Indifference Pricing in a One-period Binomial Model - City University ...

Indifference Pricing in a One-period Binomial Model
Hong-Kun Xu
Department of Applied Mathematics
National Sun Yat-sen University
Kaohsiung 80424, Taiwan
E-mail: xuhk@math.nsysu.edu.tw
March 10, 2012
Abstract
In a complete model such as the Black-Scholes model, each option payoff can be perfectly
replicated by a dynamic self-financing trading strategy, and the option price is the cost of this
replication. However, due to stochastic factors, realistic models are usually incomplete: not all
risk of every option can be eliminated by dynamic hedging. Pricing in an incomplete model gives
also rise to another difficulty which is the nonuniqueness: different martingale measures may
lead to different (no-arbitrage) prices. In this lecture, we will consider a one-period binomial
model which consists of a riskless asset and two risky assets of which one is traded and the
other is nontraded (incompleteness thus arises). We will systematically introduce the so-called
utility indifference pricing theory, including properties of indifference prices and strategies of
risk monitoring.
1

1 Introduction
In a complete market, every contingent claim can be perfectly replicated by trading in the traded
assets. That is, the payoff of the claim can be reproduced by a portfolio which consists of the
underlying stocks and bonds, and the unique price of the claim is given by the law of one price,
namely, the initial wealth of funding the replicating portfolio. However, due to frictions such as
transaction costs, nontraded assets and portfolio constraints, most markets are incomplete and
complete markets are an approximation to incomplete ones. In an incomplete market, not every
claim can be perfectly hedged. Many different option prices are consistent with the arbitrage-free
principle, each corresponding to a different martingale measure. Prices are no longer unique.
Pricing in an incomplete is complicated. There are several approaches, for example, superreplication,
minimal martingale measure, and convex risk measures. Utility indifference pricing is a
powerful valuation method in incomplete markets which originates from the premium principle of
equivalent utility of actuarial mathematics.
“The advantages of utility indifference pricing include its economic justification and incorporation
of risk aversion. It leads to a price which is non-linear in the number of units of claim,
which is in contrast to prices in complete markets and some of the alternatives mentioned above.
The indifference price reduces to the complete market price which is a necessary feature of any
good pricing mechanism. Indifference prices can also incorporate wealth dependence. This may
be desirable as the price an investor is willing to pay could well depend on the current position of
his derivative book. Although we concentrate on pricing issues here, utility indifference also gives
an explicit identification of the hedge position. This is found naturally as part of the optimization
problem. Limitations of indifference pricing methodology include the fact that explicit calculations
may be done in only a few concrete models, mainly for exponential utility. Exponential utility
has the feature that the wealth or initial endowment of the investor factors out of the problem
which makes the mathematics tractable but is also a strong assumption. Different investors with
varying initial wealths are unlikely to assign the same value to a claim. Practically, users may not
be satisfied with the concept of utility functions and unable to specify the required risk aversion
coefficient.” (Quoted from [1].)
The purpose of this lecture is to introduce the basics of the exponential utility indifference
pricing methodology for a one-period incomplete model which is composed of one riskless asset
(referred to as bond) and two risky assets (one is referred to as stock and the other is nontraded).
The contents include
• Definition of utility indifference prices and properties.
• A nonlinear indifference price valuation formula.
• Risk monitoring strategies.
• Relative indifference prices.
• Wealth, preferences and numeraires.
[The notes presented below are primarily based on and adapted from [5].]
2

2 Complete One-Period Model
We briefly recall the basics of the complete one-period binomial model.
The model consists of two tradable assets: One riskless, called bond or money market account
(MMA) and which is denoted as B , and one risky, referred to as stock and which is denoted as S.
The two assets are traded at times 0 and T > 0. The bond has a unit initial value (i.e., B 0 = 1)
and offers interest rate of r ≥ 0 for the horizon [0, T ] so that B T = 1 + r. For simplicity, we assume
that r = 0 thus, B T = 1.
