Research ArticleClosed-Form Optimal Strategies of Continuous-Time Optionswith Stochastic Differential Equations

Wei Yan

Research Institute of Petroleum Exploration and Development PetroChina Beijing 100083 China

Correspondence should be addressed to Wei Yan yanwei123456petrochinacomcn

Received 18 October 2016 Revised 15 March 2017 Accepted 24 April 2017 Published 2 July 2017

Academic Editor Pietro De Lellis

Copyright copy 2017 Wei Yan This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A continuous-time portfolio selection with options based on risk aversion utility function in financial market is studied Thedifferent price between sale and purchase of options is introduced in this paper The optimal investment-consumption problemis formulated as a continuous-time mathematical model with stochastic differential equations The prices processes follow jump-diffusion processes (Weiner process and Poisson process) Then the corresponding Hamilton-Jacobi-Bellman (HJB) equation ofthe problem is represented and its solution is obtained in different conditions The above results are applied to a special case undera Hyperbolic Absolute Risk Aversion (HARA) utility functionThe optimal investment-consumption strategies about HARA utilityfunction are also derived Finally an example and some discussions illustrating these results are also presented

1 Introduction

Pricing theory of options is an important research field In1973 Black and Scholes published an innovative paper [1]In this paper a theoretical valuation formula is derived Thewell-known Black-Scholes formula is deducedThis formularsquoscreative point does not lie in the investorrsquos preferences buta risk-neutral method with a risk-free asset It is appliedin many fields extensively However the varieties of optionsbecome more and more The theory of options pricing mustbe improved Up to now the most options pricing modelsare extended based on the Black-Scholes formula Merton[2] Jeanblanc-Picque and Pontier [3] and Lin and Ye [4]added the jump process to the option pricing formula Barronand Jensen [5] spread the formula while the interest rate hasdifference between the deposition and loan In addition thereare a lot of other pricing models such as Cox et al [6]

Nowadays the objective is to earn wealth with portfolioselection although the options pricing is still a centralresearch topic We could regard the options as a risk assetlike stocks As a risk asset options catch more and moreattention in financial market There exists options tradingin many exchanges all over the world such as CBOE(Chicago Board Option Exchange) PHLX (Philadelphia

Stock Exchange) AMEX (American Stock Exchange) PSE(Pacific Stock Exchange) NYSE (New York Stock Exchange)In most options exchange they follow the market makerrules The price of purchasing or selling an options contractis equivalent to the price at which the market maker wantsto sell or purchase it Therefore the options have the sameposition with other underlying assets However for thegeneral investment-consumption problem analysis approachthere has been few literatures involving options trading Inthis paper the options contract can be added to the optimalinvestment-consumption problem

One of the frequent questions in finance is how to allocatea certain amount of money in different assets and at whattime instant The earliest approach to consider the optimalportfolio problem is so called mean-variance criterion It waspioneered by Markowitz [7] and has been playing a criticalrole in the theory of portfolio selection It has also gainedwidespread acceptance as a practical tool for portfolio opti-mization However it is a single-period model which makesone-off decision at the beginning of the period and holds onuntil the end of the period Gradually researchers extendedthe above single-period model to continuous-time models[8 9] An optimal consumption-investment problem hasbeen formulated By applying results from stochastic control

HindawiComplexityVolume 2017 Article ID 8734235 11 pageshttpsdoiorg10115520178734235

2 Complexity

theory to the optimal portfolio problem explicit solutionshave been obtained for some special cases Many researchesabout continuous-time portfolio selection appeared [10ndash15]

In general literatures continuous-time investment-con-sumption model is modeled by assuming a market in whichthe evolution of assets prices is described by a continuousprocess However as pointed by Merton [2 16] and Jarrowand Rudd [17] the actual price of stock is not continuousbecause there is some external information affecting it Infact in addition to the continuous process based on Brownmotion the analysis of price evolution does reveal suddenand rare breaks logically accounted for by exogenous eventson information Such a behavior from probabilistic point ofview is naturally modeled by a point process This processgoverned by Brownian motion and point process is calledjump-diffusion process It is a discontinuous price processSome concrete results [2 18] prove that there are some jumpprocesses in reality A stochastic exchange rate with jump-diffusion processes is concerned in the mean-variance port-folio selection [19] However it is a linear-quadratic modelThe calculation process and results may not be complicatedWe hope that the objective function has a general form like aHyperbolic Absolute Risk Aversion (HARA) utility functionof this paper Moreover there is a nondifferentiable point inthe capital process in the following paper These lead to thecalculation process and results that aremore complicatedWewant to obtain some more general results in a sense

In financial markets the price of options has differencebetween selling and purchasing Different from the classicalMerton model we make a hypothesis on the European calloption trading price that the purchasing price is usuallybigger than selling price This simple hypothesis is onlya constraint between purchasing and selling The investorcan add some other constraints However more complicatedconstraints would prevent the derivation of closed-formanalytical solutions Therefore we add this simple constraintto the unconstrained classical Merton model

This paper is concerned with the optimal investment-consumption portfolio selection of options with differentprice between sale and purchase based on a risk aversion util-ity function in financial markets We study how the differentprices affect the strategies of investment-consumption So theproblem is formulated as a continuous-time mathematicalmodel Considering the difference price of the options thepaper is more complex in the price process than traditionalMerton model Besides these pricing processes follow jump-diffusion processes (Weiner process and Poisson process)The model contains discontinuous price processes Then thecorresponding Hamilton-Jacobi-Bellman (HJB) equation ofthe problem is represented and its solutions are obtained indifferent conditions Finally the above results are applied to aspecial case in which the objective function is a HyperbolicAbsolute Risk Aversion (HARA) utility function Underthe above objective function we can obtain a closed-formsolution We can avoid the computational complexity innumerical calculation getting the investment strategy moreeffectively and concisely

2 Mathematical Model

In this section we formulate amathematicalmodelThere are3 assets being traded on a finite horizon [0 119879] They are onebond one risky assetand its corresponding options In thispaper we only consider European call option contracts likeother kinds of similar risky assets

One asset is a bond whose price 1198780(119905) evolves accordingto the differential equation

d1198780 (119905) = 1198780 (119905) 119903 (119905) d119905 119905 isin [0 119879] 1198780 (0) = 1199040 (1)

where 119903(119905) is the interest rate of the bondSimilar to Merton [16] the price process 119865(119905) of the

risk asset should satisfy the following stochastic differentialequation governed by aWeiner process and a Poisson process

d119865 (119905) = 119865 (119905minus) [120603 (119905) d119905 + 120579 (119905) d119882(119905) + 120577 (119905) d119873(119905)] 119905 isin [0 119879]

119865 (0) = 1198910 gt 0(2)

where 120603(119905) is the appreciation rate and 120579(119905) and 120577(119905) arethe volatility coefficient tminus is 119904 lt 119905 while 119904 rarr 119905 Weconsider a financial market which is subject to uncertaintythat enters through a one-dimensional Weiner process 119882(119905)on its canonical space (Ω119882 119871119882 119875119882) and a one-dimensionalPoisson process 119873(119905) on its canonical space (Ω119873 119871119873 119875119873)with intensity equal to 120582(119905) By Jeanblanc-Picque and Pontier[3] the compensated Poisson process defined by 119872(119905) =119873(119905) minus int1198790 120582(119904)d119904 (119905 isin [0 119879]) is a martingale We denote the119875119882-augmentation of 120590(119882(119904) 0 le 119904 le 119905) by 119871 119905119882 andthe 119875119873-augmentation of 120590(119873(119904) 0 le 119904 le 119905) by 119871 119905119873Let (Ω 119871 119875) = (Ω119882 otimes Ω119873 119871119882 otimes 119871119873 119875119882 otimes 119875119873) denote theproduct space On this space119882(119905) and119873(119905) are independentMoreover the coefficients 119903(119905) 120603(119905) 120579(119905) 120577(119905) and 120582(119905) inthe stochastic differential equations are deterministic Borel-measurable and bounded

Applying Itorsquos formula [20] to the purchasing price119884(119905 119865)of the corresponding European call options we obtain

d119884 (119905 119865)= (119884119905 + 119865 (119905) 120603 (119905) 119884119865 + 12119865 (119905)2 120579 (119905)2 119884119865119865) d119905

+ 119865 (119905) 120579 (119905) 119884119865d119882(119905)+ [119884 (119905 119865 (119905) + 119865 (119905) 120577 (119905)) minus 119884 (119905 119865 (119905))] d119873(119905)

119905 isin [0 119879] 119884 (119879 119865 (119879)) = (119865 (119879) minus 119902)+

(3)

where 119884119905 = 120597119884(119905 119865)120597119905 119884119865 = 120597119884(119905 119865)120597119865 119884119865119865 = 1205972119884(119905119865)1205971198652 (119865(119879) minus 119902)+ = 119865(119879) minus 119902 if 119865(119879) ge 119902 0 if 119865(119879) lt 119902and 119902 is strike price (the rest of the partial deferential symbolsof this paper are the same as 119884119905) 119884(119905 119865) is actually a function

Complexity 3

of futures prices 119865(119905) and time 119905 119865(119905) should be a unitary partin 119884(119905 119865) When we apply Itorsquos formula we should apply itto the corresponding process 119884(119905 119865(119905)) which is defined interms of the function 119884 and the process 119865(119905)

We assume that the price of options is different whenselling or purchasing The purchasing price is usually biggerthan selling price We set the selling price as 119866(119905 119865) =120572(119905)119884(119905 119865) where 0 lt 120572(119905) lt 1 Moreover for simplicity itis assumed that 120572(119905) is a monotone nondecreasing functiondifferentiable with respect to time 119905 Then 119866(119905 119865) satisfies thefollowing equation

d119866 (119905 119865)= (119866119905 + 119865 (119905) 120603 (119905) 119866119865 + 12119865 (119905)2 120579 (119905)2 119866119865119865) d119905

+ 119865 (119905) 120579 (119905) 119866119865d119882(119905)+ [119866 (119905 119865 (119905) + 119865 (119905) 120577 (119905)) minus 119866 (119905 119865 (119905))] d119873(119905)

119905 isin [0 119879] 119866 (119879 119865 (119879)) = 120572 (119879) (119865 (119879) minus 119902)+

(4)

Substituting119866(119905 119865) = 120572(119905)119884(119905 119865) into the above equation weobtain

d119866 (119905 119865) = ( (119905) 119884 (119905 119865) + 120572 (119905) 119884119905+ 120572 (119905) 119865 (119905) 120603 (119905) 119884119865 + 12120572 (119905) 119865 (119905)2 120579 (119905)2 119884119865119865) d119905+ 120572 (119905) 119865 (119905) 120579 (119905) 119884119865d119882(119905) + 120572 (119905)sdot [119884 (119905 119865 (119905) + 119865 (119905) 120577 (119905)) minus 119884 (119905 119865 (119905))] d119873(119905)

119905 isin [0 119879] 119866 (119879 119865 (119879)) = 120572 (119879) (119865 (119879) minus 119902)+

(5)

Let 1205870(119905) 1205871(119905) and 1205872(119905) be the amount allocated tothe bond risky asset and options respectively which maybe positive or negative It is assumed that 1205960(119905) 1205961(119905) and1205962(119905) are the shares of the above derivatives mentionedrespectively and then 1205870(119905) = 1205960(119905)1198780(119905) 1205871(119905) = 1205961(119905)119865(119905)1205872(119905) = 1205962(119905)119884(119905 119865) or 1205872(119905) = 1205962(119905)119866(119905 119865)

It is assumed that an investor has initial capital 119909(119905) =1199090(119905) (119905 = 0)When an investor is purchasing option contracts (1205872(119905) gt0) we can get the differential equation as follows by consid-

ering the dynamic process of investment and consumption119888(119905) 0 le 119905 le 119879d119909 (119905) = 1205960 (119905) d1198780 (119905) + 1205961 (119905) d119865 (119905) + 1205962 (119905) d119884 (119905 119865)

minus 119888 (119905) d119905 = [119909 (119905) 119903 (119905) + 1205871 (119905) (120603 (119905) minus 119903 (119905))+ 1205872 (119905)

sdot (119884119905 + 119865 (119905) 120603 (119905) 119884119865 + (12) 1198652120579 (119905)2 119884119865119865119884 (119905 119865 (119905)) minus 119903 (119905))

minus 119888 (119905)] d119905 + [1205871 (119905) 120579 (119905) + 1205872 (119905)

sdot 119865 (119905) 120579 (119905) 119884119865119884 (119905 119865 (119905)) ] d119882(119905) + [1205871 (119905) 120577 (119905)+ 1205872 (119905)sdot 119884 (119905 119865 (119905) + 119865 (119905) 120577 (119905)) minus 119884 (119905 119865 (119905))119884 (119905 119865 (119905)) ] d119873(119905)

119905 isin [0 119879] 119909 (0) = 1199090

(6)

Set 1198871(119905) = 120603(119905) 1198872(119905) = (119884119905 + 119865(119905)120603(119905)119884119865 +(12)1198652120579(119905)2119884119865119865)(119884(119905 119865(119905))) 1205901(119905) = 120579(119905) 1205902(119905) =(119865(119905)120579(119905)119884119865)(119884(119905 119865(119905))) 1205931(119905) = 120577(119905) 1205932(119905) = (119884(119905 119865(119905) +119865(119905)120577(119905)) minus 119884(119905 119865(119905)))(119884(119905 119865(119905))) Then it can be obtainedthat

d119909 (119905) = [119909 (119905) 119903 (119905) + 1205871 (119905) (1198871 (119905) minus 119903 (119905))+ 1205872 (119905) (1198872 (119905) minus 119903 (119905)) minus 119888 (119905)] d119905 + [1205871 (119905) 1205901 (119905)+ 1205872 (119905) 1205902 (119905)] d119882(119905) + [1205871 (119905) 1205931 (119905)+ 1205872 (119905) 1205932 (119905)] d119873(119905) 119905 isin [0 119879]

119909 (0) = 1199090

(7)

When an investor is selling an option contract (1205872(119905) le 0)we can get the similar differential equation

d119909 (119905) = 1205960 (119905) d1198780 (119905) + 1205961 (119905) d119865 (119905) + 1205962 (119905) d119866 (119905 119865)minus 119888 (119905) d119905 = [119909 (119905) 119903 (119905) + 1205871 (119905) (1198871 (119905) minus 119903 (119905))

+ 1205872 (119905) (1198872 (119905) minus 119903 (119905) + (119905)120572 (119905)) minus 119888 (119905)] d119905+ [1205871 (119905) 1205901 (119905) + 1205872 (119905) 1205902] d119882(119905)+ [1205871 (119905) 1205931 (119905) + 1205872 (119905) 1205932] d119873(119905)

119905 isin [0 119879] 119909 (0) = 1199090

(8)

Remark 1 In this paper we regard options as general riskyassets just like stocks Besides it is assumed that 120572(119905) is notconstant but a function of time 119905

4 Complexity

Merging (7) and (8) it can be obtained that

d119909 (119905) = [119909 (119905) 119903 (119905) + 1205871 (119905) (1198871 (119905) minus 119903 (119905))

+ 1205872 (119905) (1198872 (119905) minus 119903 (119905)) minus ( (119905)120572 (119905) (1205872 (119905))minus)

minus 119888 (119905)] d119905 + [1205871 (119905) 1205901 (119905) + 1205872 (119905) 1205902] d119882(119905)+ [1205871 (119905) 1205931 (119905) + 1205872 (119905) 1205932] d119873(119905)

119905 isin [0 119879] 119909 (0) = 1199090

(9)

where (1205872(119905))minus = 0 if 1205872(119905) gt 0 minus1205872(119905) if 1205872(119905) le 0Then (9) can be transformed into the following equation

d119909 (119905) = [119909 (119905) 119903 (119905) + 120587 (119905)1015840 (119887 (119905) minus 119903 (119905) 12)

minus [ (119905)120572 (119905) (1205872 (119905))minus] minus 119888 (119905)] d119905 + 120587 (119905)1015840

sdot 120590 (119905) d119882(119905) + 120587 (119905)1015840 120593 (119905) d119873(119905) 119905 isin [0 119879]

119909 (0) = 1199090

(10)

where 12 denotes a 2-dimensional column vector with everyentry being 1 119887(119905) = [1198871(119905) 1198872(119905)]1015840 120587(119905) = [1205871(119905) 1205872(119905)]1015840120590(119905) = [1205901(119905) 1205902(119905)]1015840 120593(119905) = [1205931(119905) 1205932(119905)]1015840 (1198721015840denotes thetranspose of any matrix or vector119872)

The objective of an investor is to maximize the utility ofconsumption and terminal expected wealth

119869 (119909 119888 1205871 1205872)= 119864 [int119879

01198801 (119905 119888 (119905)) d119905 + 1198802 (119879 119909 (119879))]

(11)

where the utility functions 1198801(sdot) and 1198802(sdot) are risk aversionfunctions

3 HJB Equation

In this section we will deduce the HJB function correspond-ing to (10) and (11) similar to Sage and White [21]

Firstly we set the value function as follows

119881 (119905 119909) = max119888120587

119864119905119909 [int1198791199051198801 (119904 119888) d119904 + 1198802 (119879 119909)] (12)

Then it can be obtained as follows by Bellman optimalprinciple that

119881 (119905 119909) = max119888120587

119864119905119909 int119905+Δ119905119905

1198801 (119904 119888) d119904

+ int119879119905+Δ119905

1198801 (119904 119888) d119904 + 1198802 (119879 119909) = max119888120587

119864119905119909sdot int119905+Δ119905119905

1198801 (119904 119888) d119904 + 119881 (119905 + Δ119905 119909 + Δ119909)

(13)

where 119864119905119909 denotes the conditional expectation when theinitial condition is given by (119905 119909)

It can be obtained that

119881 (119905 + Δ119905 119909 + Δ119909) = 119881 (119905 119909) + 119881119905Δ119905 + 119881119909Δ119909 + 12sdot 119881119905119905 (Δ119905)2 + 12119881119909119909 (Δ119909)2 + 119881119905119909Δ119905Δ119909

Δ119909 (119905) = [119909 (119905) 119903 (119905) + 120587 (119905)1015840 (119887 (119905) minus 119903 (119905) 12)

minus [ (119905)120572 (119905) (1205872 (119905))minus] minus 119888 (119905)] Δ119905 + 120587 (119905)1015840 120590 (119905) Δ119882 (119905)+ 120587 (119905)1015840 120593 (119905) Δ119873 (119905) = [119909 (119905) 119903 (119905)+ 120587 (119905)1015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))minus [ (119905)120572 (119905) (1205872 (119905))minus] minus 119888 (119905)] Δ119905 + 120587 (119905)1015840 120590 (119905) Δ119882 (119905)+ 120587 (119905)1015840 120593 (119905) Δ119872 (119905)

(14)

Substituting the above two equations into (13) we can get

119881 (119905 119909) = max119888120587

119864119905119909 1198801 (119905 119888) Δ119905 + 119881 (119905 119909) + 119881119905Δ119905+ 119881119909Δ119909 + 12119881119905119905 (Δ119905)2 + 12119881119909119909 (Δ119909)2 + 119881119905119909Δ119905Δ119909

= max119888120587

119864119905119909 1198801 (119905 119888) Δ119905 + 119881 (119905 119909) + 119881119905Δ119905

+ 119881119909 [119909119903 (119905) + 1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

minus [ (119905)120572 (119905) (1205872)minus] minus 119888]Δ119905 + 1205871015840120590 (119905) Δ119882 (119905) + 1205871015840120593 (119905)

Complexity 5

sdot Δ119872 (119905) + 12119881119905119905 (Δ119905)2 + 12119881119909119909 [119909119903 (119905)+ 1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905)) minus [ (119905)120572 (119905) (1205872)minus]

minus 119888]Δ119905 + 1205871015840120590 (119905) Δ119882 (119905) + 1205871015840120593 (119905) Δ119872 (119905)2

+ 119881119905119909Δ119905 [119909119903 (119905) + 1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

minus [ (119905)120572 (119905) (1205872)minus] minus 119888]Δ119905 + 1205871015840120590 (119905) Δ119882 (119905) + 1205871015840120593 (119905)

sdot Δ119872 (119905) (15)

By Karatzas and Shreve [22] it can be obtained thatd119882(119905) sdot d119882(119905) = d119905 d119882(119905) sdot d119905 = 0 d119872(119905) sdot d119872(119905) =120582(119905)d119905 d119872(119905) sdot d119905 = 0

After substituting them into the above equation wesubtract from both sides of the equation 119881(119905 119909) divided byΔ119905 Finally by letting Δ119905 rarr 0 we obtain

119881119905 + 119881119909119909119903 (119905) +max119888

1198801 (119905 119888) minus 119888119881119909+max120587

119881119909 [1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

minus ( (119905)120572 (119905) (1205872)minus)] + 12119881119909119909 [1205871015840120590 (119905) 120590 (119905)1015840 120587

+ 120582 (119905) 1205871015840120593 (119905) 120593 (119905)1015840 120587] = 0

(16)

Set 119861(119905) = 119887(119905) minus 119903(119905)12 + 120582(119905)120593(119905) 119863(119905) = 120590(119905)120590(119905)1015840 +120582(119905)120593(119905)120593(119905)1015840 It is assumed that 119863(119905) is positive definiteSubstituting them into (16) it is obtained that

119881119905 + 119881119909119909119903 (119905) +max119888

1198801 (119905 119888) minus 119888119881119909+max120587

119881119909 [1205871015840119861 (119905) minus ( (119905)120572 (119905) (1205872)minus)]

+ 121198811199091199091205871015840119863 (119905) 120587 = 0(17)

with 119881(119879 119909) = 1198802(119879 119909) Equation (17) is the HJB equationcorresponding to (10) and (11)

4 Optimal Investment-Consumption Strategies

In this section the optimal investment-consumption strate-gies will be derived

41 Optimal Consumption Strategies From the HJB equation(17) and by applying a first-order optimality condition wederive that

120597 (1198801 (119905 119888) minus 119888119881119909)120597119888 = 0 (18)

That is to say 119888lowast satisfies1205971198801 (119905 119888)120597119888 = 119881119909 (19)

42 Optimal Investment Strategies We will also deduce theoptimal investment strategies from HJB function (17) Set

119891 (120587) = 119891 (1205871 1205872)= 119881119909 [1205871015840119861 (119905) minus (119905)120572 (119905) (1205872)minus] + 121198811199091199091205871015840119863 (119905) 120587

1198911 (120587) = 1198911 (1205871 1205872) = 1198811199091205871015840119861 (119905) + 121198811199091199091205871015840119863(119905) 1205871198912 (120587) = 1198912 (1205871 1205872)

= 119881119909 [1205871015840119861 (119905) + 1205872 (119905)120572 (119905)] + 121198811199091199091205871015840119863 (119905) 120587

(20)

By setting 1198611(119905) = 119861(119905) and 1198612(119905) = 119861(119905) + (0 (119905)120572(119905))1015840the above three equations can be written as follows

119891 (120587) = 119891 (1205871 1205872)= 119881119909 [12058710158401198611 (119905) minus (119905)120572 (119905) (1205872)minus]

