Martingale Measure Method for Expected Utility Maximization in ...

ANNALS OF ECONOMICS AND FINANCE

2, 445–465 (2001)

Martingale Measure Method for Expected Utility Maximization in Discrete-Time Incomplete Markets Ping Li Institute of Systems Science Academy of Mathematics and Systems Science The Chinese Academy of Sciences, Beijing, 100080 E-mail: liping [email protected]

Jianming Xia Institute of Applied Mathematics Academy of Mathematics and Systems Science The Chinese Academy of Sciences, Beijing, 100080 E-mail: jmxia [email protected]

and Jia-an Yan* Institute of Applied Mathematics Academy of Mathematics and Systems Science The Chinese Academy of Sciences, Beijing, 100080 E-mail: [email protected]

In this paper we study the expected utility maximization problem for discretetime incomplete financial markets. As shown by Xia and Yan (2000a, 2000b) in the continuous-time case, this problem can be solved by the martingale measure method. In a special discrete-time model, we explicitly work out the optimal trading strategies and the associated minimum relative entropy martingale measures and minimum Hellinger-Kakutani distance martingale c 2001 Peking University Press measures.

Key Words : Martingale measure; Incomplete market; Utility maximization; Optimal trading strategy; Relative entropy; Hellinger-Kakutani distance. JEL Classification Numbers : G11, G13.

* The work of Yan was supported by the National Natural Science Foundation of China, grant 79790130. 445 1529-7373/2001 c 2001 by Peking University Press Copyright All rights of reproduction in any form reserved.

446

PING LI, JIANMING XIA, AND JIA-AN YAN

1. INTRODUCTION If a contingent claim can be replicated by a self-financing trading strategy, then the price of the contingent claim, under the principle of “free of arbitrage opportunities”, is the cost of replication. For the problem of maximizing the expected utility of terminal wealth, Karatzas, Lehoczky, and Shreve (1987) and Karatzas, Lehoczky, Shreve and Xu (1991) developed a “replication method”. They showed that if some contingent claim that they constructed can be replicated by a self-financing trading strategy whose initial wealth is just the initial capital held by the investor, then the trading strategy is optimal. Thus the notion of replication is important in pricing contingent claims and maximizing the expected utility. In the context of a complete market, any contingent claim can be replicated by a self-financing trading strategy, thus both problems of pricing contingent claims and maximizing the expected utility are well understood. While in an incomplete market, there exist contingent claims that cannot be replicated, thus many troubles appear when dealing with the problems of pricing contingent claims and maximizing the expected utility. Karatzas, Lehoczky, Shreve and Xu (1991) studied the problem of utility maximization in an incomplete market with a diffusion model. In their model, the market consists of a bond and m stocks, and the price processes of the stocks are driven by a d-dimensional Brownian motion. When m < d, that is, the market is incomplete, they augmented the market with certain fictitious stocks so as to create a complete market. Under certain conditions on the model, they showed that one can judiciously choose fictitious stocks such that these fictitious stocks are superfluous in the optimal portfolio of the completed market. In this case, the optimal portfolio of the completed market is also optimal for the original incomplete market. In general cases, the method of “fictitious completion” is no longer applicable, since the general models are infinite-dimensional ones. From the point of mathematical view, a financial market is arbitrage-free if and only if there exists an equivalent martingale measure for the discounted price processes of the assets and the completeness of the market is equivalent to the uniqueness of the equivalent martingale measure. Thus in incomplete markets there are various equivalent martingale measures. For incomplete markets, Xia and Yan (2000a, 2000b) proposed a martingale measure method to solve the utility maximization problem and unified the so-called “numeraire portfolio approach” and Esscher transform method in the theory of pricing contingent claim. In a geometric Lévy process model, they obtained the associated explicit results. In a discrete-time incomplete market, Sch¨ al (2000a) also studied the connections between martingale measures and portfolio optimization, but the utility functions he considered are HARA utility functions Uγ with 0 ≤ γ < 1. In another paper, Schäl

MARTINGALE MEASURE METHOD FOR EXPECTED UTILITY

447

(2000b) studied the relations between arbitrage and utility maximization. The underlying utility function he considered is required to be defined on the positive half-line. In this paper we consider the same problem as that considered by Xia and Yan (2000a, 2000b), but for the discrete-time case. We first introduce the general results of utility maximization problem for an incomplete market in a general discrete-time setting. For a given utility function, we choose a class of equivalent martingale measures. Corresponding to each of the equivalent martingale measures of this class, we construct a random variable following Karatzas, Lehoczky, Shreve and Xu (1991). If one of these random variables can be replicated by a self-financing trading strategy, then the strategy is optimal and the associated martingale measure is also “optimal” in some sense. For the utility function log x (resp. γ1 (xγ − 1)(γ < 0)), the associated martingale measure minimizes the γ relative entropy (resp. the Hellinger-Kakutani distance of order γ−1 ); for 1

