15
This article was downloaded by: [University of West Florida] On: 05 October 2014, At: 06:36 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Optimization: A Journal of Mathematical Programming and Operations Research Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gopt20 Optimal martingale measure maximizing the expected total utility of consumption with applications to derivative pricing Ping Li a & Shou-Yang Wang b a School of Economics and Management, Beihang University , Beijing, 100191, China b Academy of Mathematics and System Sciences, Chinese Academy of Sciences , Beijing, 100080, China Published online: 28 Oct 2009. To cite this article: Ping Li & Shou-Yang Wang (2008) Optimal martingale measure maximizing the expected total utility of consumption with applications to derivative pricing, Optimization: A Journal of Mathematical Programming and Operations Research, 57:5, 691-703, DOI: 10.1080/02331930802355283 To link to this article: http://dx.doi.org/10.1080/02331930802355283 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,

Optimal martingale measure maximizing the expected total utility of consumption with applications to derivative pricing

Embed Size (px)

Citation preview

This article was downloaded by: [University of West Florida]On: 05 October 2014, At: 06:36Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Optimization: A Journal ofMathematical Programming andOperations ResearchPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gopt20

Optimal martingale measuremaximizing the expected total utilityof consumption with applications toderivative pricingPing Li a & Shou-Yang Wang ba School of Economics and Management, Beihang University ,Beijing, 100191, Chinab Academy of Mathematics and System Sciences, Chinese Academyof Sciences , Beijing, 100080, ChinaPublished online: 28 Oct 2009.

To cite this article: Ping Li & Shou-Yang Wang (2008) Optimal martingale measure maximizingthe expected total utility of consumption with applications to derivative pricing, Optimization:A Journal of Mathematical Programming and Operations Research, 57:5, 691-703, DOI:10.1080/02331930802355283

To link to this article: http://dx.doi.org/10.1080/02331930802355283

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,

systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

OptimizationVol. 57, No. 5, October 2008, 691–703

Optimal martingale measure maximizing the expected total utility

of consumption with applications to derivative pricing

Ping Lia* and Shou-Yang Wangb

aSchool of Economics and Management, Beihang University, Beijing 100191, China; bAcademyof Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China

(Received 29 September 2006; final version received 4 May 2007)

In this article we discuss the problem of selecting an optimal equivalentmartingale measure for discrete-time incomplete financial markets under thecriteria of maximizing the expected total utility of consumption. For a givenutility function, we choose a class of equivalent martingale measures,corresponding to each of which we construct a random variable. We show thatif one of these random variables is an admissible consumption process, then thisconsumption is optimal, and the martingale measure associated with thisconsumption process is also optimal. For a specific market model with hyperbolicabsolute risk aversion (HARA) utility functions, we work out the optimalconsumption and optimal martingale measure explicitly, and further use thisoptimal martingale measure to give the fair price for any derivative.

Keywords: martingale measure; derivative pricing; optimal consumption;incomplete market; utility maximization

1. Introduction

In this article we consider the selection of a unique equivalent martingale measure underthe criteria of maximizing the expected total utility of consumption in the setting ofdiscrete-time incomplete financial markets. For a specific market model with hyperbolicabsolute risk aversion (HARA) utility functions, we work out the optimal consumptionand optimal martingale measure explicitly, then use this optimal martingale measure togive fair prices for contingent claims.

The pricing of derivatives or contingent claims from the price dynamics of certainsecurities for complete financial markets has been studied extensively. Since in completemarkets the equivalent martingale measure is unique, accordingly one can give the uniqueprice for any contingent claim. In incomplete financial markets, there exist non-replica-table contingent claims. From the view of the martingale method for contingent claimpricing, there exist several martingale measures for the discounted price process of thebasic securities. If we adopt an arbitrage pricing approach, then these martingale measurescannot give a unique price system. Therefore, an additional criterion must be used to selectan appropriate martingale measure from the several measures with which to price

*Corresponding author. Email: liping [email protected]

ISSN 0233–1934 print/ISSN 1029–4945 online

� 2008 Taylor & Francis

DOI: 10.1080/02331930802355283

http://www.informaworld.com

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

a derivative. In recent years many different principles have been proposed for continuous-time incomplete markets, such as Follmer and Schweizer’s [7] minimal martingale measuremethod, Gerber and Shiu’s [8] Esscher transform method and the numeraire portfolioapproach initiated by Long [14] and developed by Bajeux and Portait [1,2], etc.