+ 121198811199091199091205871015840119863 (119905) 120587(21)

1198911 (120587) = 1198911 (1205871 1205872) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 (22)

1198912 (120587) = 1198912 (1205871 1205872) = 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587 (23)

thus getting the following equation

119891 (120587) = 119891 (1205871 1205872) = 1198911 (1205871 1205872) if 1205872 gt 01198912 (1205871 1205872) if 1205872 le 0 (24)

From (22) by applying the first-order unconstrainedoptimality condition we obtain

1205971198911 (120587)120597120587 = 0 997904rArr1006704120587 = minus 119881119883119881119883119883119863 (119905)minus1 1198611 (119905)

(25)

and 1198911(1006704120587) = minus((119881119883)22119881119883119883)1198611(119905)1015840119863(119905)minus11198611(119905)

6 Complexity

Similarly from (23) we obtain

1205971198912 (120587)120597120587 = 0 997904rArr = minus 119881119883119881119883119883119863(119905)minus1 1198612 (119905)

(26)

and 1198912() = minus((119881119883)22119881119883119883)1198612(119905)1015840119863(119905)minus11198612(119905)Next let us compare 1006704120587 with We have

1006704120587 minus = 119881119883119881119883119883119863 (119905)minus1 (0 (119905)120572 (119905))1015840 (27)

By Merton [16] 119881(119905 119909) is concave function because theutility functions 1198801(sdot) and 1198802(sdot) are risk aversion functionsThen 119881119909 ge 0 119881119909119909 lt 0 and 119863(119905) is positive definite So 10067041205872 minus2 le 0We will consider three cases

(1) 10067041205872 gt 0 For any 120587 le 0 we have1198912 (120587) = 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587

le 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 = 1198911 (120587)le 1198911 (1006704120587)

(28)

Therefore max119891(120587) = max1198911(120587) = 1198911(1006704120587) = minus((119881119883)22119881119883119883)1198611(119905)1015840119863(119905)minus11198611(119905) and120587lowast = 1006704120587 = minus 119881119883119881119883119883119863 (119905)minus1 1198611 (119905) (29)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909minus (119881119883)22119881119883119883 1198611 (119905)1015840119863 (119905)minus1 1198611 (119905) = 0 (30)

119881 (119879 119909) = 1198802 (119879 119909) (31)

(2) 10067041205872 le 0 le 2 The two optimization problems are

max 1198911 (120587)1205872 ge 0 119905 isin [0 119879] (32)

max 1198912 (120587)1205872 le 0 119905 isin [0 119879] (33)

Solving (32) by introducing the Lagrange multiplier 119897 wehave

1198711 (1205871 1205872 119897) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 minus 119897 sdot 1205872 (34)

where 119897 ge 0Then

1205971198711 (1205871 1205872 119897)1205971205871 = 1198811199091198611 (119905) + 119881119909119909119863 (119905) 120587 = 0 (35)

Considering 10067041205872 le 0 le 2 and the constraint of (32) itis obtained that the optimal solution of (32) satisfies 1205872 = 0Substituting it into (35) we can get the following equation

119881119909 (119887 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905))+ 119881119883119883 (12059021 (119905) + 120582 (119905) 12059321 (119905)) 1205871 = 0 (36)

Then 1205871 = minus(119881119883119881119883119883) sdot (119887(119905) minus 119903(119905) + 120582(119905)1205931(119905))(12059021(119905) +120582(119905)12059321(119905))Following the same line of argument the solution of (33)

is 1205872 = 0 1205871 = minus(119881119883119881119883119883) sdot (119887(119905) minus 119903(119905) + 120582(119905)1205931(119905))(12059021(119905) +120582(119905)12059321(119905))As the above discussion mentioned it is shown that

120587lowast = 120587 = [[minus 119881119883119881119883119883 sdot 1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905)0 ]

] (37)

Therefore max119891(120587) = 1198911(120587) = minus((119881119883)22119881119883119883) sdot ((119887(119905) minus119903(119905) + 120582(119905)1205931(119905))2)(12059021(119905) + 120582(119905)12059321(119905))Setting 119867(119905)2 = ((1198871(119905) minus 119903(119905) + 120582(119905)1205931(119905))2)(12059021(119905) +120582(119905)12059321(119905)) the above equation can be written as follows

max119891 (120587) = 1198911 (120587) = minus(119881119883)22119881119883119883119867(119905)2 (38)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909 minus (119881119883)22119881119883119883119867(119905)2 = 0 (39)

and 119881(119879 119909) = 1198802(119879 119909)(3) 2 lt 0 For any 120587 le 0 we have

1198911 (120587) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587le 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587 = 1198912 (120587)le 1198912 ()

(40)

Therefore max119891(120587) = max1198912(120587) = 1198911() = minus((119881119883)22119881119883119883)1198612(119905)1015840119863(119905)minus11198612(119905) and120587lowast = = minus 119881119883119881119883119883119863 (119905)minus1 1198612 (119905) (41)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0 (42)

and 119881(119879 119909) = 1198802(119879 119909)Next we will discuss the conditions which lead to the

above classifications

Complexity 7

1006704120587 = minus 119881119909119881119909119909119863 (119905)minus1 1198611 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]

minus1

sdot [1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)]

= minus 119881119909119881119909119909[[[[

1205801 minus 1205802120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))21205741 minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))2]]]]

= minus 119881119909119881119909119909119863 (119905)minus1 1198612 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905) + (0 (119905)120572 (119905))1015840)

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)

1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]minus1

sdot [[[

1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905) + (119905)120572 (119905)

]]]

= minus 119881119909119881119909119909[[[[[[

1205801 minus (1205802 + (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)) ( (119905) 120572 (119905)))120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2(1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) ( (119905) 120572 (119905))) minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2

]]]]]]

(43)

where

1205801 = (12059021 (119905) + 120582 (119905) 12059322 (119905)) (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)) 1205802 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205821 = (12059021 (119905) + 120582 (119905) 12059321 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205822 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905))

(44)

Then the optimal investment strategies of (10) and (11) canbe written as follows

120587lowast

=

1006704120587 if 1205741 ge 1205742120587 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(45)

5 HARA Utility Function

Generally speaking the analytical solutions to (30) (39)and (42) can not be obtained easily However the analyticalsolutions to the above HJB equations can be derived whenthe utility function is a HARA function [16] Moreover theoptimal investment-consumption strategies can be givenTherefore we assume that both utility functions of (11) areHARA utility functions that is

1198801 (119905 119910) = 1198802 (119905 119910) = 119890minus120588119905119910120581120581 (46)

where 0 lt 120581 lt 1 120588 gt 0 is a discounted factorWe can get the optimal consumption strategies from (19)

119888lowast = (119890120588119905119881119909)1(120581minus1) (47)

Then

max119888

1198801 (119905 119888) minus 119888119881119909 = 1198801 (119905 119888lowast) minus 119888lowast119881119909= 1120581119890minus120588119905 (119890120588119905119881119909)120581(120581minus1)

minus (119890120588119905119881119909)1(120581minus1) 119881119909= 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

(48)

Next we will derive the optimal investment strategies

8 Complexity

(1) 1006704120587 ge 0 The HJB function (30) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198611 (119905)1015840119863(119905)minus1 1198611 (119905) = 0(49)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)By Merton [16] letting 119881(119905 119909) = 1205731(119905)(119909120581120581) and sub-

stituting it into (49) we can obtain

1205731 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890minus120588119879

(50)

Setting

1198981 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 119899 (119905) = (1 minus 120581) 119890120588119905(120581minus1)

(51)

and substituting them into (50) we can obtain

1205731 (119905) + 1198981 (119905) 1205731 (119905) + 119899 (119905) (1205731 (119905))120581(120581minus1) = 01205731 (119879) = 119890minus120588119879 (52)

Setting 1205781(119905) = 1205731(119905)119890minusint119879119905 1198981(119904)d119904 and substituting it into(52) we can get

1205781 (119905) + 119899 (119905) (1205781 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198981(119904)d119904 = 01205781 (119879) = 119890minus120588119879

(53)

By solving it we can obtain

1205781 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(54)

Then

1205731 (119905) = 119890int119879119905 1198981(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(55)

Substituting 119881(119905 119909) = 1205731(119905)(119909120581120581) into (29) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909

(56)

(2) 1006704120587 le 0 le The HJB function (39) can be written asfollows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883119867(119905)2 = 0(57)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205731(119905)(119909120581120581) and substituting it into (57)

1205731 (119905) + 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890120588119879(58)

Setting

1198982 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] (59)

and substituting it into (58) we can obtain

1205732 (119905) + 1198982 (119905) 1205732 (119905) + 119899 (119905) (1205732 (119905))120581(120581minus1) = 01205732 (119879) = 119890minus120588119879 (60)

Setting 1205782(119905) = 1205732(119905)119890minusint119879119905 1198982(119904)d119904 and substituting it into(52) we can get

1205782 (119905) + 119899 (119905) (1205782 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198982(119904)d119904 = 01205782 (119879) = 119890minus120588119879 (61)

By solving it we can obtain

1205782 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(62)

Then

1205732 (119905) = 119890int119879119905 1198982(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(63)

Complexity 9

By substituting 119881(119905 119909) = 1205732(119905)(119909120581120581) into (37) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = [[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 10 ]

]

119888lowast = 119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909(64)

(3) le 0 The HJB function (42) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0(65)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205733(119905)(119909120581120581) and substituting it into (64)

1205733 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863(119905)minus1 1198612 (119905)] 1205733 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205733 (119905))120581(120581minus1) = 0

1205733 (119879) = 119890minus120588119879

(66)

Setting

1198983 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863 (119905)minus1 1198612 (119905)] (67)

and substituting them into (50) we can obtain

1205733 (119905) + 1198983 (119905) 1205733 (119905) + 119899 (119905) (1205733 (119905))120581(120581minus1) = 01205733 (119879) = 119890minus120588119879 (68)

Setting 1205783(119905) = 1205733(119905)119890minusint119879119905 1198983(119904)d119904 and substituting it into(68) we can get

1205783 (119905) + 119899 (119905) (1205783 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198983(119904)d119904 = 01205783 (119879) = 119890minus120588119879 (69)

By solving it we can obtain

1205783 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(70)

Then

1205733 (119905) = 119890int119879119905 1198983(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(71)

Substituting 119881(119905 119909) = 1205733(119905)(119909120581120581) into (41) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = = minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909

(72)

As the above discussion mentioned we can get the opti-mal investment-consumption strategies of (11) with HARAutility function as pointed by (46)

120587lowast =

minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909 if 1205741 ge 1205742[[[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 1

0]]] if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905)

minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(73)

119888lowast =

119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909 if 1205741 ge 1205742119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(74)

10 Complexity

From (73) and (74) it can be seen that the resultsare closed-form solutions We can avoid solving the partialdifferential equation difficulty and may obtain the optimalinvestment-consumption policies corresponding capital 119909directly

6 A Numerical Example and Discussion

In this section we will introduce a numerical exampleto illustrate model (11) with HARA utility function (46)This example is only to show how to obtain the optimalinvestment-consumption strategies Some parameters havebeen simplified to some extent We can change the assump-tions of the parameters to obtain the results which woulddeliver the different 1205741 and 1205742

(1) It is assumed that the initial capital is 1199090 = 119 times105 RMBWe only calculate the capital range [1199090 21199090]for illustrating the model The rest of the parametersare as follows

120588 = 002120581 = 05119903 = 0025120582 = 06120596 = 007120579 = 039120577 = 01

(75)

For simplicity we set 119879 = 1 In a given time 119905 = 05the futures price 119865 = 53RMB and its correspondingEurope call option price is 119884 = 477RMB We alsosuppose that119884119905 = 013119884119865 = 023119884119865119865 = minus00123 120572 =085 = 1 on this moment In the above parameters1205741 lt 1205742 le 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905))the optimal investment-consumption strategies areillustrated as Figure 1 (121119864 + 5 means 121000 thesame below) From Figure 1 we can see that we donot invest in options anymore The investor can onlyallocate hisher little capital in the futures A largenumber capital is invested in the risk-free bond

(2) We change parameters = 0005 and 120577 = 09from comparing with (1) In the case 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905)) le 1205742 the optimal investment-consumption strategies will change as shown in Fig-ure 2 Figure 2 makes us purchase futures and sell theoptions

(3) If 120577 = 01 119884119865119865 = 0005 and 120596 = 007 then1205741 ge 1205742 comparing with (1) The optimal investment-consumption strategies are illustrated in Figure 3which shows how the investors could sell futures andpurchase the options

Although this example is simple it well explains thesense of the model From the computed results the optimal

FuturesOptions

Risk-free bondConsumption

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)minus100E + 05

000E + 00

100E + 05

200E + 05

300E + 05

400E + 05

500E + 05

600E + 05

700E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 1 Optimal strategies under 1205741 lt 1205742 le 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905))

FuturesOptions

Risk-free bondConsumption

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)

Figure 2 Optimal strategies under 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905)) le 1205742

FuturesOptions

Risk-free bondConsumption

179E + 05 236E + 05121E + 05 198E + 05 217E + 05140E + 05 159E + 05

Total capital (RMB)

minus120E + 06

minus100E + 06

minus800E + 05

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

100E + 06

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 3 Optimal strategies under 1205741 ge 1205742

Complexity 11

investment-consumption strategy is a closed-form solutionthus avoiding solving partial differential equations numer-ically We remark that the numerical solution of partialdifferential equations can be a challenging task The closed-form analytical solution can greatly improve the calculationspeed and reduce the computational complexity Howeveryou need to make some assumptions of stochastic differentialequation parameters if you may also find more details Thefocus of this paper is the theoretical results rather thannumerical calculations Therefore the above example is asimulation which further describes the theoretical results

Comparing with the classical Merton model the Poissonprocess is added to the portfolio with Brownmotion processThe results are very similar to the traditionalMerton solutionoutwardly However the jump process has greatly increasedthe complexity of the solution process For utility functionssimilar to those considered in this paper relevance of capitalbetween optimal strategies can be obtained after solving theoptimal consumption-investment strategies from (73) (74)or the above examples The investors can differently allocatetheir capital by adjusting the parameters of utility function

7 Conclusions

We study the optimal investment-consumption portfolioselection of options which have a different price betweensale and purchase based on a risk aversion utility functionin financial markets These pricing processes follow aWeinerprocess and a Poisson process Then the correspondingHJB equation of the problem is derived and its solutionsare obtained in different conditions respectively Finally aspecial case when the utility function is HARA is illustratedIn addition throughout the paper it is assumed that 120572(119905) isdifferentiable with respect to time 119905 and monotone nonde-creasing Future work will be devoted to the collection of alarge number of datasets in the real market for calibrating thevarious parameters of the model

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoOption pricing when underlying stock returnsare discontinuousrdquo Journal of Financial Economics vol 3 no1-2 pp 125ndash144 1976

[3] M Jeanblanc-Picque and M Pontier ldquoOptimal portfolio for asmall investor in a market model with discontinuous pricesrdquoApplied Mathematics and Optimization vol 22 no 3 pp 287ndash310 1990

[4] J Z Lin and Z X Ye ldquoThe valuation of European contingentclaims of several stocks whose prices are governed by Brownianmotions and Poisson processesrdquo Chinese Journal of AppliedProbability vol 18 pp 167ndash172 2002

[5] EN Barron andR Jensen ldquoA stochastic control approach to thepricing of optionsrdquoMathematics of Operations Research vol 15no 1 pp 49ndash79 1990

[6] J C Cox S A Ross and M Rubinstein ldquoOption pricing asimplified approachrdquo Journal of Financial Economics vol 7 no3 pp 229ndash263 1979

[7] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952

[8] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time modelrdquo The Review of Economics andStatistics vol 51 pp 247ndash257 1969

[9] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[10] X Li X Y Zhou and A E Lim ldquoDynamic mean-varianceportfolio selection with no-shorting constraintsrdquo SIAM Journalon Control and Optimization vol 40 no 5 pp 1540ndash1555 2002

[11] A Cadenillas ldquoConsumption-investment problems with trans-action costs survey and open problemsrdquoMathematicalMethodsof Operations Research vol 51 no 1 pp 43ndash68 2000

[12] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000

[13] D Cuoco and H Liu ldquoA martingale characterization of con-sumption choices and hedging costs with margin require-mentsrdquo Mathematical Finance An International Journal ofMathematics Statistics and Financial Economics vol 10 no 3pp 355ndash385 2000

[14] A E Lim and X Y Zhou ldquoMean-variance portfolio selectionwith random parameters in a complete marketrdquoMathematics ofOperations Research vol 27 no 1 pp 101ndash120 2002

[15] I Karatzas J P Lehoczky S E Shreve and G-L Xu ldquoMar-tingale and duality methods for utility maximization in anincompletemarketrdquo SIAM Journal onControl andOptimizationvol 29 no 3 pp 702ndash730 1991

[16] R C Merton Continuous-Time Finance Basil Blackwell Cam-bridge MA USA 1990

[17] R A Jarrow and A Rudd Option Pricing vol 3 Homewood1983

[18] J C Cox and S A Ross ldquoThe valuation of options for alternativestochastic processesrdquo Journal of Financial Economics vol 3 no1-2 pp 145ndash166 1985

[19] W Yan and Y Chang ldquoModelling on optimal portfolio withexchange rate based on discontinuous stochastic processrdquoInternational Journal of Control vol 89 no 12 pp 2543ndash25482016

[20] G L Gong Introduction to Stochastic Differential EquationsPeking University Press Peking 1987

[21] A P Sage and C C White Optimum systems control vol 3Prentice-Hall Englewood Cliffs NJ USA 1977

[22] I Karatzas and S E Shreve Brownian Motion and StochasticCalculus Springer New York NY USA 2nd edition 1988

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Stochastic AnalysisInternational Journal of

Complexity 3

of futures prices 119865(119905) and time 119905 119865(119905) should be a unitary partin 119884(119905 119865) When we apply Itorsquos formula we should apply itto the corresponding process 119884(119905 119865(119905)) which is defined interms of the function 119884 and the process 119865(119905)

We assume that the price of options is different whenselling or purchasing The purchasing price is usually biggerthan selling price We set the selling price as 119866(119905 119865) =120572(119905)119884(119905 119865) where 0 lt 120572(119905) lt 1 Moreover for simplicity itis assumed that 120572(119905) is a monotone nondecreasing functiondifferentiable with respect to time 119905 Then 119866(119905 119865) satisfies thefollowing equation

d119866 (119905 119865)= (119866119905 + 119865 (119905) 120603 (119905) 119866119865 + 12119865 (119905)2 120579 (119905)2 119866119865119865) d119905

+ 119865 (119905) 120579 (119905) 119866119865d119882(119905)+ [119866 (119905 119865 (119905) + 119865 (119905) 120577 (119905)) minus 119866 (119905 119865 (119905))] d119873(119905)

119905 isin [0 119879] 119866 (119879 119865 (119879)) = 120572 (119879) (119865 (119879) minus 119902)+

(4)

Substituting119866(119905 119865) = 120572(119905)119884(119905 119865) into the above equation weobtain

d119866 (119905 119865) = ( (119905) 119884 (119905 119865) + 120572 (119905) 119884119905+ 120572 (119905) 119865 (119905) 120603 (119905) 119884119865 + 12120572 (119905) 119865 (119905)2 120579 (119905)2 119884119865119865) d119905+ 120572 (119905) 119865 (119905) 120579 (119905) 119884119865d119882(119905) + 120572 (119905)sdot [119884 (119905 119865 (119905) + 119865 (119905) 120577 (119905)) minus 119884 (119905 119865 (119905))] d119873(119905)

119905 isin [0 119879] 119866 (119879 119865 (119879)) = 120572 (119879) (119865 (119879) minus 119902)+

(5)

Let 1205870(119905) 1205871(119905) and 1205872(119905) be the amount allocated tothe bond risky asset and options respectively which maybe positive or negative It is assumed that 1205960(119905) 1205961(119905) and1205962(119905) are the shares of the above derivatives mentionedrespectively and then 1205870(119905) = 1205960(119905)1198780(119905) 1205871(119905) = 1205961(119905)119865(119905)1205872(119905) = 1205962(119905)119884(119905 119865) or 1205872(119905) = 1205962(119905)119866(119905 119865)

It is assumed that an investor has initial capital 119909(119905) =1199090(119905) (119905 = 0)When an investor is purchasing option contracts (1205872(119905) gt0) we can get the differential equation as follows by consid-

ering the dynamic process of investment and consumption119888(119905) 0 le 119905 le 119879d119909 (119905) = 1205960 (119905) d1198780 (119905) + 1205961 (119905) d119865 (119905) + 1205962 (119905) d119884 (119905 119865)

minus 119888 (119905) d119905 = [119909 (119905) 119903 (119905) + 1205871 (119905) (120603 (119905) minus 119903 (119905))+ 1205872 (119905)

sdot (119884119905 + 119865 (119905) 120603 (119905) 119884119865 + (12) 1198652120579 (119905)2 119884119865119865119884 (119905 119865 (119905)) minus 119903 (119905))

minus 119888 (119905)] d119905 + [1205871 (119905) 120579 (119905) + 1205872 (119905)

sdot 119865 (119905) 120579 (119905) 119884119865119884 (119905 119865 (119905)) ] d119882(119905) + [1205871 (119905) 120577 (119905)+ 1205872 (119905)sdot 119884 (119905 119865 (119905) + 119865 (119905) 120577 (119905)) minus 119884 (119905 119865 (119905))119884 (119905 119865 (119905)) ] d119873(119905)

119905 isin [0 119879] 119909 (0) = 1199090

(6)

Set 1198871(119905) = 120603(119905) 1198872(119905) = (119884119905 + 119865(119905)120603(119905)119884119865 +(12)1198652120579(119905)2119884119865119865)(119884(119905 119865(119905))) 1205901(119905) = 120579(119905) 1205902(119905) =(119865(119905)120579(119905)119884119865)(119884(119905 119865(119905))) 1205931(119905) = 120577(119905) 1205932(119905) = (119884(119905 119865(119905) +119865(119905)120577(119905)) minus 119884(119905 119865(119905)))(119884(119905 119865(119905))) Then it can be obtainedthat

d119909 (119905) = [119909 (119905) 119903 (119905) + 1205871 (119905) (1198871 (119905) minus 119903 (119905))+ 1205872 (119905) (1198872 (119905) minus 119903 (119905)) minus 119888 (119905)] d119905 + [1205871 (119905) 1205901 (119905)+ 1205872 (119905) 1205902 (119905)] d119882(119905) + [1205871 (119905) 1205931 (119905)+ 1205872 (119905) 1205932 (119905)] d119873(119905) 119905 isin [0 119879]

119909 (0) = 1199090

(7)

When an investor is selling an option contract (1205872(119905) le 0)we can get the similar differential equation

d119909 (119905) = 1205960 (119905) d1198780 (119905) + 1205961 (119905) d119865 (119905) + 1205962 (119905) d119866 (119905 119865)minus 119888 (119905) d119905 = [119909 (119905) 119903 (119905) + 1205871 (119905) (1198871 (119905) minus 119903 (119905))

