Discussion Papers In Economics
And Business

Utility Indifference Pricing in an Incomplete Market Modelwith Incomplete Information

Kazuhiro TAKINO
Discussion Paper 07-46

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANUtility Indifference Pricing in an Incomplete Market Modelwith Incomplete Information

Kazuhiro TAKINO
Discussion Paper 07-46

December 2007

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANUtility Indifference Pricing in an Incomplete Market Modelwith Incomplete Information∗
Kazuhiro TAKINO†
December, 2007

Abstract
In this article, we consider a derivative pricing model for the stochastic volatility model
under an incomplete information. The incomplete information in our works, supposes that thetrue value of the drift for the stock price process is a random variable, investors only havean information of its distribution. This is more practical financial market than the situation
with knowledge of the drift. There are many studies about the dynamic portfolio optimization
problem under the incomplete information. In that situation, the corresponding problem becomes a easy to treat by Separating Principle and Bayesian updating formula. We apply thesearguments to the utility indifference price approach, and present pricing method taken into account the incomplete information. On the other hand, Sircar and Zariphopoulou (2005) gives
bounds and asymptotic approximations for the indifference prices in the stochastic volatility
model. In them works, bounds include the drift parameter for the underlying price process.
However, in practice, it is able to observe the drift parameter by estimation only. Therefore, it
is meaningful to extended to the incomplete information. We derive bounds for the indifference
prices using estimated drift, and the relationship between the buyer’s indifference price andthe seller’s one.

JEL Classification. G11, G13, G14.
Key Words. Incomplete market, Incomplete information, Utility indifference price, Bayesian
updating formula, Super/sub solution for PDEs.

∗ I would like to appreciate Dr. Masamitsu Ohnishi and Dr. Hiroshi Ishijima for helpful suggestions and comments
throughout most of the paper and for their encouragement.
† Graduate School of Economics, Osaka University 1-7, Machikaneyama, Toyonaka, Osaka, 560-0043, Japan. Email: dg033tk@mail2.econ.osaka-u.ac.jp

1

Introduction

In this article, we consider a pricing model for an incomplete market model under an incomplete
information. As typical example of the incomplete market model, we work the stochastic volatility
model. The financial market in this paper, consists of the risk-free asset, one risky asset whose
volatility drives stochastically over the time horizon and a derivative written on the risky asset.
The incomplete information in our works, implies that investors don’t have knowledge for a true
value of the drift parameter of the risky-asset, and they only have an information of the distribution
for that. Therefore, one can capture the value of the drift parameter only by estimation it. Thissituation is more close to the practical market than one with precise knowledge of the drift.
Under the incomplete information, there are many works for the decision of the dynamic optimal portfolio strategy based on the utility maximization up to now. ([Detemple, 1986], [Dothan
and Feldman 1986], [Detemple and Sheldan 1991], [Gennote, 1986], [Ishijima, 1999]) The dynamic
portfolio optimization problem under the incomplete information, requires that dynamic portfolio
selection and estimation of the drift have to be done simultaneously. However, Separating Principle allows us to treat optimization and estimation separately. This changes from the optimization
problem includes uncertain drift to the problem taken into account the estimation for it. On theother hand, the utility indifference pricing approach have been studied one of pricing methods inthe incomplete market such the stochastic volatility model ([Grasselli and Hurd, 2005], [Sircar andZariphopoulou, 2005]), the financial model with constraints ([Hodges and Neuberger, 1989]) and soon. The utility indifference price criterion is defined to determines the price to be equal maximized
expected utilities for two different investment opportunities. The one is the problem without any
claims, the another one is the problem with some claims. This pricing method is arrowed us to
use various market models, although it requires us to solve expected maximization problems. So
we apply studies of the portfolio optimization under the incomplete information to the indifference
pricing approach, then we derive the price of derivatives under such circumstances.
Its procedure follows: At first, we set a financial market model with stochastic volatility under
the incomplete market model. As mentioned above, we assume that the drift parameter of underlying asset of the derivative is not clear (i.e., a random variable), but investors have an informationof the distribution of it. Next, we consider expected utility maximization problems for a portfolio
consists of both the risk-free asset and such a risky-asset, and the derivative. In this time, we solve
it using Separating Principle. The investor, under incomplete information, dynamically determine
an optimal portfolio with estimating the drift of the stock price process. The separating principle
allows the investor to treat them separately. Furthermore, Theorem 12.7 in Lipster and Shiryaev(2001) gives a dynamics of the estimator of the drift. This is called Bayesian updating formula.
These makes the optimization problem with estimation of the drift easy to solve.
We apply Hamilton-Jacobi-Bellman (HJB) equation to solve optimization problems. In ourworks, partial differential equations obtained by related HJB equations, which are non-linear type.
So we cannot explicitly represent solutions of optimization problems. Instead, we give bounds forsolutions of these problems as Sircar and Zariphopoulou (2005), then we have a bound of utility
indifference price in our financial market model. These bounds take into account the random drift
term, that is, they include estimated drift. Furthermore we derive PDEs satisfied by the indifference
prices, then we describe the relationship between the buyer’s indifference price and seller’s one byfinding super/sub solution of these PDEs.
This paper is organized as follows: In the next section, we introduce a financial market modelwith stochastic volatility model with the incomplete information. In Section 3, we introduce estimation of random drift and derive the improvement of the estimation for the drift, i.e., Bayesian1