utility function −e−x (resp. −(1 − γx) γ (γ < 0)), the associated martingale measure minimizes the relative entropy (resp. the Hellinger-Kakutani γ distance of order γ−1 ) of dual form. For a special discrete-time market model, the optimal trading strategies and the associated “optimal” equivalent martingale measures for the above two classes of utility functions are explicitly worked out. 2. THE GENERAL DISCRETE-TIME MODEL Let the time index set be {0, 1, · · · , N }, and suppose that (Ω, F, Fn , P) is a stochastic basis, where (Ω, F, P) is a probability space and (Fn )0≤n≤N is an increasing complete filtration satisfying FN = F. We put F−1 = F0 = {∅, Ω}. Assume that in the economy there are one risk-free asset (bond) and d risky assets (stocks) whose price processes are defined as follows: 1) The price of the risk-free asset at time n is Sn0 , n = 0, 1, · · · , N , which is strictly positive and predictable; 2) The price of the i-th risky asset at time n is Sni , i = 1, · · · , d. For i = 1, · · · , d, n = 0, 1, · · · , N, Sni is strictly positive and Fn -measurable. We denote Sn = (Sn1 , · · · , Snd ) and denote by βn the discount factor(Sn0 )−1 at time n. A trading strategy is a predictable Rd+1 -valued stochastic sequence ψ = (ψn )0≤n≤N , ψn = (φ0n , φn ), where φn = (φ1n , · · · , φdn ), φ0n and φin , i = 1, · · · , d, n = 0, 1, · · · , N , represent the number of units of the risk-free asset and asset i held at time n respectively. The wealth Vn (ψ) of a trading strategy ψ = {φ0 , φ} at time n is Vn (ψ) = φ0n Sn0 + φn · Sn , where φn · Sn =

448 d P


φin Sni . The discounted wealth is Ven (ψ) = βn Vn (ψ) = φ0n + φn · Sen ,

i=1

where Sen = (βn Sn1 , · · · , βn Snd ) is the vector of discounted prices. A trading strategy ψ = {φ0 , φ} is said to be self-financing, if 0 0 φ0n−1 Sn−1 + φn−1 · Sn−1 = φ0n Sn−1 + φn · Sn−1 , ∀1 ≤ n ≤ N.

It means that at time n − 1, once the price vector Sn−1 is quoted, the investor readjusts his/her positions from ψn−1 to ψn without bringing or withdrawing any wealth. It is easy to prove that a trading strategy ψ = {φ0 , φ} is self-financing if and only if Ven (ψ) = V0 (ψ) +

n X

φk · ∆Sek , ∀1 ≤ n ≤ N,

(1)

k=1

or equivalently, ∆Ven (ψ) = φn · ∆Sen , n ≥ 1, where ∆Ven (ψ) = Ven (ψ) − e Vn−1 (ψ). For any given Rd -valued predictable process ψ and any constant z, it is easy to construct a unique real-valued predictable process (φ0n ) such that (φ0n , φn ) is a self-financing strategy with initial wealth z. A probability measure Q is called an equivalent martingale measure, if it is equivalent to the historical probability measure P and if the discounted price processes (Seni ) of risky assets are Q-martingale. We denote by P the set of all equivalent martingale measures. For Q ∈ P, from (1) we can see that for any self-financing trading strategy ψ, (Ven (ψ)) is a local Q-martingale. It is well-known that there is no arbitrage in the market if and only if P is not empty. Therefore, we assume that P is not empty to exclude any arbitrage opportunity. The market is said to be complete if P is a singleton, otherwise we say that the market is incomplete. 3. THE UTILITY MAXIMIZATION IN A GENERAL SETTING 3.1. The utility maximization problem for a general utility function We take the same setting and notations as those in Xia and Yan (2000a, 2000b). Here we only give the main results for easy reference. We assume that in our model the agent has a utility function U : (DU , ∞) −→ R for wealth, −∞ ≤ DU < ∞. Throughout this paper we make the assumption that U is strictly increasing, strictly concave, continuous, and continuously differentiable, and satisfies U 0 (DU )= b lim U 0 (x) = ∞, x↓DU

U 0 (∞)= b lim U 0 (x) = 0. x→∞


449

The (continuous, strictly decreasing) inverse of the function U 0 is denoted by I : (0, ∞) −→ (DU , ∞). From the concavity of U we have the following inequality: U (I(y)) ≥ U (x) + y[I(y) − x],

∀x > DU , y > 0.

(2)

We denote by Ψzs the collection of self-financing strategies with initial capital z > 0. It is easy to show that for any ψ ∈ Ψzs , Q ∈ P, (Ven (ψ)) is a Q-local martingale. We will discuss the cases of DU > −∞ and DU = −∞, respectively. In the case of DU > −∞, for a given initial capital z > 0, we denote Ψzs (DU )={ψ b : ψ ∈ Ψzs , VN (ψ) > DU }. We consider the problem of maximizing the expected utility of terminal wealth E [U (VT (ψ))], over the class Ψzs (DU ). A strategy ψ ∈ Ψzs (DU ) which maximizes the expected utility is called optimal. It is easy to show that the wealth process of optimal trading in Ψzs (DU ) is unique. dQ strategies Q Q For Q ∈ P, we denote Zn =E b dP Fn , then (Zn ) is a strict positive martingale. Put Pn ={ b Q ∈ P : |E[βn ZnQ I(yβn ZnQ )]| < ∞, ∀y ∈ (0, ∞)},

n = 0, 1, · · · , N,

and we make the standing assumption that Pn is not empty. The following notion was initiated by Karatzas, Lehoczky, Shreve and Xu (1991). For every n = 0, 1, · · · , N and Q ∈ Pn , the function XnQ : (0, ∞) −→ (DU EQ [βn ], ∞), defined by XnQ (y)=E b βn ZnQ I(yβn ZnQ ) , 0 < y < ∞ inherits from I the property of being a continuous, strictly decreasing mapping from (0, ∞) onto (DU EQ [βn ], ∞), and hence XnQ has a (continuous, strictly decreasing) inverse YnQ from (DU EQ [βn ], ∞) onto (0, ∞). We define Q Q ξnQ (x)=I(Y b n (x)βn Zn ), 0 ≤ n ≤ N, Q ∈ Pn , x ∈ (DU EQ [βn ], ∞).