+ 1205872 (119905) (1198872 (119905) minus 119903 (119905) + (119905)120572 (119905)) minus 119888 (119905)] d119905+ [1205871 (119905) 1205901 (119905) + 1205872 (119905) 1205902] d119882(119905)+ [1205871 (119905) 1205931 (119905) + 1205872 (119905) 1205932] d119873(119905)

119905 isin [0 119879] 119909 (0) = 1199090

(8)

Remark 1 In this paper we regard options as general riskyassets just like stocks Besides it is assumed that 120572(119905) is notconstant but a function of time 119905

4 Complexity

Merging (7) and (8) it can be obtained that

d119909 (119905) = [119909 (119905) 119903 (119905) + 1205871 (119905) (1198871 (119905) minus 119903 (119905))

+ 1205872 (119905) (1198872 (119905) minus 119903 (119905)) minus ( (119905)120572 (119905) (1205872 (119905))minus)

minus 119888 (119905)] d119905 + [1205871 (119905) 1205901 (119905) + 1205872 (119905) 1205902] d119882(119905)+ [1205871 (119905) 1205931 (119905) + 1205872 (119905) 1205932] d119873(119905)

119905 isin [0 119879] 119909 (0) = 1199090

(9)

where (1205872(119905))minus = 0 if 1205872(119905) gt 0 minus1205872(119905) if 1205872(119905) le 0Then (9) can be transformed into the following equation

d119909 (119905) = [119909 (119905) 119903 (119905) + 120587 (119905)1015840 (119887 (119905) minus 119903 (119905) 12)

minus [ (119905)120572 (119905) (1205872 (119905))minus] minus 119888 (119905)] d119905 + 120587 (119905)1015840

sdot 120590 (119905) d119882(119905) + 120587 (119905)1015840 120593 (119905) d119873(119905) 119905 isin [0 119879]

119909 (0) = 1199090

(10)

where 12 denotes a 2-dimensional column vector with everyentry being 1 119887(119905) = [1198871(119905) 1198872(119905)]1015840 120587(119905) = [1205871(119905) 1205872(119905)]1015840120590(119905) = [1205901(119905) 1205902(119905)]1015840 120593(119905) = [1205931(119905) 1205932(119905)]1015840 (1198721015840denotes thetranspose of any matrix or vector119872)

The objective of an investor is to maximize the utility ofconsumption and terminal expected wealth

119869 (119909 119888 1205871 1205872)= 119864 [int119879

01198801 (119905 119888 (119905)) d119905 + 1198802 (119879 119909 (119879))]

(11)

where the utility functions 1198801(sdot) and 1198802(sdot) are risk aversionfunctions

3 HJB Equation

In this section we will deduce the HJB function correspond-ing to (10) and (11) similar to Sage and White [21]

Firstly we set the value function as follows

119881 (119905 119909) = max119888120587

119864119905119909 [int1198791199051198801 (119904 119888) d119904 + 1198802 (119879 119909)] (12)

Then it can be obtained as follows by Bellman optimalprinciple that

119881 (119905 119909) = max119888120587

119864119905119909 int119905+Δ119905119905

1198801 (119904 119888) d119904

+ int119879119905+Δ119905

1198801 (119904 119888) d119904 + 1198802 (119879 119909) = max119888120587

119864119905119909sdot int119905+Δ119905119905

1198801 (119904 119888) d119904 + 119881 (119905 + Δ119905 119909 + Δ119909)

(13)

where 119864119905119909 denotes the conditional expectation when theinitial condition is given by (119905 119909)

It can be obtained that

119881 (119905 + Δ119905 119909 + Δ119909) = 119881 (119905 119909) + 119881119905Δ119905 + 119881119909Δ119909 + 12sdot 119881119905119905 (Δ119905)2 + 12119881119909119909 (Δ119909)2 + 119881119905119909Δ119905Δ119909

Δ119909 (119905) = [119909 (119905) 119903 (119905) + 120587 (119905)1015840 (119887 (119905) minus 119903 (119905) 12)

minus [ (119905)120572 (119905) (1205872 (119905))minus] minus 119888 (119905)] Δ119905 + 120587 (119905)1015840 120590 (119905) Δ119882 (119905)+ 120587 (119905)1015840 120593 (119905) Δ119873 (119905) = [119909 (119905) 119903 (119905)+ 120587 (119905)1015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))minus [ (119905)120572 (119905) (1205872 (119905))minus] minus 119888 (119905)] Δ119905 + 120587 (119905)1015840 120590 (119905) Δ119882 (119905)+ 120587 (119905)1015840 120593 (119905) Δ119872 (119905)

(14)

Substituting the above two equations into (13) we can get

119881 (119905 119909) = max119888120587

119864119905119909 1198801 (119905 119888) Δ119905 + 119881 (119905 119909) + 119881119905Δ119905+ 119881119909Δ119909 + 12119881119905119905 (Δ119905)2 + 12119881119909119909 (Δ119909)2 + 119881119905119909Δ119905Δ119909

= max119888120587

119864119905119909 1198801 (119905 119888) Δ119905 + 119881 (119905 119909) + 119881119905Δ119905

+ 119881119909 [119909119903 (119905) + 1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

minus [ (119905)120572 (119905) (1205872)minus] minus 119888]Δ119905 + 1205871015840120590 (119905) Δ119882 (119905) + 1205871015840120593 (119905)

Complexity 5

sdot Δ119872 (119905) + 12119881119905119905 (Δ119905)2 + 12119881119909119909 [119909119903 (119905)+ 1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905)) minus [ (119905)120572 (119905) (1205872)minus]

minus 119888]Δ119905 + 1205871015840120590 (119905) Δ119882 (119905) + 1205871015840120593 (119905) Δ119872 (119905)2

+ 119881119905119909Δ119905 [119909119903 (119905) + 1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

minus [ (119905)120572 (119905) (1205872)minus] minus 119888]Δ119905 + 1205871015840120590 (119905) Δ119882 (119905) + 1205871015840120593 (119905)

sdot Δ119872 (119905) (15)

By Karatzas and Shreve [22] it can be obtained thatd119882(119905) sdot d119882(119905) = d119905 d119882(119905) sdot d119905 = 0 d119872(119905) sdot d119872(119905) =120582(119905)d119905 d119872(119905) sdot d119905 = 0

After substituting them into the above equation wesubtract from both sides of the equation 119881(119905 119909) divided byΔ119905 Finally by letting Δ119905 rarr 0 we obtain

119881119905 + 119881119909119909119903 (119905) +max119888

1198801 (119905 119888) minus 119888119881119909+max120587

119881119909 [1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

minus ( (119905)120572 (119905) (1205872)minus)] + 12119881119909119909 [1205871015840120590 (119905) 120590 (119905)1015840 120587

+ 120582 (119905) 1205871015840120593 (119905) 120593 (119905)1015840 120587] = 0

(16)

Set 119861(119905) = 119887(119905) minus 119903(119905)12 + 120582(119905)120593(119905) 119863(119905) = 120590(119905)120590(119905)1015840 +120582(119905)120593(119905)120593(119905)1015840 It is assumed that 119863(119905) is positive definiteSubstituting them into (16) it is obtained that

119881119905 + 119881119909119909119903 (119905) +max119888

1198801 (119905 119888) minus 119888119881119909+max120587

119881119909 [1205871015840119861 (119905) minus ( (119905)120572 (119905) (1205872)minus)]

+ 121198811199091199091205871015840119863 (119905) 120587 = 0(17)

with 119881(119879 119909) = 1198802(119879 119909) Equation (17) is the HJB equationcorresponding to (10) and (11)

4 Optimal Investment-Consumption Strategies

In this section the optimal investment-consumption strate-gies will be derived

41 Optimal Consumption Strategies From the HJB equation(17) and by applying a first-order optimality condition wederive that

120597 (1198801 (119905 119888) minus 119888119881119909)120597119888 = 0 (18)

That is to say 119888lowast satisfies1205971198801 (119905 119888)120597119888 = 119881119909 (19)

42 Optimal Investment Strategies We will also deduce theoptimal investment strategies from HJB function (17) Set

119891 (120587) = 119891 (1205871 1205872)= 119881119909 [1205871015840119861 (119905) minus (119905)120572 (119905) (1205872)minus] + 121198811199091199091205871015840119863 (119905) 120587

1198911 (120587) = 1198911 (1205871 1205872) = 1198811199091205871015840119861 (119905) + 121198811199091199091205871015840119863(119905) 1205871198912 (120587) = 1198912 (1205871 1205872)

= 119881119909 [1205871015840119861 (119905) + 1205872 (119905)120572 (119905)] + 121198811199091199091205871015840119863 (119905) 120587

(20)

By setting 1198611(119905) = 119861(119905) and 1198612(119905) = 119861(119905) + (0 (119905)120572(119905))1015840the above three equations can be written as follows

119891 (120587) = 119891 (1205871 1205872)= 119881119909 [12058710158401198611 (119905) minus (119905)120572 (119905) (1205872)minus]

+ 121198811199091199091205871015840119863 (119905) 120587(21)

1198911 (120587) = 1198911 (1205871 1205872) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 (22)

1198912 (120587) = 1198912 (1205871 1205872) = 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587 (23)

thus getting the following equation

119891 (120587) = 119891 (1205871 1205872) = 1198911 (1205871 1205872) if 1205872 gt 01198912 (1205871 1205872) if 1205872 le 0 (24)

From (22) by applying the first-order unconstrainedoptimality condition we obtain

1205971198911 (120587)120597120587 = 0 997904rArr1006704120587 = minus 119881119883119881119883119883119863 (119905)minus1 1198611 (119905)

(25)

and 1198911(1006704120587) = minus((119881119883)22119881119883119883)1198611(119905)1015840119863(119905)minus11198611(119905)

6 Complexity

Similarly from (23) we obtain

1205971198912 (120587)120597120587 = 0 997904rArr = minus 119881119883119881119883119883119863(119905)minus1 1198612 (119905)

(26)

and 1198912() = minus((119881119883)22119881119883119883)1198612(119905)1015840119863(119905)minus11198612(119905)Next let us compare 1006704120587 with We have

1006704120587 minus = 119881119883119881119883119883119863 (119905)minus1 (0 (119905)120572 (119905))1015840 (27)

By Merton [16] 119881(119905 119909) is concave function because theutility functions 1198801(sdot) and 1198802(sdot) are risk aversion functionsThen 119881119909 ge 0 119881119909119909 lt 0 and 119863(119905) is positive definite So 10067041205872 minus2 le 0We will consider three cases

(1) 10067041205872 gt 0 For any 120587 le 0 we have1198912 (120587) = 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587

le 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 = 1198911 (120587)le 1198911 (1006704120587)

(28)

Therefore max119891(120587) = max1198911(120587) = 1198911(1006704120587) = minus((119881119883)22119881119883119883)1198611(119905)1015840119863(119905)minus11198611(119905) and120587lowast = 1006704120587 = minus 119881119883119881119883119883119863 (119905)minus1 1198611 (119905) (29)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909minus (119881119883)22119881119883119883 1198611 (119905)1015840119863 (119905)minus1 1198611 (119905) = 0 (30)

119881 (119879 119909) = 1198802 (119879 119909) (31)

(2) 10067041205872 le 0 le 2 The two optimization problems are

max 1198911 (120587)1205872 ge 0 119905 isin [0 119879] (32)

max 1198912 (120587)1205872 le 0 119905 isin [0 119879] (33)

Solving (32) by introducing the Lagrange multiplier 119897 wehave

1198711 (1205871 1205872 119897) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 minus 119897 sdot 1205872 (34)

where 119897 ge 0Then

1205971198711 (1205871 1205872 119897)1205971205871 = 1198811199091198611 (119905) + 119881119909119909119863 (119905) 120587 = 0 (35)

Considering 10067041205872 le 0 le 2 and the constraint of (32) itis obtained that the optimal solution of (32) satisfies 1205872 = 0Substituting it into (35) we can get the following equation

119881119909 (119887 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905))+ 119881119883119883 (12059021 (119905) + 120582 (119905) 12059321 (119905)) 1205871 = 0 (36)

Then 1205871 = minus(119881119883119881119883119883) sdot (119887(119905) minus 119903(119905) + 120582(119905)1205931(119905))(12059021(119905) +120582(119905)12059321(119905))Following the same line of argument the solution of (33)

is 1205872 = 0 1205871 = minus(119881119883119881119883119883) sdot (119887(119905) minus 119903(119905) + 120582(119905)1205931(119905))(12059021(119905) +120582(119905)12059321(119905))As the above discussion mentioned it is shown that

120587lowast = 120587 = [[minus 119881119883119881119883119883 sdot 1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905)0 ]

] (37)

Therefore max119891(120587) = 1198911(120587) = minus((119881119883)22119881119883119883) sdot ((119887(119905) minus119903(119905) + 120582(119905)1205931(119905))2)(12059021(119905) + 120582(119905)12059321(119905))Setting 119867(119905)2 = ((1198871(119905) minus 119903(119905) + 120582(119905)1205931(119905))2)(12059021(119905) +120582(119905)12059321(119905)) the above equation can be written as follows

max119891 (120587) = 1198911 (120587) = minus(119881119883)22119881119883119883119867(119905)2 (38)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909 minus (119881119883)22119881119883119883119867(119905)2 = 0 (39)

and 119881(119879 119909) = 1198802(119879 119909)(3) 2 lt 0 For any 120587 le 0 we have

1198911 (120587) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587le 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587 = 1198912 (120587)le 1198912 ()

(40)

Therefore max119891(120587) = max1198912(120587) = 1198911() = minus((119881119883)22119881119883119883)1198612(119905)1015840119863(119905)minus11198612(119905) and120587lowast = = minus 119881119883119881119883119883119863 (119905)minus1 1198612 (119905) (41)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0 (42)

and 119881(119879 119909) = 1198802(119879 119909)Next we will discuss the conditions which lead to the

above classifications

Complexity 7

1006704120587 = minus 119881119909119881119909119909119863 (119905)minus1 1198611 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]

minus1

sdot [1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)]

= minus 119881119909119881119909119909[[[[

1205801 minus 1205802120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))21205741 minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))2]]]]

= minus 119881119909119881119909119909119863 (119905)minus1 1198612 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905) + (0 (119905)120572 (119905))1015840)

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)

1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]minus1

sdot [[[

1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905) + (119905)120572 (119905)

]]]

= minus 119881119909119881119909119909[[[[[[

1205801 minus (1205802 + (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)) ( (119905) 120572 (119905)))120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2(1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) ( (119905) 120572 (119905))) minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2

]]]]]]

(43)

where

1205801 = (12059021 (119905) + 120582 (119905) 12059322 (119905)) (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)) 1205802 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205821 = (12059021 (119905) + 120582 (119905) 12059321 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205822 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905))

(44)

Then the optimal investment strategies of (10) and (11) canbe written as follows

120587lowast

=

1006704120587 if 1205741 ge 1205742120587 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(45)

5 HARA Utility Function

Generally speaking the analytical solutions to (30) (39)and (42) can not be obtained easily However the analyticalsolutions to the above HJB equations can be derived whenthe utility function is a HARA function [16] Moreover theoptimal investment-consumption strategies can be givenTherefore we assume that both utility functions of (11) areHARA utility functions that is

1198801 (119905 119910) = 1198802 (119905 119910) = 119890minus120588119905119910120581120581 (46)

where 0 lt 120581 lt 1 120588 gt 0 is a discounted factorWe can get the optimal consumption strategies from (19)

119888lowast = (119890120588119905119881119909)1(120581minus1) (47)

Then

max119888

1198801 (119905 119888) minus 119888119881119909 = 1198801 (119905 119888lowast) minus 119888lowast119881119909= 1120581119890minus120588119905 (119890120588119905119881119909)120581(120581minus1)

minus (119890120588119905119881119909)1(120581minus1) 119881119909= 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

(48)

Next we will derive the optimal investment strategies

8 Complexity

(1) 1006704120587 ge 0 The HJB function (30) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198611 (119905)1015840119863(119905)minus1 1198611 (119905) = 0(49)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)By Merton [16] letting 119881(119905 119909) = 1205731(119905)(119909120581120581) and sub-

stituting it into (49) we can obtain

1205731 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890minus120588119879

(50)

Setting

1198981 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 119899 (119905) = (1 minus 120581) 119890120588119905(120581minus1)

(51)

and substituting them into (50) we can obtain

1205731 (119905) + 1198981 (119905) 1205731 (119905) + 119899 (119905) (1205731 (119905))120581(120581minus1) = 01205731 (119879) = 119890minus120588119879 (52)

Setting 1205781(119905) = 1205731(119905)119890minusint119879119905 1198981(119904)d119904 and substituting it into(52) we can get

1205781 (119905) + 119899 (119905) (1205781 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198981(119904)d119904 = 01205781 (119879) = 119890minus120588119879

(53)

By solving it we can obtain

1205781 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(54)

Then

1205731 (119905) = 119890int119879119905 1198981(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(55)

Substituting 119881(119905 119909) = 1205731(119905)(119909120581120581) into (29) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909

(56)

(2) 1006704120587 le 0 le The HJB function (39) can be written asfollows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883119867(119905)2 = 0(57)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205731(119905)(119909120581120581) and substituting it into (57)

1205731 (119905) + 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890120588119879(58)

Setting

1198982 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] (59)

and substituting it into (58) we can obtain

1205732 (119905) + 1198982 (119905) 1205732 (119905) + 119899 (119905) (1205732 (119905))120581(120581minus1) = 01205732 (119879) = 119890minus120588119879 (60)

Setting 1205782(119905) = 1205732(119905)119890minusint119879119905 1198982(119904)d119904 and substituting it into(52) we can get

1205782 (119905) + 119899 (119905) (1205782 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198982(119904)d119904 = 01205782 (119879) = 119890minus120588119879 (61)

By solving it we can obtain

1205782 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(62)

Then

1205732 (119905) = 119890int119879119905 1198982(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(63)

Complexity 9

By substituting 119881(119905 119909) = 1205732(119905)(119909120581120581) into (37) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = [[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 10 ]

]

119888lowast = 119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909(64)

(3) le 0 The HJB function (42) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0(65)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205733(119905)(119909120581120581) and substituting it into (64)

1205733 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863(119905)minus1 1198612 (119905)] 1205733 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205733 (119905))120581(120581minus1) = 0

1205733 (119879) = 119890minus120588119879

(66)

Setting

1198983 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863 (119905)minus1 1198612 (119905)] (67)

and substituting them into (50) we can obtain

1205733 (119905) + 1198983 (119905) 1205733 (119905) + 119899 (119905) (1205733 (119905))120581(120581minus1) = 01205733 (119879) = 119890minus120588119879 (68)

Setting 1205783(119905) = 1205733(119905)119890minusint119879119905 1198983(119904)d119904 and substituting it into(68) we can get

1205783 (119905) + 119899 (119905) (1205783 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198983(119904)d119904 = 01205783 (119879) = 119890minus120588119879 (69)

By solving it we can obtain

1205783 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(70)

Then

1205733 (119905) = 119890int119879119905 1198983(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(71)

Substituting 119881(119905 119909) = 1205733(119905)(119909120581120581) into (41) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = = minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909

(72)

As the above discussion mentioned we can get the opti-mal investment-consumption strategies of (11) with HARAutility function as pointed by (46)

120587lowast =

minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909 if 1205741 ge 1205742[[[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 1

0]]] if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905)

minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(73)

119888lowast =

119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909 if 1205741 ge 1205742119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(74)

10 Complexity

From (73) and (74) it can be seen that the resultsare closed-form solutions We can avoid solving the partialdifferential equation difficulty and may obtain the optimalinvestment-consumption policies corresponding capital 119909directly

6 A Numerical Example and Discussion

In this section we will introduce a numerical exampleto illustrate model (11) with HARA utility function (46)This example is only to show how to obtain the optimalinvestment-consumption strategies Some parameters havebeen simplified to some extent We can change the assump-tions of the parameters to obtain the results which woulddeliver the different 1205741 and 1205742

(1) It is assumed that the initial capital is 1199090 = 119 times105 RMBWe only calculate the capital range [1199090 21199090]for illustrating the model The rest of the parametersare as follows

120588 = 002120581 = 05119903 = 0025120582 = 06120596 = 007120579 = 039120577 = 01

(75)

For simplicity we set 119879 = 1 In a given time 119905 = 05the futures price 119865 = 53RMB and its correspondingEurope call option price is 119884 = 477RMB We alsosuppose that119884119905 = 013119884119865 = 023119884119865119865 = minus00123 120572 =085 = 1 on this moment In the above parameters1205741 lt 1205742 le 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905))the optimal investment-consumption strategies areillustrated as Figure 1 (121119864 + 5 means 121000 thesame below) From Figure 1 we can see that we donot invest in options anymore The investor can onlyallocate hisher little capital in the futures A largenumber capital is invested in the risk-free bond

(2) We change parameters = 0005 and 120577 = 09from comparing with (1) In the case 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905)) le 1205742 the optimal investment-consumption strategies will change as shown in Fig-ure 2 Figure 2 makes us purchase futures and sell theoptions

(3) If 120577 = 01 119884119865119865 = 0005 and 120596 = 007 then1205741 ge 1205742 comparing with (1) The optimal investment-consumption strategies are illustrated in Figure 3which shows how the investors could sell futures andpurchase the options

Although this example is simple it well explains thesense of the model From the computed results the optimal

FuturesOptions

Risk-free bondConsumption

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)minus100E + 05

000E + 00

100E + 05

200E + 05

300E + 05

400E + 05

500E + 05

600E + 05

700E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 1 Optimal strategies under 1205741 lt 1205742 le 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905))

FuturesOptions

Risk-free bondConsumption

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)

Figure 2 Optimal strategies under 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905)) le 1205742

FuturesOptions

Risk-free bondConsumption

179E + 05 236E + 05121E + 05 198E + 05 217E + 05140E + 05 159E + 05

Total capital (RMB)

minus120E + 06

minus100E + 06

minus800E + 05

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

100E + 06

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 3 Optimal strategies under 1205741 ge 1205742

Complexity 11

investment-consumption strategy is a closed-form solutionthus avoiding solving partial differential equations numer-ically We remark that the numerical solution of partialdifferential equations can be a challenging task The closed-form analytical solution can greatly improve the calculationspeed and reduce the computational complexity Howeveryou need to make some assumptions of stochastic differentialequation parameters if you may also find more details Thefocus of this paper is the theoretical results rather thannumerical calculations Therefore the above example is asimulation which further describes the theoretical results

Comparing with the classical Merton model the Poissonprocess is added to the portfolio with Brownmotion processThe results are very similar to the traditionalMerton solutionoutwardly However the jump process has greatly increasedthe complexity of the solution process For utility functionssimilar to those considered in this paper relevance of capitalbetween optimal strategies can be obtained after solving theoptimal consumption-investment strategies from (73) (74)or the above examples The investors can differently allocatetheir capital by adjusting the parameters of utility function

