
119878119894
119878119894+1
2
119899
sum
119894=1
119899
sum
119895=1
119890119903(119905minus119879) 120597
2119902
120597119910119894120597119910119895
1
119878119894
1
119878119895
119878119894120590119894119878119895120590119895
minus1
2
119899
sum
119894=1
119890119903(119905minus119879) 120597119902
120597119910119894
1198782
119894
1
1198782
119894
1198782
119894= 119903119891
(33)
which can be simplified as
119889119903
119889119905(119905 minus 119879) 119902 +
120597119902
120597119905+1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119902
120597119910119894120597119910119895
120590119894120590119895= 0 (34)
The final data 119891(119879 119878) = 119892(119878) can be rewritten as
119902 (119879 119878) = 119892 (1198901198781 1198901198782 119890
119878119899) (35)
Step II We introduce another variable and a new function asfollows
120591 = 119879 minus 119905 gt 0 119905 = 119879 minus 120591 120591 ge 0 119905 le 119879
119902 (119905 119910) = 119906 (119879 minus 119905 119910) or 119906 (120591 119910) = 119902 (119879 minus 120591 119910)
(36)
It can be computed that
119902119905(119905 119910) = minus119906
120591(119879 minus 119905 119910)
120597119902
120597119910119894
=120597119906
120597119910119894
(119879 minus 119905 119910)
1205972119902
120597119910119894120597119910119895
=1205972119906
120597119910119894120597119910119895
(119879 minus 119905 119910)
(37)
Substituting the above formulas into (34) we get
119889119881
119889119905(119905 minus 119879) 119906 (119879 minus 119905 119910) minus 119906
120591(119879 minus 119905 119910) +
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= 0
119906 (0 119910) = 119892 (119890119910)
(38)
Since 119903(119905 120572(119905)) is assumed as a constant 119903 (38) can be changedinto
119906120591(120591 119910) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895= 0
119906 (0 119910) = 119892 (119890119910)
(39)
Step III We claim that the unique solution of (39) is
119906 (119905 1199101 1199102 119910
119899)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0) 119889119909
119894
(40)
In fact
119906120591(120591 119910) = minus
1
2radic2120587120591120591
times
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
times(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
Abstract and Applied Analysis 7
120597119906
120597119910119894
=1
radic2120587120591int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0)
times (minus119910119894minus 119909119894
1205902
119894120591
)119889119909119894
1205972119906
120597119910119894120597119910119895
=
0 119894 = 119895
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
119890minus(119910119894minus119909119894)221205902
119894120591
times((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909i 119894 = 119895
(41)
So
119906120591(120591 119892) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= minus1
2radic2120587120591120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
minus1
2
119899
sum
119894=1
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
times 119890minus(119910119894minus119909119894)221205902
119894120591((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909119894= 0
(42)
Step IV By introducing a change of variables 119911119894= 119909119894minus 119910119894
we have 119909119894= 119911119894+ 119910119894and 119889119909
119894= 119889119911119894 where 119911
119894isin (minusinfininfin) It
follows that
119906 (120591 119910)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119911119894+119910119894 0 0) 119889119911
119894
(43)
In order to get rid of the denominator 1205902119894120591 in the exponent in
the above formula we make another change of variables as
119911119894= 120590119894radic120591119909119894 (44)
So 119889119911119894= 120590119894radic120591119889119909
119894
Recalling the relationship between 119902 and 119906 described in(36) we therefore have
119902 (119905 119910)
=1
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+119910119894 0 0) 119889119909
119894
(45)
Hence by formula (31) we have
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+ln 119878119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(46)
Since 119890ln 119878 = 119878 then
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(47)
In this way we provedTheorem 6
4 A Financial Example
