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, Schäl (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
PING LI, JIANMING XIA, AND JIA-AN YAN
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→∞
MARTINGALE MEASURE METHOD FOR EXPECTED UTILITY
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
PING LI, JIANMING XIA, AND JIA-AN YAN
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.
MARTINGALE MEASURE METHOD FOR EXPECTED UTILITY
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
, I(x) =
−∞, γ = 0
(
γ
1−x 1−γ
γ
− log x,
, γ<0
γ = 0.
When γ < 0, we have Pn = P for n = 0, 1, · · · , N and when γ = 0, PN =
γ
{ Q ∈ P : |IP (Q)| < ∞}. Put δ =
b γ−1
, then 1δ + γ1 = 1 and δ ∈ [0, 1). In this
subsection we replace the notation ξnQ (x) with ζnQ (x) (resp. ηnQ (x)) when
452
PING LI, JIANMING XIA, AND JIA-AN YAN
γ < 0 (resp. γ = 0). Thus from (3) we have


γ


Q 1−γ
1
β
−
γx
)
(Z
n
h n
i
ζnQ (x) =
1−
,
1

γ
βn E (ZnQ ) 1−γ
ηnQ (x) =
x
+ E[ZnQ log ZnQ ] − log ZnQ .
βn
(6)
(7)
Then for utility functions Wγ (x), the statement (A) above is equivalent
to the following
statement
h
i
h(D)(resp.
i (E)) when γ < 0 (resp. γ = 0):
1
1
Q∗ 1−γ
Q 1−γ
(D). E (ZN )
≥ E (ZN )
, ∀ Q ∈ P,
∗
∗
Q
Q
Q
Q
(E). E[ZN
log ZN
] ≤ E[ZN
log ZN
], ∀ Q ∈ PN .
By the results in Subsection 3.1 we know that if there exist a probability
Q∗
measure Q∗1 ∈ P and a strategy ψb1 ∈ Ψzs (DU ) such that ζN1 (z) = VN (ψb1 ),
then ψb1 is optimal for Wγ (γ < 0) and the Hellinger-Kakutani distance of
order δ of Q∗1 with respect to P is the minimum over P, that is, dδ (P, Q∗1 ) =
min dδ (P, Q), where δ satisfies 1δ + γ1 = 1; if there exist a probability measure
Q∈P
b zs such
Q∗2 ∈ P 0 with finite relative entropy IP (Q∗2 ) and a strategy ψb2 ∈ Ψ
Q∗
−x
2
that ξN (z) = VN (ψb2 ), then ψb2 is optimal for W0 = −e and the relative
0
entropy of Q∗2 with respect to P is the minimum over PN
, that is IP (Q∗2 ) =
min IP (Q).
Q∈P
4. THE UTILITY MAXIMIZATION PROBLEM FOR A
SPECIAL MARKET MODEL
4.1. The market model and the characterization of equivalent
martingale measures
In this section 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 Sn0 and Sn satisfy:
1) S00 = 1, S0 is a positive constant;
0
2) Sn0 = Sn−1
(1 + rn ) and rn is a positive constant, for n = 1, · · · , N ;
3) ∀n = 1, · · · , N, Sn = Sn−1 (1 + Rn ), where Rn is an Fn -measurable
random variable independent of Fn−1 , and Rn > −1, a.s..
Assume that for each n = 1, · · · , N, Fn = σ(R1 , · · · , Rn ), F0 = {∅, Ω}.
For notational convenience, we put F−1 = F0 .
We assume that for each n, there exist two constants dn and un such that
dn < rn < un , P(dn ≤ Rn ≤ un ) = 1, and for any ε > 0, P(Rn ≥ dn −ε) > 0
and P(Rn ≤ un − ε) > 0. Consequently, the support of the distribution of
MARTINGALE MEASURE METHOD FOR EXPECTED UTILITY
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
PING LI, JIANMING XIA, AND JIA-AN YAN
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 MARTINGALE MEASURE METHOD FOR EXPECTED UTILITY
455
That is,
h
Rn −rn
1 −r1
E (Rn − rn ) 1 + gn R1+r
,
·
·
·
,
1+r
1 h in
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
PING LI, JIANMING XIA, AND JIA-AN YAN
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
MARTINGALE MEASURE METHOD FOR EXPECTED UTILITY
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
PING LI, JIANMING XIA, AND JIA-AN YAN
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
MARTINGALE MEASURE METHOD FOR EXPECTED UTILITY
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
PING LI, JIANMING XIA, AND JIA-AN YAN
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 ])
.
MARTINGALE MEASURE METHOD FOR EXPECTED UTILITY
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
PING LI, JIANMING XIA, AND JIA-AN YAN
in (bn , ∞). Since
Z
lim hn (a) = lim
a↑∞
a↑∞
x−rn
(x − rn )e−a 1+rn Fn (dx)
ZR
x−rn
(x − rn )e−a 1+rn Fn (dx)
= lim
a↑∞
{x≥rn }
Z
x−rn
(x − rn )e−a 1+rn Fn (dx)
+ lim
a↑∞
{x<rn }
= 0 − ∞ = −∞
Thus hn (a) = 0 has a unique solution π
bn ∈ (bn , ∞) if and only if lim hn (a)
a↓bn
> 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.
MARTINGALE MEASURE METHOD FOR EXPECTED UTILITY
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
!
Rk −rk
−b
πk 1+r
k
E e
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
!
Rk −rk
−b
πk 1+r
k
E e
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
PING LI, JIANMING XIA, AND JIA-AN YAN
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
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