An equilibrium approach to indifference pricing
Mark H.A. Davis∗and Daisuke Yoshikawa†
First version: March 11, 2010,
Second version: August 5,2010,
Third version: September 9, 2010
Abstract
The utility indifference framework has received a lot of attention, because it is based on a utility
maximization principle, which is one of the most fundamental principles of economics, for pricing a
contingent claim. The price based on utility indifference framework is the maximum or minimum (in
some cases, threshold) price for each investor. Therefore, the price is the indicator for the investor
to join the market of the contingent claim. Our purpose is to expand the view of utility indifference
framework, that is, to deduce the equilibrium price in the utility indifference framework. We attain
the result that, under the setting of an exponential utility, the equilibrium price will be uniquely
evaluated by minimal entropy martingale measure.
1
Introduction
As it is well known, in the theory of the incomplete market, many kinds of frameworks have been
studied for pricing contingent claims uniquely and useful tools have been developed such as mean-variance
hedging, local-risk minimization as the pricing standard1 and Minimal Martingale Measure2 , varianceoptimal Martingale, p-optimal Martingale3 , Minimal Entropy Martingale4 as the pricing measure. Among
them, the utility indifference framework is one which has recieved a lot of attention. This framework
has been developed by many authors5 because of the consistency with economics. As basic economics
assumes that every investor acts in the market for maximizing their utilities6 , the utility indifference
framework adopts this assumption. That is, utility indifference framework assumes that an investor tries
to maximize his expected utility, whether he holds the contingent claim or not. However, if the expected
utility in the case of holding the contingent claim is larger than the expected utility in the case of not
holding the contingent claim, it is rational that the investor holds the contingent claim. Consider the
case that the investor wants to sell the contingent claim. In this case, the price under which his expected
utility selling the contingent claim is larger than the expected utility not selling the contingent claim is
clearly valuable for the investor. Utility indifference price is the price which equates the expected utility
including the contingent claim with the expected utility not including the contingent claim. That is, in
this case, the utility indifference price is the minimum price of the contingent claim for the investor as
a seller. On the other hand, when the investor wants to buy the contingent claim, if the price of the
contingent claim makes his expected utility larger than the expected utility not holding the contingent
claim, this price is advantageous for the investor. Utility indifference price makes the expected utility
holding the contingent claim equate with the expected utility not holding the contingent claim. That
is, this is the maximum price of the contingent claim for the investor as a buyer. The name of utility
∗ Imperial
College London
College London
1 Schweizer(1999)[25] is a good summary for this.
2 Follmer, et al.(1991)[8]
3 Grandits(1999)[13]
4 Frittelli(2000)[10], Goll(2001)[11], Monoyios(2007)[19]
5 e.g. Becherer(2003)[1], Sircar(2004)[26], Monoyios(2004)[18], Hugonnier, et al.(2004)[15], Hugonnier, et al.(2005)[16],
Davis(2006)[4], Monoyios(2008)[20], Biagini(2009)[2]
6 The rigorous formulation of the utility maximization in incomplete market is given by Kramkov, et al.(1999)[6],
Schachermayer(2000)[24]
† Imperial
1
“indifference” price is derived from this context. Furthermore, we can say that the utility indifference
price is “fair” for each investor in this situation.
However, although utility indifference price is fair for each investor, the price is not common for every
investor, because each investor has different preference, respectively. In fact, when a utility indifference
price is fair for an investor who wants to sell the contingent claim, if the utility indifference price is
considered expensive to other investors who want to buy the contingent claim, the price is meaningless,
because the contingent claim will not be traded in the market. On the other hand, if an offered price of
a contingent claim which is fair for an investor who wants to buy the contingent claim is too cheap for
other investors who want to sell the contingent claim, the price is meaningless, although the price is fair
for the buyer of the contingent claim.
Our purpose is simple; that is, to deduce the price available to stand in the market. We consider such
a price exists in the utility indifference framework which we call an equilibrium price.
This paper is constructed as follows. Section 2 describes the review of the utility indifference framework
and utility indifference price. In setting out the model, we adopt the exponential utility function7 . Section
3 describes the equilibrium price in the utility indifference framework. In Section 4, we apply the utility
indifference framework to a more concrete model, such that the market includes some intradable assets.
This model is expanded from the model of Davis(2006)[4]8 . In this model, we have to solve a multivariate
HJB equation and use Cole-Hopf transformation and simple Eigen-decomposition to solve it. This section
includes a numerical example and shows the situation how the price of the contingent claims is distributed
in the market in the utility indifference framework.
2
Model and the review of Utility Indifference Price