As an application we consider the European call option InTheorem 6 we have given the solution of the general B-Sequation (17) which depends on the final data (18) that is119891(119879 119904) = 119892(119904) More specific we take the final data 119892(119904) forthe European call option as
119892 (119878) = 119892 (119878minus1198961
1 119878minus1198962
2 119878
minus119896119899
119899) =
119899
sum
119894=1
(119878119894minus 119870i)+ (48)
where 119878119894gt 0 and 119870
119894gt 0 are the strike price of 119878
119894 Then we
have the following corollary fromTheorem 6
Corollary 7 For the European call option the solution to thegeneral Black-Scholes value problem (17) with the final data(48) is given by
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(49)
8 Abstract and Applied Analysis
where
minus119860119894=
(119903 minus 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radic119879 minus 119905
= 1198892
minus119860119894+ 120590119894radic119879 minus 119905 =
(119903 + 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radicT minus 119905
= 1198891
(50)
that is
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(1198892) (51)
In particular
119891 (0 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903119879
119899
sum
119894=1
119870119894Φ(1198892) (52)
Proof For a European call option we infer that
119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
gt 119870119894 (53)
Dividing (53) by 119878119894and taking the ln we get
120590119894radic119879 minus 119905119909
119894minus (119903 minus
1205902
119894
2) (119905 minus 119879) gt ln
119870119894
119878119894
(54)
that is
119909119894gt
ln (119870119894119878119894) minus (119903 minus 120590
2
1198942) (119879 minus 119905)
120590119894radic119879 minus 119905
= 119860119894 (55)
Hence from (30) and (48) it follows that
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
119860119894
119890minus1199092
1198942119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890(119903minus1205902
1198942)(119879minus119905)
119890minus1199092
1198942+120590119894radic119879minus119905119909
119894119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890minus(1205902
1198942)(119879minus119905)
times 119890minus(12)(119909
119894minus120590119894radic119879minus119905)
2
+(12)1205902
119894(119879minus119905)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894minus120590119894radic119879minus119905
119878119894119890minus11991122119889119911
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894
1
radic2120587int
minus119860119894+120590119894radic119879minus119905
minusinfin
119890minus1199092
1198942119889119909119894
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894
1
radic2120587int
minus119860119894
minusinfin
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(56)
where Φ(119905) is the probability distribution function of astandard Gaussion random variable119873(0 1) that is
Φ (119905) =1
radic2120587int
119905
minusinfin
119890minus11990922119889119909 119905 isin 119877 (57)
In this way we have proved Corollary 7
Remark 8 Theabove result is about the European call optionA similar representation to those from the above corollaryin the European put option case can be obtained by taking119892(119878) = sum
119899
119894=1(119870119894minus 119878119894)+ 119878119894gt 0 for some fixed 119870
119894gt 0
5 Conclusion
In this paper we have considered a financial market modelwith regime switching driven by geometric Levy processThis financial market model is based on the multiple riskyassets 119878
1 1198782 119878
119899driven by Levy process Ito formula and
equivalent transformation methods have been used to solvethis complicated financial market model An example of theportfolio strategy and the final value problem to applying ourmethod to the European call option has been given in the endof this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 9
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (Grants no 61075105 and no 71371046)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] B Oslashksendal Stochastic Differential Equations Springer 6thedition 2005
[4] X Guo and Q Zhang ldquoOptimal selling rules in a regimeswitching modelrdquo IEEE Transactions on Automatic Control vol50 no 9 pp 1450ndash1455 2005
[5] M Pemy Q Zhang and G G Yin ldquoLiquidation of a large blockof stock with regime switchingrdquoMathematical Finance vol 18no 4 pp 629ndash648 2008
[6] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with Markov regime switching and uncertain time-horizonrdquo Journal of Systems Science amp Complexity vol 24 no 1pp 140ndash155 2011