(3)

We can see that for ψ ∈ Ψzs (DU ), Q ∈ P, (βn Vn (ψ)) is a Q-martingale. Associated with (2) and (3) we have h i Q E U (ξN (z)) ≥ E [U (VN (ψ))] , Q ∈ PN , ψ ∈ Ψzs (DU ). (4)

450


From the discussion above we know that if there exists a probability Q∗ measure Q∗ ∈ PN and a trading strategy ψb ∈ Ψzs (DU ) such that ξN (x) = Q Q b b VN (ψ), then ψ is optimal. Since ZN is uniquely determined by ξN (x), such a Q∗ is unique. From (4) we see that Q∗ is also “optimal” in PN in the sense thath i h i Q∗ Q (A). E U (ξN (x)) ≤ E U (ξN (x)) , ∀Q ∈ PN . In the case of DU = −∞, put P 0 ={Q b ∈P:

dQ ∈ L2 (Ω, FN , P)}, dP

b zs ={ψ b : ψ ∈ Ψzs , βn Vn (ψ) ∈ L2 (Ω, Fn , P), 0 ≤ n ≤ N }, Ψ b zs , Q ∈ P 0 , it can be shown and assume that P 0 is not empty. For ψ ∈ Ψ that (βn Vn (ψ)) is a Q-martingale. For the case of DU = −∞, we consider the martingale measure in P 0 instead of P and replace the notion Pn correb zs . By the same way as in sponding to P by Pn0 and replace Ψzs (DU ) with Ψ the case of DU > −∞, we can see that if there exists a probability measure ∗ 0 b then b zs such that ξ Q (z) = VN (ψ), Q∗ ∈ PN and a trading strategy ψb ∈ Ψ N z 0 b s and the measure Q is “optimal” over P . ψb is optimal over Ψ N 3.2. The HARA utility maximization, the minimum relative entropy and the minimum Hellinger-Kakutani distance In the sequel, we assume that (Sn0 ) is deterministic, then so is (βn ). Here we consider the HARA (hyperbolic absolute risk aversion) utility functions given by 1 γ (x − 1), γ < 0, Uγ (x) = γ log x, γ = 0. For the HARA utility functions U = Uγ (x)(γ ≤ 0), we have DU = 0, I(x) = 1 γ x γ−1 and Pn = P for n = 0, 1, · · · , N . Put δ = b γ−1 ∈ [0, 1), then 1δ + γ1 = 1 for γ < 0. From (3) we have 1

ξnQ (x)

=

x(ZnQ ) γ−1 βn E[(ZnQ )δ ]

,

n = 0, 1, · · · , N.

(5)

Then for the HARA utility functions U = Uγ (γ ≤ 0), the statement (A) in Section 3.1 is equivalent to the following statement (B)(resp. (C)) when γ < 0 (resp. h γ ∗= i0): h i Q δ Q δ (B). E (ZN ) ≥ E (ZN ) , ∀Q ∈ P. ∗

Q Q (C). E[log ZN ] ≥ E[log ZN ], ∀Q ∈ P.


451

Definition 3.1. 1). Assume that probability measure Q is absolutely continuous with respect to P and δ ∈ (0, 1). Put

" Hδ (Q, P)=E b P

dQ dP

δ #

h i Q δ = EP (ZN ) ,

dδ (Q, P)=2(1 b − Hδ (Q, P)).

Hδ (Q, P) and dδ (Q, P) are called the Hellinger integral and Hellinger-Kakutani distance of order δ of P with respect to Q, respectively; 2). The relative entropy of a probability measure P with respect to Q is defined by dP dQ dP log = −EP log . IQ (P) = EQ dQ dQ dP Both the Hellinger-Kakutani distance and the relative entropy are quantitative measures of the difference between Q and P. One should be aware that in general dδ (Q, P)(resp. IQ (P)) is not equal to dδ (P, Q)(resp. IP (Q)) and it is easy to show that for δ ∈ (0, 1), dδ (Q, P) = d1−δ (P, Q). For the HARA utility functions U = Uγ (γ ≤ 0), by the results in Subsection 3.1, we know that if there exist a probability measure Q∗ ∈ P and Q∗ z b b then ψb is optimal and a strategy ψ ∈ Ψs (DU ) such that ξN (x) = VN (ψ), i) for U (x) = Uγ (γ < 0), the Hellinger-Kakutani distance of order δ of the historical measure P with respect to Q∗ is the minimum over P, that is, dδ (Q∗ , P) = min dδ (Q, P), where δ satisfies 1δ + γ1 = 1; Q∈P

ii) for U (x) = log x, the relative entropy of P with respect to Q∗ is the minimum over P, that is IQ∗ (P) = min IQ (P). Q∈P

3.3. Results of dual form In the following we consider the class of utility functions given by Wγ (x) =

1

−(1 − γx) γ , −e−x ,

γ < 0, γ = 0.