updating formula, with a change of measure. In Section 4, we examine expected utility maximization problems, and give solutions for these. In Section 5, we derive bounds of the utility
indifference prices by using results of Section 4, and we show that bounds include the estimationof the drift instead of the true drift value. Furthermore, we provide the relationship between thebuyer’s indifference price and seller’s one. As end of the paper, we give conclusions in our works.

2

Financial Market

In this section, we set a financial market model. We first define risk-free/risky asset price processes
in a stochastic volatility environment, then we introduce an admissible portfolio strategies, andderive it’s related wealth process.

2.1

Traded Assets

Let us consider the following financial market. There exist one risk free asset (typically the bankaccount) and one risky asset (typically stock). At first, as for the bank account, we assume that
one dollar in the bank account at time 0 results in
B(s) = 1
at time s ∈ [0, T ] where T > 0 is the time horizon, that is, we assume that the risk free rate is equal
to zero for simplicity without loss of generality, which implies that dB(s) = 0.
Next, we set the stock price process with a stochastic volatility as follows.
dS(s) = S(s){µds + σ(s, Y (s))dW1 (s)},S(t) = S,(2.1)

where Y is a certain state variable driven byp
1 − ρ2 dW2 (s)},Y (t) = Y,
(2.2)
£ 4¤
for (0 6)t 6 s 6 T , where µ is a random variable on (Ωµ , Fµ , ν) with E µ < ∞, E is an
expectation under P , P is defined later. Also, ν ◦ µ−1 is normally distributed with mean m0 andvariance Γ0 . And then, W = (W1 , W2 )> is two-dimensional standard Brownian motion on a filteredprobability space (ΩW , {FW,s ; s > 0}, φ), where FW,s is generated by informations of W up to s. In
this time, we can construct an filtered probability space (Ω, {Fs ; t > 0}, P ), where Ω = Ωµ × ΩW ，
Fs = Fµ ⊗ FW,s ，P = ν ⊗ φ. Furthermore, we assume that investors only have informations
obtained from the stock price up to now. That is, the information available for them is describedby Gt = σ(S(u); 0 6 u 6 t), with Gt ⊂ Ft .
We would price a European type claim already mentioned above, and it’s payoff function is supposed to g(T ) := g(T, S(T ), Y (T )). In the next section we consider utility maximization problems.
In order to ensure the existence of solutions for such problems, we set the following assumption.
dY (s) = a(s, Y (s))ds + b(s, Y (s)){ρdW1 (s) +Assmption 2.1.
• The volatility function σ(·) and the diffusion coefficient b(·, ·) are smoothand, bounded above and below away from zero.
• The drift coefficient a(·, ·) in (2.2) is Lipschitz continuous on [0, T ] × R.
• The payoff function g(T ) is a GT -measurable.

2

2.2

Wealth Process

We denote by π(s) the amount of money held in stock at time s. So that, if the investor have a
wealth X at time s, then the amount of money invested in the bank account is X − π(s). The
negative value of π(·) usually means that he/she is short-selling the stock. It is assumed that π(·)
is an adapted process, i.e., π(s) is Gs -measurable for any s ∈ [t, T ]. This assumption means thatthe investor cannot foresee the future, so π(s) is determined from the information up to time s. We
also require another technical conditions. We give them in the following definition.
Definition 2.1 (Admissible). We say that the portfolio
strategy π(s) is admissible,
if it is Gs hRi
T 22measurable, and satisfies the integrability condition E t σ (s, Y (s))π (s)ds < +∞ and the selffinancing condition. A denotes the set of all admissible policies. We also denote by X the wealthprocess (or portfolio process) corresponding to an admissible strategy π ∈ A. The self-financing
condition is defined by the following equation:dX(s) =

X(s) − π(s)
π(s)
π(s)
dS(s) +
dB(s) =
dS(s),
S(s)
B(s)
S(s)

(2.3)

for t 6 s 6 T .
Substituting (2.1) into (2.3), we have the wealth equationdX(s) = π(s){µds + σ(s, Y (s))dW1 (s)}.