The randomness of the risky S is described through a probability space (Ω, P, F), where Ω =
{ω 1 , ω 2 } with p := P(ω 1 ) > 0 and q := 1 − p = P(ω 2 ) > 0, and F = 2 Ω . S T is a random variable on
Ω and S T = S 0 ξ, with
{ u, if ω = ω1 ,
ξ(ω) =
d, if ω = ω 2 ,
where 0 < d < 1 < u which are the no-arbitrage conditions for this model.
Let now C be a contingent claim written on S and with payoff C T . For example, C T = (S T −K) +
if C represents a European call option with expiration time T and strike price of K.
One of the pricing methods is replication, and every claim in a complete model can be replicated.
As a matter of fact, let ϕ = (α, β) be a portfolio, where α is the number of shares of the stock S
and β units of the bond B. In order that ϕ replicate the claim, we must have
αS T (ω) + β = C T (ω), ω = ω 1 , ω 2 .
Solving this system yields the unique replicating portfolio
α = C T (ω 1 ) − C T (ω 2 )
S T (ω 1 ) − S T (ω 2 ) = C T (ω 1 ) − C T (ω 2 )
,
S 0 (u − d)
β = uC T (ω 2 ) − dC T (ω 1 )
.
u − d
By the law of one price, we get the price of the claim, C 0 , is the cost of the portfolio, that is,
C 0 = αS 0 + β = (u − 1)C T (ω 1 ) + (1 − d)C T (ω 2 )
. (2.1)
u − d
This shows that the claim C T can uniquely be hedged by the portfolio (α, β) determined above, and
the risk of writing the claim C T can completely by eliminated by following this hedging portfolio.
If we set
˜p = 1 − d
u − d , ˜q = u − 1
u − d ,
then we can rewrite C 0 in the form
C 0 = ˜pC T (ω 1 ) + ˜qC T (ω 2 ) = Ẽ[C T ]. (2.2)
Here Ẽ is the expectation under the probability ˜P defined by ˜P(ω 1 ) = ˜p and ˜P(ω 2 ) = ˜q. Formula
(2.2) is referred to as the risk-neutral valuation. That is, price is equal to expected payoff under
the risk-neutral probability measure (under the assumption that the interest rate is zero).
3

3 Incomplete One-period Model
In an incomplete market, not all risk can be eliminated. In other words, not every claim can be
replicated by a (dynamic) portfolio. This is the case of most practical markets due to market
frictions and nontraded assets.
3.1 Model Set-up
This is a one-period binomial model in which the trading dates are 0 and T . There are two traded
assets B and S as in the complete model. However, there is another asset Y which is nontraded
and which causes the incompleteness of the model. The riskless asset offers no interest rate so that
B 0 = B T = 1. The risky assets S and Y are given by
S T = S 0 ξ, ξ = ξ u , ξ d , 0 < ξ d < 1 < ξ u ;
Y T = Y 0 η, η = η u , η d , 0 < η d < η u .
The randomness of S and Y are described by the probability space (Ω, P, F), where
Ω = {ω 1 , ω 2 , ω 3 , ω 4 ), P(ω i ) > 0 (1 ≤ i ≤ 4), F = 2 Ω ,
S T (ω 1 ) = S 0 ξ u , Y T (ω 1 ) = S 0 η u , S T (ω 3 ) = S 0 ξ d , Y T (ω 3 ) = Y 0 η u
S T (ω 2 ) = S 0 ξ u , Y T (ω 2 ) = S 0 η d , S T (ω 3 ) = S 0 ξ d , Y T (ω 3 ) = Y 0 η d .
The filtration F coincides with the filtration F S T ,Y T
generated by S T and y T . We use F S T
to
denote the filtration generated by S T .