7 Conclusions

We study the optimal investment-consumption portfolioselection of options which have a different price betweensale and purchase based on a risk aversion utility functionin financial markets These pricing processes follow aWeinerprocess and a Poisson process Then the correspondingHJB equation of the problem is derived and its solutionsare obtained in different conditions respectively Finally aspecial case when the utility function is HARA is illustratedIn addition throughout the paper it is assumed that 120572(119905) isdifferentiable with respect to time 119905 and monotone nonde-creasing Future work will be devoted to the collection of alarge number of datasets in the real market for calibrating thevarious parameters of the model

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoOption pricing when underlying stock returnsare discontinuousrdquo Journal of Financial Economics vol 3 no1-2 pp 125ndash144 1976

[3] M Jeanblanc-Picque and M Pontier ldquoOptimal portfolio for asmall investor in a market model with discontinuous pricesrdquoApplied Mathematics and Optimization vol 22 no 3 pp 287ndash310 1990

[4] J Z Lin and Z X Ye ldquoThe valuation of European contingentclaims of several stocks whose prices are governed by Brownianmotions and Poisson processesrdquo Chinese Journal of AppliedProbability vol 18 pp 167ndash172 2002

[5] EN Barron andR Jensen ldquoA stochastic control approach to thepricing of optionsrdquoMathematics of Operations Research vol 15no 1 pp 49ndash79 1990

[6] J C Cox S A Ross and M Rubinstein ldquoOption pricing asimplified approachrdquo Journal of Financial Economics vol 7 no3 pp 229ndash263 1979

[7] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952

[8] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time modelrdquo The Review of Economics andStatistics vol 51 pp 247ndash257 1969

[9] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[10] X Li X Y Zhou and A E Lim ldquoDynamic mean-varianceportfolio selection with no-shorting constraintsrdquo SIAM Journalon Control and Optimization vol 40 no 5 pp 1540ndash1555 2002

[11] A Cadenillas ldquoConsumption-investment problems with trans-action costs survey and open problemsrdquoMathematicalMethodsof Operations Research vol 51 no 1 pp 43ndash68 2000

[12] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000

[13] D Cuoco and H Liu ldquoA martingale characterization of con-sumption choices and hedging costs with margin require-mentsrdquo Mathematical Finance An International Journal ofMathematics Statistics and Financial Economics vol 10 no 3pp 355ndash385 2000

[14] A E Lim and X Y Zhou ldquoMean-variance portfolio selectionwith random parameters in a complete marketrdquoMathematics ofOperations Research vol 27 no 1 pp 101ndash120 2002

[15] I Karatzas J P Lehoczky S E Shreve and G-L Xu ldquoMar-tingale and duality methods for utility maximization in anincompletemarketrdquo SIAM Journal onControl andOptimizationvol 29 no 3 pp 702ndash730 1991

[16] R C Merton Continuous-Time Finance Basil Blackwell Cam-bridge MA USA 1990

[17] R A Jarrow and A Rudd Option Pricing vol 3 Homewood1983

[18] J C Cox and S A Ross ldquoThe valuation of options for alternativestochastic processesrdquo Journal of Financial Economics vol 3 no1-2 pp 145ndash166 1985

[19] W Yan and Y Chang ldquoModelling on optimal portfolio withexchange rate based on discontinuous stochastic processrdquoInternational Journal of Control vol 89 no 12 pp 2543ndash25482016

[20] G L Gong Introduction to Stochastic Differential EquationsPeking University Press Peking 1987

[21] A P Sage and C C White Optimum systems control vol 3Prentice-Hall Englewood Cliffs NJ USA 1977

[22] I Karatzas and S E Shreve Brownian Motion and StochasticCalculus Springer New York NY USA 2nd edition 1988

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Complexity

Merging (7) and (8) it can be obtained that

d119909 (119905) = [119909 (119905) 119903 (119905) + 1205871 (119905) (1198871 (119905) minus 119903 (119905))

+ 1205872 (119905) (1198872 (119905) minus 119903 (119905)) minus ( (119905)120572 (119905) (1205872 (119905))minus)

minus 119888 (119905)] d119905 + [1205871 (119905) 1205901 (119905) + 1205872 (119905) 1205902] d119882(119905)+ [1205871 (119905) 1205931 (119905) + 1205872 (119905) 1205932] d119873(119905)

119905 isin [0 119879] 119909 (0) = 1199090

(9)

where (1205872(119905))minus = 0 if 1205872(119905) gt 0 minus1205872(119905) if 1205872(119905) le 0Then (9) can be transformed into the following equation

d119909 (119905) = [119909 (119905) 119903 (119905) + 120587 (119905)1015840 (119887 (119905) minus 119903 (119905) 12)

minus [ (119905)120572 (119905) (1205872 (119905))minus] minus 119888 (119905)] d119905 + 120587 (119905)1015840

sdot 120590 (119905) d119882(119905) + 120587 (119905)1015840 120593 (119905) d119873(119905) 119905 isin [0 119879]

119909 (0) = 1199090

(10)

where 12 denotes a 2-dimensional column vector with everyentry being 1 119887(119905) = [1198871(119905) 1198872(119905)]1015840 120587(119905) = [1205871(119905) 1205872(119905)]1015840120590(119905) = [1205901(119905) 1205902(119905)]1015840 120593(119905) = [1205931(119905) 1205932(119905)]1015840 (1198721015840denotes thetranspose of any matrix or vector119872)

The objective of an investor is to maximize the utility ofconsumption and terminal expected wealth

119869 (119909 119888 1205871 1205872)= 119864 [int119879

01198801 (119905 119888 (119905)) d119905 + 1198802 (119879 119909 (119879))]

(11)

where the utility functions 1198801(sdot) and 1198802(sdot) are risk aversionfunctions

3 HJB Equation

In this section we will deduce the HJB function correspond-ing to (10) and (11) similar to Sage and White [21]

Firstly we set the value function as follows

119881 (119905 119909) = max119888120587

119864119905119909 [int1198791199051198801 (119904 119888) d119904 + 1198802 (119879 119909)] (12)

Then it can be obtained as follows by Bellman optimalprinciple that

119881 (119905 119909) = max119888120587

119864119905119909 int119905+Δ119905119905

1198801 (119904 119888) d119904

+ int119879119905+Δ119905

1198801 (119904 119888) d119904 + 1198802 (119879 119909) = max119888120587

119864119905119909sdot int119905+Δ119905119905

1198801 (119904 119888) d119904 + 119881 (119905 + Δ119905 119909 + Δ119909)

(13)

where 119864119905119909 denotes the conditional expectation when theinitial condition is given by (119905 119909)

It can be obtained that

119881 (119905 + Δ119905 119909 + Δ119909) = 119881 (119905 119909) + 119881119905Δ119905 + 119881119909Δ119909 + 12sdot 119881119905119905 (Δ119905)2 + 12119881119909119909 (Δ119909)2 + 119881119905119909Δ119905Δ119909

Δ119909 (119905) = [119909 (119905) 119903 (119905) + 120587 (119905)1015840 (119887 (119905) minus 119903 (119905) 12)

minus [ (119905)120572 (119905) (1205872 (119905))minus] minus 119888 (119905)] Δ119905 + 120587 (119905)1015840 120590 (119905) Δ119882 (119905)+ 120587 (119905)1015840 120593 (119905) Δ119873 (119905) = [119909 (119905) 119903 (119905)+ 120587 (119905)1015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))minus [ (119905)120572 (119905) (1205872 (119905))minus] minus 119888 (119905)] Δ119905 + 120587 (119905)1015840 120590 (119905) Δ119882 (119905)+ 120587 (119905)1015840 120593 (119905) Δ119872 (119905)

(14)

Substituting the above two equations into (13) we can get

119881 (119905 119909) = max119888120587

119864119905119909 1198801 (119905 119888) Δ119905 + 119881 (119905 119909) + 119881119905Δ119905+ 119881119909Δ119909 + 12119881119905119905 (Δ119905)2 + 12119881119909119909 (Δ119909)2 + 119881119905119909Δ119905Δ119909

= max119888120587

119864119905119909 1198801 (119905 119888) Δ119905 + 119881 (119905 119909) + 119881119905Δ119905

+ 119881119909 [119909119903 (119905) + 1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

minus [ (119905)120572 (119905) (1205872)minus] minus 119888]Δ119905 + 1205871015840120590 (119905) Δ119882 (119905) + 1205871015840120593 (119905)

Complexity 5

sdot Δ119872 (119905) + 12119881119905119905 (Δ119905)2 + 12119881119909119909 [119909119903 (119905)+ 1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905)) minus [ (119905)120572 (119905) (1205872)minus]

minus 119888]Δ119905 + 1205871015840120590 (119905) Δ119882 (119905) + 1205871015840120593 (119905) Δ119872 (119905)2

+ 119881119905119909Δ119905 [119909119903 (119905) + 1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

minus [ (119905)120572 (119905) (1205872)minus] minus 119888]Δ119905 + 1205871015840120590 (119905) Δ119882 (119905) + 1205871015840120593 (119905)

sdot Δ119872 (119905) (15)

By Karatzas and Shreve [22] it can be obtained thatd119882(119905) sdot d119882(119905) = d119905 d119882(119905) sdot d119905 = 0 d119872(119905) sdot d119872(119905) =120582(119905)d119905 d119872(119905) sdot d119905 = 0

After substituting them into the above equation wesubtract from both sides of the equation 119881(119905 119909) divided byΔ119905 Finally by letting Δ119905 rarr 0 we obtain

119881119905 + 119881119909119909119903 (119905) +max119888

1198801 (119905 119888) minus 119888119881119909+max120587

119881119909 [1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

minus ( (119905)120572 (119905) (1205872)minus)] + 12119881119909119909 [1205871015840120590 (119905) 120590 (119905)1015840 120587

+ 120582 (119905) 1205871015840120593 (119905) 120593 (119905)1015840 120587] = 0

(16)

Set 119861(119905) = 119887(119905) minus 119903(119905)12 + 120582(119905)120593(119905) 119863(119905) = 120590(119905)120590(119905)1015840 +120582(119905)120593(119905)120593(119905)1015840 It is assumed that 119863(119905) is positive definiteSubstituting them into (16) it is obtained that

119881119905 + 119881119909119909119903 (119905) +max119888

1198801 (119905 119888) minus 119888119881119909+max120587

119881119909 [1205871015840119861 (119905) minus ( (119905)120572 (119905) (1205872)minus)]

+ 121198811199091199091205871015840119863 (119905) 120587 = 0(17)

with 119881(119879 119909) = 1198802(119879 119909) Equation (17) is the HJB equationcorresponding to (10) and (11)

4 Optimal Investment-Consumption Strategies

In this section the optimal investment-consumption strate-gies will be derived

41 Optimal Consumption Strategies From the HJB equation(17) and by applying a first-order optimality condition wederive that

120597 (1198801 (119905 119888) minus 119888119881119909)120597119888 = 0 (18)

That is to say 119888lowast satisfies1205971198801 (119905 119888)120597119888 = 119881119909 (19)

42 Optimal Investment Strategies We will also deduce theoptimal investment strategies from HJB function (17) Set

119891 (120587) = 119891 (1205871 1205872)= 119881119909 [1205871015840119861 (119905) minus (119905)120572 (119905) (1205872)minus] + 121198811199091199091205871015840119863 (119905) 120587

1198911 (120587) = 1198911 (1205871 1205872) = 1198811199091205871015840119861 (119905) + 121198811199091199091205871015840119863(119905) 1205871198912 (120587) = 1198912 (1205871 1205872)

= 119881119909 [1205871015840119861 (119905) + 1205872 (119905)120572 (119905)] + 121198811199091199091205871015840119863 (119905) 120587

(20)

By setting 1198611(119905) = 119861(119905) and 1198612(119905) = 119861(119905) + (0 (119905)120572(119905))1015840the above three equations can be written as follows

119891 (120587) = 119891 (1205871 1205872)= 119881119909 [12058710158401198611 (119905) minus (119905)120572 (119905) (1205872)minus]

+ 121198811199091199091205871015840119863 (119905) 120587(21)

1198911 (120587) = 1198911 (1205871 1205872) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 (22)

1198912 (120587) = 1198912 (1205871 1205872) = 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587 (23)

thus getting the following equation

119891 (120587) = 119891 (1205871 1205872) = 1198911 (1205871 1205872) if 1205872 gt 01198912 (1205871 1205872) if 1205872 le 0 (24)

From (22) by applying the first-order unconstrainedoptimality condition we obtain

1205971198911 (120587)120597120587 = 0 997904rArr1006704120587 = minus 119881119883119881119883119883119863 (119905)minus1 1198611 (119905)

(25)

and 1198911(1006704120587) = minus((119881119883)22119881119883119883)1198611(119905)1015840119863(119905)minus11198611(119905)

6 Complexity

Similarly from (23) we obtain

1205971198912 (120587)120597120587 = 0 997904rArr = minus 119881119883119881119883119883119863(119905)minus1 1198612 (119905)

(26)

and 1198912() = minus((119881119883)22119881119883119883)1198612(119905)1015840119863(119905)minus11198612(119905)Next let us compare 1006704120587 with We have

1006704120587 minus = 119881119883119881119883119883119863 (119905)minus1 (0 (119905)120572 (119905))1015840 (27)

By Merton [16] 119881(119905 119909) is concave function because theutility functions 1198801(sdot) and 1198802(sdot) are risk aversion functionsThen 119881119909 ge 0 119881119909119909 lt 0 and 119863(119905) is positive definite So 10067041205872 minus2 le 0We will consider three cases

(1) 10067041205872 gt 0 For any 120587 le 0 we have1198912 (120587) = 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587

le 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 = 1198911 (120587)le 1198911 (1006704120587)

(28)

Therefore max119891(120587) = max1198911(120587) = 1198911(1006704120587) = minus((119881119883)22119881119883119883)1198611(119905)1015840119863(119905)minus11198611(119905) and120587lowast = 1006704120587 = minus 119881119883119881119883119883119863 (119905)minus1 1198611 (119905) (29)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909minus (119881119883)22119881119883119883 1198611 (119905)1015840119863 (119905)minus1 1198611 (119905) = 0 (30)

119881 (119879 119909) = 1198802 (119879 119909) (31)

(2) 10067041205872 le 0 le 2 The two optimization problems are

max 1198911 (120587)1205872 ge 0 119905 isin [0 119879] (32)

max 1198912 (120587)1205872 le 0 119905 isin [0 119879] (33)

Solving (32) by introducing the Lagrange multiplier 119897 wehave

1198711 (1205871 1205872 119897) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 minus 119897 sdot 1205872 (34)

where 119897 ge 0Then

1205971198711 (1205871 1205872 119897)1205971205871 = 1198811199091198611 (119905) + 119881119909119909119863 (119905) 120587 = 0 (35)

Considering 10067041205872 le 0 le 2 and the constraint of (32) itis obtained that the optimal solution of (32) satisfies 1205872 = 0Substituting it into (35) we can get the following equation

119881119909 (119887 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905))+ 119881119883119883 (12059021 (119905) + 120582 (119905) 12059321 (119905)) 1205871 = 0 (36)

Then 1205871 = minus(119881119883119881119883119883) sdot (119887(119905) minus 119903(119905) + 120582(119905)1205931(119905))(12059021(119905) +120582(119905)12059321(119905))Following the same line of argument the solution of (33)

is 1205872 = 0 1205871 = minus(119881119883119881119883119883) sdot (119887(119905) minus 119903(119905) + 120582(119905)1205931(119905))(12059021(119905) +120582(119905)12059321(119905))As the above discussion mentioned it is shown that

120587lowast = 120587 = [[minus 119881119883119881119883119883 sdot 1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905)0 ]

] (37)

Therefore max119891(120587) = 1198911(120587) = minus((119881119883)22119881119883119883) sdot ((119887(119905) minus119903(119905) + 120582(119905)1205931(119905))2)(12059021(119905) + 120582(119905)12059321(119905))Setting 119867(119905)2 = ((1198871(119905) minus 119903(119905) + 120582(119905)1205931(119905))2)(12059021(119905) +120582(119905)12059321(119905)) the above equation can be written as follows

max119891 (120587) = 1198911 (120587) = minus(119881119883)22119881119883119883119867(119905)2 (38)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909 minus (119881119883)22119881119883119883119867(119905)2 = 0 (39)

and 119881(119879 119909) = 1198802(119879 119909)(3) 2 lt 0 For any 120587 le 0 we have

1198911 (120587) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587le 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587 = 1198912 (120587)le 1198912 ()

(40)

Therefore max119891(120587) = max1198912(120587) = 1198911() = minus((119881119883)22119881119883119883)1198612(119905)1015840119863(119905)minus11198612(119905) and120587lowast = = minus 119881119883119881119883119883119863 (119905)minus1 1198612 (119905) (41)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0 (42)

and 119881(119879 119909) = 1198802(119879 119909)Next we will discuss the conditions which lead to the

above classifications

Complexity 7

1006704120587 = minus 119881119909119881119909119909119863 (119905)minus1 1198611 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]

minus1

sdot [1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)]

= minus 119881119909119881119909119909[[[[

1205801 minus 1205802120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))21205741 minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))2]]]]

= minus 119881119909119881119909119909119863 (119905)minus1 1198612 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905) + (0 (119905)120572 (119905))1015840)

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)

1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]minus1

sdot [[[

1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905) + (119905)120572 (119905)

]]]

= minus 119881119909119881119909119909[[[[[[

1205801 minus (1205802 + (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)) ( (119905) 120572 (119905)))120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2(1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) ( (119905) 120572 (119905))) minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2

]]]]]]

(43)

where

1205801 = (12059021 (119905) + 120582 (119905) 12059322 (119905)) (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)) 1205802 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205821 = (12059021 (119905) + 120582 (119905) 12059321 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205822 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905))

(44)

Then the optimal investment strategies of (10) and (11) canbe written as follows

120587lowast

=

1006704120587 if 1205741 ge 1205742120587 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(45)

5 HARA Utility Function

Generally speaking the analytical solutions to (30) (39)and (42) can not be obtained easily However the analyticalsolutions to the above HJB equations can be derived whenthe utility function is a HARA function [16] Moreover theoptimal investment-consumption strategies can be givenTherefore we assume that both utility functions of (11) areHARA utility functions that is

1198801 (119905 119910) = 1198802 (119905 119910) = 119890minus120588119905119910120581120581 (46)

where 0 lt 120581 lt 1 120588 gt 0 is a discounted factorWe can get the optimal consumption strategies from (19)

119888lowast = (119890120588119905119881119909)1(120581minus1) (47)

Then

max119888

1198801 (119905 119888) minus 119888119881119909 = 1198801 (119905 119888lowast) minus 119888lowast119881119909= 1120581119890minus120588119905 (119890120588119905119881119909)120581(120581minus1)

minus (119890120588119905119881119909)1(120581minus1) 119881119909= 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

(48)

Next we will derive the optimal investment strategies

8 Complexity

(1) 1006704120587 ge 0 The HJB function (30) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198611 (119905)1015840119863(119905)minus1 1198611 (119905) = 0(49)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)By Merton [16] letting 119881(119905 119909) = 1205731(119905)(119909120581120581) and sub-

stituting it into (49) we can obtain

1205731 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890minus120588119879

(50)

Setting

1198981 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 119899 (119905) = (1 minus 120581) 119890120588119905(120581minus1)

(51)

and substituting them into (50) we can obtain

1205731 (119905) + 1198981 (119905) 1205731 (119905) + 119899 (119905) (1205731 (119905))120581(120581minus1) = 01205731 (119879) = 119890minus120588119879 (52)

Setting 1205781(119905) = 1205731(119905)119890minusint119879119905 1198981(119904)d119904 and substituting it into(52) we can get

1205781 (119905) + 119899 (119905) (1205781 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198981(119904)d119904 = 01205781 (119879) = 119890minus120588119879

(53)

By solving it we can obtain

1205781 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(54)

Then

1205731 (119905) = 119890int119879119905 1198981(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(55)

Substituting 119881(119905 119909) = 1205731(119905)(119909120581120581) into (29) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909

(56)

(2) 1006704120587 le 0 le The HJB function (39) can be written asfollows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883119867(119905)2 = 0(57)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205731(119905)(119909120581120581) and substituting it into (57)

1205731 (119905) + 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890120588119879(58)

Setting

1198982 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] (59)

and substituting it into (58) we can obtain

1205732 (119905) + 1198982 (119905) 1205732 (119905) + 119899 (119905) (1205732 (119905))120581(120581minus1) = 01205732 (119879) = 119890minus120588119879 (60)

Setting 1205782(119905) = 1205732(119905)119890minusint119879119905 1198982(119904)d119904 and substituting it into(52) we can get

1205782 (119905) + 119899 (119905) (1205782 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198982(119904)d119904 = 01205782 (119879) = 119890minus120588119879 (61)

By solving it we can obtain

1205782 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(62)

Then

1205732 (119905) = 119890int119879119905 1198982(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(63)

Complexity 9

By substituting 119881(119905 119909) = 1205732(119905)(119909120581120581) into (37) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = [[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 10 ]

]

119888lowast = 119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909(64)

(3) le 0 The HJB function (42) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0(65)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205733(119905)(119909120581120581) and substituting it into (64)

1205733 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863(119905)minus1 1198612 (119905)] 1205733 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205733 (119905))120581(120581minus1) = 0

1205733 (119879) = 119890minus120588119879

(66)

Setting

1198983 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863 (119905)minus1 1198612 (119905)] (67)

and substituting them into (50) we can obtain

1205733 (119905) + 1198983 (119905) 1205733 (119905) + 119899 (119905) (1205733 (119905))120581(120581minus1) = 01205733 (119879) = 119890minus120588119879 (68)

Setting 1205783(119905) = 1205733(119905)119890minusint119879119905 1198983(119904)d119904 and substituting it into(68) we can get

1205783 (119905) + 119899 (119905) (1205783 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198983(119904)d119904 = 01205783 (119879) = 119890minus120588119879 (69)

By solving it we can obtain

1205783 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(70)

Then

1205733 (119905) = 119890int119879119905 1198983(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(71)

Substituting 119881(119905 119909) = 1205733(119905)(119909120581120581) into (41) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = = minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909

(72)

As the above discussion mentioned we can get the opti-mal investment-consumption strategies of (11) with HARAutility function as pointed by (46)

120587lowast =

minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909 if 1205741 ge 1205742[[[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 1

0]]] if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905)

minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(73)

119888lowast =

119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909 if 1205741 ge 1205742119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(74)

10 Complexity

From (73) and (74) it can be seen that the resultsare closed-form solutions We can avoid solving the partialdifferential equation difficulty and may obtain the optimalinvestment-consumption policies corresponding capital 119909directly

6 A Numerical Example and Discussion

In this section we will introduce a numerical exampleto illustrate model (11) with HARA utility function (46)This example is only to show how to obtain the optimalinvestment-consumption strategies Some parameters havebeen simplified to some extent We can change the assump-tions of the parameters to obtain the results which woulddeliver the different 1205741 and 1205742

(1) It is assumed that the initial capital is 1199090 = 119 times105 RMBWe only calculate the capital range [1199090 21199090]for illustrating the model The rest of the parametersare as follows

120588 = 002120581 = 05119903 = 0025120582 = 06120596 = 007120579 = 039120577 = 01

(75)