3

(2.4)

Incomplete Information and Estimation for Drift

In this section, we consider the estimation of the drift µ. In order to use the indifference priceapproach, we examine utility maximization problems for two different investment opportunities
respectively. The one is the problem without a claim, the another one is the problem with a claim.
We suppose that the preference of the market participant is an exponential type, i.e., the utility
function is u(x) = −e−γx where γ is a risk-aversion parameter. Therefore, the former problem ispresented byhi
U (t, X(t), Y (t)) = sup Et −e−γX(T ) ,(3.1)
π(t)∈Aand the latter one is formulated ashi
V (t, S(t), X(t), Y (t); η) = sup Et −e−γ(X(T )+ηg(T )) ,(3.2)

π(t)∈A

where η = ±1, where Et denotes the expectation under P conditioned with informations up to t,
and η(= ±1) implies the position of the claim, namely η = 1 is a long position for the claim andη = −1 is short one.
We adopt Separating Principle [7] to solve these optimization problems. The separating principle
allows us to estimate the drift µ and optimize the expected utility separately. Then this method
stands in solving the above problems. In particularly, as shown latter, we utilize the continuous
Bayesian updating formula of Lipster-Shiryaev [14].
As mentioned above, the investors cannot know a true value of the drift µ. Given an information
Gt , the investor estimates µ asm(t) := E[µ|Gt ],¯ ih2¯Γ(t) := E (µ − m(t)) ¯ Gt ,

3

(3.3)
(3.4)

where Γ(t) implies the estimation error. From Theorem 12.7 in [14], using infinitesimal observations
dS(s), we can improve the estimation for µ and its error as½¾
Γ(s) dS(s)dm(s) := − 2− m(s)ds ,
(3.5)
σ
S(s)
µ
¶2
Γ(s)
dΓ(s) := −
ds
(3.6)
σ
for t 6 s 6 T . Now we set a Doleans-Dade exponential as follows
µ
¶
Z
Z s
Z s
ª
1 s© 2E(θ, ζ)(s) = exp −θ (u) + ζ 2 (u) du −θ(u)dW1 (u) −ζ(u)dW2 (u) ,2 0
00
for t 6 s 6 T , where θ(u) :=(3.7)µ − m(u)ρµand ζ(u) := p.
2σ(u, Y (u))
1 − ρ σ(u, Y (u))

Definition 3.1. We assume that (3.7) is a P -martingale. Then we define an equivalent martingale
measure (EMM) P̃ by¯dP̃ ¯¯(3.8)¯ = E(θ, ζ)(s).
dP ¯
GsFrom Girsanov theorem,µW̃ (s) :=W̃1 (s)W̃2 (s)

¶

µ
:=

Rs
¶
W̃1 (s) + R0 θ(u)dusW̃2 (s) + 0 ζ(u)du(3.9)

is a two-dimensional standard Brownian motion under P̃ . Therefore, under measure P̃ , the stock
price process, it’s variance process and the wealth process introduced in the above, are rewritten
the form taken into account of Bayes updating. That isdS(s)= m(s)ds + σ(s, Y (s))dW̃1 (s),
S(s)
·
¸
³
´
p
ρb(s, Y (s))m(s)dY (s) = a(s, Y (s)) +ds + b(s, Y (s)) ρdW̃1 (s) + 1 − ρ2 dW̃2 (s) ,σ(s, Y (s))nodX(s) = π(s) m(s)ds + σ(s, Y (s))dW̃1 (s) .
Also, m satisfies
dm(s) = −(3.10)(3.11)(3.12)

Γ(s)dW̃1 (s).
σ(s, Y (s))

In the rest of the paper, we discuss based on these dynamics. And also, the value function (3.1)
and (3.2) are written asih(3.13)U (t, m(t), X(t)) = sup Ẽt −e−γX(T ) ,π(t)∈Aih
V (t, m(t), S(t), X(t); η) = sup Ẽt −e−γ(X(T )+ηg(T )) ,π(t)∈A

where Ẽt denotes the expectation under P̃ conditioned with informations up to t.
4(3.14)

4

Optimization Problems

4.1

Operators

To facilitate the presentation, we define the following operators: for any smooth function f ,µ¶ρb(t, Y (t))m(t)hm,Y iA
f := a(t, Y (t)) +fY
σ(t, Y (t))µ¶21ρΓ(t)b(t, Y (t))Γ(t)
1
fmm + b2 (t, Y (T ))fY Y −fmY ,+2 σ(t, Y (t))
2
σ(t, Y (t))µ¶ρb(t, Y (t))m(t)hm,S,Y iA
f := m(t)S(t)fS + a(t, Y (t)) +fY
σ(t, Y (t))
µ
¶2
1
Γ(t)
1
1
+
fmm + σ 2 (t, Y (t))S 2 fSS + b2 (t, Y (T ))fY Y
2 σ(t, Y (t))22ρΓ(t)b(t, Y (t))− Γ(t)S(t)fmS −fmY + ρσ(t, Y (t))b(t, Y (t))S(t)fSY ,σ(t, Y (t))·
12hm,Y iH(fX , fXX , fmX , fXY ) := max π(t)m(t)fX + (π(t)) σ 2 (t, Y (t))fXX2π(t)∈A¸− π(t)Γ(t)fmX + π(t)ρσ(t, Y (t))b(t, Y (t))fXY ,·H