We consider the mathematical framework in given probability space (Ω, F, F, P ), where F := (Ft )0≤t≤T ,
F := FT and F0 is trivial. Stochastic process X ∈ Rd is defined as semimartingale, and the expected
value of X is given by the probability measure P . Consider the FT -measurable random variable (or
‘random endowment’) B which will generate some payoff at time T (we also write it as BT for specifying
the time T ). The random variable B is assumed unbounded from below (Delbaen et al.(2002)[5]) . We
assume that for some fixed α, ∈ (0, ∞),
E[e(α+)B ] < ∞ and E[e−B ] < ∞.
This assumption guarantees that B is Lebesgue integrable for all martingale measures for which the
relative entropy is able to be defined (Becherer(2003)[1]). Utility indifference price is defined as the
value or price of the random endowment B which equates the maximized expected utility of the terminal
wealth without the random endowment and the maximized expected utility of the terminal wealth with
the random endowment. The maximized expected utility with the random endowment is described as
follows,
[
∫
T
sup E U (x + pq +
θ∈Θ
]
θt> dXt
− BT q) ,
(1)
0
where let q > 0 be the quantity of B, x + pq be initial capital (x is the money market account, p is
the price of the random endowment B), θ := {θt ; t ∈ [0, T ]} ∈ Θ be the Rd -valued admissible trading
strategy9 , Θ be the set of X-integrable and predictable processes. We can interpret X as a discount price
process of tradable assets. In this paper, we set the quantity q as positive to make demand and supply
to be clearly divided. In fact, by this setting, the relation between demand curve and supply curve is
distinctly depicted as seen in Figure 1.
The maximized expected utility without the random endowment is described as follows,
[
]
∫
T
sup E U (x +
θ∈Θ
θt> dXt ) .
(2)
0
7 HARA utility function or CRRA type utility are also available to use and it is also one of the representative way; e.g.
Zariphopoulou(2001)[27], Monoyios(2010)[?]
8 Davis(2006)[4] uses the model in which there are two assets; the one is tradable asset and the other is intradable one.
∫
9 That is, θ ∈ L(X) and for some constant c ∈ R, t θ > dX ≥ c, t ∈ [0, T ].
s
0 s
2
If the price p is the utility indifference price, then it satisfies as follows,
[
]
[
]
∫ T
∫ T
>
>
sup E U (x +
θt dXt ) = sup E U (x + pq +
θt dXt − BT q) .
θ∈Θ
θ∈Θ
0
(3)
0
We call this utility indifference price as utility indifference sell price, because it implies that the investor
with the utility U (·) sells the random endowment by the price p. This definition implies that the utility
indifference sell price is the minimum price of the random endowment B; that is, if the market price of
the random endowment B is less than utility indifference sell price, then the investor doesn’t sell the
random endowment.
It is natural to define the utility indifference buy price p which satisfies as follows,
[
∫
T
sup E U (x +
θ∈Θ
]
[
θt> dXt ) = sup E U (x − pq +
θ∈Θ
0
∫
T
]
θt> dXt + BT q) .
(4)
0
Consider the problem (2) which is the left hand side of (3)(4). Using the function V (·) which is the
dual function of U (·), this is rewritten as follows,
[
∫
sup E U (x +
θ∈Θ
0
T
]
θt> dXt )
{ [ (
(
)]}
)
∫ T
dQ
dQ
∗ >
E V η
=
inf
+η
x+
(θt ) dXt
η∈R+ ,Q∈M
dP
dP
0
)]
}
{ [ (
dQ
=
inf
+ ηx
E V η
η∈R+ ,Q∈M
dP
where θ∗ = {θt∗ ; t ∈ [0, T ]} ∈ Θ is the solution of the problem (2), M is a set of Θ-martingale measures
satisfying H(Q|P ) < ∞, and H(Q|P ) is relative entropy of Q with respect to P , which is defined as
follows,
{ ∫ dQ dQ
QP
dP ln dP dP
H(Q|P ) :=
+∞
otherwise.
Note that relative entropy is always non negative (c.f. Theorem 1.4.1 of Ihara(1993)[17]).
Hereafter, we assume that
M 6= ∅,
and we specify the utility function U (·) as
U (x) = −e−γx ,
where γ ∈ R+ is risk-aversion. We write the utility indifference sell price and the utility indifference buy
price as ps (BT ; q) and pb (BT ; γ), respectively.
( ( )
)
If the utility function is exponential, then V (y) = γy ln γy − 1 . So, the above equation is calculated
in continuation,
)]
}
{ [ (
dQ
+ ηx
inf
E V η
η∈R+ ,Q∈M
dP
{
( ) [
]
[
(
)]
[
]
}
η
η
dQ
η
dQ
dQ
η
dQ
=
inf
ln
E
+ E
ln
− E
+ ηx
η∈R+ ,Q∈M γ
γ
dP
γ
dP
dP
γ
dP
{
( )
}
η
η
η
η
= inf
ln
− + ηx +
inf H[Q|P ]
η∈R+
γ
γ
γ
γ Q∈M
{
( )
}
η
η
η
η
0
= inf
ln
− + ηx + H[Q |P ] .
η∈R+
γ
γ
γ
γ
0
On line 3, we use η ∈ R+ . We write the solution of inf Q∈M H[Q|P ] as Q
we call the minimal
{ ∈M
( which
)
}
entropy martingale measure (hereafter MEMM). The solution of inf η∈R γη ln γη − γη + ηx + γη H[Q0 |P ]
is given as follows,
0
1
η 0 = γe−γ (x+ γ H[Q |P ]) .
3
Therefore, the maximized utility is described as follows,
[
]
[ (
(
)]
)
∫ T
∫ T
0
0
dQ
dQ
sup E U (x +
θt> dXt )
= E V η0
+ η0
x+
(θt0 )> dXt
dP
dP
θ∈Θ
0
0
= −e−γ (x+ γ H[Q
1
0
|P ])
.
(5)
Likewise, we can solve the utility maximization problem (1) which is the right hand side of (3).
]
[
∫ T
sup E U (x + pq +
θt> dXt − BT q)
θ∈Θ
0
)]}
{ [ (
(
)
∫ T
>
dQ
dQ
ˆ
θt dXt − BT q
=
inf
E V η
+η
x + pq +
η∈R+ ,Q∈M
dP
dP
0
( )
}
{
{
}
η
η
η
η
Q
= inf
ln
inf H[Q|P ] − γqE [BT ]
− + η (x + pq) +
η∈R+
γ
γ
γ
γ Q∈M
{
( )
)}
η
η
η
η(
Q̂
= inf
ln
− + η (x + pq) +
H[Q̂|P ] − γqE [BT ]
,
η∈R+
γ
γ
γ
γ
(6)
{
}
where θ̂ := {θ̂t ; t ∈ [0, T ]} ∈ Θ is the solution of (1), Q̂ is the solution of inf Q∈M H[Q|P ] − γqEQ [BT ] .
Note that the existence of Q̂ is shown by Proposition 2.2 of Becherer(2003)[1]. Then, the optimal η is
given as follows,
(
)
η̂ = γe
−γ x+ γ1 H[Q̂|P ]+q(p−EQ̂ [BT ])
.
Therefore, the maximized utility is given such that,
[
]
∫ T
(
)
−γ x+ γ1 H[Q̂|P ]+pq−qEQ̂ [BT ]
>
.
sup E U (x + pq +
θt dXt − qBT ) = −e
θ∈Θ
(7)