[7] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with regime switching and a stochastic cash flowrdquoInsuranceMathematics amp Economics vol 50 no 3 pp 371ndash3842012
[8] J Kallsen ldquoOptimal portfolios for exponential Levy processesrdquoMathematical Methods of Operations Research vol 51 no 3 pp357ndash374 2000
[9] D Applebaum Levy Processes and Stochastic Calculus vol 93Cambridge University Press Cambridge UK 2004
[10] N Vandaele and M Vanmaele ldquoA locally risk-minimizinghedging strategy for unit-linked life insurance contracts ina Levy process financial marketrdquo Insurance Mathematics ampEconomics vol 42 no 3 pp 1128ndash1137 2008
[11] C Weng ldquoConstant proportion portfolio insurance under aregime switching exponential Levy processrdquo Insurance Math-ematics amp Economics vol 52 no 3 pp 508ndash521 2013
[12] N Bauerle and A Blatter ldquoOptimal control and dependencemodeling of insurance portfolios with Levy dynamicsrdquo Insur-ance Mathematics amp Economics vol 48 no 3 pp 398ndash4052011
[13] K C Yuen and C Yin ldquoOn optimality of the barrier strategyfor a general Levy risk processrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 1700ndash1707 2011
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 7
120597119906
120597119910119894
=1
radic2120587120591int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
119892 (0 0 119890119909119894 0 0)
times (minus119910119894minus 119909119894
1205902
119894120591
)119889119909119894
1205972119906
120597119910119894120597119910119895
=
0 119894 = 119895
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
119890minus(119910119894minus119909119894)221205902
119894120591
times((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909i 119894 = 119895
(41)
So
119906120591(120591 119892) minus
1
2
119899
sum
119894=1
119899
sum
119895=1
1205972119906
120597119910119894120597119910119895
120590119894120590119895
= minus1
2radic2120587120591120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0) 119889119909
119894
+1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119909119894 0 0)
(119910119894minus 119909119894)2
21205902
1198941205912
119889119909119894
minus1
2
119899
sum
119894=1
1
radic2120587120591int
infin
minusinfin
119892 (1198901199091 119890
119909119899)
1205902
119894
times 119890minus(119910119894minus119909119894)221205902
119894120591((119910119894minus 119909119894)2
1205904
1198941205912
minus1
1205902
119894120591)1205902
119894119889119909119894= 0
(42)
Step IV By introducing a change of variables 119911119894= 119909119894minus 119910119894
we have 119909119894= 119911119894+ 119910119894and 119889119909
119894= 119889119911119894 where 119911
119894isin (minusinfininfin) It
follows that
119906 (120591 119910)
=1
radic2120587120591
119899
sum
119894=1
int
infin
minusinfin
119890minus(119910119894minus119909119894)221205902
119894120591
120590119894
times 119892 (0 0 119890119911119894+119910119894 0 0) 119889119911
119894
(43)
In order to get rid of the denominator 1205902119894120591 in the exponent in
the above formula we make another change of variables as
119911119894= 120590119894radic120591119909119894 (44)
So 119889119911119894= 120590119894radic120591119889119909
119894
Recalling the relationship between 119902 and 119906 described in(36) we therefore have
119902 (119905 119910)
=1
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+119910119894 0 0) 119889119909
119894
(45)
Hence by formula (31) we have
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119890120590119894radic119879minus119905119909
119894+ln 119878119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(46)
Since 119890ln 119878 = 119878 then
119891 (119905 119878)
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
minusinfin
119890minus1199092
1198942
times 119892 (0 0 119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
0 0) 119889119909119894
(47)
In this way we provedTheorem 6
4 A Financial Example
As an application we consider the European call option InTheorem 6 we have given the solution of the general B-Sequation (17) which depends on the final data (18) that is119891(119879 119904) = 119892(119904) More specific we take the final data 119892(119904) forthe European call option as
119892 (119878) = 119892 (119878minus1198961
1 119878minus1198962
2 119878
minus119896119899
119899) =
119899
sum
119894=1