The similar problem for discrete-time incomplete markets was studied later than andnot so much as that for continuous-time case. Elliott and Madan [6] constructed anequivalent martingale measure using the extended Girsanov principle. Buhlmann et al. [3]obtained a martingale measure by Esscher transforms. Following Follmer and Schweizer’smethod, Li and Xia [12] gave the minimal martingale measure for a specific discrete-timefinancial market. In Li et al. [13] they got an equivalent martingale measure based on thecriterion of utility maximization of terminal wealth and related the optimal equivalentmartingale measure to minimum relative entropy and minimum Hellinger–Kakutanidistance.

Here we will get an optimal equivalent martingale measure by maximizing the expectedtotal utility of consumption rather than terminal wealth. There have been many authorsthat surveyed the problem of utility maximization of consumption. Among them,El Karoui and Jeanblanc-Picque [5] considered the model of optimal consumption withlabour income. Carr et al. [4] derived the optimal consumption and investment plans whenderivative’s underlying asset price process is a pure jump Levy process. He and Pearson [9]studied the optimal consumption problem with short-sale constraint. Shreve and Soner[15] discussed the optimal portfolio-consumption problem with transaction costs.

For a given utility function, we choose a class of equivalent martingale measures.Corresponding to each of them, we construct a random variable in a similar way to that in[13]. We show that if one of these variables is an admissible consumption process, then it isan optimal consumption process, and the martingale measure associated with thisconsumption process is an optimal and unique equivalent martingale measure.

This article is organized as follows. Section 2 introduces the general market model anda characterization for admissible consumption processes is given. In Section 3 we discussthe problem of optimal consumption for the general model and give the relationshipbetween equivalent martingale measure and optimal consumption process. Based on theresults of Li et al. [13] we discuss the optimal consumption and optimal martingalemeasure for HARA utility functions in Section 4, and work out explicitly the optimalmartingale measure for a specific market model with HARA utility functions in Section 5.Then in Section 6 we use the optimal martingale measure obtained in Section 5 to give theprice of any derivative. Section 7 concludes the article.

2. The market model in a general setting

Let the time index set be {0, 1, . . . ,N}. Suppose that (�,F , (F n), IP) is a stochastic basis,where (�,F , IP) is a probability space and (F n)0�n�N is an increasing complete filtrationsatisfying FN¼F . We put F�1¼F 0¼ {;,�} for notational convenience.

Assume that in the economy there are one risk-free asset (bond) and d risky assets(stocks) whose price processes are defined as below:

(1) The price process of the risk-free asset ðS0nÞ satisfies

S0n ¼ S0

n�1ð1þ rnÞ, n ¼ 1, . . . ,N, S00 ¼ 1,

where rn is non-negative and predictable;

692 P. Li and S.-Y. Wang

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

(2) For each i¼ 1, . . . , d, the price process of the i-th risky asset ðSinÞ1�n�N satisfies

Sin ¼ Si

n�1ð1þ RinÞ, n ¼ 1, . . . ,N, Si

0 is a positive constant,

where Rin is F n-measurable and Ri

n 4�1 a.s.

Denote Sn b¼ ðS1n, . . . ,Sd

nÞ and let the discount factor be

�n b¼ ðS0nÞ�1z ¼

Ynk¼1

ð1þ rkÞ�1:

Thus, the discounted price process of the i-th risky asset ~Sin b¼ �nSi

n satisfies

� ~Sin ¼

~Sin�1

Rin � rn1þ rn

, 1 � n � N:

A portfolio process � ¼ ð�nÞ, �n ¼ ð�1n, . . . , �dnÞ is an IRd-valued F n-predictable process,

where �in represents the amount of an investor’s wealth invested in the i-th risky asset at

time n. In the sequel these amounts are not required to be non-negative. In other words,

both short-selling of risky assets and borrowing of risk-free asset are allowed.A consumption process C¼ (Cn)1�n�N is a non-negative F n-adapted process satisfyingPN

n¼1 Cn 51: The quantity Cn represents the rate of consumption at time n. We denote

by ðV x,�,Cn Þ0�n�N the wealth process of a portfolio-consumption pair (�,C) with initial

capital x40, and denote by ~Vx,�,C the discounted one. Hence, ðV x,�,Cn Þ satisfies