hm,S,Y i

12(π(t)) σ 2 (t, Y (t))fXX2¸2− π(t)Γ(t)fmX + π(t)σ (t, Y (t))SfSX + π(t)ρσ(t, Y (t))b(t, Y (t))fXY ,(fX , fXX , fmX , fSX , fXY ) := max π(t)m(t)fX +π(t)∈A

µ
¶2 2
µ
¶2 2
1
m(t)
fX
1
Γ(t)
fmX
1
f2
J
(fX , fXX , fmX , fXY ) := −−− ρ2 b2 (t, Y (t)) XY
2 σ(t, Y (t))
fXX
2 σ(t, Y (t))
fXX
2
fXX
Γ(t)m(t) fmX fX
ρb(t, Y (t))m(t) fXY fX
ρΓ(t)b(t, Y (t)) fmX fXY+−+,
σ(t, Y (t))2 fXX
σ(t, Y (t))
fXXσ(t, Y (t))
fXX
µ
¶2 2
µ
¶2 2
1
m(t)
fX
1
Γ(t)
fmX
hm,S,Y iJ(fX , fXX , fmX , fSX , fXY ) := −−2 σ(t, Y (t))
fXX
2 σ(t, Y (t))
fXX
2
2
1 2
f
1
f
Γ(t)m(t)ff
fX fSX
mX X− σ (t, Y (t))S 2 SX − ρ2 b2 (t, Y (t)) XY +− m(t)S(t)
2
2
fXX
2
fXX
σ(t, Y (t)) fXX
fXX
ρb(t, Y (t))m(t) fXY fX
fmX fSXρΓ(t)b(t, Y (t)) fmX fXY−+ Γ(t)S(t)+σ(t, Y (t))
fXXfXX
σ(t, Y (t))
fXXfSX fXY,− ρσ(t, Y (t))b(t, Y (t))S(t)fXXhm,Y i

Lhm,Y i f := Ahm,Y i f +Γ(t)m(t)ρb(t, Y (t))m(t)
fm −
fY ,
2
σ (t, Y (T ))σ(t, Y (t))

Lhm,S,Y i f := Ahm,S,Y i f − m(t)S(t)fS +5

Γ(t)m(t)ρb(t, Y (t))m(t)
fm −
fY ,
2
σ (t, Y (T ))σ(t, Y (t))

¶2 2
Γ(t)
fm
ρΓ(t)b(t, Y (t)) fm fY1
f2−+ ρ2 b2 (t, Y (t)) Y ,σ(t, Y (t))
fσ(t, Y (t))
f
2
f
µ
¶2 2
1
Γ(t)
fm
1
f2
1
f2
Mhm,S,Y i (fm , fS , fY , f ) :=+ σ 2 (t, Y (t))S 2 (t) S + ρ2 b2 (t, Y (t)) Y
2 σ(t, Y (t))
f
2
f
2
f
ρΓ(t)b(t, Y (t)) fm fY
fS fY
fm fS
−
+ ρσ(t, Y (t))b(t, Y (t))S(t).
− Γ(t)S(t)fσ(t, Y (t))
ff
Mhm,Y i (fm , fY , f ) :=

4.2

1
2

µ

Optimizations

Recall that, the utility maximization problem without any claims is
hi
U (t, m(t), X(t), Y (t)) = sup Ẽt −e−γX(T ) .

(4.1)

π∈A

The HJB equation of U is Ut + Ahm,Y i U + Hhm,Y i (UX , UXX , UmX , UXY ) = 0,

(4.2)
−γX(T )U (T, m(T ), X(T ), Y (T )) = −e.

The maximum in (4.2) is achieved atπ∗ = −

m(t)
UX
Γ(t)
UmX
ρb(t, Y (T )) UXY
+ 2
−
.
2
σ (t, Y (t)) UXX
σ (t, Y (t)) UXX
σ(t, Y (t)) UXX(4.3)

Plugging (4.3) into (4.2) gives us the following non-linear partial differential equation (PDE): Ut + Ahm,Y i U + J hm,Y i (UX , UXX , UmX , UXY ) = 0,(4.4)
U (T, m(T ), X(T ), Y (T )) = −e−γX(T ) .