(119878119894minus 119870i)+ (48)
where 119878119894gt 0 and 119870
119894gt 0 are the strike price of 119878
119894 Then we
have the following corollary fromTheorem 6
Corollary 7 For the European call option the solution to thegeneral Black-Scholes value problem (17) with the final data(48) is given by
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(49)
8 Abstract and Applied Analysis
where
minus119860119894=
(119903 minus 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radic119879 minus 119905
= 1198892
minus119860119894+ 120590119894radic119879 minus 119905 =
(119903 + 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radicT minus 119905
= 1198891
(50)
that is
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(1198892) (51)
In particular
119891 (0 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903119879
119899
sum
119894=1
119870119894Φ(1198892) (52)
Proof For a European call option we infer that
119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
gt 119870119894 (53)
Dividing (53) by 119878119894and taking the ln we get
120590119894radic119879 minus 119905119909
119894minus (119903 minus
1205902
119894
2) (119905 minus 119879) gt ln
119870119894
119878119894
(54)
that is
119909119894gt
ln (119870119894119878119894) minus (119903 minus 120590
2
1198942) (119879 minus 119905)
120590119894radic119879 minus 119905
= 119860119894 (55)
Hence from (30) and (48) it follows that
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
119860119894
119890minus1199092
1198942119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890(119903minus1205902
1198942)(119879minus119905)
119890minus1199092
1198942+120590119894radic119879minus119905119909
119894119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890minus(1205902
1198942)(119879minus119905)
times 119890minus(12)(119909
119894minus120590119894radic119879minus119905)
2
+(12)1205902
119894(119879minus119905)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894minus120590119894radic119879minus119905
119878119894119890minus11991122119889119911
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894
1
radic2120587int
minus119860119894+120590119894radic119879minus119905
minusinfin
119890minus1199092
1198942119889119909119894
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894
1
radic2120587int
minus119860119894
minusinfin
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(56)
where Φ(119905) is the probability distribution function of astandard Gaussion random variable119873(0 1) that is
Φ (119905) =1
radic2120587int
119905
minusinfin
119890minus11990922119889119909 119905 isin 119877 (57)
In this way we have proved Corollary 7
Remark 8 Theabove result is about the European call optionA similar representation to those from the above corollaryin the European put option case can be obtained by taking119892(119878) = sum
119899
119894=1(119870119894minus 119878119894)+ 119878119894gt 0 for some fixed 119870
119894gt 0
5 Conclusion
In this paper we have considered a financial market modelwith regime switching driven by geometric Levy processThis financial market model is based on the multiple riskyassets 119878
1 1198782 119878
119899driven by Levy process Ito formula and
equivalent transformation methods have been used to solvethis complicated financial market model An example of theportfolio strategy and the final value problem to applying ourmethod to the European call option has been given in the endof this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 9
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (Grants no 61075105 and no 71371046)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] B Oslashksendal Stochastic Differential Equations Springer 6thedition 2005
[4] X Guo and Q Zhang ldquoOptimal selling rules in a regimeswitching modelrdquo IEEE Transactions on Automatic Control vol50 no 9 pp 1450ndash1455 2005
[5] M Pemy Q Zhang and G G Yin ldquoLiquidation of a large blockof stock with regime switchingrdquoMathematical Finance vol 18no 4 pp 629ndash648 2008
[6] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with Markov regime switching and uncertain time-horizonrdquo Journal of Systems Science amp Complexity vol 24 no 1pp 140ndash155 2011
[7] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with regime switching and a stochastic cash flowrdquoInsuranceMathematics amp Economics vol 50 no 3 pp 371ndash3842012