V x,�,Cn ¼ xþ

Xdi¼1

Xnk¼1

�ikRik þ

Xnk¼1

V x,�,C

k�1 �Xdi¼1

�ik

!rk �

Xnk¼1

Ck

¼ xþXnk¼1

Vx,�,Ck�1 rk þ

Xdi¼1

Xnk¼1

�ikðRik � rkÞ �

Xnk¼1

Ck

or equivalently,

� ~V x,�,Cn ¼d

i¼1 �n�inðR

in � rnÞ � �nCn,

�nVx,�,Cn ¼ xþ

Pdi¼1

Pnk¼1

�k�ikðR

ik � rkÞ �

Pnk¼1

�kCk: ð2:1Þ

A consumption process C is called admissible (with initial capital x) if there exists

a portfolio process � such that for each n¼ 0, 1, . . . ,N, V x,�,Cn � 0, a.s. We denote by bAðxÞ

the class of all admissible consumption processes.A probability measure lQ is called an equivalent martingale measure if it is equivalent to

the historical probability measure IP and the discounted price process ð ~SinÞ of risky asset i is

a lQ-martingale for each i¼ 1, . . . , d. We denote by P the set of all equivalent martingale

measures and assume that P is not empty to exclude any arbitrage opportunity.

The market is said to be complete if P is a singleton, otherwise the market is said to be

incomplete.In the sequel, we denote by (a, b) the inner product of vectors a and b. We quote the

following result in [10] as a lemma.

Optimization 693

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

LEMMA 2.1 Assume that B is a non-negative FN-measurable random variable, and

sup lQ2PE lQ(B)51. Put

Jn ¼ esssup lQ2PE lQ½BjF n�, 0 � n � N:

Then 8 lQ2P J¼ (Jn)0�n�N is a lQ-supermartingale, and

Jn ¼ J0 þXnk¼1

ð�k,� ~SkÞ � An, 0 � n � N, ð2:2Þ

where (�n) is a d-dimensional predictable process such that Mn ¼ J0 þPn

k¼1ð�k,�~SkÞ is a

lQ-martingale, and (An) is a d-dimensional integrable adapted increasing process with initial

value 0.

Then we can obtain the following proposition which gives a characterization for

admissible consumption:

PROPOSITION 2.2 A consumption process C is admissible with initial capital x if and only if

IE lQ

XNn¼1

�nCn

" #� x, lQ 2 P: ð2:3Þ

Furthermore, if there exists a lQ02P such that

IE lQ0

XNn¼1

�nCn

" #¼ x, ð2:4Þ

then the portfolio process � satisfying Vx.�,C� 0 is unique under the meaning of equivalence

and satisfies

IE lQ0

Xnk¼1

�kCk

�����F n

" #¼ xþ

Xdi¼1

Xnk¼1

�k�ikðR

ik � rkÞ, ð2:5Þ

�nVx,�,Cn ¼ IE lQ0

XNk¼nþ1

�kCk

�����F n

" #: ð2:6Þ

In particular,

V x,�,CN ¼ 0, a.s. ð2:7Þ

Proof Assume that C2 bA(x), that is, there exists a portfolio process � such that Vx,�,C� 0.

Put

Mn ¼ �nVx,�,Cn þ

Xnk¼1

�kCk, 0 � n � N:

From (2.1) we have

Mn ¼ xþXdi¼1

Xnk¼1

�k�ikðR

ik � rkÞ, 0 � n � N:

694 P. Li and S.-Y. Wang

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

If lQ2P then for each i ¼ 1, . . . , d, IE lQðRinÞ ¼ rn. Thus, (Mn) is a non-negative lQ-local

martingale and therefore a lQ-martingale. Hence,

IE lQ

XNn¼1

�nCn

" #� IE lQ

XNn¼1

�nCn þ �NVx,�,CN

" #¼ IE lQðMN Þ � x, lQ 2 P, ð2:8Þ

which implies (2.3).Conversely, assume that C is a consumption process satisfying (2.3). Put

Jn ¼ esssup lQ2PIE lQ

XNk¼1

�kCkjF n

" #, 0 � n � N:

By Lemma 2.1, we know that (Jn) is a non-negative lQ-supermartingale, and there

exist a predictable process (�n) and an adapted zero-initial-valued increasing process A

such that

Jn ¼ J0 þXnk¼1

Xdi¼1

�ik�~Sik � An ¼ J0 þ

Xnk¼1

Xdi¼1

�ik�kSik�1ðR

ik � rkÞ � An:

Put �in b¼�inSin�1, then ð�

inÞ is a portfolio process and

Jn ¼ J0 þXnk¼1

Xdi¼1

�k�ikðR

ik � rkÞ � An:

Therefore,

�nVx,�,Cn ¼ xþ

Xdi¼1

Xnk¼1

�k�ikðR

ik � rkÞ �

Xnk¼1

�kCk ¼ x� J0 þ Jn �Xnk¼1

�kCk þ An � 0,

which implies that C is admissible.Now we assume that there exists a measure lQ02P such that (2.4) is true and suppose

that C2 bA(x), then from (2.8), Equation (2.7) holds for the portfolio process � created

above. Next we proceed to prove the uniqueness of �.Suppose that ~� is another portfolio process such that Vx, ~�,C � 0. Put

M0n b¼ �nV x, ~�,Cn þ

Xnk¼1

�kCk ¼ xþXdi¼1

Xnk¼1

�k ~�ikðRik � rkÞ, 0 � n � N:

Then (Mn) and ðM0nÞ are both lQ0-martingales. Obviously, MN ¼M0N ¼

PNk¼1 �kCk. Thus,

Mn ¼M0n, that is,Xdi¼1

Xnk¼1

�k�ikðR

ik � rkÞ ¼

Xdi¼1

Xnk¼1

�k ~�ikðRik � rkÞ, 0 � n � N,

which shows the equivalence of � and ~�. (2.5) and (2.6) follow from the fact that (Mn) is

a lQ0-martingale. g

Remark 2.1 It is easy to see from above proposition that if the consumption process C

and C0 are both admissible, so is the consumption process "Cþ (1� ") C0 for any "2 [0, 1].

Optimization 695

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

3. Optimal consumption and martingale measure under the general setting

In our model the agent is assumed to have a utility function U : (0,1)! IR, for

consumption, where U is strictly increasing, strictly concave, continuously differentiable

and satisfies

U0ð0Þb¼ limx#0

U0ðxÞ ¼ 1, U0ð1Þb¼ limx!1

U0ðxÞ ¼ 0:

The quantity U(Cn) represents the amount of utility the agent gains by consuming at rate

Cn at time n.Recall that Li et al. [13] discussed the problem of maximizing the expected utility of

terminal wealth IE[U(VN)] and gave the relationship between the optimal martingale

measure and the optimal trading strategy. Here we will find the optimal martingale

measure by maximizing the expected total utility of consumption IE½PN

n¼1 UðCnÞ�.Specifically, for a given utility function U and a given initial capital x40, we consider

the following problem:

maxC2bAðxÞ IE

XNn¼1

UðCnÞ

" #:

An admissible consumption process bC is called optimal if for any other C 2 bAðxÞ,IE

XNn¼1

UðbCnÞ

" #� IE

XNn¼1

UðCnÞ

" #:

From the strict concavity of U and Remark 2.1, we can easily prove the uniqueness of

optimal consumption process.From Proposition 2.2 we can get the following theorem which gives a property for

optimal consumption.

THEOREM 3.1: If C is the optimal consumption process for the initial capital x, then

suplQ2P

IE lQ

XNn¼1

�nCn

" #¼ x:

Proof Let y ¼ sup lQ2P IE lQ½PN

n¼1 �nCn�, thus y� 0. By Proposition 2.2 it suffices to show

that y� x. If y5x, by Proposition 2.2, there exists a portfolio process � such that

Vx,�,C� 0, and

�nVy,�,Cn ¼ yþ

Xdi¼1

Xnk¼1

�k�ikðR

ik � rkÞ �

Xnk¼1

�kCk:

Let bC ¼ xy C and b� ¼ x

y �, we have

�nVx,b�,bCn ¼ xþ

Xdi¼1

Xnk¼1

�kbxtikðRik � rkÞ �

Xnk¼1

�kbCk ¼x

y�nV

y,�,Cn � 0,

which implies bC 2 bAðxÞ. On the other hand, since bCn ¼xy Cn 4Cn and U is strict

increasing,

IE U

XNn¼1

�nbCn

!" #4 IE U

XNn¼1

�nCn

!" #:

696 P. Li and S.-Y. Wang

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

This contradicts the optimality of C. Therefore, y� x. g

The (continuous, strictly decreasing) inverse of the function U0 is denoted byI : (0,1)! (0,1). From the concavity of U, we have the following inequality:

UðIð yÞÞ � UðxÞ þ y½Ið yÞ � x�, 8x4 0, y4 0: ð3:1Þ

For lQ2P, if we denote Z lQn b¼ IE

hd lQdIP jF n

i, then ðZQ

n Þ is a strict positive martingale.Put

P0 ¼

(lQ 2 P : IE

XNn¼1

�nZlQn Ið y�nZ

lQn Þ

" #51, y4 0

)and assume that P0 is non-empty. The following notion was initiated by Karatzas et al.[11]. For every lQ2P0, the function X lQ defined by:

X lQð yÞb¼ IEXNn¼1

�nZlQn Ið y�nZ

lQn Þ

" #, 05 y51,

is a continuous, strictly decreasing mapping from (0,1) onto (0,1), and so X lQ hasa continuous, strictly decreasing inverse Y lQ from (0,1) onto (0,1).