Since Uγ (−Wγ (x)) = −x(γ ≤ 0), we say that Wγ (x) is the dual utility function of Uγ (x). For Wγ (x)(γ ≤ 0) we have DU =

1 γ,

γ 0, P(Rn ≥ dn −ε) > 0 and P(Rn ≤ un − ε) > 0. Consequently, the support of the distribution of


453

Rn is [dn , un ]. Readers can refer to Li and Yan (1999) for the economic meaning of the above conditions. From the above 2) and 3) we have Sn0 =

n Y

(1 + rk ),

Sn = S0

k=1

n Y

(1 + Rk ),

∀n = 1, · · · , N.

(8)

k=1

Recalling that βn = (Sn0 )−1 , we have Sen = βn Sn , ∆Sen = Sen − Sen−1 = Sen−1

R n − rn 1 + Rn − 1 = Sen−1 . 1 + rn 1 + rn

For a self-financing trading strategy ψ = {φ0 , φ}, we put πn0 =

0 φ0n Sn−1 φn Sn−1 , πn = , Vn−1 Vn−1

n = 1, · · · , N.

Then πn0 and πn represent the proportion of the wealth Vn−1 invested in the risk-free asset and the risky asset at time n, respectively. Since 0 0 Vn−1 = φ0n−1 Sn−1 + φn−1 Sn−1 = φ0n Sn−1 + φn Sn−1 ,

we have πn0 = 1 − πn . We also call (πn0 , πn ) a portfolio at time n. Thus R n − rn R n − rn = πn Ven−1 . Ven − Ven−1 = ∆Ven = φn ∆Sen = φn Sen−1 1 + rn 1 + rn Consequently, Vn = Vn−1 [1 + rn + πn (Rn − rn )]. 1+rn n When πn takes value in − u1+r , , Vn (ψ) is strictly positive. n −rn rn −dn z,π We denote by (Vn ) the solution of the equation z,π Rn − rn ∆Venz,π = πn Ven−1 , 1 + rn

Then Vnz,π = Vn (ψ) = z

n Q

V0z,π = z.

[1 + rk + πk (Rk − rk )].

k=1

From (8) we have Sen = βn Sn = S0

n n Y Y 1 + Rk R k − rk = S0 1+ . 1 + rk 1 + rk

k=1

k=1

(9)

454


Let (Mn ) be a local martingale with M0 = 0. Since Mn ∈ Fn , from the Doob Representation Theorem we know that there exists an n-variate Borel-measurable function gn (x1 , · · · , xn ) such that Mn = gn

R1 − r1 R n − rn ,··· , 1 + r1 1 + rn

, n = 1, · · · , N.

(10)

From E(∆Mn |Fn−1 ) = 0 we know that

Mn−1 = E gn

Rn − rn R1 − r1 ,··· , Fn−1 , 1 + r1 1 + rn

n = 1, · · · , N.

Therefore, Mn

n X R1 − r1 R k − rk = gk ,··· , 1 + r1 1 + rk k=1 R 1 − r1 Rk − rk − E gk ,··· , Fk−1 . 1 + r1 1 + rk

(11)

dQ If Q is a probability measure and is equivalent to P, we put Zn =E b dP Fn , n P ∆Zk then (Zn ) is a strictly positive martingale. Put Mn = Zk−1 , then (Mn ) is a local martingale and Zn =

n Q

k=1

(1 + ∆Mk ). From the above

k=1

analysis we know that there exists an n-variate Borel-measurable function gn (x1 , · · · , xn ) such that (10) holds. Hence the density process (Zn )1≤n≤N associated with g = (gn )1≤n≤N is Zg (n) = b

n h Q

Rk −rk 1 −r1 1 + gk R1+r , · · · , 1+rk 1 k=1 i Rk −rk R1 −r1 − E gk 1+r1 , · · · , 1+rk Fk−1 ,

(12) 1 ≤ n ≤ N.

Specifically, if Q is an equivalent martingale measure, that is, (Sen ) is a Q-martingale, then EQ (Rn |Fn−1 ) = rn . Thus E[Zg (n)(Rn − rn )|Fn−1 ] = 0. From (12) we have R 1 − r1 R n − rn Zg (n − 1)E (Rn − rn ) 1 + gn ,··· , 1 + r1 1 + rn R 1 − r1 Rn − rn Fn−1 = 0. − E gn ,··· , F n−1 1 + r1 1 + rn


455

That is, h Rn −rn 1 −r1 E (Rn − rn ) 1 + gn R1+r , · · · , 1+r 1 h i n i Rn −rn 1 −r1 − E gn R1+r , · · · , F F = 0. n−1 n−1 1+rn 1

(13)

In the following we denote by G the collection of all g =(g b n )1≤n≤N , where gn is an n-variate Borel-measurable function and (Zg (n))1≤n≤N is a strictly positive martingale and denote the probability measure associated with Q dQ g ∈ G by Qg , that is, Zg (n) = E dPg Fn . The random variable ξn g is denoted for short by ξng . Then from the above analysis we obtain the following theorem which gives a characterization for equivalent martingale measures. Theorem 4.1. A probability measure Q is an equivalent martingale measure if and only if Q = Qg for some g ∈ G such that (13) holds.