From Section 3 in Pham [16] or Theorem 2.5 in Sircar and Zariphopoulou [17], U is given by
U (t, m(t), X(t), Y (t)) = −e−γX(t) F (t, m(t), Y (t))(4.5)

where F ∈ C1,2,2 ([0, T ]×R×R) is the unique bounded solution of the quasi linear partial differential
equation

µ
¶2
1
m(t)

hm,Y
i

 Ft + L+ Mhm,Y i (Fm , FY , F ),F =2 σ(t, Y (t))
(4.6)



F (T, m(T ), Y (T )) = 1.
Note that (4.6) is obtained by substituting (4.5) into (4.4), and F is a non-negative because U isan indirect utility function of u.
Next we consider the optimization problem with a claim. This problem is formulated as
i
h
(4.7)
V (t, m(t), S(t), X(t), Y (t); η) = sup Ẽt −e−γ(X(T )+ηg(T )) ,π∈A6where g(T ) := g (T, S(T ), Y (T )), and recall that η(= ±1) implies the position of the claim. The
HJB equation of (4.7) is Vt + Ahm,S,Y i U + Hhm,S,Y i (VX , VXX , VmX , VSX , VXY ) = 0,
(4.8)
V (T, m(T ), S(T ), X(T ), Y (T ); η) = −e−γ(X(T )+ηg(T )) .
The maximum in (4.8) is achieved atπ∗ = −

m(t)
VX
Γ(t)
VmX
VSX
ρb(t, Y (T )) VXY
+ 2
−S
−
.
σ 2 (t, Y (t)) VXX
σ (t, Y (t)) VXX
VXX
σ(t, Y (t)) VXX(4.9)

Plugging (4.9) into (4.8) gives us the following PDE:
 Vt + Ahm,S,Y i V + J hm,S,Y i (VX , VXX , VmX , VSX , VXY ) = 0,
(4.10)V (T, m(T ), S(T ), X(T ), Y (T ); η) = −e−γ(X(T )+ηg(T )) .

From Pham [16], V is given by
V (t, m(t), S(t), X(t), Y (t); η) = −e−γX(t) G (t, m(t), S(t), Y (t); η)(4.11)

where G ∈ C1,2,2,2 ([0, T ] × R × R+ × R) is the unique bounded solution of the quasi linear partial
differential equation

µ
¶2
m(t)
1

hm,S,Yi Gt + L+ Mhm,S,Y i (Gm , GS , GY , G),G=
2 σ(t, Y (t))(4.12)−γηg(T )G (T, m(T ), S(T ), Y (T ); η) = e.
PDE (4.12) is also obtained by substituting (4.11) into (4.10), and G must be a non-negative if V
maintain a property of the indirect utility function for u.

5

Indifference Prices and Bounds

5.1

Bounds for the Utility Indifference Prices

In this section, we provide utility indifference prices and their bounds. We derive bounds of indifference prices using the method of super/sub solution of non-linear PDE as Sircar and Zariphopoulou[17]. At first we define the utility indifference price.
Definition 5.1 (Utility Indifference Pirce). The buyer’s utility indifference price pb (t) :=pb (t, m(t), S(t), Y (t)) at time t is defined as the solution ofU (t, m(t), X(t), Y (t)) = V (t, m(t), S(t), X(t) − pb (t), Y (t); 1) .
While the seller’s utility indifference price ps (t) := ps (t, m(t), S(t), Y (t)) at time t is defined as thesolution ofU (t, m(t), X(t), Y (t)) = V (t, m(t), S(t), X(t) + ps (t), Y (t); −1) .
7

In the previous subsection, we derived solutions for utility maximization problems with orwithout a claim and corresponding PDEs. By consider a super/sub solutions for these PDEs, we
introduce bounds of the buyer’s/seller’s indifference price respectively. Before that, we additionallyconstruct an EMM. Define
µ
¶
Z s
Z
1 s 2
Ẽ(λ, 0)(s) = exp −λ (u)du −λ(u)dW̃1 (u)
(5.1)
2 0
0
where λ(s) :=m(s). From Assumption 2.1, this is a P̃ -martingale.
σ(s, Y (s))

Definition 5.2. We define an equivalent measure Q according to¯
dQ ¯¯= Ẽ (0, λ) (s).
dP̃ ¯Gs
In this time,µM (s) :=

M1 (s)
M2 (s)

¶

µ
:=

Rs
¶
W̃1 (s) + 0 λ(u)duW̃2 (s)(5.2)

is a two-dimensional standard Brownian motion under Q. Under this, dynamics becomedS(s)= σ(s, Y (s))dM1 (s),
S(s)

(5.3)

´
p
dY (s) = a(s, Y (s))ds + b(s, Y (s)) ρdM1 (s) + 1 − ρ2 dM2 (s) ,dm(s) =³

Γ(s)m(s)Γ(s)−
dM1 (s).
σ 2 (s, Y (s) σ(s, Y (s))

(5.4)
(5.5)

Proposition 5.1. The solution F of (4.6) has a bound. That isF̄ (t, m(t), Y (t)) < F (t, m(t), Y (t)) < F̂ (t, m(t), Y (t))(5.6)

where F̄ and F̂ are given byÃ"F̄ (t, m(t), Y (t)) = exp EtQ"F̂ (t, m(t), Y (t)) = EtQ

Ã

1
−
2

1
exp −
2

Z Tµ
t

Z Tµ
t

m(s)
σ(s, Y (s))m(s)σ(s, Y (s))

#!