3.2 Utility Indifference Prices
Let C T be a claim written on S and Y . Due to incompleteness, C T may, in general, have different
(no-arbitrage) prices if different valuation methods are applied. Here we are to introduce utility
indifference prices under the exponential utility:
U(x) = −e −γx , x ∈ R,
where γ > 0. Note that U is twice continuously differentiable, U ′ > 0 and U ′′ < 0 so that U is
strictly increasing and strictly concave (these are the definition of a utility function).
Let ϕ = (α, β) be a portfolio consisting of α shares of the traded risky asset S and β units of
the riskless asset B. Its current value X 0 = x is given by x = αS 0 + β, and its value at time T is
given by
X T = αS T + β = x + α(S T − S 0 ).
[X T is a wealth that can be generated from the initial fortune x.] The value function of the claim
C T (in terms of the exponential utility function U) is defined as
V C T
(x) = sup
α
[ ]
E P [U(X T − C T )] = sup E P −e −γ(X T −C T )
α
[ ]
= e −γx sup E P −e −γα(S T −S 0 )+γC T
. (3.1)
α
4

Definition 3.1. The indifference price of the claim C T = c(S T , Y T ) is defined as the amount ν(C T )
for which the two values V C T
and V 0 , respectively defined in (3.1) for C T and 0, coincide. That
is, ν(C T ) is a solution to the equation
V 0 (x) = V C T
(x + ν(C T )) (3.2)
for all initial wealth x. This definition actually says that the investor is indifferent between paying
nothing and not having the claim C T and paying ν(C T ) now to receive the claim C T at time T .
Remark 3.2. The classical no-arbitrage pricing theory in a complete model gives the price of a
claim C T as C(C T ) = E˜P[C T ] (assuming zero rate of interest), where ˜P is the risk-neutral measure.
That is, price is a linear functional of the expected (discounted) payoff under the (unique) riskneutral
equivalent martingale measure. This is however no longer true in an incomplete model:
Prices of claims in an incomplete market are no longer linear functional of the expected discounted
payoff under a unique equivalent martingale measure. The valuation functional is nonlinear and
the pricing equivalent measure is not unique. We will look for possible formula as follows (recall
we assume zero rate of interest):
ν(C T ) = E Q [C T ], (3.3)
where E Q is a nonlinear valuation functional and Q is an equivalent martingale measure. An
advantage of this formula is that prices are expressed in one measure. However, for (3.3) to hold,
some regularity property must hold.
Remark 3.3. Consider two special cases: (a) C T = c(S T ); (b) C T = c(Y T ). In the case of (a), the
randomness of the nontraded asset Y has no affects on the claim C T at all. Therefore, the classical
risk-neutral valuation method remains applicable and ν(c(S T )) = E˜P[c(S T )]. So the indifference
price is reduced to the no-arbitrage price.
In the case of (b), we further assume that S T and Y T are independent under P so that the value
function V C T
= V c(Y T ) is reduced to
It is also straightforward from (3.1) that
Combining (3.4) and (3.5) gives that
[ ]
V c(Y T ) (x) = e −γx E P [e γc(Y T ) ] sup E P −e −γα(S T −S 0 )
. (3.4)
α
[ ]
V 0 (x) = e −γx sup E P −e −γα(S T −S 0 )
. (3.5)
α
By Definition 3.1, ν(c(Y T ) is the solution to the following system:
V c(Y T ) (x) = V 0 (x)E P [e γc(Y T ) ]. (3.6)
V 0 (x) = V c(Y T ) (x + ν(c(Y T ))
= V 0 (x + ν(c(Y T ))E P [e γc(Y T ) ]
= V 0 (x)e −γν(c(Y T )) E P [e γc(Y T ) ].
It turns out that
e −γν(c(Y T )) E P [e γc(Y T ) ] = 1.
5

Consequently, we get
ν(c(Y T )) = 1 γ log EP [e γc(Y T ) ]. (3.7)
The indifference price is reduced to the classical actuarial valuation principle, the so-called certainty
equivalent value, which is nonlinear in the payoff and uses the historical measure P as the pricing
measure.