For n¼ 1, . . . ,N, lQ2P0, x40, we define the following random variable:

CQ

n ðxÞb¼ IðY lQðxÞ�nZlQn Þ:

Then by the definition,

IEPNn¼1

�nZlQn C

Q

n ðxÞ

� �¼ IE

PNn¼1

�nZlQn IðY

lQðxÞ�nZlQn Þ

� �¼ X lQðY lQðxÞÞ ¼ x, x 2 ð0,1Þ:

ð3:2Þ

For any admissible consumption process C with initial capital x, from (3.1) we have

UðCQ

n ðxÞÞ � UðCnÞ þ Y lQðxÞ�nZlQn ðC

Q

n ðxÞ � CnÞ:

Associating with (2.3) and (3.2) we have

IEXNn¼1

UðCQ

n ðxÞÞ

" #� IE

XNn¼1

UðCnÞ

" #þ Y lQðxÞ IE

XNn¼1

�nZlQn C

Q

n ðxÞ

" #� IE

XNn¼1

�nZlQn Cn

" # !

¼ IEXNn¼1

UðCnÞ

" #þ Y lQðxÞ x� IE lQ

XNn¼1

�nCn

" # !� IE

XNn¼1

UðCnÞ

" #:

Consequently, for a given x40, if there exists a probability measure blQ 2 P0 such

that ðCblQn ðxÞÞ is an admissible consumption process for the initial capital x, then ðC

blQn ðxÞÞ is

optimal. In this case we also say blQ is optimal. SinceZ lQn is uniquely determined byCQ

n ðxÞ andthe optimal consumption process is unique, the optimal martingale measure is also unique.

From Proposition 2.2 and the above analysis, we have the following theorem:

THEOREM 3.2: Let lQ 2 P0. If

IE lQ

XNn¼1

�nCblQn ðxÞ

" #� x, lQ 2 P,

then ðCblQn ðxÞÞ is optimal for the initial capital x.

Optimization 697

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

Remark 3.3 This theorem means that if the expected total consumption doesnot exceedthe initial capital, then the consumption is optimal.

4. Optimal consumption for HARA utility functions

In the following we will consider the optimal consumption for a class of widely used utilityfunctions – HARA utility functions, which are given below:

U�ðxÞ ¼1� ðx

� � 1Þ, �5 0,

logx, � ¼ 0:

(For this kind of utility function we have

U0ðxÞ ¼ x��1, IðxÞ ¼ x1��1, �b¼ �

� � 12 ½0, 1Þ,

where � satisfies 1� þ

1� ¼ 1 for �50. Thus for all lQ2P and y2 (0,1), we have

CQ

n ðxÞ ¼ ðYlQðxÞ�nZ

lQn Þ

1��1 ð4:1Þ

and

X lQð yÞ ¼ IEXNn¼1

�nZlQn ð y�nZ

lQn Þ

1��1

" #

¼ IEXNn¼1

y1��1ð�nZ

lQn Þ

" #¼ y

1��1

XNn¼1

IE ð�nZlQn Þ

�� �

� y1��1N51

which shows that P0¼P.A trading strategy is a predictable IRdþ1-valued stochastic sequence ¼ ð nÞ0�n�N,

n ¼ ð�0n,�

1n, . . . ,�dnÞ. �

0n and �

in, n ¼ 0, 1, . . . ,N, i ¼ 1, . . . , d, represent the number of units

of the risk-free asset and risky asset i held at time n, respectively. The wealth process ofa trading strategy is denoted by Vn( ).

A trading strategy ¼ {�0,�} is said to be self-financing if

�0n�1S0n�1 þ �n�1 � Sn�1 ¼ �

0nS

0n�1 þ �n � Sn�1, 81 � n � N:

It means that at time n� 1, once the price vector Sn�1 is quoted, the investor readjustshis/her positions from n�1 to n without bringing in or withdrawing any wealth. It is easyto prove that a trading strategy ¼ {�0,�} is self-financing if and only if

~V nð Þb¼ �nV nð Þ ¼ V 0ð Þ þXnk¼1

ð�k,� ~SkÞ, 81 � n � N: ð4:2Þ

A trading strategy is said to be admissible if Vn( )� 0 for any n¼ 0, 1, . . . ,N. Wedenote by A(x) the collection of all self-financing and admissible trading strategies. It canbe shown that for any 2A(x) and lQ2P, (�nVn( )) is a lQ-martingale.