[8] J Kallsen ldquoOptimal portfolios for exponential Levy processesrdquoMathematical Methods of Operations Research vol 51 no 3 pp357ndash374 2000
[9] D Applebaum Levy Processes and Stochastic Calculus vol 93Cambridge University Press Cambridge UK 2004
[10] N Vandaele and M Vanmaele ldquoA locally risk-minimizinghedging strategy for unit-linked life insurance contracts ina Levy process financial marketrdquo Insurance Mathematics ampEconomics vol 42 no 3 pp 1128ndash1137 2008
[11] C Weng ldquoConstant proportion portfolio insurance under aregime switching exponential Levy processrdquo Insurance Math-ematics amp Economics vol 52 no 3 pp 508ndash521 2013
[12] N Bauerle and A Blatter ldquoOptimal control and dependencemodeling of insurance portfolios with Levy dynamicsrdquo Insur-ance Mathematics amp Economics vol 48 no 3 pp 398ndash4052011
[13] K C Yuen and C Yin ldquoOn optimality of the barrier strategyfor a general Levy risk processrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 1700ndash1707 2011
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
8 Abstract and Applied Analysis
where
minus119860119894=
(119903 minus 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radic119879 minus 119905
= 1198892
minus119860119894+ 120590119894radic119879 minus 119905 =
(119903 + 1205902
1198942) (119879 minus 119905) + ln (119878
119894119870119894)
120590119894radicT minus 119905
= 1198891
(50)
that is
119891 (119905 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(1198892) (51)
In particular
119891 (0 119878) =
119899
sum
119894=1
119878119894Φ(1198891) minus 119890minus119903119879
119899
sum
119894=1
119870119894Φ(1198892) (52)
Proof For a European call option we infer that
119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
gt 119870119894 (53)
Dividing (53) by 119878119894and taking the ln we get
120590119894radic119879 minus 119905119909
119894minus (119903 minus
1205902
119894
2) (119905 minus 119879) gt ln
119870119894
119878119894
(54)
that is
119909119894gt
ln (119870119894119878119894) minus (119903 minus 120590
2
1198942) (119879 minus 119905)
120590119894radic119879 minus 119905
= 119860119894 (55)
Hence from (30) and (48) it follows that
119891 (119905 119878) =119890minus119903(119879minus119905)
radic2120587
times
119899
sum
119894=1
int
infin
119860119894
119890minus1199092
1198942119878119894119890120590119894radic119879minus119905119909
119894minus(119903minus120590
2
1198942)(119905minus119879)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890(119903minus1205902
1198942)(119879minus119905)
119890minus1199092
1198942+120590119894radic119879minus119905119909
119894119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894
119878119894119890minus(1205902
1198942)(119879minus119905)
times 119890minus(12)(119909
119894minus120590119894radic119879minus119905)
2
+(12)1205902
119894(119879minus119905)
119889119909119894
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=1
radic2120587
119899
sum
119894=1
int
infin
119860119894minus120590119894radic119879minus119905
119878119894119890minus11991122119889119911
minus119890minus119903(119879minus119905)
radic2120587
119899
sum
119894=1
119870119894int
infin
119860119894
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894
1
radic2120587int
minus119860119894+120590119894radic119879minus119905
minusinfin
119890minus1199092
1198942119889119909119894
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894
1
radic2120587int
minus119860119894
minusinfin
119890minus1199092
1198942119889119909119894
=
119899
sum
119894=1
119878119894Φ(minus119860
119894+ 120590119894radic119879 minus 119905)
minus 119890minus119903(119879minus119905)
119899
sum
119894=1
119870119894Φ(minus119860
119894)
(56)
where Φ(119905) is the probability distribution function of astandard Gaussion random variable119873(0 1) that is
Φ (119905) =1
radic2120587int
119905
minusinfin
119890minus11990922119889119909 119905 isin 119877 (57)
In this way we have proved Corollary 7
Remark 8 Theabove result is about the European call optionA similar representation to those from the above corollaryin the European put option case can be obtained by taking119892(119878) = sum
119899
119894=1(119870119894minus 119878119894)+ 119878119894gt 0 for some fixed 119870
119894gt 0
5 Conclusion
In this paper we have considered a financial market modelwith regime switching driven by geometric Levy processThis financial market model is based on the multiple riskyassets 119878
1 1198782 119878