0
From the definition of the utility indifference price and (5)(7), ps (BT ; q) satisfies,
−e−γ (x+ γ H[Q
1
0
|P ])
= −e
(
)
−γ x+ γ1 H[Q̂|P ]+ps (BT ,γ)q−qEQ̂ [BT ]
Consequently, it holds that
)
1 (
EQ̂ [BT ] −
H(Q̂|P ) − H(Q0 |P )
γq
{
}
)
1 (
= sup EQ [BT ] −
H(Q|P ) − H(Q0 |P ) .
γq
Q∈M
ps (BT ; q) =
(8)
This result is consistent with the main result of Delbaen et al(2002)[5].
Remark 2.1
From the definition of dual function, for any θ ∈ Θ, it holds,
[
]
∫ T
>
E U (x + pq +
θt dXt − BT q)
0
(
)]}
{ [ (
)
∫ T
dQ
dQ
+η
x + pq +
θt> dXt − BT q
.
≤
inf
E V η
η∈R+ ,Q∈M
dP
dP
0
In section 7 of Davis(2006)[4], it is pointed out that the equality of the above equation holds if and
only if
(
)
∫ T
1
η̂ dQ̂
>
x + pq +
θt dXt − BT q = − ln
.
(9)
γ
γ dP
0
4
Here, we show that the condition (9) is equivalent θ = θ̂, which means that the equality of (6) holds.
Q
1
As seen above, η̂ is given by η̂ = γe−γ (x+pq+ γ inf Q∈M {H[Q|P ]−γqE [BT ]}) . It implies that
(
)
(
)
1
1
1
η̂ dQ̂
dQ̂
Q̂
= x + pq + H[Q̂|P ] − qE [BT ] − ln
− ln
γ
γ dP
γ
γ
dP
Therefore, (9) is calculated as follows,
∫
x + pq +
T
1
1
= x + pq + H[Q̂|P ] − qEQ̂ [BT ] − ln
γ
γ
θt> dXt − BT q
0
∫T
↔ e−γ (
0
θt> dXt +q(p−BT ))
[
]
∫T >
↔ E e−γ ( 0 θt dXt +q(p−BT ))
From line 2 and line 3,
dQ̂
dP
dQ̂
dP
Q̂
)
[BT ]) dQ̂
dP
−H[Q̂|P ]−γq(p−E [BT ]) dQ̂
Q̂
= E e
dQ̂
dP
dP
]
= e−H[Q̂|P ]−γq(p−E
Q̂
[BT ])
has to the form as follows,
∫T
=
= e−H[Q̂|P ]−γq(p−E
[
(
e−γ (
0
θt> dXt +q(p−BT ))
e−H[Q̂|P ]+γq(p−EQ̂ [BT ])
∫T
>
e−γ ( 0 θt dXt +q(p−BT ))
]
= [
∫T >
E e−γ ( 0 θt dXt +q(p−BT ))
(10)
Note that, from the convexity of the expected utility, θ̂, the solution of (1), is unique and it has to
satisfy,
)]
(∫
[
∫T >
T
E −e−γ(x+pq+ 0 θ̂t dXt −BT q) 0 θt0 dXt = 0, for all θ0 ∈ Θ.
(11)
It implies that the martingale property generated by θ̂. That is, from the uniqueness of θ̂, it holds θ = θ̂.
2
Remark 2.2
Although the equation (11) is an expanded result of Section 3.1 of Frittelli(2000)[10], where the discrete
model is discussed, we can directly deduce it. If θ ∈ Θ, for any κ > 0, it holds κθ ∈ Θ. Since θ̂ is the
solution of the problem (1), it holds for any θ ∈ Θ and any κ > 0,
]
[
]
[
∫T >
∫T >
0
E −e−γ(x+pq+ 0 θ̂t dXt −BT q) ≥ E −e−γ(x+pq+ 0 (θ̂t +κθt )dXt −BT q)
]
[
∫T
∫T >
0
= E −e−γ 0 κθt dXt e−γ(x+pq+ 0 θ̂t dXt −BT q)
[(
]
)
∫ T
∫ T
)
(
∫T >
1
= E
1 − γκ
θt0 dXt + γ 2 κ2
(θt0 )2 dhXit
−e−γ(x+pq+ 0 θ̂t dXt −BT q)
0
0 2
]
[(
[(∫
)
]
)
∫ T
T
)
(
)
(
∫
∫
1
0
−γ(x+pq+ 0T θ̂t> dXt −BT q)
0 2
−γ(x+pq+ 0T θ̂t> dXt −BT q)
≤E
↔ E
θt dXt
e
.
γκ
(θt ) dhXit
e
2
0
0
Likewise, for same θ0 ,
]
]
[
[
∫T >
∫T >
0
E −e−γ(x+pq+ 0 θ̂t dXt −BT q) ≥ E −e−γ(x+pq+ 0 (θ̂t −κθt )dXt −BT q)
[(
]
)
∫ T
∫
)
(
∫
1 2 2 T 0 2
0
−γ(x+pq+ 0T θ̂t> dXt −BT q)
= E
1 + γκ
θt dXt + γ κ
(θt ) dhXit
−e
2
0
0
]
[(
[( ∫
)
]
)
∫ T
T
)
(
)
(
∫
∫
1
0
−γ(x+pq+ 0T θ̂t> dXt −BT q)
0 2
−γ(x+pq+ 0T θ̂t> dXt −BT q)
≤E
↔ E
−
θt dXt
e
.
γκ
(θt ) dhXit
e
2
0
0
5
That is, for some M ∈ R, it holds,
[∫
]
T
(
)
−γ(x+pq+∫0T θ̂t> dXt −BT q)
0
E θ dX e
≤ κM.
0 t t
Since the above inequality holds for any κ, martingale property is proved.
2
We can also consider the problem (4). Left hand side of (4) is same as (5). Right hand side of (4) is
given as follows,
[
]
∫ T
)
(
−γ x+ γ1 H[Q̃|P ]−pq+qEQ̃ [BT ]
>
,
sup E U (x − pq +
θt dXt + BT q) = −e
θ∈Θ
0
{
}
where Q̃ is the solution of the problem inf Q∈M H[Q|P ] + γqEQ [BT ] . Therefore, utility indifference
buy price is given as follows,
pb (BT ; q)
3
)
1 (
EQ̃ [BT ] +
H(Q̃|P ) − H(Q0 |P )
γq
{
}
)
1 (
0
Q
H(Q|P ) − H(Q |P )
= inf E [BT ] +
Q∈M
γq
{
}
)
1 (
Q
0
= − sup −E [BT ] −
H(Q|P ) − H(Q |P ) .
γq
Q∈M
=
(12)
The equilibrium of utility indifference price
In this section, we consider the equilibrium of the market. First, we give an definition of the equilibrium
on the market of the random endowment B (c.f. Mas-Collel et.al. [21]).
Definition 3.1
Let an economy specify the investors’ preferences which is described by the utility function U :=
{Ui (·); Ui (x) := −e−γi x , i = 1, · · · , I + J}. An allocation q s := {qis , i = 1, · · · , I}, q b := {qjb , j = 1, · · · , J}
and a price p of the random endowment B constitutes a price equilibrium if there is an assignment such
that
1. Offer price condition: For any investor with utility function {Ui , i = 1, · · · , I}, when the investor
sells qis -units of the random endowment, (p, qis ) is preferred to all other allocations (p, (qis )0 ); that
is, an expected utility corresponding to the allocation (p, qis ) is larger than another expected utility
corresponding to the allocation (p, (qis )0 ).
2. Bid price condition: For any investor with utility function {UI+j , j = 1, · · · , J}, when the
investor buys qjb -units of the random endowment, (p, qjb ) is preferred to all other allocations (p, (qjb )0 );
that is, an expected utility corresponding to the allocation (p, qjb ) is larger than another expected
utility corresponding to the allocation (p, (qjb )0 ).
∑I
∑J
3. Market cleared condition i=1 qis = j=1 qjb .
Theorem 3.2
If investors in the market of the random endowment BT act according to the utility maximization, there
is an equilibrium price p∗ , such that
0
p∗ = EQ [BT ] .
Before proving above Theorem, we show a few Lemmas and define the set of the Radon-Nikodým
derivative of equivalent martingale measure for physical measure P , that is,
{
}
dQ
N :=
(ω)ω ∈ Ω, Q ∈ M .
dP
6
Furthermore, we define the set of the terminal wealth for strategy Θ,
{
}
∫ T
G := G(θ)θ ∈ Θ, G(θ) =
θt> dXt < ∞ .
0
Let H be a Hilbert space. It is clear that the set G ⊂ H. We set G(Θ) ∈ G as terminal wealths via all
admissible strategies Θ for a fixed ω ∈ Ω.
Lemma 3.3