For a HARA utility function U¼U�(� � 0), put

ðY lQn ðxÞÞ

11�� b¼ 1

xIE ð�nZ

lQn Þ

�� �

, ð4:3Þ

� lQn ðxÞb¼ ðY lQ

n ðxÞ�nZlQn Þ

1��1, n ¼ 1, . . . ,N, lQ 2 P: ð4:4Þ

698 P. Li and S.-Y. Wang

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

Then we have the following theorem which gives the optimal consumption process for

HARA utility functions:

THEOREM 4.1: If there exists a probability measure lQ*2P such that the variable � lQ�

n ðxÞ can

be replicated by a trading strategy b 2 AðxÞ, that is, � lQ�

n ðxÞ ¼ V nðb Þ, then ðC lQ� ðxÞÞ defined

by (4.1) is an optimal consumption process for utility functions U¼U�(� � 0).

Proof: Since � lQ�

n ðxÞ ¼ V nðb Þ, lQ*2P, b 2 AðxÞ, we have

IE lQ �n�lQ�

n ðxÞh i

¼ x, lQ 2 P, n ¼ 1, . . . ,N:

By (3.2) and (4.3),

x ¼ X lQðY lQðxÞÞ ¼ IEXNn¼1

�nZlQn IðY

lQðxÞ�nZlQn Þ

" #

¼ IEXNn¼1

ð�nZlQn Þ

�ðY lQðxÞÞ

1��1

" #¼ xðY lQðxÞÞ

1��1

XNn¼1

ðY lQn ðxÞÞ

11�� :

Thus

ðY lQðxÞÞ1��1

XNn¼1

ðY lQn ðxÞÞ

11�� ¼ 1: ð4:5Þ

From (4.1), (4.4) and (4.5), we have

IE lQ

XNn¼1

�nClQ�

n ðxÞ

" #¼ IE lQ

XNn¼1

�nðYlQðxÞ�nZ

lQ�

n Þ1��1

" #

¼ ðY lQðxÞÞ1��1

XNn¼1

IE lQ½�n�lQ�

n ðxÞ�ðYlQ�

n ðxÞÞ1

1��

¼ xðY lQðxÞÞ1��1

XNn¼1

ðY lQ�

n ðxÞÞ1

1�� ¼ x:

Then we can conclude from Theorem 3.2 that ðC lQ� ðxÞÞ is an optimal consumption

process. g

5. Optimal martingale measure for a specific model

In this section we will work out the optimal consumption process for a specific market

model with HARA utility functions.We consider a discrete-time incomplete financial market in which there are only two

assets: one risk-free asset (bond) and one risky asset (stock) whose price processes ðS0nÞ and

(Sn) satisfy the conditions (1) and (2) in Section 2. But here we further assume that for

n¼ 1, . . . ,N, rn is a positive constant, Rn is an F n-measurable random variable

independent of F n�1.We assume that for each n, the support of Rn is [dn, un], where dn and un are two real

numbers. Since there doesn’t exist any arbitrage opportunity in the market, it can be seen

from the analysis of He et al. [10] that dn5rn5un.

Optimization 699

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

For a self-financing trading strategy b¼f�0,�g, we put

�0n b¼ �0nS0n�1

V n�1, �n b¼ �nSn�1

V n�1, n ¼ 1, . . . ,N:

From the property of self-financing, we have �0n þ �n ¼ 1: Thus, �0n and �n represent theproportion of the wealth Vn�1 invested in the risk-free asset and the risky asset at time n,

respectively.Recalling that ~Sn ¼ �nSn and ~V n ¼ �nV n, we have

� ~V n ¼�nV n�1

Sn�1� ~Sn, V 0 ¼ x: ð5:1Þ

We denote by Vx,�n the solution of Equation (5.1).

LEMMA 5.1: For a fixed initial capital x40 and a given probability measure blQ 2 P, thefollowing two statements are equivalent:

(1) There exists a trading strategy b 2 AðxÞ such that �NVNðb Þ ¼PNn¼1 �nC

blQn ðxÞ;

(2) IE lQ½PN

n¼1 �nCblQn ðxÞ� ¼ x, 8 lQ2P.