119899driven by Levy process Ito formula and
equivalent transformation methods have been used to solvethis complicated financial market model An example of theportfolio strategy and the final value problem to applying ourmethod to the European call option has been given in the endof this paper
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 9
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (Grants no 61075105 and no 71371046)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] B Oslashksendal Stochastic Differential Equations Springer 6thedition 2005
[4] X Guo and Q Zhang ldquoOptimal selling rules in a regimeswitching modelrdquo IEEE Transactions on Automatic Control vol50 no 9 pp 1450ndash1455 2005
[5] M Pemy Q Zhang and G G Yin ldquoLiquidation of a large blockof stock with regime switchingrdquoMathematical Finance vol 18no 4 pp 629ndash648 2008
[6] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with Markov regime switching and uncertain time-horizonrdquo Journal of Systems Science amp Complexity vol 24 no 1pp 140ndash155 2011
[7] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with regime switching and a stochastic cash flowrdquoInsuranceMathematics amp Economics vol 50 no 3 pp 371ndash3842012
[8] J Kallsen ldquoOptimal portfolios for exponential Levy processesrdquoMathematical Methods of Operations Research vol 51 no 3 pp357ndash374 2000
[9] D Applebaum Levy Processes and Stochastic Calculus vol 93Cambridge University Press Cambridge UK 2004
[10] N Vandaele and M Vanmaele ldquoA locally risk-minimizinghedging strategy for unit-linked life insurance contracts ina Levy process financial marketrdquo Insurance Mathematics ampEconomics vol 42 no 3 pp 1128ndash1137 2008
[11] C Weng ldquoConstant proportion portfolio insurance under aregime switching exponential Levy processrdquo Insurance Math-ematics amp Economics vol 52 no 3 pp 508ndash521 2013
[12] N Bauerle and A Blatter ldquoOptimal control and dependencemodeling of insurance portfolios with Levy dynamicsrdquo Insur-ance Mathematics amp Economics vol 48 no 3 pp 398ndash4052011
[13] K C Yuen and C Yin ldquoOn optimality of the barrier strategyfor a general Levy risk processrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 1700ndash1707 2011
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
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (Grants no 61075105 and no 71371046)
References
[1] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952
[2] D Li andW-L Ng ldquoOptimal dynamic portfolio selection mul-tiperiod mean-variance formulationrdquo Mathematical Financevol 10 no 3 pp 387ndash406 2000
[3] B Oslashksendal Stochastic Differential Equations Springer 6thedition 2005
[4] X Guo and Q Zhang ldquoOptimal selling rules in a regimeswitching modelrdquo IEEE Transactions on Automatic Control vol50 no 9 pp 1450ndash1455 2005
[5] M Pemy Q Zhang and G G Yin ldquoLiquidation of a large blockof stock with regime switchingrdquoMathematical Finance vol 18no 4 pp 629ndash648 2008
[6] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with Markov regime switching and uncertain time-horizonrdquo Journal of Systems Science amp Complexity vol 24 no 1pp 140ndash155 2011
[7] H Wu and Z Li ldquoMulti-period mean-variance portfolioselection with regime switching and a stochastic cash flowrdquoInsuranceMathematics amp Economics vol 50 no 3 pp 371ndash3842012
[8] J Kallsen ldquoOptimal portfolios for exponential Levy processesrdquoMathematical Methods of Operations Research vol 51 no 3 pp357ndash374 2000
[9] D Applebaum Levy Processes and Stochastic Calculus vol 93Cambridge University Press Cambridge UK 2004
[10] N Vandaele and M Vanmaele ldquoA locally risk-minimizinghedging strategy for unit-linked life insurance contracts ina Levy process financial marketrdquo Insurance Mathematics ampEconomics vol 42 no 3 pp 1128ndash1137 2008
[11] C Weng ldquoConstant proportion portfolio insurance under aregime switching exponential Levy processrdquo Insurance Math-ematics amp Economics vol 52 no 3 pp 508ndash521 2013
[12] N Bauerle and A Blatter ldquoOptimal control and dependencemodeling of insurance portfolios with Levy dynamicsrdquo Insur-ance Mathematics amp Economics vol 48 no 3 pp 398ndash4052011
[13] K C Yuen and C Yin ldquoOn optimality of the barrier strategyfor a general Levy risk processrdquo Mathematical and ComputerModelling vol 53 no 9-10 pp 1700ndash1707 2011
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