¶2

ds

,

!#

¶2

ds

(5.7)
,

where EtQ denotes the expectation under Q conditioned with informations up to time t.
Proof. Let us consider a lower bound. By F > 0 and ρ2 < 1, it holds thatFt + Lhm,Y i

¶2
µ
¶2 2
m(t)
1
Γ(t)
Fm
1
F2
F+
+ b2 (t, Y (t)) Y
σ(t, Y (t))
2 σ(t, Y (t))F
2F
ρΓ(t)b(t, Y (t)) Fm FY.
−
σ(t, Y (t))
F

1
F <
2

µ

8

(5.8)

So F is a super solution of F̄ which solves toF̄t + Lhm,Y i

¶2
µ
¶2 2
m(t)
1
Γ(t)
F̄m
1
F̄ 2
F̄ +
+ b2 (t, Y (t)) Y
σ(t, Y (t))
2 σ(t, Y (t))2F̄F̄ρΓ(t)b(t, Y (t)) F̄m F̄Y−,
σ(t, Y (t))
F̄

1
F̄ =
2

µ

(5.9)

and F̄ (T, m(T ), Y (T )) = 1. That is
F (t, m(t), Y (t)) > F̄ (t, m(t), Y (t)) .
We transform F̄ as
F

1,2,2

where φ ∈ C(5.10)FF̄ (t, m(t), Y (t)) = eφ (t,m(t),Y (t)) ,F(5.11)
F([0, T ] × R × R+ ) and φ (T, m(T ), Y (T )) = 0. Then φ solves to

µ
¶2
Γ(t)
1

F
hm,Y
i
F

 φt + Lφ =,2 σ(t, Y (t))
 Fφ (T, m(T ), Y (T )) .

Therefore, by Feynman-Kac formula and under Q, we have
"
¶2 #
Z µ
m(s)
1 T
Q
F
ds .
φ (t, m(t), Y (t)) = Et −2 tσ(s, Y (s))(5.12)(5.13)The lower bound is obtained from (5.10), (5.11) and (5.13).
We turn to derive an upper bound of F . From (4.6) it holds thatFt + Lhm,Y i F >

1
2

µ

Γ(t)
σ(t, Y (t))¶2
F.

It is following that F is a sub solution of F̂ , that is, F < F̂ . Where F̂ is a solution of

µ
¶2
1
Γ(t)


F̂ ,
 F̂t + Lhm,Y i F̂ =2 σ(t, Y (t))
F̂ (T, m(T ), Y (T )) = 1.
Again, by using Feynman-Kac formula to (5.14), we obtain the upper bound F̂ as
"
Ã
¶2 !#
Z µ
1 T
m(s)
Q
F̂ (t, m(t), Y (t)) = Et exp −ds.
2 t
σ(s, Y (s))

On the other hand, the bound of G is given by the following proposition.

9(5.14)

Proposition 5.2. The solution G of (4.12) has a bound. That isḠ (t, m(t), S(t), Y (t); η) < G (t, m(t), S(t), Y (t); η) < Ĝ (t, m(t), S(t), Y (t); η)(5.15)

where Ḡ and Ĝ are given by
¶2 #!
Z µ
1 T
m(s)
ds
,
Ḡ (t, m(t), S(t), Y (t); η) = exp−γηg(T ) −2 tσ(s, Y (s))
"
Ã
¶2 !#
Z µ
m(s)
1 T
Q
Ĝ (t, m(t), S(t), Y (t); η) = Et exp −γηg(T ) −ds,
2 t
σ(s, Y (s))
Ã

"

EtQ

(5.16)