Proof

(1) )(2) is obvious from the fact that (�nVn( )) is a lQ-martingale for any lQ2P,

2A(x).(2) )(1) Let

Jn ¼ esssup lQ2PIE lQ

XNn¼1

�nCblQn ðxÞjF n

" #, n ¼ 1, . . . ,N:

From condition (2), J0¼x40 and Jn is strictly positive. From Lemma 2.1, there exist

a predictable process b� and an adapted increasing process A with A0¼ 0 such that

Jn ¼ J0 þXnk¼1

ðb�k,� ~SkÞ � An, n ¼ 1, . . . ,N:

Thus

Jn þ An ¼ xþXnk¼1

ðb�k,� ~SkÞ, n ¼ 1, . . . ,N,

is a non-negative lQ-martingale. So we have

IE lQ JN þ AN½ � ¼ x: ð5:2Þ

On the other hand,

JN ¼XNn¼1

�nCblQn ðxÞ,

then

IE lQ JN þ AN½ � ¼ xþ IE lQ AN½ �: ð5:3Þ

700 P. Li and S.-Y. Wang

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

It follows from (5.2) and (5.3) that P lQ[AN]¼ 0, thus AN¼ 0, which implies that A is a nullprocess. So we have

Jn ¼ xþXnk¼1

ðb�k,� ~SkÞ, n ¼ 1, . . . ,N:

It is easy to see that there exists a real-valued predictable process b�0 such that b ¼ fb�0,b�g isa self-financing trading strategy with initial capital x. That is, �Vðb Þ ¼ J4 0 from (4.2).Thus b 2 AðxÞ. Particularly,

�NVNðb Þ ¼ JN ¼XNn¼1

�nCblQn ðxÞ:

g

The above lemma shows that if there exists a trading strategy such that its discountedterminal wealth is all consumed, then the expected total consumption is just the initialcapital and vice versa.

Now we recall some results in [13] which will be used in the following. Interestedreaders can refer to the article for details.

For each n¼ 1, . . . ,N, let Fn be the distribution function of Rn and put

fnðaÞ ¼

ZIR

x� rn

ð1þ rn þ aðx� rnÞÞ1��

FnðdxÞ, a 2 �1þ rnun � rn

,1þ rnrn � dn

� �:

Assume that

lima#� 1þrn

un�rn

fnðaÞ4 0 and lima" 1þrn

rn�dn

fnðaÞ5 0:

Then from Lemma 4.1 in [13] we know that equation fn(a)¼ 0 has a unique solution ��n in�

1þrnun�rn

, 1þrnrn�dn

, and hence, the wealth process Vx,�� is strictly positive.

Define an n-variate Borel-measurable function g* as follows:

g�nðx1, . . . ,xnÞ ¼ð1þ ��nxnÞ

��1

IE½ð1þ ��nxnÞ��1�, 1 � n � N,

and denote by lQg� the probability measure associated with g* through the followingequation:

ZlQg�

n b¼IE d lQg�

dIP

����F n

� �¼Ynk¼1

1þ gkR1 � r11þ r1

, . . . ,Rk � rk1þ rk

� �� IE gk

R1 � r11þ r1

, . . . ,Rk � rk1þ rk

� �����F k�1

� �� �:

The random variable �n(x) associated with lQg� through (4.4) is denoted by �lQg�

n ðxÞ.From Theorem 4.2 in [13] we know that lQg� 2 P and �

lQg�

n ðxÞ ¼ V x,��

n for HARA utilityfunctions U¼U�(� � 0). Then from (4.1) and Theorem 4.1,

ClQg�

n ðxÞ ¼ ðYlQg� ðxÞ�nZ

lQg�

n Þ1��1 ð5:4Þ

is an optimal consumption process, and lQg� is the optimal martingale measure for initialcapital x and HARA utility functions U�(� � 0). In next section we will use this martingalemeasure to price derivatives.

Optimization 701

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

6. Derivative pricing under the optimal martingale measure

A derivative is an FN-measurable non-negative random variable. The following theorem isknown as the Martingale Measure Method for Derivative Pricing:

THEOREM 6.1: Let lQ2P, (F n)0�n�N be an information set, �n is the discount factor, thenthe price of any derivative � is

Cn ¼ ��1n E lQ½�N�jF n�: ð6:1Þ

It is easy to show that the process (�nCn) is a lQ-martingale. In other words, at thisprice there does not exist any arbitrage opportunity for both the buyer and seller of thederivative. In this sense, we say Cn is the ‘fair’ price of derivative �. This method of pricingderivatives is called the arbitrage pricing method, and equation (6.1) is called therisk-neutral valuation formula.