For MEMM Q0 , Radon-Nikodým derivative
dQ0
dP (ω)
is given as follows,
e−γG(θ )
dQ0
= ∫ −γG(θ0 ) ,
dP
e
dP
0
(13)
where γ is a risk-aversion and θ0 is the solution of the problem (2) in the case that the utility function
is specified as U (x) := −e−γx .
Proof
MEMM is given as the solution of the problem
inf {H(Q|P )} .
(14)
Q∈M
Since Q ∈ M, for any admissible strategy θ ∈ Θ, it holds that
∫
G(θ)(ω) dQ
dP (ω)dP (ω) = 0, for all
Ω
dQ
dP
∈ N.
It means that dQ
dP ∈ N has Θ-martingale property and this martingale property is the constraint
of the optimization problem. Therefore, for a unique Lagrange multiplier λ ∈ H, we can rewrite the
optimization problem (14) as follows,
}
{∫
∫
dQ
dQ
dQ
inf
(ω) ln
(ω)dP (ω) + hλ, G(Θ)i(ω)
(ω)dP (ω) ,
(15)
dQ
dP
dP
Ω
Ω dP
dP ∈N ,λ∈H
∫
dQ
where we used the Frechét differentiability of the relative entropy functional Ω dQ
dP (ω) ln dP (ω)dP (ω)
∫
dQ
10
and the continuous linear functional Ω G(θ)(ω) dP (ω)dP (ω) in order to apply the Lagrange multiplier
0
method (Favretti(2005)[7], Cochrane and Magcgregor(1978)[3]
and Botelho(2009)[?]). For some dQ
dP ∈ N ,
∫
∈ R+ and some function η(ω) such that Ω η(ω)dP (ω) = 0, we set as follows,
dQ
dQ0
(ω) =
(ω) + η(ω).
dP
dP
Let λ∗ be the optimizer for the problem (15). By a variational method using
the first order condition of the optimization is deduced as follows,
{
}
∫
dQ
∗
η(ω) ln
(ω) + 1 + hλ , G(Θ)i(ω) dP (ω) = 0.
dP
Ω
dQ
dP (ω)
=
dQ0
dP (ω)
∗
Therefore, for any ω ∈ Ω, ln dQ
dP (ω) + 1 + hλ , G(Θ)i(ω) = constant. Since it is required that
dQ
dP has the form as follows,
∗
e−hλ ,G(Θ)i(ω)
dQ
(ω) = ∫ −hλ∗ ,G(Θ)i(ω)
.
dP
e
dP (ω)
Ω
∫
+ η(ω),
dQ
dP dP
= 1,
The optimizer λ∗ has to satisfy the martingale property,
∫
Ω
10 From
G(θ0 )(ω)
dQ
(ω)dP (ω) =
dP
∫
∗
e−hλ ,G(Θ)i(ω) dP (ω)
= 0 for all θ0 ∈ Θ.
−hλ∗ ,G(Θ)i(ω) dP (ω)
e
Ω
G(θ0 )(ω) ∫
Ω
the definition of G(Θ), the linear functional
∫
ω∈Ω
G(θ)(ω) dQ
(ω)dP (ω) is bounded. Therefore, it is continuous.
dP
7
Since hλ∗ , G(Θ)i is given by the linear combination of λ∗ and G(Θ), it holds hλ∗ , G(Θ)i ∈ G. That is, for
some θ∗ ∈ Θ, it holds that hλ∗ , G(Θ)i = G(θ∗ ).
Since the solution of (1), say θ0 , has to satisfy the following equation,
[
]
0
E −γG(θ0 )e−γ (x+G(θ )) = 0 for all θ0 ∈ Θ,
it holds
θ∗ = γθ0 .
Therefore,
dQ
e−γG(θ )(ω)
(ω) = ∫
.
dP
e−γG(θ0 )(ω) dP (ω)
ω∈Ω
0
Q.E.D.
0
Proposition 3.2 of Becherer(2003)[1] shows that limγ↓0 ps (BT ; q) = EQ [BT ]. This is the case of
utility indifference sell price. We can easily show that the case of utility indiffernce buy price, too; that is
0
limγ↓0 pb (BT ; q) = EQ [BT ]. Furthremore, we can show the convergence of the utility indifference price
on the quantity q.
Lemma 3.4
On the utility indifference price based on the exponential utility U (x) = −e−γx , it holds that
0
lim ps (BT ; q) = lim pb (BT ; q) = EQ [BT ] .
q↓0
Proof
q↓0
Consider the problem (12) for the utility indifference buy price, that is,
{
}
)
1 (
b
Q
0
p (BT ; q) = inf E [BT ] +
H(Q|P ) − H(Q |P ) .
Q∈M
γq
Let Q̃ be the solution of this problem. Likewise the Lemma 3.3, the Radon-Nikodým derivative
given as follows,
dQ̃
dP
is
dQ̃
e−γq(B(ω)+hλ,G(Θ)i(ω))
,
(ω) = ∫ −γq(B(ω)+hλ,G(Θ)i(ω))
dP
e
dP (ω)
Ω
where unique optimizer λ ∈ H and G(Θ) ∈ G are given for satisfying as follows,
∫
e−γq(B(ω)+hλ,G(Θ)i(ω))
G(θ0 )(ω) ∫
ω∈Ω
Ω
e−γq(B(ω)+hλ,G(Θ)i(ω)) dP (ω)
dP (ω) = 0, for all θ0 ∈ Θ.
For the utility maximization problem (4), the optimal solution θ̃ satisfies
0, for all θ0 ∈ Θ. From the uniqueness of λ, it holds that
∫
Ω
G(θ0 )(ω)e−γ(qB(ω)+G(θ̃)(ω)) dP (ω) =
qhλ, G(Θ)i = G(θ̃).
That is,
e−γ(qB(ω)+G(θ̃)(ω))
dQ̃
(ω) = ∫
.
dP
e−γ(qB(ω)+G(θ̃)(ω)) dP (ω)
Ω
Next, we show that the convergence Q̃ → Q0 , when q → 0.
[
]
[
]
∫T >
∫T >
lim sup E −e−γ ( 0 θt dXt +BT q)
= lim E −e−γ ( 0 θ̃t dXt +BT q)
q↓0 θ∈Θ
q↓0
[
]
∫T 0 >
≥ lim E −e−γ ( 0 (θt ) dXt +BT q)
q↓0
[
]
∫T 0 >
= E −e−γ ( 0 (θt ) dXt ) .
8
On the other hand, using duality,
[
]
∫T >
lim sup E −e−γ ( 0 θt dXt +BT q)
q↓0 θ∈Θ
{
The solution of inf η∈R+
Therefore,
η
γ
{ [ (
(∫
)]}
)
T
>
dQ
dQ
= lim
inf
E V η
+η
θˆt dXt + BT q
q↓0 η∈R+ ,Q∈M
dP
dP
0
{
}
{
}
η η
η
η
= lim inf
ln − +
inf H[Q|P ] + γqEQ [BT ]
q↓0 η∈R+
γ γ
γ
γ Q∈M
{
)}
0
η η
η
η(
≤ lim inf
ln − +
H[Q0 |P ] + γqEQ [BT ]
q↓0 η∈R+
γ γ
γ
γ
ln γη −
η
γ
+
η
γ
(
0
H[Q0 |P ] + γqEQ [BT ]
[
]
∫T >
lim sup E −e−γ ( 0 θt dXt +BT q)
≤
q↓0 θ∈Θ
=
lim −e
−γ
(
1
γ
1
0
is given by, γe
0
H[Q0 |P ]+qEQ [BT ])
q↓0
e−γ ( γ H[Q
)}
|P ])
−γ
(
1
γ
0
H[Q0 |P ]+qEQ [BT ])
)
[
]
∫T >
= sup E −e−γ 0 θt dXt .
θ∈Θ
That is,
[
]
[
]
∫T >
∫T 0 >
lim E −e−γ ( 0 θ̃t dXt +BT q) = E −e−γ 0 (θt ) dXt .
q↓0
From the uniqueness of the solution of the utility maximization problem, when q ↓ 0,
qB + G(θ̃) → G(θ0 ).
Therefore, when q → 0, the measure Q̃ converges to MEMM, that is, Q̃ → Q0 , which implies that
H(Q̃|P ) → H(Q0 |P ).
0
|P )
Since Q0 is MEMM, H(Q|P ) ≥ H(Q0 |P ) for any Q ∈ M. So, H(Q|P )−H(Q
≥ 0, for q > 0.
q
H(Q|P )−H(Q0 |P )
> 0. Then, for any Q ∈ M and q > 0, some > 0 exists, such
q
H(Q|P )−H(Q0 |P )
that
> . So, H(Q|P ) > H(Q0 |P ) + q. This implies that, for Q̃, limq↓0 H(Q̃|P ) >
q
H(Q0 |P ) + limq↓0 q. That is, limq↓0 H(Q̃|P ) > H(Q0 |P ). However, limq↓0 H(Q̃|P ) = H(Q0 |P ). There0
|P )
fore, limq↓0 H(Q̃|P )−H(Q
= 0 The case of the utility indifference buy price is proved. By the same way,
q
Assume that limq↓0
we can show the case of the utility indifference sell price.
Q.E.D.
Frittelli(2000b)[9] and Rouge(2000)[23] show that the utility indifference sell price is non-decreasing,
when the risk-aversion increases. Since pb (BT ; q) = −ps (−BT ; q) (Becherer(2003)[1]), it is easily shown
that utility indifference buy price is non-increasing, when the risk-aversion increases.