where η = ±1.
Proof. Let us consider a lower bound. By G > 0 and ρ2 < 1, it holds that
µ
¶2
µ
¶2 2
1
m(t)
1
Γ(t)
Gm
1
G2
hm,S,Y i
G++ σ 2 (t, Y (t))S 2 (t) S
Gt + LG<
2 σ(t, Y (t))
2 σ(t, Y (t))
G
2
G
1 2
G2Y
Gm GS
ρΓ(t)b(t, Y (t)) Gm GY
GS GY+ b (t, Y (t))− Γ(t)S(t)−− ρσ(t, Y (t))b(t, Y (t))S(t).
2G
G
σ(t, Y (t))
GG
(5.17)So G is a super solution of Ḡ which solves to
µ
¶2
µ
¶2 2
1
1
m(t)
Γ(t)
Ḡm
1
Ḡ2
Ḡ +
Ḡt + Lhm,Y i Ḡ =+ σ 2 (t, Y (t))S 2 (t) S
2 σ(t, Y (t))
2 σ(t, Y (t))
2
Ḡ
Ḡ
2
1
Ḡ
Ḡm ḠSρΓ(t)b(t, Y (t)) Ḡm ḠY
ḠS ḠY+ b2 (t, Y (t)) Y − Γ(t)S(t)−− ρσ(t, Y (t))b(t, Y (t))S(t),2
σ(t, Y (t))
Ḡ
Ḡ
Ḡ
Ḡ
(5.18)and Ḡ (T, m(T ), S(T ), Y (T ); η) = e−γηg(T ) . That is
G (t, m(t), S(t), Y (t); η) > Ḡ (t, m(t), S(t), Y (t); η) .
We transform Ḡ as

G

Ḡ (t, m(t), S(t), Y (t); η) = eφ (t,m(t),S(t),Y (t);η) ,(5.19)(5.20)

where φG ∈ C1,2,2 ([0, T ] × R × R++ × R+ ) and φG (T, m(T ), Y (T ); η) = −γηg(T ). Then φG solves
to

µ
¶2
1
Γ(t)

G
hm,S,YiG φt + Lφ =,2 σ(t, Y (t))(5.21)
 G
φ (T, m(T ), Y (T ); η) = −γηg(T ).
Therefore, by Feynman-Kac formula and under Q, we have
"
¶2 #
Z µ
1 T
m(s)
Q
G
φ (t, m(t), S(t), Y (t); η) = Et −γηg(T ) −ds .
2 t
σ(s, Y (s))
The lower bound is obtained from (5.19), (5.20) and (5.22).
10(5.22)

We turn to derive an upper bound of G. From (4.12) it holds thatGt + Lhm,S,Y i

1
G>
2

µ

Γ(t)
σ(t, Y (t))¶2
G.

It is following that G is a sub solution of Ĝ, that is, G < Ĝ. Where Ĝ is a solution of

µ
¶2
Γ(t)
1

hm,S,Y iĜ,Ĝ = Ĝt + L
2 σ(t, Y (t))
Ĝ (T, m(T ), Y (T )) = 1.

(5.23)

Again using Feynman-Kac formula to (5.23), we obtain the upper bound Ĝ as
"
Ã
¶2 !#
Z µ
1 T
m(s)
Q
ds
.
Ĝ (t, m(t), S(t), Y (t); η) = Et exp −γηg(T ) −2 tσ(s, Y (s))

From the above calculations, we derive bounds for indifference prices. Recall that pb (t, m(t), S(t), Y (t))
and ps (t, m(t), S(t), Y (t)) respectively denote the buyer’s indifference price and the seller’s one attime t. By (5.1), (5.6) and (5.15), it holds that
1F̄ (t)1F̂ (t)ln< pb < ln,γ Ĝ(t; 1)γ Ḡ(t; 1)
1 Ḡ(t; −1)
1 Ĝ(t; −1)ln< ps < ln,γγF̄ (t)F̂ (t)(5.24)

where F (t) ≡ F (t, m(t), Y (t)) and G(t; η) ≡ G (t, m(t), S(t), Y (t); η). Substituting (5.7) and (5.16)
into both of (5.24), we obtain the following theorem.
Theorem 5.1. (i) The buyer’s indifference price of the claim g(T ), has a bound as follows.
Ã
"
¶2 #
Z µ
1
1 T
m(s)
Q
Et −
ds
γ
2 t
σ(s, Y (s))"Ã
¶2 !#!
Z µ
1 T
m(s)
Q
− ln Et exp −γg(T ) −ds
2 t
σ(s, Y (s))
1γÃ< pb (t) <
"
Ã

¶2

!#

m(s)
ds
σ(s,
Y (s))
t
"
¶2 #!
Z µ
1 T
m(s)
Q
− Et −γg(T ) −ds
2 t
σ(s, Y (s))

ln EtQ exp −

1
2

Z Tµ

11

(5.25)

(ii) The seller’s indifference price lies in a bound as follows.
Ã
"
¶2 #
Z µ
1
1 T
m(s)
Q
Et γg(T ) −dsγ
2 tσ(s, Y (s))"Ã
¶2 !#!
Z µ
m(s)
1 T
Q
ds
− ln Et exp −2 tσ(s, Y (s))
1γÃ< ps (t) <"Ã
ln EtQ exp γg(T ) −
"
− EtQ

−

1
2

Z Tµ
t

1
2

Z Tµ
t

m(s)
σ(s, Y (s))m(s)σ(s, Y (s))
#!
¶

!#

¶2

(5.26)ds

2

ds

From Theorem 5.1, under the incomplete information, we observe that bounds of utility indifference prices include estimated drift term. Compared to Theorem 3.1 in [17], it is replaced µ withm(s) = Es [µ].