For simplicity we set 119879 = 1 In a given time 119905 = 05the futures price 119865 = 53RMB and its correspondingEurope call option price is 119884 = 477RMB We alsosuppose that119884119905 = 013119884119865 = 023119884119865119865 = minus00123 120572 =085 = 1 on this moment In the above parameters1205741 lt 1205742 le 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905))the optimal investment-consumption strategies areillustrated as Figure 1 (121119864 + 5 means 121000 thesame below) From Figure 1 we can see that we donot invest in options anymore The investor can onlyallocate hisher little capital in the futures A largenumber capital is invested in the risk-free bond

(2) We change parameters = 0005 and 120577 = 09from comparing with (1) In the case 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905)) le 1205742 the optimal investment-consumption strategies will change as shown in Fig-ure 2 Figure 2 makes us purchase futures and sell theoptions

(3) If 120577 = 01 119884119865119865 = 0005 and 120596 = 007 then1205741 ge 1205742 comparing with (1) The optimal investment-consumption strategies are illustrated in Figure 3which shows how the investors could sell futures andpurchase the options

Although this example is simple it well explains thesense of the model From the computed results the optimal

FuturesOptions

Risk-free bondConsumption

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)minus100E + 05

000E + 00

100E + 05

200E + 05

300E + 05

400E + 05

500E + 05

600E + 05

700E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 1 Optimal strategies under 1205741 lt 1205742 le 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905))

FuturesOptions

Risk-free bondConsumption

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)

Figure 2 Optimal strategies under 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905)) le 1205742

FuturesOptions

Risk-free bondConsumption

179E + 05 236E + 05121E + 05 198E + 05 217E + 05140E + 05 159E + 05

Total capital (RMB)

minus120E + 06

minus100E + 06

minus800E + 05

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

100E + 06

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 3 Optimal strategies under 1205741 ge 1205742

Complexity 11

investment-consumption strategy is a closed-form solutionthus avoiding solving partial differential equations numer-ically We remark that the numerical solution of partialdifferential equations can be a challenging task The closed-form analytical solution can greatly improve the calculationspeed and reduce the computational complexity Howeveryou need to make some assumptions of stochastic differentialequation parameters if you may also find more details Thefocus of this paper is the theoretical results rather thannumerical calculations Therefore the above example is asimulation which further describes the theoretical results

Comparing with the classical Merton model the Poissonprocess is added to the portfolio with Brownmotion processThe results are very similar to the traditionalMerton solutionoutwardly However the jump process has greatly increasedthe complexity of the solution process For utility functionssimilar to those considered in this paper relevance of capitalbetween optimal strategies can be obtained after solving theoptimal consumption-investment strategies from (73) (74)or the above examples The investors can differently allocatetheir capital by adjusting the parameters of utility function

7 Conclusions

We study the optimal investment-consumption portfolioselection of options which have a different price betweensale and purchase based on a risk aversion utility functionin financial markets These pricing processes follow aWeinerprocess and a Poisson process Then the correspondingHJB equation of the problem is derived and its solutionsare obtained in different conditions respectively Finally aspecial case when the utility function is HARA is illustratedIn addition throughout the paper it is assumed that 120572(119905) isdifferentiable with respect to time 119905 and monotone nonde-creasing Future work will be devoted to the collection of alarge number of datasets in the real market for calibrating thevarious parameters of the model

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoOption pricing when underlying stock returnsare discontinuousrdquo Journal of Financial Economics vol 3 no1-2 pp 125ndash144 1976

[3] M Jeanblanc-Picque and M Pontier ldquoOptimal portfolio for asmall investor in a market model with discontinuous pricesrdquoApplied Mathematics and Optimization vol 22 no 3 pp 287ndash310 1990

[4] J Z Lin and Z X Ye ldquoThe valuation of European contingentclaims of several stocks whose prices are governed by Brownianmotions and Poisson processesrdquo Chinese Journal of AppliedProbability vol 18 pp 167ndash172 2002

[5] EN Barron andR Jensen ldquoA stochastic control approach to thepricing of optionsrdquoMathematics of Operations Research vol 15no 1 pp 49ndash79 1990

[6] J C Cox S A Ross and M Rubinstein ldquoOption pricing asimplified approachrdquo Journal of Financial Economics vol 7 no3 pp 229ndash263 1979

[7] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952

[8] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time modelrdquo The Review of Economics andStatistics vol 51 pp 247ndash257 1969

[9] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[10] X Li X Y Zhou and A E Lim ldquoDynamic mean-varianceportfolio selection with no-shorting constraintsrdquo SIAM Journalon Control and Optimization vol 40 no 5 pp 1540ndash1555 2002

[11] A Cadenillas ldquoConsumption-investment problems with trans-action costs survey and open problemsrdquoMathematicalMethodsof Operations Research vol 51 no 1 pp 43ndash68 2000

[12] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000

[13] D Cuoco and H Liu ldquoA martingale characterization of con-sumption choices and hedging costs with margin require-mentsrdquo Mathematical Finance An International Journal ofMathematics Statistics and Financial Economics vol 10 no 3pp 355ndash385 2000

[14] A E Lim and X Y Zhou ldquoMean-variance portfolio selectionwith random parameters in a complete marketrdquoMathematics ofOperations Research vol 27 no 1 pp 101ndash120 2002

[15] I Karatzas J P Lehoczky S E Shreve and G-L Xu ldquoMar-tingale and duality methods for utility maximization in anincompletemarketrdquo SIAM Journal onControl andOptimizationvol 29 no 3 pp 702ndash730 1991

[16] R C Merton Continuous-Time Finance Basil Blackwell Cam-bridge MA USA 1990

[17] R A Jarrow and A Rudd Option Pricing vol 3 Homewood1983

[18] J C Cox and S A Ross ldquoThe valuation of options for alternativestochastic processesrdquo Journal of Financial Economics vol 3 no1-2 pp 145ndash166 1985

[19] W Yan and Y Chang ldquoModelling on optimal portfolio withexchange rate based on discontinuous stochastic processrdquoInternational Journal of Control vol 89 no 12 pp 2543ndash25482016

[20] G L Gong Introduction to Stochastic Differential EquationsPeking University Press Peking 1987

[21] A P Sage and C C White Optimum systems control vol 3Prentice-Hall Englewood Cliffs NJ USA 1977

[22] I Karatzas and S E Shreve Brownian Motion and StochasticCalculus Springer New York NY USA 2nd edition 1988

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Complexity 5

sdot Δ119872 (119905) + 12119881119905119905 (Δ119905)2 + 12119881119909119909 [119909119903 (119905)+ 1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905)) minus [ (119905)120572 (119905) (1205872)minus]

minus 119888]Δ119905 + 1205871015840120590 (119905) Δ119882 (119905) + 1205871015840120593 (119905) Δ119872 (119905)2

+ 119881119905119909Δ119905 [119909119903 (119905) + 1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

minus [ (119905)120572 (119905) (1205872)minus] minus 119888]Δ119905 + 1205871015840120590 (119905) Δ119882 (119905) + 1205871015840120593 (119905)

sdot Δ119872 (119905) (15)

By Karatzas and Shreve [22] it can be obtained thatd119882(119905) sdot d119882(119905) = d119905 d119882(119905) sdot d119905 = 0 d119872(119905) sdot d119872(119905) =120582(119905)d119905 d119872(119905) sdot d119905 = 0

After substituting them into the above equation wesubtract from both sides of the equation 119881(119905 119909) divided byΔ119905 Finally by letting Δ119905 rarr 0 we obtain

119881119905 + 119881119909119909119903 (119905) +max119888

1198801 (119905 119888) minus 119888119881119909+max120587

119881119909 [1205871015840 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

minus ( (119905)120572 (119905) (1205872)minus)] + 12119881119909119909 [1205871015840120590 (119905) 120590 (119905)1015840 120587

+ 120582 (119905) 1205871015840120593 (119905) 120593 (119905)1015840 120587] = 0

(16)

Set 119861(119905) = 119887(119905) minus 119903(119905)12 + 120582(119905)120593(119905) 119863(119905) = 120590(119905)120590(119905)1015840 +120582(119905)120593(119905)120593(119905)1015840 It is assumed that 119863(119905) is positive definiteSubstituting them into (16) it is obtained that

119881119905 + 119881119909119909119903 (119905) +max119888

1198801 (119905 119888) minus 119888119881119909+max120587

119881119909 [1205871015840119861 (119905) minus ( (119905)120572 (119905) (1205872)minus)]

+ 121198811199091199091205871015840119863 (119905) 120587 = 0(17)

with 119881(119879 119909) = 1198802(119879 119909) Equation (17) is the HJB equationcorresponding to (10) and (11)

4 Optimal Investment-Consumption Strategies

In this section the optimal investment-consumption strate-gies will be derived

41 Optimal Consumption Strategies From the HJB equation(17) and by applying a first-order optimality condition wederive that

120597 (1198801 (119905 119888) minus 119888119881119909)120597119888 = 0 (18)

That is to say 119888lowast satisfies1205971198801 (119905 119888)120597119888 = 119881119909 (19)

42 Optimal Investment Strategies We will also deduce theoptimal investment strategies from HJB function (17) Set

119891 (120587) = 119891 (1205871 1205872)= 119881119909 [1205871015840119861 (119905) minus (119905)120572 (119905) (1205872)minus] + 121198811199091199091205871015840119863 (119905) 120587

1198911 (120587) = 1198911 (1205871 1205872) = 1198811199091205871015840119861 (119905) + 121198811199091199091205871015840119863(119905) 1205871198912 (120587) = 1198912 (1205871 1205872)

= 119881119909 [1205871015840119861 (119905) + 1205872 (119905)120572 (119905)] + 121198811199091199091205871015840119863 (119905) 120587

(20)

By setting 1198611(119905) = 119861(119905) and 1198612(119905) = 119861(119905) + (0 (119905)120572(119905))1015840the above three equations can be written as follows

119891 (120587) = 119891 (1205871 1205872)= 119881119909 [12058710158401198611 (119905) minus (119905)120572 (119905) (1205872)minus]

+ 121198811199091199091205871015840119863 (119905) 120587(21)

1198911 (120587) = 1198911 (1205871 1205872) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 (22)

1198912 (120587) = 1198912 (1205871 1205872) = 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587 (23)

thus getting the following equation

119891 (120587) = 119891 (1205871 1205872) = 1198911 (1205871 1205872) if 1205872 gt 01198912 (1205871 1205872) if 1205872 le 0 (24)

From (22) by applying the first-order unconstrainedoptimality condition we obtain

1205971198911 (120587)120597120587 = 0 997904rArr1006704120587 = minus 119881119883119881119883119883119863 (119905)minus1 1198611 (119905)

(25)

and 1198911(1006704120587) = minus((119881119883)22119881119883119883)1198611(119905)1015840119863(119905)minus11198611(119905)

6 Complexity

Similarly from (23) we obtain

1205971198912 (120587)120597120587 = 0 997904rArr = minus 119881119883119881119883119883119863(119905)minus1 1198612 (119905)

(26)

and 1198912() = minus((119881119883)22119881119883119883)1198612(119905)1015840119863(119905)minus11198612(119905)Next let us compare 1006704120587 with We have

1006704120587 minus = 119881119883119881119883119883119863 (119905)minus1 (0 (119905)120572 (119905))1015840 (27)

By Merton [16] 119881(119905 119909) is concave function because theutility functions 1198801(sdot) and 1198802(sdot) are risk aversion functionsThen 119881119909 ge 0 119881119909119909 lt 0 and 119863(119905) is positive definite So 10067041205872 minus2 le 0We will consider three cases

(1) 10067041205872 gt 0 For any 120587 le 0 we have1198912 (120587) = 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587

le 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 = 1198911 (120587)le 1198911 (1006704120587)

(28)

Therefore max119891(120587) = max1198911(120587) = 1198911(1006704120587) = minus((119881119883)22119881119883119883)1198611(119905)1015840119863(119905)minus11198611(119905) and120587lowast = 1006704120587 = minus 119881119883119881119883119883119863 (119905)minus1 1198611 (119905) (29)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909minus (119881119883)22119881119883119883 1198611 (119905)1015840119863 (119905)minus1 1198611 (119905) = 0 (30)

119881 (119879 119909) = 1198802 (119879 119909) (31)

(2) 10067041205872 le 0 le 2 The two optimization problems are

max 1198911 (120587)1205872 ge 0 119905 isin [0 119879] (32)

max 1198912 (120587)1205872 le 0 119905 isin [0 119879] (33)

Solving (32) by introducing the Lagrange multiplier 119897 wehave

1198711 (1205871 1205872 119897) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 minus 119897 sdot 1205872 (34)

where 119897 ge 0Then

1205971198711 (1205871 1205872 119897)1205971205871 = 1198811199091198611 (119905) + 119881119909119909119863 (119905) 120587 = 0 (35)

Considering 10067041205872 le 0 le 2 and the constraint of (32) itis obtained that the optimal solution of (32) satisfies 1205872 = 0Substituting it into (35) we can get the following equation

119881119909 (119887 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905))+ 119881119883119883 (12059021 (119905) + 120582 (119905) 12059321 (119905)) 1205871 = 0 (36)

Then 1205871 = minus(119881119883119881119883119883) sdot (119887(119905) minus 119903(119905) + 120582(119905)1205931(119905))(12059021(119905) +120582(119905)12059321(119905))Following the same line of argument the solution of (33)

is 1205872 = 0 1205871 = minus(119881119883119881119883119883) sdot (119887(119905) minus 119903(119905) + 120582(119905)1205931(119905))(12059021(119905) +120582(119905)12059321(119905))As the above discussion mentioned it is shown that

120587lowast = 120587 = [[minus 119881119883119881119883119883 sdot 1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905)0 ]

] (37)

Therefore max119891(120587) = 1198911(120587) = minus((119881119883)22119881119883119883) sdot ((119887(119905) minus119903(119905) + 120582(119905)1205931(119905))2)(12059021(119905) + 120582(119905)12059321(119905))Setting 119867(119905)2 = ((1198871(119905) minus 119903(119905) + 120582(119905)1205931(119905))2)(12059021(119905) +120582(119905)12059321(119905)) the above equation can be written as follows

max119891 (120587) = 1198911 (120587) = minus(119881119883)22119881119883119883119867(119905)2 (38)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909 minus (119881119883)22119881119883119883119867(119905)2 = 0 (39)

and 119881(119879 119909) = 1198802(119879 119909)(3) 2 lt 0 For any 120587 le 0 we have

1198911 (120587) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587le 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587 = 1198912 (120587)le 1198912 ()

(40)

Therefore max119891(120587) = max1198912(120587) = 1198911() = minus((119881119883)22119881119883119883)1198612(119905)1015840119863(119905)minus11198612(119905) and120587lowast = = minus 119881119883119881119883119883119863 (119905)minus1 1198612 (119905) (41)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0 (42)

and 119881(119879 119909) = 1198802(119879 119909)Next we will discuss the conditions which lead to the

above classifications

Complexity 7

1006704120587 = minus 119881119909119881119909119909119863 (119905)minus1 1198611 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]

minus1

sdot [1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)]

= minus 119881119909119881119909119909[[[[

1205801 minus 1205802120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))21205741 minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))2]]]]

= minus 119881119909119881119909119909119863 (119905)minus1 1198612 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905) + (0 (119905)120572 (119905))1015840)

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)

1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]minus1

sdot [[[

1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905) + (119905)120572 (119905)

]]]

= minus 119881119909119881119909119909[[[[[[

1205801 minus (1205802 + (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)) ( (119905) 120572 (119905)))120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2(1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) ( (119905) 120572 (119905))) minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2

]]]]]]

(43)

where

1205801 = (12059021 (119905) + 120582 (119905) 12059322 (119905)) (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)) 1205802 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205821 = (12059021 (119905) + 120582 (119905) 12059321 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205822 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905))

(44)

Then the optimal investment strategies of (10) and (11) canbe written as follows

120587lowast

=

1006704120587 if 1205741 ge 1205742120587 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(45)

5 HARA Utility Function

Generally speaking the analytical solutions to (30) (39)and (42) can not be obtained easily However the analyticalsolutions to the above HJB equations can be derived whenthe utility function is a HARA function [16] Moreover theoptimal investment-consumption strategies can be givenTherefore we assume that both utility functions of (11) areHARA utility functions that is

1198801 (119905 119910) = 1198802 (119905 119910) = 119890minus120588119905119910120581120581 (46)

where 0 lt 120581 lt 1 120588 gt 0 is a discounted factorWe can get the optimal consumption strategies from (19)

119888lowast = (119890120588119905119881119909)1(120581minus1) (47)

Then

max119888

1198801 (119905 119888) minus 119888119881119909 = 1198801 (119905 119888lowast) minus 119888lowast119881119909= 1120581119890minus120588119905 (119890120588119905119881119909)120581(120581minus1)

minus (119890120588119905119881119909)1(120581minus1) 119881119909= 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

(48)

Next we will derive the optimal investment strategies

8 Complexity

(1) 1006704120587 ge 0 The HJB function (30) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198611 (119905)1015840119863(119905)minus1 1198611 (119905) = 0(49)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)By Merton [16] letting 119881(119905 119909) = 1205731(119905)(119909120581120581) and sub-

stituting it into (49) we can obtain

1205731 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890minus120588119879

(50)

Setting

1198981 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 119899 (119905) = (1 minus 120581) 119890120588119905(120581minus1)

(51)

and substituting them into (50) we can obtain

1205731 (119905) + 1198981 (119905) 1205731 (119905) + 119899 (119905) (1205731 (119905))120581(120581minus1) = 01205731 (119879) = 119890minus120588119879 (52)

Setting 1205781(119905) = 1205731(119905)119890minusint119879119905 1198981(119904)d119904 and substituting it into(52) we can get

1205781 (119905) + 119899 (119905) (1205781 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198981(119904)d119904 = 01205781 (119879) = 119890minus120588119879

(53)

By solving it we can obtain

1205781 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(54)

Then

1205731 (119905) = 119890int119879119905 1198981(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(55)

Substituting 119881(119905 119909) = 1205731(119905)(119909120581120581) into (29) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909

(56)

(2) 1006704120587 le 0 le The HJB function (39) can be written asfollows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883119867(119905)2 = 0(57)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205731(119905)(119909120581120581) and substituting it into (57)

1205731 (119905) + 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890120588119879(58)

Setting

1198982 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] (59)

and substituting it into (58) we can obtain

1205732 (119905) + 1198982 (119905) 1205732 (119905) + 119899 (119905) (1205732 (119905))120581(120581minus1) = 01205732 (119879) = 119890minus120588119879 (60)

Setting 1205782(119905) = 1205732(119905)119890minusint119879119905 1198982(119904)d119904 and substituting it into(52) we can get

1205782 (119905) + 119899 (119905) (1205782 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198982(119904)d119904 = 01205782 (119879) = 119890minus120588119879 (61)

By solving it we can obtain

1205782 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(62)

Then

1205732 (119905) = 119890int119879119905 1198982(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(63)

Complexity 9

By substituting 119881(119905 119909) = 1205732(119905)(119909120581120581) into (37) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = [[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 10 ]

]

119888lowast = 119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909(64)

(3) le 0 The HJB function (42) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0(65)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205733(119905)(119909120581120581) and substituting it into (64)

1205733 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863(119905)minus1 1198612 (119905)] 1205733 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205733 (119905))120581(120581minus1) = 0

1205733 (119879) = 119890minus120588119879

(66)

Setting

1198983 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863 (119905)minus1 1198612 (119905)] (67)

and substituting them into (50) we can obtain

1205733 (119905) + 1198983 (119905) 1205733 (119905) + 119899 (119905) (1205733 (119905))120581(120581minus1) = 01205733 (119879) = 119890minus120588119879 (68)

Setting 1205783(119905) = 1205733(119905)119890minusint119879119905 1198983(119904)d119904 and substituting it into(68) we can get

1205783 (119905) + 119899 (119905) (1205783 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198983(119904)d119904 = 01205783 (119879) = 119890minus120588119879 (69)

By solving it we can obtain

1205783 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(70)

Then

1205733 (119905) = 119890int119879119905 1198983(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(71)

Substituting 119881(119905 119909) = 1205733(119905)(119909120581120581) into (41) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = = minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909

(72)

As the above discussion mentioned we can get the opti-mal investment-consumption strategies of (11) with HARAutility function as pointed by (46)

120587lowast =

minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909 if 1205741 ge 1205742[[[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 1

0]]] if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905)

minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(73)

119888lowast =

119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909 if 1205741 ge 1205742119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(74)

10 Complexity

From (73) and (74) it can be seen that the resultsare closed-form solutions We can avoid solving the partialdifferential equation difficulty and may obtain the optimalinvestment-consumption policies corresponding capital 119909directly

6 A Numerical Example and Discussion

In this section we will introduce a numerical exampleto illustrate model (11) with HARA utility function (46)This example is only to show how to obtain the optimalinvestment-consumption strategies Some parameters havebeen simplified to some extent We can change the assump-tions of the parameters to obtain the results which woulddeliver the different 1205741 and 1205742

(1) It is assumed that the initial capital is 1199090 = 119 times105 RMBWe only calculate the capital range [1199090 21199090]for illustrating the model The rest of the parametersare as follows

120588 = 002120581 = 05119903 = 0025120582 = 06120596 = 007120579 = 039120577 = 01

(75)

For simplicity we set 119879 = 1 In a given time 119905 = 05the futures price 119865 = 53RMB and its correspondingEurope call option price is 119884 = 477RMB We alsosuppose that119884119905 = 013119884119865 = 023119884119865119865 = minus00123 120572 =085 = 1 on this moment In the above parameters1205741 lt 1205742 le 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905))the optimal investment-consumption strategies areillustrated as Figure 1 (121119864 + 5 means 121000 thesame below) From Figure 1 we can see that we donot invest in options anymore The investor can onlyallocate hisher little capital in the futures A largenumber capital is invested in the risk-free bond

(2) We change parameters = 0005 and 120577 = 09from comparing with (1) In the case 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905)) le 1205742 the optimal investment-consumption strategies will change as shown in Fig-ure 2 Figure 2 makes us purchase futures and sell theoptions

(3) If 120577 = 01 119884119865119865 = 0005 and 120596 = 007 then1205741 ge 1205742 comparing with (1) The optimal investment-consumption strategies are illustrated in Figure 3which shows how the investors could sell futures andpurchase the options

Although this example is simple it well explains thesense of the model From the computed results the optimal

FuturesOptions

Risk-free bondConsumption

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)minus100E + 05

000E + 00

100E + 05

200E + 05

300E + 05

400E + 05

500E + 05

600E + 05

700E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 1 Optimal strategies under 1205741 lt 1205742 le 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905))

FuturesOptions

Risk-free bondConsumption

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)

Figure 2 Optimal strategies under 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905)) le 1205742

FuturesOptions

Risk-free bondConsumption

179E + 05 236E + 05121E + 05 198E + 05 217E + 05140E + 05 159E + 05

Total capital (RMB)

minus120E + 06

minus100E + 06

minus800E + 05

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

100E + 06

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 3 Optimal strategies under 1205741 ge 1205742