Lemma 3.5
For the increasing buying (selling) quantity of random endowment B, the utility indifference buy price
is non-increasing (the utility indifference sell price is non-decreasing).
Proof It is sufficient to prove the case of the utility indifference buy price, since we can use same way
in the case of the utility indifference sell price.
The utility indifference buy price for the risk-aversion γ and the quantity q is given by,
}
{
)
1 (
Q
0
b
H(Q|P ) − H(Q |P )
p (BT ; q) = inf E [B] +
Q∈M
γq
)
1 (
H(Q̃|P ) − H(Q0 |P ) ,
= EQ̃ [B] +
γq
9
)
.
{
}
where Q̃ is the solution of the problem inf Q∈M H[Q|P ] + γqEQ [BT ] . For > 0, consider the utility
indifference buy price p0 for the risk-aversion γ and the quantity q + . It is given by,
{
}
(
)
1
0
Q
0
p = inf E [B] +
H(Q|P ) − H(Q |P )
Q∈M
γ(q + )
(
)
1
≤ EQ̃ [B] +
H(Q̃|P ) − H(Q0 |P )
γ(q + )
)
)
(
)
1 (
1 (
1
= EQ̃ [B] +
H(Q̃|P ) − H(Q0 |P ) −
H(Q̃|P ) − H(Q0 |P ) +
H(Q̃|P ) − H(Q0 |P )
γq
γq
γ(q + )
(
)(
)
1
1
−
H(Q̃|P ) − H(Q0 |P )
= pb (B; q) −
γq γ(q + )
≤ pb (B; q)
Q.E.D.
Corollary 3.6
The lower bound of a utility indifference buy price is given as follows,
pb (BT ; q) ≥
[
]
1
ln EQ0 eγqBT .
γq
The upper bound of a utility indifference sell price is given as follows,
ps (BT ; q) ≤
Proof
[
]
1
ln EQ0 e−γqBT .
γq
From the definition of the utility indifference price, it holds
]
]
[
[
∫T >
∫T 0 >
s
E e−γ(x+ 0 (θt ) dXt ) = E e−γ(x+p (BT ;q)q+ 0 θ̂t dXt −BT q) .
where θ0 and θ̂ are solutions of left-hand side and right hand side of (3), respectively.
From above equation,
[
]
[
]
∫T 0 >
∫T 0 >
s
E e−γ(x+ 0 (θt ) dXt ) ≥ E e−γ(x+p (BT ;q)q+ 0 (θt ) dXt −BT q)
[
]
∫
−γ(x+ 0T (θt0 )> dXt −BT q)
E
e
[
]
s
[
] = EQ0 eγBT q
↔ eγp (BT ;q)q ≥
∫
−γ(x+ 0T (θt0 )> dXt )
E e
↔ ps (BT ; q) ≥
[
]
1
ln EQ0 eγqBT .
γq
Likewise, we can show the case of utility indifference buy price.
Q.E.D.
Proof of Theorem 3.2 Assume that there are I + J investors with risk-aversion {γi ; i = 1, · · · , I + J}.
Let the allocation q s := {qis ; i = 1, · · · , I, for some i, qis > 0}, q b := {qjb ; j = 1, · · · , J, for some j, qjb > 0}
be selling quantities and buying quantities, respectively.
0
Assume that some p > EQ [BT ] and the allocation (q s , q b ) is the equilibrium. From Lemma 3.4
and Lemma 3.5, the lower (upper) bound of the utility indifference sell price (utility indifference buy
0
price) is EQ [BT ]. Therefore, any allocation (q s , q b ) is not preferred for the buy-side of the random endowment to not buying the random endowment. That is, the allocation (q s , q b ) is impossible.
0
Conversely, if p < EQ [BT ], then any allocation (q s , q b ) is not preferred for the sell-side of the ran0
dom endowment to not selling the random endowment. Therefore, the price p∗ = EQ [BT ] is only
available price in which some allocation is available. Furthermore, the available allocation is given as
{(qis )∗ ; i = 1, · · · , I, for all i, (qis )∗ = 0}, {(qjb )∗ ; j = 1, · · · , J, for all j, (qjb )∗ = 0}.
Q.E.D.
10
Investor’s information bias In the previous part, every investor has the same information which is
represented as a filtration F. However, in the real market, there might be some information bias; that
is, some investors have much information, and some investors have less information on the market. We
consider the information bias by the sub-filtration (Gombani et al.(2007)[12], Becherer(2003)[1]).
We define filtrations such that these satisfy the usual conditions of completeness and right-continuity;
F0 := {Ft0 ; t ∈ [0, T ]} and Ii := {Iti ; t ∈ [0, T ]}, i ∈ N. At time 0, these are trivial σ-algebras. Furthermore, we assume that σ-algebras Ft0 , It0 , It1 , · · · are independent under P , where σ-algebra Ft0 is
generated by the observable price processes. Therefore, the information generated by Ft0 is common for
all investors. Other independent σ-algebra It0 , · · · is generated by the additional information
which is
∨∞
not necessarily observable by all investors. Let I := {It ; t ∈ [0, T ]} be given by It := i=0 Iti . As the
∨
same way, we construct the other filtration Î with IT = ÎT and such that It ⊆ Ît . Let F := F0 I and
∨
F̂ := F0 Î. Furthermore, we set M(F) and M(F̂) to be the set of martingale measures corresponding
to the different filtrations.
The proof of Proposition 4.7 of Becherer(2003)[1] shows that
M(F) ⊇ M(F̂).
For any H ∈ {F, F̂},
Following Becherer(2003)[1], we assume a unique Q∗ ∈ M exists such that its density
measurable, i.e.
}
{
dQ
0
∗
is FT − measurable .
{Q } := Q ∈ M
dP
dQ∗
dP
is FT0 -
Becherer(2003)[1] proves that the measure Q∗ minimizes the relative entropy over M(H). That is, Q∗ is
MEMM and this is common for every filtration.
From these results, (8) and (12), it holds that
ps (BT ; F)
≥ ps (BT ; F̂)
pb (BT ; F)
≤ pb (BT ; F̂)
Since MEMM Q∗ is common for every investor, it implies that the equilibrium price of the random
∗
endowment B is given by p∗ = EQ [BT ], even if there is information bias among investors.
4
Applying to the multi assets basis model
In this section, we apply the above results to the basis model of Davis(2006)[4] and show how the utility
indifference buy (sell) price are distributed in the market. We consider the market which is constructed
of tradable assets X = {Xti , 0 ≤ t ≤ ∞, i = 1, · · · , d} and intradable assets Y = {Ytj , 0 ≤ t ≤ ∞, j =
1, · · · , k}. For simplicity, let X and Y be discounted price processes. On the given probability space,
we assume stochastic processes for these assets as below; let wi = {wti , 0 ≤ t ≤ ∞}, i = 1, · · · , d + k
be Brownian Motion, and E[dwti dwtj ] = ρi,j dt for constant ρi,j > 0, i, j = 1, · · · , d + k, and all other