Remark 3.4. Consider the case where the claim is decomposed as
C T = c 1 (S T ) + c 2 (Y T ).
In this case, one could be led to a wrongful idea to price this claim as first pricing the claim c 1 (S T )
by the classical no-arbitrage risk-neutral valuation method, and then pricing the claim c 2 (Y T ) by
the actuarial certainty equivalent value principle, and finally put them together as the price of the
claim C T = c 1 (S T ) + c 2 (Y T ). In general,
unless S T and Y T are independent.
3.3 A Nonlinear Pricing Formula
ν(c 1 (S T ) + c 2 (Y T )) ≠ ν(c 1 (S T )) + ν(c 2 (S T ))
Theorem 3.5. Let Q be a equivalent measure under which the traded asset S is a martingale,
and at the same time, the conditional distribution of the nontraded asset, given the traded one, is
preserved with respect to the historical measure P, that is,
Q[Y T |S T ] = P[Y T |S T ]. (3.8)
Let C T = c(S T , Y T ) be the claim to be priced under the exponential preferences with risk aversion
coefficient γ. Then the indifference price of C T is given by
[ ]
1
ν(C T ) = E Q [C T ] = E Q γ log EQ [e γC T
|S T ] . (3.9)
Proof. We prove the formula (3.9) by computing the price ν(C T ) from its definition (3.1) and then
verifying it equal to the right side of (3.9). Set
By definition of the value function, we get
c i = C T (ω i ) = c(S T (ω i ), Y T (ω i )), i = 1, 2, 3, 4.
[ ]
V C T
(x) = e −γx sup E P −e −γα(S T −S 0 )+γC T
α
4∑
= e −γx sup −p i e −γα(S T (ω i )−S 0 )+γC T (ω i )
α
i=1
{
= e −γx sup −e −γαS 0(ξ u−1) (p 1 e γc 1
+ p 2 e γc 2
)
α
}
−e −γαS 0(ξ d−1) (p 3 e γc 3
+ p 4 e γc 4
)
≡ e −γx sup g(α).
α
6

We have
g ′ (α) = γS 0 (ξ u − 1)e −γαS 0(ξ u −1) (p 1 e γc 1
+ p 2 e γc 2
)
The unique solution to the equation g ′ (α) = 0 is
Putting
and noticing
we get the value function
It turns out that
α ∗ =
+ γS 0 (ξ d − 1)e −γαS 0(ξ d −1) (p 3 e γc 3
+ p 4 e γc 4
).
1
γS 0 (ξ u − ξ d ) log (ξu − 1)(p 1 e γc 1
+ p 2 e γc 2
)
(1 − ξ d )(p 3 e γc 3 + p4 e γc . (3.10)
4 )
q = 1 − ξd
ξ u − ξ d , thus 1 − q = ξu − 1
ξ u − ξ d , (3.11)
γα ∗ S 0 (ξ u − 1) = ξu − 1
ξ u − ξ d log (ξu − 1)(p 1 e γc 1
+ p 2 e γc 2
)
(1 − ξ d )(p 3 e γc 3 + p4 e γc 4 )
,
γα ∗ S 0 (ξ d − 1) = ξd − 1
ξ u − ξ d log (ξu − 1)(p 1 e γc 1
+ p 2 e γc 2
)
(1 − ξ d )(p 3 e γc 3 + p4 e γc 4 )
,
V C T
(x) = e −γx g(α ∗ )
{
= e −γx −e −γα∗ S 0 (ξ u−1) (p 1 e γc 1
+ p 2 e γc 2
)
}
−e −γα∗ S 0 (ξ d−1) (p 3 e γc 3
+ p 4 e γc 4
)
(
= e
{−(p −γx 1 e γc 1
+ p 2 e γc 2 (ξ u − 1)(p 1 e γc 1
+ p 2 e γc 2
)
) −(1−q)
)
(1 − ξ d )(p 3 e γc 3 + p4 e γc 4 )
(
−(p 3 e γc 3
+ p 4 e γc 4 (ξ u − 1)(p 1 e γc 1
+ p 2 e γc 2
)
) q }
)
(1 − ξ d )(p 3 e γc 3 + p4 e γc 4 )
= −e −γx (p 1 e γc 1
+ p 2 e γc 2
) q (p 3 e γc 3
+ p 4 e γc 4
) 1−q
[ ( ) q 1−q ( ) ]
1 − q
q
×
+
.