4.2. The HARA Utility Maximization for the Special Model In this subsection, we will explicitly work out the optimal trading strategy, the minimum relative entropy martingale measure and the minimum Hellinger-Kakutani distance martingale measure for the HARA utility functions U = Uγ (x)(γ ≤ 0). Lemma 4.1. For R n = 1, · · · , N, let Fn be the distribution function of Rn and assume that R |x|Fn (dx) < ∞. For given γ ≤ 0 we put

Z fn (a) = R

x − rn Fn (dx), a ∈ (1 + rn + a(x − rn ))1−γ

−

1 + rn 1 + rn , u n − rn rn − d n

.(14)

If E[Rn ] = rn , then fn (a) = 0 has a unique solution πn∗ = 0. In other n case, fn (a) = 0 has a unique solution πn∗ in (− u1+r , 1+rn ) if and only if n −rn rn −dn lim

n a↓− u1+r n −rn

fn (a) > 0,

lim

n a↑ r1+r n −dn

fn (a) < 0.

(15)

Proof. See the appendix. 1+rn n In the following we assume that (15) holds and πn∗ ∈ − u1+r , is n −rn rn −dn ∗

the unique solution of the equation fn (a) = 0, (Vnz,π ) is the wealth process n Q ∗ corresponding to π ∗ . Then Vnz,π = z [1 + rk + πk∗ (Rk − rk )]. k=1

456


Theorem 4.2. Let

gn∗ (x1 , · · · , xn ) =

(1 + πn∗ xn )γ−1 , 1 ≤ n ≤ N. E[(1 + πn∗ xn )γ−1 ] ∗

∗

g Then g ∗ ∈ G, Qg∗ ∈ P and ξN (z) = VNz,π for the utility function Uγ (x)(γ ≤ 0).

Proof. See the appendix. By Theorem 4.2 and the results in Subsection 3.2 we know that the strategy corresponding to π ∗ is optimal for the HARA utility functions U = Uγ (γ ≤ 0) and Qg∗ is just the associated optimal martingale measure. That is, when γ < 0, Qg∗ minimizes the Hellinger-Kakutani distance (of order δ) dδ (Q, P) over P, where δ satisfies 1δ + γ1 = 1; when γ = 0, Qg∗ minimizes the relative entropy IQ (P) over P. 4.3. Results of dual form In this subsection we will work out the optimal strategies, the minimum relative entropy martingale measure and the minimum Hellinger-Kakutani distance martingale measure of dual form for the utility function Wγ 0 (γ 0 ≤ 0) given by 1 0 γ0 , −(1 − γ x) γ 0 < 0, 0 Wγ (x) = −x −e , γ 0 = 0. We will first consider the case of γ 0 < 0. In this case, DU = 0

0

γ γ 0 −1 ,

γ γ−1

1 γ0 .

Put

then + = 0. Let γ = + = 1, then δ = = 1 − δ0 . δ = Under the notation of Section 3.3 and Section 4.1 and the condition of Theorem 4.2 we put   γ0  0 0  1−γ ∗ βn − γ x (Zg∗ (n)) 1 · . ζng (x) = 0 1 − 1 γ  βn E[(Zgb(n) ) 1−γ 0 ]  1 δ0

1 γ0

1 1 γ0 , δ

1 γ

Then from Theorem 4.2 we have γ0

1

∗

∗

(Zg∗ (n)) 1−γ 0 (Zg∗ (n)) γ−1 βn ξng (z) βn Vnz,π i h = = = . 1 E [(Zg∗ (n))δ ] z z E (Zg∗ (n) ) 1−γ 0 Thus ∗

ζng (z) =

∗ 1 βn + 1 − Vnz,π . 0 0 γ γz


457

b defined as follows: Now we construct self-financing strategy ψb = {φb0 , φ} let ψ = (φ0 , φ) be the strategy corresponding to the portfolio π ∗ , we put φb0n = 1 − γβ0Nz φ0n + βγN0 , φbn = (1 − γβ0Nz )φn . Then b = βN + βn 1 − βN V z,π∗ , n = 1, · · · , N. Ven (ψ) n γ0 γ0z Specifically ∗ 1 βN b VN (ψ) = 0 + 1 − 0 VNz,π . γ γz ∗ b > 10 . Consequently, ψb ∈ Ψz (DU ) Recalling that Vnz,π > 0, thus Ven (ψ) s γ ∗ b = ζ g (z). Then from the results in Section 3.3 we know that ψb and VN (ψ) N 1 is optimal for Wγ0 (x) = −(1 − γ 0 x) γ 0 (γ 0 < 0) and the martingale measure Qg∗ minimizes the Hellinger-Kakutani distance (of order δ 0 ) dδ0 (P, Q) over P. For utility function W0 (x) = −e−x , we have DU = −∞. And for g = (gn ) ∈ G,

ηng (x) =

x + E[Zg (n) log Zg (n)] − log Zg (n). βn

Lemma 4.2. For n = 1, · · · , N, let Fn be the distribution function of Rn and R assume that |x|Fn (dx) < ∞. Put R

Z

x−rn

(x − rn )e−a 1+rn Fn (dx), 1 ≤ n ≤ N

hn (a) =

(16)

R

Then there exists some bn (−∞ ≤ bn ≤ 0) such that hn (a) is +∞ for a ≤ bn and finite for a > bn . Furthermore, hn (a) = 0 has a unique solution π bn in (bn , ∞) if and only if lim hn (a) > 0. a↓bn