coefficients be constant11 .
(
dXt /Xt
dYt /Yt
)
= mdt + Σdwt ,
where dXt = (dXt1 /Xt1 , · · · , dXtd /Xtd )> , dYt /Yt = (dYt1 /Yt1 , · · · , dYtk /Ytk )> ,
m = (µ1 , · · · , µd+k )> , dwt = (dwt1 , · · · , dwtd+k )> , Σ = (σi,j )i,j=1,··· ,d+k and σi,j =
(16)
{
σi
0
if i = j,
.
otherwise
We consider the contingent claim BT on the intradable assets Y such that,
BT := h(YT ), .
11 We can formulate the model such that the volatility matrix Σ is not a diagonal matrix. Such a model setting directly
express that each asset is driven by every Brownian motions {wi , i = 1, · · · , n + k}. However, under the setting which sets
the volatility matrix as the diagonal matrix, each assets is driven by every Brownian motion via the correlation between
each Brownian motions. So, in this paper, we adopt the diagonal volatility matrix.
11
where {hi (·), i = 1, · · · , k} are the European pay-off functions and {bi ∈ R, i = 1, · · · , k}. Many
authors12 considered the case of k = 1. We expand the case of k ≥ 1.
We try to reformulate from Brownian motions {wi , i = 1, · · · , d + k} dependent each other to independent Brownian motions {w̃i , i = 1, · · · , d + k}. For the correlation defined as dhwi , wj it = ρi,j dt, the
correlation matrix is set by R := (ρi,j )i,j=1,··· ,d+k . We can decompose it (by Cholesky decomposition) as
follows,
R = AA> ,
where A := (ai,j )i,j=1,··· ,d+k is lower triangular matrix; ai,j = 0 if i < j. Using this matrix, for ini
dependent Brownian motions w0 = {w0 t ; t ∈ [0, T ], i = 1, · · · , d + k}, we can set the relationship as
follows,
dwt = Adwt0
Therefore, the dynamics of underlying assets is rewritten as follows,
(
)
dXt /Xt
= mdt + ΣAdw0 ,
dYt /Yt
(17)
Next, we find the local martingale measure for the process X. Generally, if Q is a probability measure
equivalent to P on FT , then there exist adapted processes g := {gti ; t ∈ [0, T ], i = 1, · · · , d + k} ∈ G,
∫T
where g is square integrable on the time satisfying 0 gti dt < ∞, a.s. and G is the set of all adapted
processes of square integrable on the time, and
(d+k ∫
)
d+k
∑∫ T
∑ T
1
dQ
(gti )2 dt .
= exp
gti d(w0 )it −
dP
2
0
0
i=1
i=1
)>
(
For gt := gt1 , · · · , gtd+k , we can find that the transformation
dw̃t = dwt0 − gt dt
where w̃ = {w̃ti ; t ∈ [0, T ], i = 1, · · · , d+k} is a Brownian motion under the measure Q. We give (d+k, 1)vector g1 = (g 1 , · · · , g d , 0, · · · , 0)> by g1 = −A−1 Σ−1 m1 , where (d + k, 1) vector m1 is defined as m1 :=
(µ1 , · · · , µd , 0, · · · , 0)> . It is clear that g1 is uniquely defined and we set {gti ; i = 1, · · · , d} as gti ≡ g i ,
which makes the asset X local martingale under the measure Q. That is, except for (gtd+1 , · · · , gtd+k ),
every factor of gt is given by g1 .
For i = 1, · · · , k, Y is rewritten as follows,
dYti /Yti
i
= (µd+i + σd+i a>
d+i gt )dt + σd+i dWt ,
where ad+i := (ad+i,1 , · · · , ad+i,d+i , 0, · · · , 0)> is (d + i)-th row vector of A, and W i = {Wti , t ∈ [0, ∞)}
is defined as follows,
dWti = (ad+i,1 dw̃t1 + · · · + ad+i,d+i dw̃td+i ).
That is, W i is a Brownian motion under Q such that dhW i , W j it = a>
d+i ad+j dt = ρd+i,d+j dt. PDE on
Y is solved as,
YTi = yi e
∫T
t
1 2
(µd+i +σd+i (a>
d+i gt )+ 2 σd+i )dt+
∫T
t
σd+i dWti
, Yti = yi ∈ R
First, we deduce the MEMM, that is, we consider the problem inf Q∈M H[Q|P ].
H[Q0 |P ] = inf H[Q|P ]
Q∈M
∫
∫
dQ
dQ dQ
ln
ln
dP = inf
dQ
= inf
Q∈M
Q∈M
dP
dP
dP
( d
)
d+k ∫
d+k
∑ ∫ T
1∑ T i 2
1 ∑ i 2
i 2
= i
inf
(gt ) dt = i
inf
(g ) T +
(gt ) dt
gt ∈G,i=d+1,··· ,d+k 2
gt ∈G,i=d+1,··· ,d+k 2
0
i=1 0
i=1
i=d+1
=
12 e.g.
d
1∑ i 2
(g ) T
2 i=1
Henderson(2002)[14], Musiela and Zariphopoulou(2004)[22], Monoyios(2004)[18]
12
Next, we deduce the utility indifference sell price.
{
}
1
1
EQ [BT ] −
(H(Q|P )) +
H(Q0 |P )
γq
γq
Q∈M
[ (
)]}
{
d
dQ
1 1∑ i 2
1 Q
Q
= sup E [h(YT )] −
E ln
+
(g ) T
γq
dP
γq 2 i=1
Q∈M
{
(d+k ∫
)}
d+k ∫ T
1 ∑ T i i 1 ∑
Q
i 2
= sup E
h(YT ) −
g dw̃ +
(gt ) dt
γq i=1 0 t t 2
Q∈M
0
ps (BT ; q) = sup
i=d+1
We define the variable Z := {Z 1 , · · · , Z k }, where Z i := {Zti , t ∈ [0, ∞)} is given as follows,
Zti :=
and for i = 1, · · · , k,
1
ln Yti ,
σd+i
f (z1 , · · · , zk ) := h(eσd+1 z1 , · · · , eσd+k zk ).
Then, we can construct, for s ∈ (t, T ),
(
)
1
1 2
>
dZsi =
µd+i − σd+i + σd+i ad+i gt ds + dWsi , Zti = zi ∈ R
σd+i
2
We think a value function V (t, z) as,
{
V (t, z)
:=
=
)}
(d+k ∫
d+k ∫ T
1 ∑ T i i 1 ∑
i 2
(gs ) ds
sup E
f (ZT ) −
g dw̃ +
γq i=1 t s s 2
Q∈M
i=d+1 t
{
)
}
(d+k
d+k
1 ∑ i i
1 ∑ i 2
Q
sup E
−
(gt ) dt + V (t+, z+)
g dw̃ −
γq i=1 t t
2γq
Q∈M
Q
i=d+1
where z = (z1 , · · · , zk )> . The HJB equation is deduced as follows,
1
Vt +
2
∑
Vzi zj ρd+i,d+j
i,j=1,··· ,k
(
)
k
∑
Vzi
1 2
+
µd+i − σd+i
+
σ
2
i=1 d+i
sup
g j ∈G,
j=d+1,··· ,d+k
{ k
∑
Vzi a>
d+i gt
i=1
g > g2
− 2
2γq
}
= 0, (18)
V (T, z) = f (z),
where g2 = (gtd+1 , · · · , gtd+k )> . The optimal g2∗ is given by,
g2∗ = γq Ã> Vz ,
where Vz := (Vz1 , · · · , Vzk )> and à := (ad+i,d+j )i,j=1,··· ,k is lower triangular matrix; ad+i,d+j = 0, if
i < j.
Therefore, (18) is rewritten as,
∑
1
1 ∑∑
Vzi zj ρd+i,d+j +
Vzi
2 i=1 j=1
σ
d+i
i=1
k
Vt +
k
k
where R̃ := (ρ̃d+i,d+j )i,j=1,··· ,k is given by
Since R̃ is symmetric,
(
1 2
µd+i − σd+i
+ σd+i a>
d+i g1
2
R̃ = ÃÃ> .
R̃ = EΛE > ,
13
)
+
γq >
V R̃Vz = 0,
2 z
where Λ = (λi,j )i,j=1,···
{ ,k and E := (e1 , · · · , ek ). The components of Λ and E are eigenvalues and
λi , if i = j,
>
eigenvectors; λi,j =
are eigenvalues and {ei = (e1,i , · · · , ek,i ) , i = 1, · · · , k} are
0 otherwise
eigenvectors such that,
{
1 if i = j
e>
e
=
i j
0 if i 6= j.
Define z0 := (z10 , · · · , zk0 )> as follows,
 