5.2

Utility Indifference Pricing Equation

It is not clear the relationship of the buyer’s indifference price and the seller’s one from (5.25)
and (5.26) under the assumption that the buyer of the derivative and seller have same preference(i.e., the risk-aversion parameters for the buyer and seller are equal). In this section, we derive
PDEs satisfied by the indifference prices1 , then we describe the relationship between the buyer’s
indifference price and seller’s one by finding super/sub solution of these PDEs.
Assmption 5.1. We assume that FY /F is smooth and bounded, where F is given by (4.5).
Proposition 5.3. (i) The buyer’s indifference price pb is a unique and bounded solution of thepricing equation
FY b
1
pbt + Lhm,S,Y i pb + (1 − ρ2 )b2
p = b2 γ(1 − ρ2 )(pbY )2(5.27)F Y
2with pb (T ) = g(T ).
(ii) The seller’s indifference price ps is a unique and bounded solution of the pricing equation
pst + Lhm,S,Y i ps + (1 − ρ2 )b2

FY s
1
pY = − b2 γ(1 − ρ2 )(psY )2F
2(5.28)

with ps (T ) = g(T ).
Proof. We only show (i). The second statement is derived by the same way. From Definition 5.1,
(4.5) and (4.11), we havebG(t, m, S, y) = e−γp F (t, m, y).
(5.29)
Substituting (5.29) into (4.12) together with (4.6), it holds that (5.27). The uniqueness and boundedness follow by proof of Theorem 2.6 in [17].
1 Sircar and Zariphopoulou [17] call this PDE the indifference pricing equation.

12

p
FY
From Assumption 2.1 and 5.1, κ(s) := − 1 − ρ2 b(s, Y (s))
is smooth and bounded. Thus,F
it holds that
µ
¶
Z
Z s
1 s 2
M
E (0, κ)(s) := exp −κ (u)ds −κ(u)dM2 (u)(5.30)
2 00
is a Q-martingale.
Definition 5.3. We define an equivalent measure Q̂ according to¯dQ̂ ¯¯¯ = E M (0, κ) (s)dQ ¯Gsfor t 6 s 6 T .
In this time,µM̂ (s) :=M̂1 (s)M̂2 (s)¶µ:=M1R(s)s
M2 (s) + 0 κ(u)du¶(5.31)

is a two-dimensional standard Brownian motion under Q̂. Under this measure, the dynamics of Y
become·¸³´pFYdY (s) = a(s, Y (s)) + (1 − ρ2 )b2 (s, Y (s))
ds + b(s, Y (s)) ρdM̂1 (s) + 1 − ρ2 dM̂2 (s) .
F
Proposition 5.4. The buyer’s indifference price and seller’s price have the following relationship:pb (t) < ps (t)for 0 6 t < T .
Proof. We observe thatpbt + Lhm,S,Y i pb + (1 − ρ2 )b2

FY b
1
pY = b2 γ(1 − ρ2 )(pbY )2 > 0.
F
2

So pb is a sub solution of q, i.e., pb (t, m, S, y) < q(t, m, S, y) where q is a solution of
qt + Lhm,S,Y i q + (1 − ρ2 )b2

FY
qY = 0Fwith q(T ) = g(T ). Under Q̂-measure, Feynman-Kac formula gives the following representation:q(t, m, S, y) = EtQ̂ [g(T )].
On the other hand, since it holds thatpst + Lhm,S,Y i ps + (1 − ρ2 )b2

FY s1
p = − b2 γ(1 − ρ2 )(psY )2 < 0,F Y
2hence ps is a super solution of q, i.e., ps (t, m, S, y) > q(t, m, S, y). Therefore, we havepb (t, m, S, y) < EtQ̂ [g(T )] < ps (t, m, S, y)
as required.
Remark 5.1. The result of Proposition 5.4 is not restricted to the incomplete information model. It
might hold same relationship for other incomplete market models with respect to appropriate EMMs.
13

6

Summary

In this paper, we considered the utility indifference pricing for the derivative written on the stockwith stochastic volatility under incomplete information, i.e., the drift parameter of the stock is
assumed to be a normally distributed random variable. In order to derive the indifference price, weshould solve expected utility maximization problems, although Separating principle and Bayesian
updating formula make them easy to treat. By using HJB equation for optimization problems, we
obtained non-linear PDEs. Since it is so hard to find an explicit solution for such PDEs, we provide
bounds of indifference prices by solve super/sub solutions of corresponding PDEs. Furthermore,
under the incomplete information, bounds of utility indifference prices include estimated drift term
instead the true drift parameter µ. This is the main result in this paper. Also we found therelationship between the buyer’s indifference price and the seller’s one by the indifference pricing
equations.

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15