Complexity 11

investment-consumption strategy is a closed-form solutionthus avoiding solving partial differential equations numer-ically We remark that the numerical solution of partialdifferential equations can be a challenging task The closed-form analytical solution can greatly improve the calculationspeed and reduce the computational complexity Howeveryou need to make some assumptions of stochastic differentialequation parameters if you may also find more details Thefocus of this paper is the theoretical results rather thannumerical calculations Therefore the above example is asimulation which further describes the theoretical results

Comparing with the classical Merton model the Poissonprocess is added to the portfolio with Brownmotion processThe results are very similar to the traditionalMerton solutionoutwardly However the jump process has greatly increasedthe complexity of the solution process For utility functionssimilar to those considered in this paper relevance of capitalbetween optimal strategies can be obtained after solving theoptimal consumption-investment strategies from (73) (74)or the above examples The investors can differently allocatetheir capital by adjusting the parameters of utility function

7 Conclusions

We study the optimal investment-consumption portfolioselection of options which have a different price betweensale and purchase based on a risk aversion utility functionin financial markets These pricing processes follow aWeinerprocess and a Poisson process Then the correspondingHJB equation of the problem is derived and its solutionsare obtained in different conditions respectively Finally aspecial case when the utility function is HARA is illustratedIn addition throughout the paper it is assumed that 120572(119905) isdifferentiable with respect to time 119905 and monotone nonde-creasing Future work will be devoted to the collection of alarge number of datasets in the real market for calibrating thevarious parameters of the model

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoOption pricing when underlying stock returnsare discontinuousrdquo Journal of Financial Economics vol 3 no1-2 pp 125ndash144 1976

[3] M Jeanblanc-Picque and M Pontier ldquoOptimal portfolio for asmall investor in a market model with discontinuous pricesrdquoApplied Mathematics and Optimization vol 22 no 3 pp 287ndash310 1990

[4] J Z Lin and Z X Ye ldquoThe valuation of European contingentclaims of several stocks whose prices are governed by Brownianmotions and Poisson processesrdquo Chinese Journal of AppliedProbability vol 18 pp 167ndash172 2002

[5] EN Barron andR Jensen ldquoA stochastic control approach to thepricing of optionsrdquoMathematics of Operations Research vol 15no 1 pp 49ndash79 1990

[6] J C Cox S A Ross and M Rubinstein ldquoOption pricing asimplified approachrdquo Journal of Financial Economics vol 7 no3 pp 229ndash263 1979

[7] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952

[8] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time modelrdquo The Review of Economics andStatistics vol 51 pp 247ndash257 1969

[9] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[10] X Li X Y Zhou and A E Lim ldquoDynamic mean-varianceportfolio selection with no-shorting constraintsrdquo SIAM Journalon Control and Optimization vol 40 no 5 pp 1540ndash1555 2002

[11] A Cadenillas ldquoConsumption-investment problems with trans-action costs survey and open problemsrdquoMathematicalMethodsof Operations Research vol 51 no 1 pp 43ndash68 2000

[12] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000

[13] D Cuoco and H Liu ldquoA martingale characterization of con-sumption choices and hedging costs with margin require-mentsrdquo Mathematical Finance An International Journal ofMathematics Statistics and Financial Economics vol 10 no 3pp 355ndash385 2000

[14] A E Lim and X Y Zhou ldquoMean-variance portfolio selectionwith random parameters in a complete marketrdquoMathematics ofOperations Research vol 27 no 1 pp 101ndash120 2002

[15] I Karatzas J P Lehoczky S E Shreve and G-L Xu ldquoMar-tingale and duality methods for utility maximization in anincompletemarketrdquo SIAM Journal onControl andOptimizationvol 29 no 3 pp 702ndash730 1991

[16] R C Merton Continuous-Time Finance Basil Blackwell Cam-bridge MA USA 1990

[17] R A Jarrow and A Rudd Option Pricing vol 3 Homewood1983

[18] J C Cox and S A Ross ldquoThe valuation of options for alternativestochastic processesrdquo Journal of Financial Economics vol 3 no1-2 pp 145ndash166 1985

[19] W Yan and Y Chang ldquoModelling on optimal portfolio withexchange rate based on discontinuous stochastic processrdquoInternational Journal of Control vol 89 no 12 pp 2543ndash25482016

[20] G L Gong Introduction to Stochastic Differential EquationsPeking University Press Peking 1987

[21] A P Sage and C C White Optimum systems control vol 3Prentice-Hall Englewood Cliffs NJ USA 1977

[22] I Karatzas and S E Shreve Brownian Motion and StochasticCalculus Springer New York NY USA 2nd edition 1988

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Complexity

Similarly from (23) we obtain

1205971198912 (120587)120597120587 = 0 997904rArr = minus 119881119883119881119883119883119863(119905)minus1 1198612 (119905)

(26)

and 1198912() = minus((119881119883)22119881119883119883)1198612(119905)1015840119863(119905)minus11198612(119905)Next let us compare 1006704120587 with We have

1006704120587 minus = 119881119883119881119883119883119863 (119905)minus1 (0 (119905)120572 (119905))1015840 (27)

By Merton [16] 119881(119905 119909) is concave function because theutility functions 1198801(sdot) and 1198802(sdot) are risk aversion functionsThen 119881119909 ge 0 119881119909119909 lt 0 and 119863(119905) is positive definite So 10067041205872 minus2 le 0We will consider three cases

(1) 10067041205872 gt 0 For any 120587 le 0 we have1198912 (120587) = 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587

le 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 = 1198911 (120587)le 1198911 (1006704120587)

(28)

Therefore max119891(120587) = max1198911(120587) = 1198911(1006704120587) = minus((119881119883)22119881119883119883)1198611(119905)1015840119863(119905)minus11198611(119905) and120587lowast = 1006704120587 = minus 119881119883119881119883119883119863 (119905)minus1 1198611 (119905) (29)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909minus (119881119883)22119881119883119883 1198611 (119905)1015840119863 (119905)minus1 1198611 (119905) = 0 (30)

119881 (119879 119909) = 1198802 (119879 119909) (31)

(2) 10067041205872 le 0 le 2 The two optimization problems are

max 1198911 (120587)1205872 ge 0 119905 isin [0 119879] (32)

max 1198912 (120587)1205872 le 0 119905 isin [0 119879] (33)

Solving (32) by introducing the Lagrange multiplier 119897 wehave

1198711 (1205871 1205872 119897) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587 minus 119897 sdot 1205872 (34)

where 119897 ge 0Then

1205971198711 (1205871 1205872 119897)1205971205871 = 1198811199091198611 (119905) + 119881119909119909119863 (119905) 120587 = 0 (35)

Considering 10067041205872 le 0 le 2 and the constraint of (32) itis obtained that the optimal solution of (32) satisfies 1205872 = 0Substituting it into (35) we can get the following equation

119881119909 (119887 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905))+ 119881119883119883 (12059021 (119905) + 120582 (119905) 12059321 (119905)) 1205871 = 0 (36)

Then 1205871 = minus(119881119883119881119883119883) sdot (119887(119905) minus 119903(119905) + 120582(119905)1205931(119905))(12059021(119905) +120582(119905)12059321(119905))Following the same line of argument the solution of (33)

is 1205872 = 0 1205871 = minus(119881119883119881119883119883) sdot (119887(119905) minus 119903(119905) + 120582(119905)1205931(119905))(12059021(119905) +120582(119905)12059321(119905))As the above discussion mentioned it is shown that

120587lowast = 120587 = [[minus 119881119883119881119883119883 sdot 1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905)0 ]

] (37)

Therefore max119891(120587) = 1198911(120587) = minus((119881119883)22119881119883119883) sdot ((119887(119905) minus119903(119905) + 120582(119905)1205931(119905))2)(12059021(119905) + 120582(119905)12059321(119905))Setting 119867(119905)2 = ((1198871(119905) minus 119903(119905) + 120582(119905)1205931(119905))2)(12059021(119905) +120582(119905)12059321(119905)) the above equation can be written as follows

max119891 (120587) = 1198911 (120587) = minus(119881119883)22119881119883119883119867(119905)2 (38)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909 minus (119881119883)22119881119883119883119867(119905)2 = 0 (39)

and 119881(119879 119909) = 1198802(119879 119909)(3) 2 lt 0 For any 120587 le 0 we have

1198911 (120587) = 11988111990912058710158401198611 (119905) + 121198811199091199091205871015840119863 (119905) 120587le 11988111990912058710158401198612 (119905) + 121198811199091199091205871015840119863 (119905) 120587 = 1198912 (120587)le 1198912 ()

(40)

Therefore max119891(120587) = max1198912(120587) = 1198911() = minus((119881119883)22119881119883119883)1198612(119905)1015840119863(119905)minus11198612(119905) and120587lowast = = minus 119881119883119881119883119883119863 (119905)minus1 1198612 (119905) (41)

The HJB function (17) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1198801 (119905 119888lowast) minus 119888lowast119881119909minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0 (42)

and 119881(119879 119909) = 1198802(119879 119909)Next we will discuss the conditions which lead to the

above classifications

Complexity 7

1006704120587 = minus 119881119909119881119909119909119863 (119905)minus1 1198611 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]

minus1

sdot [1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)]

= minus 119881119909119881119909119909[[[[

1205801 minus 1205802120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))21205741 minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))2]]]]

= minus 119881119909119881119909119909119863 (119905)minus1 1198612 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905) + (0 (119905)120572 (119905))1015840)

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)

1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]minus1

sdot [[[

1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905) + (119905)120572 (119905)

]]]

= minus 119881119909119881119909119909[[[[[[

1205801 minus (1205802 + (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)) ( (119905) 120572 (119905)))120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2(1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) ( (119905) 120572 (119905))) minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2

]]]]]]

(43)

where

1205801 = (12059021 (119905) + 120582 (119905) 12059322 (119905)) (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)) 1205802 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205821 = (12059021 (119905) + 120582 (119905) 12059321 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205822 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905))

(44)

Then the optimal investment strategies of (10) and (11) canbe written as follows

120587lowast

=

1006704120587 if 1205741 ge 1205742120587 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(45)

5 HARA Utility Function

Generally speaking the analytical solutions to (30) (39)and (42) can not be obtained easily However the analyticalsolutions to the above HJB equations can be derived whenthe utility function is a HARA function [16] Moreover theoptimal investment-consumption strategies can be givenTherefore we assume that both utility functions of (11) areHARA utility functions that is

1198801 (119905 119910) = 1198802 (119905 119910) = 119890minus120588119905119910120581120581 (46)

where 0 lt 120581 lt 1 120588 gt 0 is a discounted factorWe can get the optimal consumption strategies from (19)

119888lowast = (119890120588119905119881119909)1(120581minus1) (47)

Then

max119888

1198801 (119905 119888) minus 119888119881119909 = 1198801 (119905 119888lowast) minus 119888lowast119881119909= 1120581119890minus120588119905 (119890120588119905119881119909)120581(120581minus1)

minus (119890120588119905119881119909)1(120581minus1) 119881119909= 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

(48)

Next we will derive the optimal investment strategies

8 Complexity

(1) 1006704120587 ge 0 The HJB function (30) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198611 (119905)1015840119863(119905)minus1 1198611 (119905) = 0(49)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)By Merton [16] letting 119881(119905 119909) = 1205731(119905)(119909120581120581) and sub-

stituting it into (49) we can obtain

1205731 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890minus120588119879

(50)

Setting

1198981 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 119899 (119905) = (1 minus 120581) 119890120588119905(120581minus1)

(51)

and substituting them into (50) we can obtain

1205731 (119905) + 1198981 (119905) 1205731 (119905) + 119899 (119905) (1205731 (119905))120581(120581minus1) = 01205731 (119879) = 119890minus120588119879 (52)

Setting 1205781(119905) = 1205731(119905)119890minusint119879119905 1198981(119904)d119904 and substituting it into(52) we can get

1205781 (119905) + 119899 (119905) (1205781 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198981(119904)d119904 = 01205781 (119879) = 119890minus120588119879

(53)

By solving it we can obtain

1205781 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(54)

Then

1205731 (119905) = 119890int119879119905 1198981(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(55)

Substituting 119881(119905 119909) = 1205731(119905)(119909120581120581) into (29) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909

(56)

(2) 1006704120587 le 0 le The HJB function (39) can be written asfollows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883119867(119905)2 = 0(57)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205731(119905)(119909120581120581) and substituting it into (57)

1205731 (119905) + 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890120588119879(58)

Setting

1198982 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] (59)

and substituting it into (58) we can obtain

1205732 (119905) + 1198982 (119905) 1205732 (119905) + 119899 (119905) (1205732 (119905))120581(120581minus1) = 01205732 (119879) = 119890minus120588119879 (60)

Setting 1205782(119905) = 1205732(119905)119890minusint119879119905 1198982(119904)d119904 and substituting it into(52) we can get

1205782 (119905) + 119899 (119905) (1205782 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198982(119904)d119904 = 01205782 (119879) = 119890minus120588119879 (61)

By solving it we can obtain

1205782 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(62)

Then

1205732 (119905) = 119890int119879119905 1198982(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(63)

Complexity 9

By substituting 119881(119905 119909) = 1205732(119905)(119909120581120581) into (37) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = [[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 10 ]

]

119888lowast = 119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909(64)

(3) le 0 The HJB function (42) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0(65)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205733(119905)(119909120581120581) and substituting it into (64)

1205733 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863(119905)minus1 1198612 (119905)] 1205733 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205733 (119905))120581(120581minus1) = 0

1205733 (119879) = 119890minus120588119879

(66)

Setting

1198983 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863 (119905)minus1 1198612 (119905)] (67)

and substituting them into (50) we can obtain

1205733 (119905) + 1198983 (119905) 1205733 (119905) + 119899 (119905) (1205733 (119905))120581(120581minus1) = 01205733 (119879) = 119890minus120588119879 (68)

Setting 1205783(119905) = 1205733(119905)119890minusint119879119905 1198983(119904)d119904 and substituting it into(68) we can get

1205783 (119905) + 119899 (119905) (1205783 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198983(119904)d119904 = 01205783 (119879) = 119890minus120588119879 (69)

By solving it we can obtain

1205783 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(70)

Then

1205733 (119905) = 119890int119879119905 1198983(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(71)

Substituting 119881(119905 119909) = 1205733(119905)(119909120581120581) into (41) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = = minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909

(72)

As the above discussion mentioned we can get the opti-mal investment-consumption strategies of (11) with HARAutility function as pointed by (46)

120587lowast =

minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909 if 1205741 ge 1205742[[[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 1

0]]] if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905)

minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(73)

119888lowast =

119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909 if 1205741 ge 1205742119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(74)

10 Complexity

From (73) and (74) it can be seen that the resultsare closed-form solutions We can avoid solving the partialdifferential equation difficulty and may obtain the optimalinvestment-consumption policies corresponding capital 119909directly

6 A Numerical Example and Discussion

In this section we will introduce a numerical exampleto illustrate model (11) with HARA utility function (46)This example is only to show how to obtain the optimalinvestment-consumption strategies Some parameters havebeen simplified to some extent We can change the assump-tions of the parameters to obtain the results which woulddeliver the different 1205741 and 1205742

(1) It is assumed that the initial capital is 1199090 = 119 times105 RMBWe only calculate the capital range [1199090 21199090]for illustrating the model The rest of the parametersare as follows

120588 = 002120581 = 05119903 = 0025120582 = 06120596 = 007120579 = 039120577 = 01

(75)

For simplicity we set 119879 = 1 In a given time 119905 = 05the futures price 119865 = 53RMB and its correspondingEurope call option price is 119884 = 477RMB We alsosuppose that119884119905 = 013119884119865 = 023119884119865119865 = minus00123 120572 =085 = 1 on this moment In the above parameters1205741 lt 1205742 le 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905))the optimal investment-consumption strategies areillustrated as Figure 1 (121119864 + 5 means 121000 thesame below) From Figure 1 we can see that we donot invest in options anymore The investor can onlyallocate hisher little capital in the futures A largenumber capital is invested in the risk-free bond

(2) We change parameters = 0005 and 120577 = 09from comparing with (1) In the case 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905)) le 1205742 the optimal investment-consumption strategies will change as shown in Fig-ure 2 Figure 2 makes us purchase futures and sell theoptions

(3) If 120577 = 01 119884119865119865 = 0005 and 120596 = 007 then1205741 ge 1205742 comparing with (1) The optimal investment-consumption strategies are illustrated in Figure 3which shows how the investors could sell futures andpurchase the options

Although this example is simple it well explains thesense of the model From the computed results the optimal

FuturesOptions

Risk-free bondConsumption

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)minus100E + 05

000E + 00

100E + 05

200E + 05

300E + 05

400E + 05

500E + 05

600E + 05

700E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 1 Optimal strategies under 1205741 lt 1205742 le 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905))

FuturesOptions

Risk-free bondConsumption

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)

Figure 2 Optimal strategies under 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905)) le 1205742

FuturesOptions

Risk-free bondConsumption

179E + 05 236E + 05121E + 05 198E + 05 217E + 05140E + 05 159E + 05

Total capital (RMB)

minus120E + 06

minus100E + 06

minus800E + 05

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

100E + 06

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 3 Optimal strategies under 1205741 ge 1205742

Complexity 11

investment-consumption strategy is a closed-form solutionthus avoiding solving partial differential equations numer-ically We remark that the numerical solution of partialdifferential equations can be a challenging task The closed-form analytical solution can greatly improve the calculationspeed and reduce the computational complexity Howeveryou need to make some assumptions of stochastic differentialequation parameters if you may also find more details Thefocus of this paper is the theoretical results rather thannumerical calculations Therefore the above example is asimulation which further describes the theoretical results

Comparing with the classical Merton model the Poissonprocess is added to the portfolio with Brownmotion processThe results are very similar to the traditionalMerton solutionoutwardly However the jump process has greatly increasedthe complexity of the solution process For utility functionssimilar to those considered in this paper relevance of capitalbetween optimal strategies can be obtained after solving theoptimal consumption-investment strategies from (73) (74)or the above examples The investors can differently allocatetheir capital by adjusting the parameters of utility function

7 Conclusions

We study the optimal investment-consumption portfolioselection of options which have a different price betweensale and purchase based on a risk aversion utility functionin financial markets These pricing processes follow aWeinerprocess and a Poisson process Then the correspondingHJB equation of the problem is derived and its solutionsare obtained in different conditions respectively Finally aspecial case when the utility function is HARA is illustratedIn addition throughout the paper it is assumed that 120572(119905) isdifferentiable with respect to time 119905 and monotone nonde-creasing Future work will be devoted to the collection of alarge number of datasets in the real market for calibrating thevarious parameters of the model

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoOption pricing when underlying stock returnsare discontinuousrdquo Journal of Financial Economics vol 3 no1-2 pp 125ndash144 1976

[3] M Jeanblanc-Picque and M Pontier ldquoOptimal portfolio for asmall investor in a market model with discontinuous pricesrdquoApplied Mathematics and Optimization vol 22 no 3 pp 287ndash310 1990

[4] J Z Lin and Z X Ye ldquoThe valuation of European contingentclaims of several stocks whose prices are governed by Brownianmotions and Poisson processesrdquo Chinese Journal of AppliedProbability vol 18 pp 167ndash172 2002

[5] EN Barron andR Jensen ldquoA stochastic control approach to thepricing of optionsrdquoMathematics of Operations Research vol 15no 1 pp 49ndash79 1990

[6] J C Cox S A Ross and M Rubinstein ldquoOption pricing asimplified approachrdquo Journal of Financial Economics vol 7 no3 pp 229ndash263 1979

[7] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952

[8] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time modelrdquo The Review of Economics andStatistics vol 51 pp 247ndash257 1969

[9] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[10] X Li X Y Zhou and A E Lim ldquoDynamic mean-varianceportfolio selection with no-shorting constraintsrdquo SIAM Journalon Control and Optimization vol 40 no 5 pp 1540ndash1555 2002

[11] A Cadenillas ldquoConsumption-investment problems with trans-action costs survey and open problemsrdquoMathematicalMethodsof Operations Research vol 51 no 1 pp 43ndash68 2000

[12] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000

[13] D Cuoco and H Liu ldquoA martingale characterization of con-sumption choices and hedging costs with margin require-mentsrdquo Mathematical Finance An International Journal ofMathematics Statistics and Financial Economics vol 10 no 3pp 355ndash385 2000

[14] A E Lim and X Y Zhou ldquoMean-variance portfolio selectionwith random parameters in a complete marketrdquoMathematics ofOperations Research vol 27 no 1 pp 101ndash120 2002

[15] I Karatzas J P Lehoczky S E Shreve and G-L Xu ldquoMar-tingale and duality methods for utility maximization in anincompletemarketrdquo SIAM Journal onControl andOptimizationvol 29 no 3 pp 702ndash730 1991

[16] R C Merton Continuous-Time Finance Basil Blackwell Cam-bridge MA USA 1990

[17] R A Jarrow and A Rudd Option Pricing vol 3 Homewood1983

[18] J C Cox and S A Ross ldquoThe valuation of options for alternativestochastic processesrdquo Journal of Financial Economics vol 3 no1-2 pp 145ndash166 1985

[19] W Yan and Y Chang ldquoModelling on optimal portfolio withexchange rate based on discontinuous stochastic processrdquoInternational Journal of Control vol 89 no 12 pp 2543ndash25482016

[20] G L Gong Introduction to Stochastic Differential EquationsPeking University Press Peking 1987

[21] A P Sage and C C White Optimum systems control vol 3Prentice-Hall Englewood Cliffs NJ USA 1977

[22] I Karatzas and S E Shreve Brownian Motion and StochasticCalculus Springer New York NY USA 2nd edition 1988

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Stochastic AnalysisInternational Journal of

Complexity 7

1006704120587 = minus 119881119909119881119909119909119863 (119905)minus1 1198611 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905))

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]

minus1

sdot [1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)]

= minus 119881119909119881119909119909[[[[

1205801 minus 1205802120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))21205741 minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 120582 (119905) 1205902 (119905) 1205931 (119905))2]]]]

= minus 119881119909119881119909119909119863 (119905)minus1 1198612 (119905) = minus 119881119909119881119909119909 (120590 (119905) 120590 (119905)1015840 + 120582 (119905) 120593 (119905) 120593 (119905)1015840)minus1 (119887 (119905) minus 119903 (119905) 12 + 120582 (119905) 120593 (119905) + (0 (119905)120572 (119905))1015840)

= minus 119881119909119881119909119909 [12059021 (119905) + 120582 (119905) 12059321 (119905) 1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)

1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905) 12059022 (119905) + 120582 (119905) 12059322 (119905) ]minus1

sdot [[[

1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905) + (119905)120572 (119905)

]]]

= minus 119881119909119881119909119909[[[[[[

1205801 minus (1205802 + (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905)) ( (119905) 120572 (119905)))120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2(1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) ( (119905) 120572 (119905))) minus 1205742120582 (119905) (1205901 (119905) 1205932 (119905) minus 1205902 (119905) 1205931 (119905))2

]]]]]]