Proof. See the appendix. In the following we assume that lim hn (a) > 0, for n = 1, · · · , N . Then a↓bn

hn (a) = 0 has a unique solution π bn ∈ (bn , ∞). For n = 1, · · · , N, define βn π bk φnk = βk−1 , k = 1, · · · , n, and denote by ψ n the corresponding selfSk−1 financing strategy with the initial wealth z. Then the discounted wealth of

458


ψ n is Ven (ψ n ) = z + = z+

n X k=1 n X

φnk ∆Sek = z +

n X

φnk βk−1 Sk−1

k=1

R k − rk 1 + rk

n

βn π bk

k=1

X R k − rk Rk − rk . = z + βn π bk 1 + rk 1 + rk k=1

Thus n

Vn (ψ n ) =

X R k − rk z + π bk . βn 1 + rk k=1

Theorem 4.3. Let

gbn (x1 , · · · , xn ) =

e−bπn xn , 1 ≤ n ≤ N. E[e−bπn xn ]

b zs , η gb (z) = VN (ψ N ) for the utility Then gb ∈ G, Qgb ∈ P 0 and ψ N ∈ Ψ N −x function W0 (x) = −e . Proof. See the appendix. By Theorem 4.3 and the results in Section 3.3 we know that the strategy ψ N is optimal for the utility function −e−x and Qgb is just the equivalent 0 martingale measure minimizing the relative entropy IP (Q) over PN . 5. CONCLUSION For the problem of maximizing the expected utility of terminal wealth, Karatzas, Lehoczky, and Shreve (1987) and Karatzas, Lehoczky, Shreve and Xu (1991) developed a “replication method” and gave satisfactory results for complete markets. For an incomplete market, Karatzas, Lehoczky, Shreve and Xu (1991) studied this problem in a market with a diffusion model using the method of “fictitious completion”. In their model, the market consists of a bond and m stocks, and the price processes of the stocks are driven by a d-dimensional Brownian motion. When m < d, that is, the market is incomplete, they augmented the market with certain fictitious stocks so as to create a complete market and showed that the optimal portfolio of the completed market is also optimal for the original incomplete market. But for general infinite-dimensional market models, the “fictitious completion” method is no longer applicable. For continuous-time incomplete markets, Xia and Yan (2000a, 2000b) proposed a martingale measure


459

method to solve the utility maximization problem and obtained the associated results for a geometric Lévy process model. In this paper we study the utility maximization problem for discretetime incomplete markets using the method of martingale measure method. We first introduce the general results of utility maximization problem for a general discrete-time incomplete market. For a given utility function we construct a random variable following Karatzas, Lehoczky, Shreve and Xu (1991). If one of these variables can be replicated by a self-financing trading strategy, then the strategy is optimal and the associated martingale measure is also “optimal” in some sense. For the utility function log x (resp. γ1 (xγ − 1)(γ < 0)), the associated martingale measure minimizes the γ relative entropy (resp. the Hellinger-Kakutani distance of order γ−1 ); for 1

utility function −e−x (resp. −(1 − γx) γ (γ < 0)), the associated martingale measure minimizes the relative entropy (resp. the Hellinger-Kakutani disγ ) of dual form. The major characteristics of this paper tance of order γ−1 is that for a special discrete-time market model, we work out explicitly the optimal trading strategies and the associated “optimal” equivalent martingale measures for the above two classes of utility functions. APPENDIX 1+rn n Proof of Lemma 4.1. Since for a ∈ − u1+r , and x ∈ [dn , un ] n −rn rn −dn we have (1 + rn + a(x − rn ))γ−1 ≤ (1 + rn − a(rn − dn ))γ−1 ∧ (1 + rn + a(un − rn ))γ−1 , n we see that fn (a) is well-defined. Let a ∈ − u1+r , 1+rn , and ε > 0 n −rn rn −dn 1+rn n , . Since such that a + ε ∈ − u1+r n −rn rn −dn (x − rn ) (1 + rn + (a + ε)(x − rn ))γ−1 − (1 + rn + a(x − rn ))γ−1 < 0, for x ∈ [dn , un ], it follows that fn (a + ε) − fn (a) Z = (x − rn ) (1 + rn + (a + ε)(x − rn ))γ−1 R − (1 + rn + a(x − rn ))γ−1 Fn (dx) < 0. 1+rn n , . By the Thus fn (a) is a strict decreasing function in − u1+r n −rn rn −dn 1+rn n , . monotone convergence theorem, fn (a) is also continuous in − u1+r n −rn rn −dn Then the conclusion follows.