1 − q
q
V C T
(x) = −e −γx 1
q q (1 − q) 1−q (p 1e γc 1
+ p 2 e γc 2
) q (p 3 e γc 3
+ p 4 e γc 4
) 1−q . (3.12)
When C T = 0 (i.e., c 1 = · · · = c 4 = 0), we get
V 0 (x) = −e −γx (
p1 + p 2
q
To determine ν(C T ), we solve the equation
) q ( )
p3 + p 1−q
4
. (3.13)
1 − q
V 0 (x) = V C T
(x + ν(C T )) (3.14)
7

which is reduced to (via (3.12) and (3.13))
Consequently,
(p 1 + p 2 ) q (p 3 + p 4 ) 1−q = e −γν(C T ) (p 1 e γc 1
+ p 2 e γc 2
) q (p 3 e γc 3
+ p 4 e γc 4
) 1−q .
ν(C T ) = q 1 γ log p 1e γc 1
+ p 2 e γc 2
p 1 + p 2
+ (1 − q) 1 γ log p 3e γc 3
+ p 4 e γc 4
p 3 + p 4
. (3.15)
Note that the terms inside the log functions of (3.15) can be expressed as the conditional expectations
of e γC T
under the measure P as follows.
and
where
p 1 e γc 1
+ p 2 e γc 2
p 1 + p 2
= E P [e γC T
|A] (3.16)
p 3 e γc 3
+ p 4 e γc 4
p 3 + p 4
= E P [e γC T
|A c ], (3.17)
A = {ω 1 , ω 2 } = {ω : S T (ω) = S 0 ξ u },
A c = {ω 3 , ω 4 } = {ω : S T (ω) = S 0 ξ d }.
Next we are going to seek the desired measure Q with distributions
Q(ω i ) = q i , i = 1, 2, 3, 4.
That is, we must designate the values of q i for i = 1, 2, 3, 4 in such a way that q 1 + q 2 = q and
q 3 + q 4 = 1 − q, where q is given in (3.11).
Consider the conditional probability P(Y T = Y 0 η u |S T = S 0 ξ u ). Using condition (3.8), we get
Q(Y T = Y 0 η u |S T = S 0 ξ u ) = P(Y T = Y 0 η u |S T = S 0 ξ u )
which is reduced to
Similarly, we have
Q(ω 1 , ω 3 |ω 1 , ω 2 ) = P(ω 1 , ω 3 |ω 1 , ω 2 )
⇒
q 1
q 1 + q 2
= p 1
p 1 + p 2
.
q 2
q 1 + q 2
= p 2
p 1 + p 2
,
These can be rewritten as (noting q = q 1 + q 2 )
q 3
q 3 + q 4
= p 3
p 3 + p 4
,
q 4
q 3 + q 4
= p 4
p 3 + p 4
.