e1,1
z1
 ..   ..
 . = .
ek,1
zk
 0 
z1
e1,k
..   ..  = Ez0 ,
.  . 
ek,k
zk0
···
..
.
···
>
Let V 0 (t, z10 , · · · , zk0 ) be V (t, z1 , · · · , zk ) := V 0 (t, e>
1 z, · · · , ek z). Then, it holds that Vzi = ei,1 Vz10 +
ei,k Vzk0 = ẽi Vz , where ẽi := (ei,1 , · · · , ei,k )> . That is, Vz = EVz0 . And, it also holds
Vzi ,zj
(
)
(
)
= ei,1 ej,1 Vz010 ,z10 + ej,k Vz010 ,z0 + ei,k ej,1 Vz00 ,z10 + ej,k Vz00 ,z0
k
k
k k



Vz00 ,z0 · · · Vz00 ,z0
ej,1
1 k
 . 
 1. 1 .
.


.
.
.
= (ei,1 , · · · , ei,k ) 
.
.
.
  .. 
0
0
ej,k
Vz0 ,z0 · · · Vz0 ,z0
1
k
0
= ẽ>
i Vz0 ,z0 ẽj =
k ∑
k
∑
k
k
ei,l Vz00 ,zm
0 ej,m .
l
l=1 m=1
Therefore HJB equation (18) is rewritten again,
Vt0 +
(
)
k
k
k ∑
k
k ∑
k
∑
∑
1 ∑∑
1 2
γq 0 >
1
ρd+i,d+j
µd+i − σd+i
+ σd+i a>
g
(Vz0 ) ΛVz00 = 0
ei,l Vz00 ,zm
ei,l Vz00
0 ej,m +
d+i 1 +
l
l σ
2 i=1 j=1
2
2
d+i
m=1
i=1
l=1
l=1
V 0 (T, z10 , · · · , zk0 ) = f (Ez0 ),
0
=
Vt0
(
)
k
k ∑
k
k
k
k
∑
∑
∑
1∑∑ 0
1
1 2
γq 0 >
0
>
+
ei,l ρd+i,d+j ej,m +
Vz0
ei,l
Vz0 ,zm
µd+i − σd+i + σd+i ad+i g1 +
(Vz0 ) ΛVz0
0
l
l
2
σ
2
2
d+i
m=1
i=1 j=1
i=1
l=1
l=1
=
=
Vt0 +
Vt0 +
k
k
1∑∑
2
l
l=1 m=1
k
k
1 ∑∑
2
>
Vz00 ,zm
0 el R d em +
k
∑
Vz00 e>
l α+
l
l=1
Vz0i0 ,zj0 e>
i Rd ej +
i=1 j=1
k
∑
Vz0i0 e>
i α+
i=1
γq 0 >
(V 0 ) ΛVz00
2 z
γq 0 >
(V 0 ) ΛVz00 .
2 z
e>
(
)
∑k
i Rd ei
1
2
>
0
0
i
0
γqλi
where αi := σd+i
µd+i − 21 σd+i
+ σd+i a>
d+i g1 and α = (α1 , · · · , αk ) . Let V (t, z ) :=
i=1 ln U (t, zi )
Then, we can calculate in continuation,
0=
)
(
k
∑
1 >
e>
i Rd ei
i
i
>
i
U
+
e
R
e
U
+
e
αU
0
0
0
d i zi zi
t
i
zi ,
γqλi U i
2 i
i=1
k
∑
ln U i (T, zi0 )
i=1
14
e>
i Rd ei
γqλi
= f (Ez0 ).
Setting some functions {ci (t); i = 1, · · · , k} (independent of {yi ; i = 1, · · · , k}) as such
(
)
1 >
e>
i R d ei
i
i
>
i
Ut + ei Rd ei Uzi0 zi0 + ei αUzi0 = ci (t), i = 1, · · · , k,
γqλi U i
2
∑k
and i=1 ci (t) = 0 for any t ∈ [0, T ]. In addition, we set
−
e>
i α
>
U i (t, zi0 ) = e ei Rd ei
τ = (T − t)
zi0 − 12
2
(e>
i α)
e> Rd ei
i
τ−
γqλi
e> Rd ei
i
∫τ
0
ci (T −s)ds i
u (τ, zi0 )
so the HJB equation is rewritten as, ∀i = 1, · · · , k,
uiτ
e>
i Rd ei i
uzi0 zi0
2
=
(19)
And, terminal condition is described as,
k
∑
(
ln e
e> α
i
− γqλ
z0 i
i i
u
(0, zi0 )
e>
i Rd ei
γqλi
)
= f (Ez0 ).
i=1
That is,
k
∏
u
i
(0, zi0 )
e>
i Rd ei
λi
=
e
e>
0
i α 0
λi zi +γqf (Ez )
.
i=1
i=1
k
∑
e> Rd ei
i
i=1
k
∏
λi
ln ui (0, zi0 ) = γqf (Ez0 ) +
k
∑
e> R d ei
i
i=1
λi
zi0 .
Note that, the interval is changed as,
{t ≤ T } → {τ ≥ 0}.
We have to solve these partial differential equations,
u1τ
=
e>
1 Rd e1 1
uz10 z10
2
..
.
k
∏
ui (0, zi0 )
ukτ
=
e>
i Rd ei
λi
=
i=1
e>
k Rd ek k
uz 0 z 0
k k
2
k
∏ e>
i α z 0 +γqf (Ez0 )
e λi i
.
i=1
∫
e> R e
It is well known that the solution of uiτ = i 2 d i uiz0 z0 is given as ui (τ, zi0 ) = K i (zi0 − ỹi , τ )u(0, ỹi )dỹi ,
i i
where (as it is also well known that) the fundamental solution K(·) is given as follows,
K
i
(zi0 , τ )
1
√
e
2πe>
i R d ei τ
=
−
(zi0 )2
2e> Rd ei τ
i
However, the terminal condition is given by the combination of variables z. Therefore, it is difficult to
express each u(0, ỹi ) independently. So, we consider the relationship below,
k
∏
ui (τ, zi0 )
e>
i Rd ei
λi
=
i=1
k (∫
∏
K i (zi0 − ỹi , τ )u(0, ỹi )dỹi
i Rd ei
) e>
λ
i
i=1
≤
∫ ∏
k
K i (zi0 − ỹi , τ )
i=1
15
e>
i Rd ei
γqλi
G(ỹ1 , · · · , ỹk )dỹ,
e>
∫ ∏k
∏k
i Rd ei
i
where G(ỹ1 , · · · , ỹk ) = i=1 ui (0, ỹi ) λi expresses terminal condition for ui . Furthermore,
i=1 K (zi −
ỹi , τ )G(ỹ)dỹ is given as follows,
∫ ∏
k
K i (zi0 − ỹi , τ )
e>
i Rd ei
λi
G(ỹ1 , · · · , ỹk )dỹ
(
∫ ∏
k
=
i=1
i=1
(
∫ ∏
k
=
i=1
∫ ∏
k (
=
i=1
For ȳ = (ȳ1 , · · · , ȳk )> , we define ȳσ =
tion as follows,
above equation
=
∫ ∏
k (
i=1
(
ln ȳ1
σd+1 , · · ·
1
√
2πτ
1
√
e
2πτ
1
√
e
2πτ
1
√
2πτ
ȳk
, σlnd+k
i Rd ei
) e>
λ
i
e
−
−
(zi0 −ỹi )2
2e> Rd ei τ
i
i Rd ei
) e>
λ
i
k
∏
ui (0, ỹi )
e>
i Rd ei
λi
dỹ
i=1
(zi0 −ỹi )2
− >
2e Rd ei τ
i
i Rd ei
) e>
λ
i
k
∏
e
e>
i α
λi ỹi +γqf (Eỹ)
dỹ
i=1
i Rd ei
) e>
λ
−
i
e
(zi0 −ỹi )2
2λi τ
+
e>
i α
λi ỹi +γqf (Eỹ)
dỹ
)
:= Eỹ. Then, we can calculate in continua-
2
(zi0 −e>
i ȳσ )
2λi τ
+
e>
i α >
λi ei ȳσ +γqh(ȳ)
J(ȳ)dȳ,
where J(ȳ) is Jacobian such that,
J(ȳ) =
1
e11 σd+1
ȳ1
..
.
e1k 1
σd+1 ȳ1
···
..
.
···
1
ek1 σd+k
ȳk
..
.
1
ekk σd+k
ȳk
e11
..
= .
e1k
k
k
∏ 1
∏
1
=
σ ȳ
σ ȳ
i=1 d+i i
i=1 d+i i
· · · ek1
..
..
.
.
· · · ekk
Therefore,
k
∏
ui (τ, zi0 )
e>
i Rd ei
λi
≤
∫ ∏
k
i=1
K i (zi0 − ỹi , τ )
e>
i Rd ei
λi
G(ỹ1 , · · · , ỹk )dỹ
i=1
=
∫ ∏
k (
i=1
1
√
2πτ
i Rd ei
) e>
λ
i
e
−
2
(zi0 −e>
i ȳσ )
2λi τ
+
e>
i α >
λi ei ȳσ +γqh(ȳ)
1
σd+i ȳi
dȳ
(
)
1
1
Let C(t, y1 , · · · , yk ) = V t, σd+1
ln y1 , · · · , σd+k
ln yk .
Then, it is clear that ps (BT ; q) = C(t, y).
s
p (BT ; q) =
(
Define yσ :=
ln y1
σd+1 , · · ·
y1
, σlnd+k
)>
(
C(t, y) = V t,
1
σd+1
ln y1 , · · · ,
1
σd+k
)
ln yk .
>
. From the definition of V (t, z1 , · · · , zk ) = V 0 (t, e>
1 z, · · · , ek z), it
16
(
)
1
1
>
holds V t, σd+1
ln y1 , · · · , σd+k
ln yk := V 0 (t, e>
1 yσ , · · · , ek yσ ). We can calculate in continuation,
ps (BT ; q)
=
=
>
V 0 (t, e>
1 yσ , · · · , ek yσ )
k
∑
ln U i (t, e>
i yσ )
e>
i Rd ei
γqλi
=
k
∑
(
ln e
e> α
− >i
e Rd ei
i
>
2
1 (ei α)
e>
i yσ − 2 e> R e
i
d
i
γqλ
τ− > i
e Rd ei
i
∫τ
0
i Rd ei
) e>
γqλ
c (T −s)ds i
i
u (τ, e>
i yσ )
i
i=1
i=1
)
k (
>
2
∫
e>
e>
∏
i α e> y − 1 (ei α) τ − τ ci (T −s)ds
i Rd ei
− γqλ
i
>
σ
i
2 γqλi
γqλi
0
i
= ln
e
u (τ, ei yσ )
i=1
= ln
)
k (
e> α
(e> α)2
e>
∏
i Rd ei
− i e> y − 1 i
τ
γqλi
e γqλi i σ 2 γqλi ui (τ, e>
y
)
i σ
i=1
≤ ln
k
∏