460


Proof of Theorem 4.2. From (12) we have γ−1 k −rk 1 + πk∗ R1+r k Zg∗ (n) = γ−1 R −r ∗ k k k=1 E 1 + πk 1+rk n Y

=

n Y

γ−1

(1 + rk + πk∗ (Rk − rk )) h i. γ−1 ∗ k=1 E (1 + rk + πk (Rk − rk ))

k Since ∀k = 1, · · · , n, πk∗ ∈ − u1+r , 1+rk , we have 1+rk +πk∗ (Rk −rk ) > k −rk rk −dk 0, thus Zg∗ (n) is strictly positive. Obviously (Zg∗ (n)) is a martingale, thus g ∗ ∈ G. On the other hand, since πn∗ is the solution of fn (a) = 0, we have fn (πn∗ ) = 0, that is E[(Rn − rn )(1 + rn + πn∗ (Rn − rn ))γ−1 ] = 0. Consequently (13) holds for g ∗ . Thus from Theorem 4.1 we know that Qg∗ ∈ P. ∗ ∗ Therefore, (Venz,π ) is a Qg∗ -martingale, that is, E[Zg∗ (n)βn Vnz,π ] = z. n Q ∗ Recall that Vnz,π = z [1 + rk + πk∗ (Rk − rk )]. Thus k=1

∗

Zg∗ (n)Vnz,π = z

n Y k=1

(1 + rk + πk∗ (Rk − rk ))γ . E[(1 + rk + πk∗ (Rk − rk ))γ−1 ]

Therefore, n Y ∗ 1 1 E[(1 + rk + πk∗ (Rk − rk ))γ ] . = E(Zg∗ (n)Vnz,π ) = βn z E[(1 + rk + πk∗ (Rk − rk ))γ−1 ] k=1

On the other hand, for δ =

Zgδ∗ (n) =

γ γ−1

and n = 1, · · · , N , we have

n Y

(1 + rk + πk∗ (Rk − rk ))γ

k=1

(E[(1 + rk + πk∗ (Rk − rk ))γ−1 ])

,

δ

E[Zgδ∗ (n)] =

n Y

E[(1 + rk + πk∗ (Rk − rk ))γ ] δ

k=1

(E[(1 + rk + πk∗ (Rk − rk ))γ−1 ])

.


461

Thus 1

∗ ξng (z)

z (Zg∗ (n)) γ−1 = βn E[Zgδ∗ (n)] =

=

n z Y 1 + rk + πk∗ (Rk − rk ) 1 ∗ βn γ−1 ]) γ−1 k=1 (E[(1 + rk + πk (Rk − rk )) δ n Y E[(1 + rk + πk∗ (Rk − rk ))γ−1 ] × E[(1 + rk + πk∗ (Rk − rk ))γ ] k=1 n Y

z βn

k=1

(1 + rk + πk∗ (Rk − rk ))E[(1 + rk + πk∗ (Rk − rk ))γ−1 ] E[(1 + rk + πk∗ (Rk − rk ))γ ]

n z Y = · (1 + rk + πk∗ (Rk − rk )) · βn βn k=1

= z

n Y

∗

(1 + rk + πk∗ (Rk − rk )) = Vnz,π ,

n = 1, 2, · · · , N.

k=1 ∗

∗

g Specifically, ξN (z) = VNz,π . Thus the conclusion follows. Proof of Lemma 4.2. According to the assumption that Rn > −1, we n −rn > −1. Since know that R1+r n

x−rn (x − rn )e−a 1+rn I{| x−rn |≤1} ≤ e|a| (1 + rn ), 1+rn

Thus Z x−rn (x − rn )e−a 1+rn I{| x−rn |≤1} Fn (dx) < ∞. 1+rn

R

On the other hand, it is clear that there exists some bn (−∞ ≤ bn ≤ 0) R x−rn such that R (x − rn )e−a 1+rn I{| x−rn |>1} Fn (dx) is +∞ for a < bn and finite 1+rn

for a > bn . Thus the first property of hn (a) follows. For any a ∈ (bn , ∞) and ε > 0 we have x−rn

x−rn

(x − rn )e−(a+ε) 1+rn − (x − rn )e−a 1+rn < 0,

∀x ∈ R.

Consequently hn (a + ε) − hn (a) < 0, which means that hn (a) is strictly decrease in (bn , ∞). By monotone convergence theorem, hn is also continuous

462


in (bn , ∞). Since Z lim hn (a) = lim

a↑∞

a↑∞

x−rn

(x − rn )e−a 1+rn Fn (dx)

ZR

x−rn


= lim

a↑∞

{x≥rn }

Z

x−rn


+ lim

a↑∞

{x 0. Proof of Theorem 4.3. Similar to Theorem 4.2, from (12) we have n Y

−b π

Rk −rk

e k 1+rk . Zgb(n) = Rk −rk −b πk 1+r k=1 E e k Obviously, (Zgb(n)) is a strictly positive martingale, thus gb ∈ G. Note that π bk ∈ (bk , ∞), Rk ∈ [dn , un ], a.s., thus Zgb(N ) is P-square-integrable. Since π bn is the solution of hn (a) = 0, we have hn (b πn ) = 0. That is, h i Rn −rn E (Rn − rn )e−bπn 1+rn = 0. Thus (13) holds for gb. From Theorem 4.1 we know that Qgb is an equivalent martingale measure and Qgb ∈ P 0 . Then (Ven (ψ n )) is a Qgb-martingale, that is, EQgb [βn Vn (ψ n )] = E[Zgb(n)βn Vn (ψ n )] = z. Therefore, " !# n X z R − r z k k = E[Zgb(n)Vn (ψ n )] = E Zgb(n) + π bk βn βn 1 + rk k=1   " !# Rk −rk n n −b πk 1+r X  z Y  k e R − r k k  = E π bk  βn ·  + E Zgb(n) Rk −rk 1 + rk −b πk 1+r k=1 E e k=1 k " !# n X z Rk − rk = + E Zgb(n) π bk . βn 1 + rk k=1

Then we have " E Zgb(n)

n X k=1

R k − rk π bk 1 + rk

!# = 0.