From Section 5 we know that lQg� is the optimal equivalent martingale measure, so wecan use this martingale measure to give the price for any derivative. Specifically, we cangive the price of a derivative of the form f(SN), where f(x) is a non-negative measurablefunction:

Cn ¼ ��1n IE lQg�

½�Nf ðSN ÞjF n�:

We know that the European call and put options and futures are all derivatives of the formf(SN), so they can be priced according to the above pricing formula.

7. Conclusion

In this article, we discussed the problem of selecting an optimal martingale measure bymaximizing the expected total utility of consumption for discrete-time incomplete financialmarkets. We first introduced the general results on utility maximization for a generaldiscrete-time incomplete market. For a given utility function, we constructed a randomvariable following Karatzas et al. [11]. If one of these variables which is associated witha certain equivalent martingale measure is an admissible consumption process, then thisconsumption process is optimal, and the martingale measure associated with it is alsooptimal and unique. For a specific market model with HARA utility functions, we workedout explicitly the Radon–Nikodym derivative of the optimal martingale measure and theoptimal consumption process based on the results in [13]. Finally we gave the fair price fora derivative using the optimal martingale measure.

Both of this article and [13] investigate the problem of selecting an optimal martingalemeasure for discrete-time incomplete financial markets, but the latter one discussed it fromthe point of view ofmaximizing the expected utility of terminal wealthwithout consumption,while the former one from the point of view maximizing the expected total utility ofconsumption. One can also combine the two to consider the maximization of the total utilityof both wealth and consumption, which will be more difficult and needs further effort.

Acknowledgement

This work is supported by the National Natural Science Foundation of China, Grant Nos. 70371006and 70501003, and the National Basic Research Program of China (973 Program, GrantNo. 2007CB814306).

702 P. Li and S.-Y. Wang

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4

References

[1] I. Bajieux and R. Portait, The numeraire portfolio: a new approach to continuous time finance,Working paper, George Washington University and E.S.S.E.C, 1995.

[2] I. Bajieux and R. Portait, Pricing contingent claims in incomplete markets using the numeraireportfolio, Working paper, George Washington University and E.S.S.E.C, 1995.

[3] H. Buhlmann, F. Delbaen, P. Embrechts, A.N. Shiryaev, On Esscher transforms in discrete

finance models, Astin Bulletin, 2005.[4] P. Carr, X. Jin, and D.B. Madan, Optimal investment in derivative securities, Finance Stoch. 5

(2001), pp. 33–59.[5] N. El Karoui and M. Jeanblanc-Picque, Optimization of consumption with labor income, Finance

Stoch. 2 (1998), pp. 409–440.[6] R.J. Elliott and D.B. Madan, A discrete time equivalent martingale measure, Math. Finance. 8

(1998), pp. 127–152.

[7] H. Follmer and M. Schweizer, Hedging of contingent claims under incomplete information,in Applied Stochastic Analysis, Vol. 5, M.H.A. Davis and R.J. Elliott, eds., New YorkStochastic Monographs, Gordon Breach, 1991, pp. 389–414.

[8] H.U. Gerber and E.S.W. Shiu, Option pricing by Esscher transform, Trans. Soc. Actuaries 46(1994), pp. 51–92.

[9] H. He and N.D. Pearson, Consumption and portfolio policies with incomplete markets and short-

sale constrains: the finite dimensional case, J. Econ. Theory 54 (1991), pp. 259–304.[10] S.W. He, J.J. Li, and J.M. Xia, Financial market of finite discrete time model, Adv. Math. 28(1)

(1999), pp. 1–28.[11] L. Karatzas, J.P. Lehoczky, and S.E. Shreve, Optimal portfolio and consumption decisions for

a ‘small investor’ on a finite horizon, SIAM J. Control Optim. 25(6) (1987), pp. 1557–1586.[12] P. Li and J.M. Xia, Minimal martingale measures for discrete-time incomplete financial markets,

Acta Math. Appl. Sinica 18(2) (2002), pp. 349–4352.

[13] P. Li, J.M. Xia, and J.A. Yan, Martingale measure method for utility maximization in discrete-time financial markets, Ann. Econ. Finance 2(2) (2001), pp. 445–465.

[14] J.B. Long, The numeraire portfolio, J. Finan. Econ. 26(1) (1990), pp. 29–69.

[15] S.E. Shreve and H.M. Soner, Optimal investment and consumption with transaction costs, Ann.Appl. Probab. 4 (1994), pp. 609–692.

Optimization 703

Dow

nloa

ded

by [

Uni

vers

ity o

f W

est F

lori

da]

at 0

6:36

05

Oct

ober

201

4