e>
i α
− γqλ
i

e
>
2
1 (ei α)
e>
i yσ − 2 γqλi
τ

∫ (
√
i=1

= ln
k
∏

2
(e>
i α)
− 12 γqλ
i

e
τ

∫ (
i=1
= ln
k
∏


∫ (
i=1
= ln
k
∏


∫ (
i=1
1
√
2πτ
1
√
2πτ
1
√
2πτ
i Rd ei
) e>
λ
1
2πτ
i Rd ei
) e>
λ
i
e
i Rd ei
) e>
λ
i
e
e
(e> (yσ −ȳσ +τ α))2
− i
2λi τ
i
1
e>
i α (e> y −e> ȳ )+γqh(ȳ)
− λ
σ
σ
i
i
i
1
σd+i ȳi
e
i Rd ei
) e>
λ
e>
i α e> ȳ +γqh(ȳ)
+ λ
σ
i
i
σd+i ȳi
(e> yσ −e> ȳσ )2
− i 2λ τi
i
>
2
(e> yσ −e>
i ȳσ +τ ei α)
− i
2λi τ
i
(e> yσ −e> ȳσ )2
− i 2λ τi
i
1 
 γq

dȳ 
1 
 γq

dȳ 
1
 γq
1
+γqh(ȳ)
σd+i ȳi
dȳ
1
 γq
+γqh(ȳ)
1
σd+i ȳi
dȳ
Here,
k
∑
i=1
)2
1 ( >
ei (ȳσ − yσ − τ α)
γqλi τ
=
k
∑
i=1
1
(ȳσ − yσ − τ α)> ei e>
i (ȳσ − yσ − τ α)
γqλi τ


e1,i
k
∑
1  . 
=
 ..  (e1,i , · · · , ek,i )
λ
i=1 i
ek,i


e1,i e1,i · · · e1,i ek,i
k
∑
1 

..
..
..
=


.
.
.
λ
i
i=1
ek,i e1,i · · · e1,i ek,i
k
∑
1
ei e>
i
λ
i
i=1
= E > Λ−1 E = R̃−1
Therefore,
∏
1
ln
γq i=1
k
ps (BT ; q)
≤
(
1
√
2πτ
=
∏ 1
1
√
ln
γq i=1 2πτ
=
∏
1
ln
γq i=1
k
k
=
k
∏
(
i
e−
(ȳσ −yσ −τ α)> R̃−1 (ȳσ −yσ −τ α)
+γqh(ȳ)
2τ
∏k
i=1
e>
i Rd ei
λi
1
√
2πτ
√
1
ln
2πτ
γq i=1
i Rd ei ∫
) e>
λ
∫
σd+i ȳi
dȳ
∏k √
(ȳσ −yσ −τ α)> R̃−1 (ȳσ −yσ −τ α)
(2π)k/2 i=1 λi τ
+h(ȳ)
2τ
e−
dȳ
∏k √
k/2
(2π)
λ
τ
σ
ȳ
i
d+i i
i=1
i Rd ei
) e>
λ
e>
i Rd ei
λi
1
i
((2π)
−1 √
∫
λi
k/2
∫
k √
∏
1
γq
λi τ )
i=1
(2π)
1
e
√
λi τ σd+i ȳi
i=1
∏k
k/2
−
(ȳσ −yσ −τ α)> R̃−1 (ȳσ −yσ −τ α)
+h(
2τ
(ȳσ −yσ −τ α)> R̃−1 (ȳσ −yσ −τ α)
1
+h(ȳ)
2τ
e−
dȳ
√
∏
k
k/2
(2π)
λi τ σd+i ȳi
i=1
17
where g is multivariate lognormal density.
Since pb (BT ; q) = −ps (−BT ; q), the utility indifference buy price is given as follows,
pb (BT ; q) = −
k
∑
i=1
[
]
i
i
1
ln E e−γqλi bi h (ỸT ) .
γqλi
(20)
Furthermore, the equilibrium price is deduced from above equation as follows,
p∗ (BT ; γ) =
=
=
]
[
i
i
1
ln E eγqλi bi h (ỸT )
q↓0
q↓0
γqλi
i=1
[
( )
]
γqλi bi hi (ỸTi )
i
i
k
∑
1 E λi bi h ỸT e
]
[
lim
i
i
q↓0
λ
E eγqλi bi h (ỸT )
i=1 i
lim ps (BT ; q) = lim
k
∑
k
∑
[ ( )]
bi E hi ỸTi .
(21)
i=1
4.1
Numerical Example
In this section, we give a numerical example which shows the situation described in the Theorem 3.2. We
consider the random endowment BT such that,
BT := h1 (YT1 ) + · · · + h3 (YT3 ),
where {hi (·); i = 1, 2, 3} is the pay-off function of put option. Basic parameters required to calculate
(??)(20)(21) are given in Table 1. In this case, there are three intradable assets. The price of the random
endowment on the three intradable assets are given by the other three tradable assets. The exercise prices
of each put option are common. We calculate the utility indifference price of this random endowment by
using the results (??)(20)(21), where we use Monte Carlo simulation through 500,000 sample paths to
calculate expectation. We assume that there are four sellers of the random endowment with risk-aversion
0.1, 0.5, 1.0, and 2.0 and there are three buyers of the random endowment with risk-aversion 0.1, 0.3 and
1.5.
The result is given in Table 2. Each column shows the utility indifference price for corresponding
risk-aversion and the quantity of the random endowment. We can see that the utility indifference sell
price is increasing with the quantity and the risk-aversion. On the other hand, utility indifference buy
price is decreasing with the quantity and the risk-aversion. Figure 1 shows this situation more clearly.
The lines on this figure made by the Bezier interpolation. Furthermore, from this result, we can construct
the aggregate demand curve and the aggregate supply curve. In fact, when we calculate each investor’s
selling quantities of the random endowment to some utility indifference sell price and sum up these
quantities, the summed quantity represents the market aggregate supply of the random endowment for
this utility indifference sell price. After we do this calculation for all utility indifference sell prices, we can
construct the aggregate supply curve. From Figure 1, we can easily estimate that the aggregate supply
curve will be increasing with the quantity of the random endowment. Likewise, the aggregate demand
curve can be deduced, and the form will be decreasing with the quantity. The fact that the form of the
aggregate supply curve is increasing and the form of the aggregate demand curve is decreasing matches
to our intuition of economics. However, the intersection of the supply curve and demand curve is the
point where the quantity of the random endowment is zero. Therefore, the equilibrium is zero for some
positive equilibrium price (27.31 in this numerical example). Theorem 3.2 claims such a situation.
18
Table 1: Market Parameters
assets number
6
price
volatility
drift
100
0.25
0.05
tradable assets interest rate expiry
3
0.05
1
tradable asset
100
100
100
0.3
0.35
0.3
0.07
0.09
0.12
correlation matrix
1
0.96 0.95 0.94 0.86
0.96
1
0.95 0.94 0.85
0.95 0.95
1
0.94 0.86
0.94 0.94 0.94
1
0.86
0.86 0.86 0.86 0.86
1
0.85 0.85 0.85 0.85 0.85
exercise price payoff type
100
put option
intradable asset
100
100
0.3
0.3
0.12
0.12
0.85
0.85
0.85
0.85
0.85
1
Table 2: Utility indifference prices
quantity
0
1
2
3
4
5
6
7
8
9
9.9
0.1
27.3199
33.1102
40.4708
48.1325
55.7899
62.8009
69.3489
75.4546
81.414
86.849
90.6584
utility indifference sell price
0.5
1
2
27.3199
27.3199
27.3199
62.9274
91.5274
126.672
91.2545
125.633
160.014
112.077
147.066
174.636
123.873
155.735
184.607
137.994
168.231
191.151
144.735
174.035
195.016
151.537
180.033
198.685
157.430
182.150
194.370
162.681
185.395
202.827
165.611
189.259
203.211
19
utility indifference buy price
0.1
0.3
1.5
27.3199
27.3199
27.3199
23.0138
17.4881
7.06278
19.8002
12.8162
4.01197
17.4197
10.1038
2.78318
15.548
8.31115
2.12865
14.05 5
7.06659
1.72495
12.8385
6.14092
1.45109
11.7399
5.40671
1.24903
10.883
4.85598
1.09712
10.1240
4.38175
0.97976
9.49419
4.04483
0.89277
Figure 1: Utility indifference prices of each investor
250
sell
sell
sell
sell
buy
buy
buy
200
price
price
price
price
price
price
price
1
2
3
4
1
2
3
Price
150
100
50
0
0
1
2
3
4
5
Quantity
6
7
8
9
10
“Sell price 1” depicts that the relation between the price of the random endowment and the supply of
the random endowment by the investor with risk-aversion 0.1. Likewise, “sell price 2”, “sell price 3” and
“sell price 4” depicts this relation by the investor with risk-aversion 0.5, 1 and 2, respectively. And “buy
price 1”, “buy price 2” and “buy price 3” depicts that the relation between the price and the demand
by the investor with risk-aversion 0.1, 0.3 and 1.5.
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