(43)

where

1205801 = (12059021 (119905) + 120582 (119905) 12059322 (119905)) (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)) 1205802 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205821 = (12059021 (119905) + 120582 (119905) 12059321 (119905))

sdot (1198872 (119905) minus 119903 (119905) + 120582 (119905) 1205932 (119905)) 1205822 = (1205901 (119905) 1205902 (119905) + 120582 (119905) 1205931 (119905) 1205932 (119905))

sdot (1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905))

(44)

Then the optimal investment strategies of (10) and (11) canbe written as follows

120587lowast

=

1006704120587 if 1205741 ge 1205742120587 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(45)

5 HARA Utility Function

Generally speaking the analytical solutions to (30) (39)and (42) can not be obtained easily However the analyticalsolutions to the above HJB equations can be derived whenthe utility function is a HARA function [16] Moreover theoptimal investment-consumption strategies can be givenTherefore we assume that both utility functions of (11) areHARA utility functions that is

1198801 (119905 119910) = 1198802 (119905 119910) = 119890minus120588119905119910120581120581 (46)

where 0 lt 120581 lt 1 120588 gt 0 is a discounted factorWe can get the optimal consumption strategies from (19)

119888lowast = (119890120588119905119881119909)1(120581minus1) (47)

Then

max119888

1198801 (119905 119888) minus 119888119881119909 = 1198801 (119905 119888lowast) minus 119888lowast119881119909= 1120581119890minus120588119905 (119890120588119905119881119909)120581(120581minus1)

minus (119890120588119905119881119909)1(120581minus1) 119881119909= 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

(48)

Next we will derive the optimal investment strategies

8 Complexity

(1) 1006704120587 ge 0 The HJB function (30) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198611 (119905)1015840119863(119905)minus1 1198611 (119905) = 0(49)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)By Merton [16] letting 119881(119905 119909) = 1205731(119905)(119909120581120581) and sub-

stituting it into (49) we can obtain

1205731 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890minus120588119879

(50)

Setting

1198981 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 119899 (119905) = (1 minus 120581) 119890120588119905(120581minus1)

(51)

and substituting them into (50) we can obtain

1205731 (119905) + 1198981 (119905) 1205731 (119905) + 119899 (119905) (1205731 (119905))120581(120581minus1) = 01205731 (119879) = 119890minus120588119879 (52)

Setting 1205781(119905) = 1205731(119905)119890minusint119879119905 1198981(119904)d119904 and substituting it into(52) we can get

1205781 (119905) + 119899 (119905) (1205781 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198981(119904)d119904 = 01205781 (119879) = 119890minus120588119879

(53)

By solving it we can obtain

1205781 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(54)

Then

1205731 (119905) = 119890int119879119905 1198981(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(55)

Substituting 119881(119905 119909) = 1205731(119905)(119909120581120581) into (29) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909

(56)

(2) 1006704120587 le 0 le The HJB function (39) can be written asfollows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883119867(119905)2 = 0(57)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205731(119905)(119909120581120581) and substituting it into (57)

1205731 (119905) + 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890120588119879(58)

Setting

1198982 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] (59)

and substituting it into (58) we can obtain

1205732 (119905) + 1198982 (119905) 1205732 (119905) + 119899 (119905) (1205732 (119905))120581(120581minus1) = 01205732 (119879) = 119890minus120588119879 (60)

Setting 1205782(119905) = 1205732(119905)119890minusint119879119905 1198982(119904)d119904 and substituting it into(52) we can get

1205782 (119905) + 119899 (119905) (1205782 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198982(119904)d119904 = 01205782 (119879) = 119890minus120588119879 (61)

By solving it we can obtain

1205782 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(62)

Then

1205732 (119905) = 119890int119879119905 1198982(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(63)

Complexity 9

By substituting 119881(119905 119909) = 1205732(119905)(119909120581120581) into (37) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = [[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 10 ]

]

119888lowast = 119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909(64)

(3) le 0 The HJB function (42) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0(65)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205733(119905)(119909120581120581) and substituting it into (64)

1205733 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863(119905)minus1 1198612 (119905)] 1205733 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205733 (119905))120581(120581minus1) = 0

1205733 (119879) = 119890minus120588119879

(66)

Setting

1198983 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863 (119905)minus1 1198612 (119905)] (67)

and substituting them into (50) we can obtain

1205733 (119905) + 1198983 (119905) 1205733 (119905) + 119899 (119905) (1205733 (119905))120581(120581minus1) = 01205733 (119879) = 119890minus120588119879 (68)

Setting 1205783(119905) = 1205733(119905)119890minusint119879119905 1198983(119904)d119904 and substituting it into(68) we can get

1205783 (119905) + 119899 (119905) (1205783 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198983(119904)d119904 = 01205783 (119879) = 119890minus120588119879 (69)

By solving it we can obtain

1205783 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(70)

Then

1205733 (119905) = 119890int119879119905 1198983(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(71)

Substituting 119881(119905 119909) = 1205733(119905)(119909120581120581) into (41) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = = minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909

(72)

As the above discussion mentioned we can get the opti-mal investment-consumption strategies of (11) with HARAutility function as pointed by (46)

120587lowast =

minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909 if 1205741 ge 1205742[[[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 1

0]]] if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905)

minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(73)

119888lowast =

119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909 if 1205741 ge 1205742119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(74)

10 Complexity

From (73) and (74) it can be seen that the resultsare closed-form solutions We can avoid solving the partialdifferential equation difficulty and may obtain the optimalinvestment-consumption policies corresponding capital 119909directly

6 A Numerical Example and Discussion

In this section we will introduce a numerical exampleto illustrate model (11) with HARA utility function (46)This example is only to show how to obtain the optimalinvestment-consumption strategies Some parameters havebeen simplified to some extent We can change the assump-tions of the parameters to obtain the results which woulddeliver the different 1205741 and 1205742

(1) It is assumed that the initial capital is 1199090 = 119 times105 RMBWe only calculate the capital range [1199090 21199090]for illustrating the model The rest of the parametersare as follows

120588 = 002120581 = 05119903 = 0025120582 = 06120596 = 007120579 = 039120577 = 01

(75)

For simplicity we set 119879 = 1 In a given time 119905 = 05the futures price 119865 = 53RMB and its correspondingEurope call option price is 119884 = 477RMB We alsosuppose that119884119905 = 013119884119865 = 023119884119865119865 = minus00123 120572 =085 = 1 on this moment In the above parameters1205741 lt 1205742 le 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905))the optimal investment-consumption strategies areillustrated as Figure 1 (121119864 + 5 means 121000 thesame below) From Figure 1 we can see that we donot invest in options anymore The investor can onlyallocate hisher little capital in the futures A largenumber capital is invested in the risk-free bond

(2) We change parameters = 0005 and 120577 = 09from comparing with (1) In the case 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905)) le 1205742 the optimal investment-consumption strategies will change as shown in Fig-ure 2 Figure 2 makes us purchase futures and sell theoptions

(3) If 120577 = 01 119884119865119865 = 0005 and 120596 = 007 then1205741 ge 1205742 comparing with (1) The optimal investment-consumption strategies are illustrated in Figure 3which shows how the investors could sell futures andpurchase the options

Although this example is simple it well explains thesense of the model From the computed results the optimal

FuturesOptions

Risk-free bondConsumption

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)minus100E + 05

000E + 00

100E + 05

200E + 05

300E + 05

400E + 05

500E + 05

600E + 05

700E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 1 Optimal strategies under 1205741 lt 1205742 le 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905))

FuturesOptions

Risk-free bondConsumption

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)

Figure 2 Optimal strategies under 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905)) le 1205742

FuturesOptions

Risk-free bondConsumption

179E + 05 236E + 05121E + 05 198E + 05 217E + 05140E + 05 159E + 05

Total capital (RMB)

minus120E + 06

minus100E + 06

minus800E + 05

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

100E + 06

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 3 Optimal strategies under 1205741 ge 1205742

Complexity 11

investment-consumption strategy is a closed-form solutionthus avoiding solving partial differential equations numer-ically We remark that the numerical solution of partialdifferential equations can be a challenging task The closed-form analytical solution can greatly improve the calculationspeed and reduce the computational complexity Howeveryou need to make some assumptions of stochastic differentialequation parameters if you may also find more details Thefocus of this paper is the theoretical results rather thannumerical calculations Therefore the above example is asimulation which further describes the theoretical results

Comparing with the classical Merton model the Poissonprocess is added to the portfolio with Brownmotion processThe results are very similar to the traditionalMerton solutionoutwardly However the jump process has greatly increasedthe complexity of the solution process For utility functionssimilar to those considered in this paper relevance of capitalbetween optimal strategies can be obtained after solving theoptimal consumption-investment strategies from (73) (74)or the above examples The investors can differently allocatetheir capital by adjusting the parameters of utility function

7 Conclusions

We study the optimal investment-consumption portfolioselection of options which have a different price betweensale and purchase based on a risk aversion utility functionin financial markets These pricing processes follow aWeinerprocess and a Poisson process Then the correspondingHJB equation of the problem is derived and its solutionsare obtained in different conditions respectively Finally aspecial case when the utility function is HARA is illustratedIn addition throughout the paper it is assumed that 120572(119905) isdifferentiable with respect to time 119905 and monotone nonde-creasing Future work will be devoted to the collection of alarge number of datasets in the real market for calibrating thevarious parameters of the model

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoOption pricing when underlying stock returnsare discontinuousrdquo Journal of Financial Economics vol 3 no1-2 pp 125ndash144 1976

[3] M Jeanblanc-Picque and M Pontier ldquoOptimal portfolio for asmall investor in a market model with discontinuous pricesrdquoApplied Mathematics and Optimization vol 22 no 3 pp 287ndash310 1990

[4] J Z Lin and Z X Ye ldquoThe valuation of European contingentclaims of several stocks whose prices are governed by Brownianmotions and Poisson processesrdquo Chinese Journal of AppliedProbability vol 18 pp 167ndash172 2002

[5] EN Barron andR Jensen ldquoA stochastic control approach to thepricing of optionsrdquoMathematics of Operations Research vol 15no 1 pp 49ndash79 1990

[6] J C Cox S A Ross and M Rubinstein ldquoOption pricing asimplified approachrdquo Journal of Financial Economics vol 7 no3 pp 229ndash263 1979

[7] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952

[8] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time modelrdquo The Review of Economics andStatistics vol 51 pp 247ndash257 1969

[9] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[10] X Li X Y Zhou and A E Lim ldquoDynamic mean-varianceportfolio selection with no-shorting constraintsrdquo SIAM Journalon Control and Optimization vol 40 no 5 pp 1540ndash1555 2002

[11] A Cadenillas ldquoConsumption-investment problems with trans-action costs survey and open problemsrdquoMathematicalMethodsof Operations Research vol 51 no 1 pp 43ndash68 2000

[12] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000

[13] D Cuoco and H Liu ldquoA martingale characterization of con-sumption choices and hedging costs with margin require-mentsrdquo Mathematical Finance An International Journal ofMathematics Statistics and Financial Economics vol 10 no 3pp 355ndash385 2000

[14] A E Lim and X Y Zhou ldquoMean-variance portfolio selectionwith random parameters in a complete marketrdquoMathematics ofOperations Research vol 27 no 1 pp 101ndash120 2002

[15] I Karatzas J P Lehoczky S E Shreve and G-L Xu ldquoMar-tingale and duality methods for utility maximization in anincompletemarketrdquo SIAM Journal onControl andOptimizationvol 29 no 3 pp 702ndash730 1991

[16] R C Merton Continuous-Time Finance Basil Blackwell Cam-bridge MA USA 1990

[17] R A Jarrow and A Rudd Option Pricing vol 3 Homewood1983

[18] J C Cox and S A Ross ldquoThe valuation of options for alternativestochastic processesrdquo Journal of Financial Economics vol 3 no1-2 pp 145ndash166 1985

[19] W Yan and Y Chang ldquoModelling on optimal portfolio withexchange rate based on discontinuous stochastic processrdquoInternational Journal of Control vol 89 no 12 pp 2543ndash25482016

[20] G L Gong Introduction to Stochastic Differential EquationsPeking University Press Peking 1987

[21] A P Sage and C C White Optimum systems control vol 3Prentice-Hall Englewood Cliffs NJ USA 1977

[22] I Karatzas and S E Shreve Brownian Motion and StochasticCalculus Springer New York NY USA 2nd edition 1988

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Stochastic AnalysisInternational Journal of

8 Complexity

(1) 1006704120587 ge 0 The HJB function (30) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198611 (119905)1015840119863(119905)minus1 1198611 (119905) = 0(49)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)By Merton [16] letting 119881(119905 119909) = 1205731(119905)(119909120581120581) and sub-

stituting it into (49) we can obtain

1205731 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890minus120588119879

(50)

Setting

1198981 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198611 (119905)1015840119863 (119905)minus1 1198611 (119905)] 119899 (119905) = (1 minus 120581) 119890120588119905(120581minus1)

(51)

and substituting them into (50) we can obtain

1205731 (119905) + 1198981 (119905) 1205731 (119905) + 119899 (119905) (1205731 (119905))120581(120581minus1) = 01205731 (119879) = 119890minus120588119879 (52)

Setting 1205781(119905) = 1205731(119905)119890minusint119879119905 1198981(119904)d119904 and substituting it into(52) we can get

1205781 (119905) + 119899 (119905) (1205781 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198981(119904)d119904 = 01205781 (119879) = 119890minus120588119879

(53)

By solving it we can obtain

1205781 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(54)

Then

1205731 (119905) = 119890int119879119905 1198981(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198981(V)dVd119904)1minus120581

(55)

Substituting 119881(119905 119909) = 1205731(119905)(119909120581120581) into (29) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909

(56)

(2) 1006704120587 le 0 le The HJB function (39) can be written asfollows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883119867(119905)2 = 0(57)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205731(119905)(119909120581120581) and substituting it into (57)

1205731 (119905) + 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] 1205731 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205731 (119905))120581(120581minus1) = 0

1205731 (119879) = 119890120588119879(58)

Setting

1198982 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)119867 (119905)2] (59)

and substituting it into (58) we can obtain

1205732 (119905) + 1198982 (119905) 1205732 (119905) + 119899 (119905) (1205732 (119905))120581(120581minus1) = 01205732 (119879) = 119890minus120588119879 (60)

Setting 1205782(119905) = 1205732(119905)119890minusint119879119905 1198982(119904)d119904 and substituting it into(52) we can get

1205782 (119905) + 119899 (119905) (1205782 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198982(119904)d119904 = 01205782 (119879) = 119890minus120588119879 (61)

By solving it we can obtain

1205782 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(62)

Then

1205732 (119905) = 119890int119879119905 1198982(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198982(V)dVd119904)1minus120581

(63)

Complexity 9

By substituting 119881(119905 119909) = 1205732(119905)(119909120581120581) into (37) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = [[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 10 ]

]

119888lowast = 119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909(64)

(3) le 0 The HJB function (42) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0(65)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205733(119905)(119909120581120581) and substituting it into (64)

1205733 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863(119905)minus1 1198612 (119905)] 1205733 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205733 (119905))120581(120581minus1) = 0

1205733 (119879) = 119890minus120588119879

(66)

Setting

1198983 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863 (119905)minus1 1198612 (119905)] (67)

and substituting them into (50) we can obtain

1205733 (119905) + 1198983 (119905) 1205733 (119905) + 119899 (119905) (1205733 (119905))120581(120581minus1) = 01205733 (119879) = 119890minus120588119879 (68)

Setting 1205783(119905) = 1205733(119905)119890minusint119879119905 1198983(119904)d119904 and substituting it into(68) we can get

1205783 (119905) + 119899 (119905) (1205783 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198983(119904)d119904 = 01205783 (119879) = 119890minus120588119879 (69)

By solving it we can obtain

1205783 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(70)

Then

1205733 (119905) = 119890int119879119905 1198983(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(71)

Substituting 119881(119905 119909) = 1205733(119905)(119909120581120581) into (41) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = = minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909

(72)

As the above discussion mentioned we can get the opti-mal investment-consumption strategies of (11) with HARAutility function as pointed by (46)

120587lowast =

minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909 if 1205741 ge 1205742[[[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 1

0]]] if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905)

minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(73)

119888lowast =

119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909 if 1205741 ge 1205742119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(74)

10 Complexity

From (73) and (74) it can be seen that the resultsare closed-form solutions We can avoid solving the partialdifferential equation difficulty and may obtain the optimalinvestment-consumption policies corresponding capital 119909directly

6 A Numerical Example and Discussion

In this section we will introduce a numerical exampleto illustrate model (11) with HARA utility function (46)This example is only to show how to obtain the optimalinvestment-consumption strategies Some parameters havebeen simplified to some extent We can change the assump-tions of the parameters to obtain the results which woulddeliver the different 1205741 and 1205742

(1) It is assumed that the initial capital is 1199090 = 119 times105 RMBWe only calculate the capital range [1199090 21199090]for illustrating the model The rest of the parametersare as follows

120588 = 002120581 = 05119903 = 0025120582 = 06120596 = 007120579 = 039120577 = 01

(75)

For simplicity we set 119879 = 1 In a given time 119905 = 05the futures price 119865 = 53RMB and its correspondingEurope call option price is 119884 = 477RMB We alsosuppose that119884119905 = 013119884119865 = 023119884119865119865 = minus00123 120572 =085 = 1 on this moment In the above parameters1205741 lt 1205742 le 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905))the optimal investment-consumption strategies areillustrated as Figure 1 (121119864 + 5 means 121000 thesame below) From Figure 1 we can see that we donot invest in options anymore The investor can onlyallocate hisher little capital in the futures A largenumber capital is invested in the risk-free bond

(2) We change parameters = 0005 and 120577 = 09from comparing with (1) In the case 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905)) le 1205742 the optimal investment-consumption strategies will change as shown in Fig-ure 2 Figure 2 makes us purchase futures and sell theoptions

(3) If 120577 = 01 119884119865119865 = 0005 and 120596 = 007 then1205741 ge 1205742 comparing with (1) The optimal investment-consumption strategies are illustrated in Figure 3which shows how the investors could sell futures andpurchase the options

Although this example is simple it well explains thesense of the model From the computed results the optimal

FuturesOptions

Risk-free bondConsumption

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)minus100E + 05

000E + 00

100E + 05

200E + 05

300E + 05

400E + 05

500E + 05

600E + 05

700E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 1 Optimal strategies under 1205741 lt 1205742 le 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905))

FuturesOptions

Risk-free bondConsumption

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)

Figure 2 Optimal strategies under 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905)) le 1205742

FuturesOptions

Risk-free bondConsumption

179E + 05 236E + 05121E + 05 198E + 05 217E + 05140E + 05 159E + 05

Total capital (RMB)

minus120E + 06

minus100E + 06

minus800E + 05

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

100E + 06

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 3 Optimal strategies under 1205741 ge 1205742

Complexity 11

investment-consumption strategy is a closed-form solutionthus avoiding solving partial differential equations numer-ically We remark that the numerical solution of partialdifferential equations can be a challenging task The closed-form analytical solution can greatly improve the calculationspeed and reduce the computational complexity Howeveryou need to make some assumptions of stochastic differentialequation parameters if you may also find more details Thefocus of this paper is the theoretical results rather thannumerical calculations Therefore the above example is asimulation which further describes the theoretical results

Comparing with the classical Merton model the Poissonprocess is added to the portfolio with Brownmotion processThe results are very similar to the traditionalMerton solutionoutwardly However the jump process has greatly increasedthe complexity of the solution process For utility functionssimilar to those considered in this paper relevance of capitalbetween optimal strategies can be obtained after solving theoptimal consumption-investment strategies from (73) (74)or the above examples The investors can differently allocatetheir capital by adjusting the parameters of utility function

7 Conclusions

We study the optimal investment-consumption portfolioselection of options which have a different price betweensale and purchase based on a risk aversion utility functionin financial markets These pricing processes follow aWeinerprocess and a Poisson process Then the correspondingHJB equation of the problem is derived and its solutionsare obtained in different conditions respectively Finally aspecial case when the utility function is HARA is illustratedIn addition throughout the paper it is assumed that 120572(119905) isdifferentiable with respect to time 119905 and monotone nonde-creasing Future work will be devoted to the collection of alarge number of datasets in the real market for calibrating thevarious parameters of the model

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoOption pricing when underlying stock returnsare discontinuousrdquo Journal of Financial Economics vol 3 no1-2 pp 125ndash144 1976

[3] M Jeanblanc-Picque and M Pontier ldquoOptimal portfolio for asmall investor in a market model with discontinuous pricesrdquoApplied Mathematics and Optimization vol 22 no 3 pp 287ndash310 1990

[4] J Z Lin and Z X Ye ldquoThe valuation of European contingentclaims of several stocks whose prices are governed by Brownianmotions and Poisson processesrdquo Chinese Journal of AppliedProbability vol 18 pp 167ndash172 2002

[5] EN Barron andR Jensen ldquoA stochastic control approach to thepricing of optionsrdquoMathematics of Operations Research vol 15no 1 pp 49ndash79 1990

[6] J C Cox S A Ross and M Rubinstein ldquoOption pricing asimplified approachrdquo Journal of Financial Economics vol 7 no3 pp 229ndash263 1979

[7] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952

[8] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time modelrdquo The Review of Economics andStatistics vol 51 pp 247ndash257 1969

[9] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[10] X Li X Y Zhou and A E Lim ldquoDynamic mean-varianceportfolio selection with no-shorting constraintsrdquo SIAM Journalon Control and Optimization vol 40 no 5 pp 1540ndash1555 2002

[11] A Cadenillas ldquoConsumption-investment problems with trans-action costs survey and open problemsrdquoMathematicalMethodsof Operations Research vol 51 no 1 pp 43ndash68 2000

[12] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000

[13] D Cuoco and H Liu ldquoA martingale characterization of con-sumption choices and hedging costs with margin require-mentsrdquo Mathematical Finance An International Journal ofMathematics Statistics and Financial Economics vol 10 no 3pp 355ndash385 2000

[14] A E Lim and X Y Zhou ldquoMean-variance portfolio selectionwith random parameters in a complete marketrdquoMathematics ofOperations Research vol 27 no 1 pp 101ndash120 2002

[15] I Karatzas J P Lehoczky S E Shreve and G-L Xu ldquoMar-tingale and duality methods for utility maximization in anincompletemarketrdquo SIAM Journal onControl andOptimizationvol 29 no 3 pp 702ndash730 1991

[16] R C Merton Continuous-Time Finance Basil Blackwell Cam-bridge MA USA 1990

[17] R A Jarrow and A Rudd Option Pricing vol 3 Homewood1983

[18] J C Cox and S A Ross ldquoThe valuation of options for alternativestochastic processesrdquo Journal of Financial Economics vol 3 no1-2 pp 145ndash166 1985

[19] W Yan and Y Chang ldquoModelling on optimal portfolio withexchange rate based on discontinuous stochastic processrdquoInternational Journal of Control vol 89 no 12 pp 2543ndash25482016

[20] G L Gong Introduction to Stochastic Differential EquationsPeking University Press Peking 1987

[21] A P Sage and C C White Optimum systems control vol 3Prentice-Hall Englewood Cliffs NJ USA 1977