463

Thus n Y

E[Zgb(n) log(Zgb(n))] + log

! Rk −rk −b πk 1+r k E e

k=1





= E Zgb(n) log 

n Y

e

−b πk

k=1 E[e

" = E Zgb(n)

n X k=1

n Y

+ log



Rk −rk −b πk 1+r k Rk −rk 1+rk


n Y

 + log ]

k=1

# Rk −rk R k − rk −b πk 1+r k −b πk − log E e 1 + rk ! R −r

−b π E e k

k k 1+rk

k=1

"

n X

Rk − rk = E − Zgb(n) π bk 1 + rk k=1 ! n Y Rk −rk −b πk 1+r k + log E e

!

n Y

− Zgb(n) log

!# Rk −rk −b πk 1+r k E e

k=1

k=1

" = E −Zgb(n) log

n Y

!# Rk −rk −b πk 1+r k + log E e

k=1

= − log

n Y


n Y k=1

! Rk −rk −b πk 1+r k Zgb(n) + log E e

n Y

k=1

! Rk −rk −b πk 1+r k E e = 0.

k=1

Thus for n = 1, · · · , N,

ηngb (z) =

z + E[Zgb (n) log Zgb (n)] − log Zgb (n) βn 

n Y z = + E[Zgb (n) log Zgb (n)] − log   βn

k=1



R −r

k k −b π e k 1+rk

Rk −rk −b πk 1+r k

E e

  

n Rk −rk X z Rk − rk −b π + E[Zgb (n) log Zgb (n)] − −b πk − log E e k 1+rk βn 1 + rk k=1 ! n n Rk −rk X Rk − rk Y z −b πk 1+r k = + E[Zgb (n) log Zgb (n)] + π bk + log E e βn 1 + rk

=

k=1

n

X Rk − rk z = + π bk = Vn (ψ n ). βn 1 + rk k=1

k=1

464


n P g b k −rk , we Particularly, ηN (z) = VN (ψ N ). Since Ven (ψ N ) = z + βN π bk R1+r k k=1

b z . Thus the conclusion follows. know that ψ N ∈ Ψ s REFERENCES Bajieux, I. and R. Portait, 1995b, Pricing contingent claims in incomplete markets using the numeraire portfolio. Working paper. George Washington Univ. and E.S.S.E.C. B¨ uhlmann, H., F. Delbaen, P. Embrechts, and A. N. Shiryaev, essch transforms in discrete financial market. Preprint. Chan, T., 1999, Pricing contingent claims on stocks driven by L´ evy processes. The Annals of Applied Probability 9(2), 504-528. Davis H. A. M., 1997, Option pricing in incomplete markets. In:Mathematics of Derivative Securities, Dempster, A. H. and Pliska, S. R. (eds.). Publications of the Newton Institute, Cambridge Univ. Press, 216-226. Elliott, R. J. and D. B. Madan, 1998, A discrete time equivalent martingale measure. Mathematical Finance 8, 127-152. Gerber, H. U. and E. S. W. Shiu, 1994, Option pricing by Esscher transform. Transactions Society of Actuaries 46, 51-92. He, S. W., J. J. Li, and J.M. Xia, 1999, Financial market of finite discrete-time model. Advances in Mathematics (China) 28(1), 1-28. Kar1 Karatzas, L., 1989, Optimization problem in the theory of continuous trading. SIAM Journal on Control and Optimization 27, 1221-1259. Karatzas, L., J. P. Lehoczky, and S. E. Shreve, 1987, Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM Journal on Control and Optimization 25, 1557-1586. Karatzas, L., J. P. Lehoczky, S. E. Shreve, and G. L. Xu, 1991, Martingale and duality methods for utility maximization in incomplete markets. SIAM Journal on Control and Optimization 29, 702-730. Li, P. and J. A. Yan, 1999, The growth optimal portfolio in a discrete-time financial market. Preprint. Long, J. B., 1990, The numeraire portfolio. Journal of Financial Economics 26(1), 29-69. Musiela, M. and M. Rutkowsky, 1997, Martingale Methods in Financial Modelling: Theory and Applications, Karatzas, I., Yor, M., eds, Berlin-Heidelberg-New York. Sch¨ al, M., 2000a, Portfolio optimization and martingale measures. Mathematical Finance 10(2), 289-303. Sch¨ al, M., 2000b, Price systems constructed by optimal dynamic portfolios. Mathematical Methods of Operations Research 51(3). Xia, J. M. and J. A. Yan, 2000a, Martingale measure method for expected utility maximization and valuation in incomplete markets. Preprint. Xia, J.M. and J.A. Yan, 2000b, The utility maximization approach to a martingale measure constructed via esscher transform. Preprint. Yan, J. A., 1998, Introduction to maritngale methods in option pricing, Lecture Notes in Mathematics, 4, Liu Bie Ju Centre for Mathematical Sciences, City Univ. of Hong Kong.


465

Yan, J. A., 1998, Lectures on Measure Theory, Chinese Science Press. Yan J. A., Q. Zhang, and S. G. Zhang, 2000, Growth optimal portfolio in a market driven by a jump-diffusion-like process or a l´ evy process. Annals of Economics and Finance 1(1), 101-116.