[22] I Karatzas and S E Shreve Brownian Motion and StochasticCalculus Springer New York NY USA 2nd edition 1988

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Stochastic AnalysisInternational Journal of

Complexity 9

By substituting 119881(119905 119909) = 1205732(119905)(119909120581120581) into (37) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = 1006704120587 = [[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 10 ]

]

119888lowast = 119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909(64)

(3) le 0 The HJB function (42) can be written as follows

119881119905 + 119881119909119909119903 (119905) + 1 minus 120581120581 119890120588119905(120581minus1) (119881119909)120581(120581minus1)

minus (119881119883)22119881119883119883 1198612 (119905)1015840119863(119905)minus1 1198612 (119905) = 0(65)

and 119881(119879 119909) = 119890minus120588119879(119909120581120581)Similar to (1) we can obtain by letting119881(119905 119909) = 1205733(119905)(119909120581120581) and substituting it into (64)

1205733 (119905)+ 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863(119905)minus1 1198612 (119905)] 1205733 (119905)+ (1 minus 120581) 119890120588119905(120581minus1) (1205733 (119905))120581(120581minus1) = 0

1205733 (119879) = 119890minus120588119879

(66)

Setting

1198983 (119905) = 120581 [119903 (119905) + 12 (1 minus 120581)1198612 (119905)1015840119863 (119905)minus1 1198612 (119905)] (67)

and substituting them into (50) we can obtain

1205733 (119905) + 1198983 (119905) 1205733 (119905) + 119899 (119905) (1205733 (119905))120581(120581minus1) = 01205733 (119879) = 119890minus120588119879 (68)

Setting 1205783(119905) = 1205733(119905)119890minusint119879119905 1198983(119904)d119904 and substituting it into(68) we can get

1205783 (119905) + 119899 (119905) (1205783 (119905))120581(120581minus1) 119890(1(120581minus1)) int119879119905 1198983(119904)d119904 = 01205783 (119879) = 119890minus120588119879 (69)

By solving it we can obtain

1205783 (119905) = (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(70)

Then

1205733 (119905) = 119890int119879119905 1198983(119904)d119904 (119890120588119879(120581minus1)

+ 11 minus 120581 int119879119905119899 (119904) 119890(1(120581minus1)) int119879119904 1198983(V)dVd119904)1minus120581

(71)

Substituting 119881(119905 119909) = 1205733(119905)(119909120581120581) into (41) and (47)we can get the optimal investment-consumption strategies asfollows

120587lowast = = minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909119888lowast = 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909

(72)

As the above discussion mentioned we can get the opti-mal investment-consumption strategies of (11) with HARAutility function as pointed by (46)

120587lowast =

minus119863 (119905)minus1 1198611 (119905)120581 minus 1 119909 if 1205741 ge 1205742[[[minus1198871 (119905) minus 119903 (119905) + 120582 (119905) 1205931 (119905)12059021 (119905) + 120582 (119905) 12059321 (119905) sdot 119909120581 minus 1

0]]] if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905)

minus119863 (119905)minus1 1198612 (119905)120581 minus 1 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(73)

119888lowast =

119890120588119905(120581minus1) (1205731 (119905))1(120581minus1) 119909 if 1205741 ge 1205742119890120588119905(120581minus1) (1205732 (119905))1(120581minus1) 119909 if 1205741 lt 1205742 le 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) 119890120588119905(120581minus1) (1205733 (119905))1(120581minus1) 119909 if 1205741 + (12059021 (119905) + 120582 (119905) 12059321 (119905)) (119905)120572 (119905) le 1205742

(74)

10 Complexity

From (73) and (74) it can be seen that the resultsare closed-form solutions We can avoid solving the partialdifferential equation difficulty and may obtain the optimalinvestment-consumption policies corresponding capital 119909directly

6 A Numerical Example and Discussion

In this section we will introduce a numerical exampleto illustrate model (11) with HARA utility function (46)This example is only to show how to obtain the optimalinvestment-consumption strategies Some parameters havebeen simplified to some extent We can change the assump-tions of the parameters to obtain the results which woulddeliver the different 1205741 and 1205742

(1) It is assumed that the initial capital is 1199090 = 119 times105 RMBWe only calculate the capital range [1199090 21199090]for illustrating the model The rest of the parametersare as follows

120588 = 002120581 = 05119903 = 0025120582 = 06120596 = 007120579 = 039120577 = 01

(75)

For simplicity we set 119879 = 1 In a given time 119905 = 05the futures price 119865 = 53RMB and its correspondingEurope call option price is 119884 = 477RMB We alsosuppose that119884119905 = 013119884119865 = 023119884119865119865 = minus00123 120572 =085 = 1 on this moment In the above parameters1205741 lt 1205742 le 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905))the optimal investment-consumption strategies areillustrated as Figure 1 (121119864 + 5 means 121000 thesame below) From Figure 1 we can see that we donot invest in options anymore The investor can onlyallocate hisher little capital in the futures A largenumber capital is invested in the risk-free bond

(2) We change parameters = 0005 and 120577 = 09from comparing with (1) In the case 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905)) le 1205742 the optimal investment-consumption strategies will change as shown in Fig-ure 2 Figure 2 makes us purchase futures and sell theoptions

(3) If 120577 = 01 119884119865119865 = 0005 and 120596 = 007 then1205741 ge 1205742 comparing with (1) The optimal investment-consumption strategies are illustrated in Figure 3which shows how the investors could sell futures andpurchase the options

Although this example is simple it well explains thesense of the model From the computed results the optimal

FuturesOptions

Risk-free bondConsumption

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)minus100E + 05

000E + 00

100E + 05

200E + 05

300E + 05

400E + 05

500E + 05

600E + 05

700E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 1 Optimal strategies under 1205741 lt 1205742 le 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905))

FuturesOptions

Risk-free bondConsumption

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)

Figure 2 Optimal strategies under 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905)) le 1205742

FuturesOptions

Risk-free bondConsumption

179E + 05 236E + 05121E + 05 198E + 05 217E + 05140E + 05 159E + 05

Total capital (RMB)

minus120E + 06

minus100E + 06

minus800E + 05

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

100E + 06

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 3 Optimal strategies under 1205741 ge 1205742

Complexity 11

investment-consumption strategy is a closed-form solutionthus avoiding solving partial differential equations numer-ically We remark that the numerical solution of partialdifferential equations can be a challenging task The closed-form analytical solution can greatly improve the calculationspeed and reduce the computational complexity Howeveryou need to make some assumptions of stochastic differentialequation parameters if you may also find more details Thefocus of this paper is the theoretical results rather thannumerical calculations Therefore the above example is asimulation which further describes the theoretical results

Comparing with the classical Merton model the Poissonprocess is added to the portfolio with Brownmotion processThe results are very similar to the traditionalMerton solutionoutwardly However the jump process has greatly increasedthe complexity of the solution process For utility functionssimilar to those considered in this paper relevance of capitalbetween optimal strategies can be obtained after solving theoptimal consumption-investment strategies from (73) (74)or the above examples The investors can differently allocatetheir capital by adjusting the parameters of utility function

7 Conclusions

We study the optimal investment-consumption portfolioselection of options which have a different price betweensale and purchase based on a risk aversion utility functionin financial markets These pricing processes follow aWeinerprocess and a Poisson process Then the correspondingHJB equation of the problem is derived and its solutionsare obtained in different conditions respectively Finally aspecial case when the utility function is HARA is illustratedIn addition throughout the paper it is assumed that 120572(119905) isdifferentiable with respect to time 119905 and monotone nonde-creasing Future work will be devoted to the collection of alarge number of datasets in the real market for calibrating thevarious parameters of the model

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoOption pricing when underlying stock returnsare discontinuousrdquo Journal of Financial Economics vol 3 no1-2 pp 125ndash144 1976

[3] M Jeanblanc-Picque and M Pontier ldquoOptimal portfolio for asmall investor in a market model with discontinuous pricesrdquoApplied Mathematics and Optimization vol 22 no 3 pp 287ndash310 1990

[4] J Z Lin and Z X Ye ldquoThe valuation of European contingentclaims of several stocks whose prices are governed by Brownianmotions and Poisson processesrdquo Chinese Journal of AppliedProbability vol 18 pp 167ndash172 2002

[5] EN Barron andR Jensen ldquoA stochastic control approach to thepricing of optionsrdquoMathematics of Operations Research vol 15no 1 pp 49ndash79 1990

[6] J C Cox S A Ross and M Rubinstein ldquoOption pricing asimplified approachrdquo Journal of Financial Economics vol 7 no3 pp 229ndash263 1979

[7] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952

[8] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time modelrdquo The Review of Economics andStatistics vol 51 pp 247ndash257 1969

[9] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[10] X Li X Y Zhou and A E Lim ldquoDynamic mean-varianceportfolio selection with no-shorting constraintsrdquo SIAM Journalon Control and Optimization vol 40 no 5 pp 1540ndash1555 2002

[11] A Cadenillas ldquoConsumption-investment problems with trans-action costs survey and open problemsrdquoMathematicalMethodsof Operations Research vol 51 no 1 pp 43ndash68 2000

[12] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000

[13] D Cuoco and H Liu ldquoA martingale characterization of con-sumption choices and hedging costs with margin require-mentsrdquo Mathematical Finance An International Journal ofMathematics Statistics and Financial Economics vol 10 no 3pp 355ndash385 2000

[14] A E Lim and X Y Zhou ldquoMean-variance portfolio selectionwith random parameters in a complete marketrdquoMathematics ofOperations Research vol 27 no 1 pp 101ndash120 2002

[15] I Karatzas J P Lehoczky S E Shreve and G-L Xu ldquoMar-tingale and duality methods for utility maximization in anincompletemarketrdquo SIAM Journal onControl andOptimizationvol 29 no 3 pp 702ndash730 1991

[16] R C Merton Continuous-Time Finance Basil Blackwell Cam-bridge MA USA 1990

[17] R A Jarrow and A Rudd Option Pricing vol 3 Homewood1983

[18] J C Cox and S A Ross ldquoThe valuation of options for alternativestochastic processesrdquo Journal of Financial Economics vol 3 no1-2 pp 145ndash166 1985

[19] W Yan and Y Chang ldquoModelling on optimal portfolio withexchange rate based on discontinuous stochastic processrdquoInternational Journal of Control vol 89 no 12 pp 2543ndash25482016

[20] G L Gong Introduction to Stochastic Differential EquationsPeking University Press Peking 1987

[21] A P Sage and C C White Optimum systems control vol 3Prentice-Hall Englewood Cliffs NJ USA 1977

[22] I Karatzas and S E Shreve Brownian Motion and StochasticCalculus Springer New York NY USA 2nd edition 1988

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Complexity

From (73) and (74) it can be seen that the resultsare closed-form solutions We can avoid solving the partialdifferential equation difficulty and may obtain the optimalinvestment-consumption policies corresponding capital 119909directly

6 A Numerical Example and Discussion

In this section we will introduce a numerical exampleto illustrate model (11) with HARA utility function (46)This example is only to show how to obtain the optimalinvestment-consumption strategies Some parameters havebeen simplified to some extent We can change the assump-tions of the parameters to obtain the results which woulddeliver the different 1205741 and 1205742

(1) It is assumed that the initial capital is 1199090 = 119 times105 RMBWe only calculate the capital range [1199090 21199090]for illustrating the model The rest of the parametersare as follows

120588 = 002120581 = 05119903 = 0025120582 = 06120596 = 007120579 = 039120577 = 01

(75)

For simplicity we set 119879 = 1 In a given time 119905 = 05the futures price 119865 = 53RMB and its correspondingEurope call option price is 119884 = 477RMB We alsosuppose that119884119905 = 013119884119865 = 023119884119865119865 = minus00123 120572 =085 = 1 on this moment In the above parameters1205741 lt 1205742 le 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905))the optimal investment-consumption strategies areillustrated as Figure 1 (121119864 + 5 means 121000 thesame below) From Figure 1 we can see that we donot invest in options anymore The investor can onlyallocate hisher little capital in the futures A largenumber capital is invested in the risk-free bond

(2) We change parameters = 0005 and 120577 = 09from comparing with (1) In the case 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905)) le 1205742 the optimal investment-consumption strategies will change as shown in Fig-ure 2 Figure 2 makes us purchase futures and sell theoptions

(3) If 120577 = 01 119884119865119865 = 0005 and 120596 = 007 then1205741 ge 1205742 comparing with (1) The optimal investment-consumption strategies are illustrated in Figure 3which shows how the investors could sell futures andpurchase the options

Although this example is simple it well explains thesense of the model From the computed results the optimal

FuturesOptions

Risk-free bondConsumption

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)minus100E + 05

000E + 00

100E + 05

200E + 05

300E + 05

400E + 05

500E + 05

600E + 05

700E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 1 Optimal strategies under 1205741 lt 1205742 le 1205741 + (12059021(119905) +120582(119905)12059321(119905))((119905)120572(119905))

FuturesOptions

Risk-free bondConsumption

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

Inve

stmen

t-con

sum

ptio

n (R

MB)

140E + 05 159E + 05 179E + 05 198E + 05 217E + 05 236E + 05121E + 05

Total capital (RMB)

Figure 2 Optimal strategies under 1205741 + (12059021(119905) + 120582(119905)12059321(119905))((119905)120572(119905)) le 1205742

FuturesOptions

Risk-free bondConsumption

179E + 05 236E + 05121E + 05 198E + 05 217E + 05140E + 05 159E + 05

Total capital (RMB)

minus120E + 06

minus100E + 06

minus800E + 05

minus600E + 05

minus400E + 05

minus200E + 05

000E + 00

200E + 05

400E + 05

600E + 05

800E + 05

100E + 06

Inve

stmen

t-con

sum

ptio

n (R

MB)

Figure 3 Optimal strategies under 1205741 ge 1205742

Complexity 11

investment-consumption strategy is a closed-form solutionthus avoiding solving partial differential equations numer-ically We remark that the numerical solution of partialdifferential equations can be a challenging task The closed-form analytical solution can greatly improve the calculationspeed and reduce the computational complexity Howeveryou need to make some assumptions of stochastic differentialequation parameters if you may also find more details Thefocus of this paper is the theoretical results rather thannumerical calculations Therefore the above example is asimulation which further describes the theoretical results

Comparing with the classical Merton model the Poissonprocess is added to the portfolio with Brownmotion processThe results are very similar to the traditionalMerton solutionoutwardly However the jump process has greatly increasedthe complexity of the solution process For utility functionssimilar to those considered in this paper relevance of capitalbetween optimal strategies can be obtained after solving theoptimal consumption-investment strategies from (73) (74)or the above examples The investors can differently allocatetheir capital by adjusting the parameters of utility function

7 Conclusions

We study the optimal investment-consumption portfolioselection of options which have a different price betweensale and purchase based on a risk aversion utility functionin financial markets These pricing processes follow aWeinerprocess and a Poisson process Then the correspondingHJB equation of the problem is derived and its solutionsare obtained in different conditions respectively Finally aspecial case when the utility function is HARA is illustratedIn addition throughout the paper it is assumed that 120572(119905) isdifferentiable with respect to time 119905 and monotone nonde-creasing Future work will be devoted to the collection of alarge number of datasets in the real market for calibrating thevarious parameters of the model

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoOption pricing when underlying stock returnsare discontinuousrdquo Journal of Financial Economics vol 3 no1-2 pp 125ndash144 1976

[3] M Jeanblanc-Picque and M Pontier ldquoOptimal portfolio for asmall investor in a market model with discontinuous pricesrdquoApplied Mathematics and Optimization vol 22 no 3 pp 287ndash310 1990

[4] J Z Lin and Z X Ye ldquoThe valuation of European contingentclaims of several stocks whose prices are governed by Brownianmotions and Poisson processesrdquo Chinese Journal of AppliedProbability vol 18 pp 167ndash172 2002

[5] EN Barron andR Jensen ldquoA stochastic control approach to thepricing of optionsrdquoMathematics of Operations Research vol 15no 1 pp 49ndash79 1990

[6] J C Cox S A Ross and M Rubinstein ldquoOption pricing asimplified approachrdquo Journal of Financial Economics vol 7 no3 pp 229ndash263 1979

[7] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952

[8] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time modelrdquo The Review of Economics andStatistics vol 51 pp 247ndash257 1969

[9] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[10] X Li X Y Zhou and A E Lim ldquoDynamic mean-varianceportfolio selection with no-shorting constraintsrdquo SIAM Journalon Control and Optimization vol 40 no 5 pp 1540ndash1555 2002

[11] A Cadenillas ldquoConsumption-investment problems with trans-action costs survey and open problemsrdquoMathematicalMethodsof Operations Research vol 51 no 1 pp 43ndash68 2000

[12] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000

[13] D Cuoco and H Liu ldquoA martingale characterization of con-sumption choices and hedging costs with margin require-mentsrdquo Mathematical Finance An International Journal ofMathematics Statistics and Financial Economics vol 10 no 3pp 355ndash385 2000

[14] A E Lim and X Y Zhou ldquoMean-variance portfolio selectionwith random parameters in a complete marketrdquoMathematics ofOperations Research vol 27 no 1 pp 101ndash120 2002

[15] I Karatzas J P Lehoczky S E Shreve and G-L Xu ldquoMar-tingale and duality methods for utility maximization in anincompletemarketrdquo SIAM Journal onControl andOptimizationvol 29 no 3 pp 702ndash730 1991

[16] R C Merton Continuous-Time Finance Basil Blackwell Cam-bridge MA USA 1990

[17] R A Jarrow and A Rudd Option Pricing vol 3 Homewood1983

[18] J C Cox and S A Ross ldquoThe valuation of options for alternativestochastic processesrdquo Journal of Financial Economics vol 3 no1-2 pp 145ndash166 1985

[19] W Yan and Y Chang ldquoModelling on optimal portfolio withexchange rate based on discontinuous stochastic processrdquoInternational Journal of Control vol 89 no 12 pp 2543ndash25482016

[20] G L Gong Introduction to Stochastic Differential EquationsPeking University Press Peking 1987

[21] A P Sage and C C White Optimum systems control vol 3Prentice-Hall Englewood Cliffs NJ USA 1977

[22] I Karatzas and S E Shreve Brownian Motion and StochasticCalculus Springer New York NY USA 2nd edition 1988

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 11

investment-consumption strategy is a closed-form solutionthus avoiding solving partial differential equations numer-ically We remark that the numerical solution of partialdifferential equations can be a challenging task The closed-form analytical solution can greatly improve the calculationspeed and reduce the computational complexity Howeveryou need to make some assumptions of stochastic differentialequation parameters if you may also find more details Thefocus of this paper is the theoretical results rather thannumerical calculations Therefore the above example is asimulation which further describes the theoretical results

Comparing with the classical Merton model the Poissonprocess is added to the portfolio with Brownmotion processThe results are very similar to the traditionalMerton solutionoutwardly However the jump process has greatly increasedthe complexity of the solution process For utility functionssimilar to those considered in this paper relevance of capitalbetween optimal strategies can be obtained after solving theoptimal consumption-investment strategies from (73) (74)or the above examples The investors can differently allocatetheir capital by adjusting the parameters of utility function

7 Conclusions

We study the optimal investment-consumption portfolioselection of options which have a different price betweensale and purchase based on a risk aversion utility functionin financial markets These pricing processes follow aWeinerprocess and a Poisson process Then the correspondingHJB equation of the problem is derived and its solutionsare obtained in different conditions respectively Finally aspecial case when the utility function is HARA is illustratedIn addition throughout the paper it is assumed that 120572(119905) isdifferentiable with respect to time 119905 and monotone nonde-creasing Future work will be devoted to the collection of alarge number of datasets in the real market for calibrating thevarious parameters of the model

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this article

References

[1] F Black and M Scholes ldquoThe pricing of options corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973

[2] R C Merton ldquoOption pricing when underlying stock returnsare discontinuousrdquo Journal of Financial Economics vol 3 no1-2 pp 125ndash144 1976

[3] M Jeanblanc-Picque and M Pontier ldquoOptimal portfolio for asmall investor in a market model with discontinuous pricesrdquoApplied Mathematics and Optimization vol 22 no 3 pp 287ndash310 1990

[4] J Z Lin and Z X Ye ldquoThe valuation of European contingentclaims of several stocks whose prices are governed by Brownianmotions and Poisson processesrdquo Chinese Journal of AppliedProbability vol 18 pp 167ndash172 2002

[5] EN Barron andR Jensen ldquoA stochastic control approach to thepricing of optionsrdquoMathematics of Operations Research vol 15no 1 pp 49ndash79 1990

[6] J C Cox S A Ross and M Rubinstein ldquoOption pricing asimplified approachrdquo Journal of Financial Economics vol 7 no3 pp 229ndash263 1979

[7] H Markowitz ldquoPortfolio selectionrdquo Journal of Finance vol 7pp 77ndash91 1952

[8] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuous-time modelrdquo The Review of Economics andStatistics vol 51 pp 247ndash257 1969

[9] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[10] X Li X Y Zhou and A E Lim ldquoDynamic mean-varianceportfolio selection with no-shorting constraintsrdquo SIAM Journalon Control and Optimization vol 40 no 5 pp 1540ndash1555 2002

[11] A Cadenillas ldquoConsumption-investment problems with trans-action costs survey and open problemsrdquoMathematicalMethodsof Operations Research vol 51 no 1 pp 43ndash68 2000

[12] X Y Zhou and D Li ldquoContinuous-time mean-variance portfo-lio selection a stochastic LQ frameworkrdquo Applied Mathematicsand Optimization vol 42 no 1 pp 19ndash33 2000

[13] D Cuoco and H Liu ldquoA martingale characterization of con-sumption choices and hedging costs with margin require-mentsrdquo Mathematical Finance An International Journal ofMathematics Statistics and Financial Economics vol 10 no 3pp 355ndash385 2000

[14] A E Lim and X Y Zhou ldquoMean-variance portfolio selectionwith random parameters in a complete marketrdquoMathematics ofOperations Research vol 27 no 1 pp 101ndash120 2002

[15] I Karatzas J P Lehoczky S E Shreve and G-L Xu ldquoMar-tingale and duality methods for utility maximization in anincompletemarketrdquo SIAM Journal onControl andOptimizationvol 29 no 3 pp 702ndash730 1991

[16] R C Merton Continuous-Time Finance Basil Blackwell Cam-bridge MA USA 1990

[17] R A Jarrow and A Rudd Option Pricing vol 3 Homewood1983

[18] J C Cox and S A Ross ldquoThe valuation of options for alternativestochastic processesrdquo Journal of Financial Economics vol 3 no1-2 pp 145ndash166 1985

[19] W Yan and Y Chang ldquoModelling on optimal portfolio withexchange rate based on discontinuous stochastic processrdquoInternational Journal of Control vol 89 no 12 pp 2543ndash25482016

[20] G L Gong Introduction to Stochastic Differential EquationsPeking University Press Peking 1987

[21] A P Sage and C C White Optimum systems control vol 3Prentice-Hall Englewood Cliffs NJ USA 1977

[22] I Karatzas and S E Shreve Brownian Motion and StochasticCalculus Springer New York NY USA 2nd edition 1988

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of