Next we have
p i
q i = q (i = 1, 2), q i = (1 − q) (i = 3, 4). (3.18)
p 1 + p 2 p 3 + p 4
p i
log E Q [e γC T
|S T ] = (log E Q [e γC T
|S T ])I A + (log E Q [e γC T
|S T ])I A c
= (log E P [e γC T
|S T ])I A + (log E P [e γC T
|S T ])I A c
(
= log p 1e γc 1
+ p 2 e γc ) (
2
I A + log p 3e γc 3
+ p 4 e γc )
4
I A c.
p 1 + p 2 p 3 + p 4
8

It turns out that
[ ]
1
E Q γ log EQ [e γC T
|S T ]
[ ( 1
= E Q log p 1e γc 1
+ p 2 e γc )
2
I A + 1 (
γ p 1 + p 2 γ
= 1 (
log p 1e γc 1
+ p 2 e γc )
2
Q(A) + 1 γ p 1 + p 2 γ
= q 1 (
log p 1e γc 1
+ p 2 e γc )
2
+ (1 − q) 1 γ p 1 + p 2 γ
= ν(C T ).
The proof is complete.
log p 3e γc 3
+ p 4 e γc )
4
I A c
p 3 + p 4
(
log p 3e γc 3
+ p 4 e γc )
4
p 3 + p 4
(
log p 3e γc 3
+ p 4 e γc )
4
p 3 + p 4
]
Q(A c )
Remark 3.6. From the formula (3.9), one finds that the indifference valuation is carried out in a
two-step nonlinear procedure and under a single measure. The first step takes risk preference into
the valuation procedure: the original payoff is distorted to the preference adjusted payoff, known
as the conditional certainty equivalent:
˜C T = 1 γ log EQ [e γC T
|S T ]. (3.19)
The second step is of the classical no-arbitrage pricing nature: the price of the claim is obtained
by applying the measure Q to the distorted payoff ˜C T ; that is,
ν(C T ) = E Q [ ˜C T ] = E Q [C T ].
Another feature of the formula (3.9) is its single pricing measure: Only one measure is used
throughout the whole pricing procedure.
3.4 Comparison to Another Pricing Formula
Recall that the minimal relative entropy measure ˜Q is defined as the minimizer of the relative
entropy, that is,
H(˜Q|P) = min H(Q|P), (3.20)
Q∈Q e
where Q e is the set of equivalent (to P) martingale measures and
[ ]
dQ
H(Q|P) = E P dQ
log . (3.21)
dP dP
The following is a formula of indifference price of a claim C T in terms of Q e .
Theorem 3.7. The indifference price of a claim C T is given by
(
ν(C T ) = sup E Q [C T ] − 1 )
[H(Q|P) − H(˜Q|P)] . (3.22)
Q∈Q e
γ
9

The formula (3.22) has some advantages. For instance, it is valid for general models and arbitrary
payoffs. It however has some shortcomings. For instance, it needs to solve a new optimization
problem which is not required in its arbitrage-free counterpart. It also yields a pricing measure
that undesirably depends on the specific payoff. Moreover, this formula considerably obstructs the
analysis and study of certain important aspects of indifference valuation.
We next look at the relation between pricing measure Q used in the formula (3.9) and the minimum
entropy measure ˜Q used in the formula (3.22). To see this, consider an arbitrary martingale
measure ̂Q ∈ Q e defined by ̂Q(ω i ) = ˆq i for i = 1, 2, 3, 4. Then
H( ̂Q|P) =
4∑
i=1
ˆq i log ˆq i
p i
.
We must minimize H( ̂Q|P) over the set
{(ˆq 1 , ˆq 2 , ˆq 3 , ˆq 4 ) :
4∑
ˆq i = 1, ˆq i ≥ 0}.
i=1
Applying the Lagrangian method, we find that the minimizing elementary probabilities ˜q i for
i = 1, 2, 3, 4 satisfy the conditions
˜q 1
= ˜q 2 ˜q 3
, = ˜q 4
.
p 1 p 2 p 3 p 4
If we set q = ˜q 1 + ˜q 2 , then ˜q 3 + ˜q 4 = 1 − q and we arrive at
p i
˜q i = q , i = 1, 2, ˜q i = (1 − q) , i = 3, 4.
p 1 + p 2 p 3 + p 4
These are the same as given in (3.18). Therefore, ˜Q = Q, that is, the pricing measure Q used in
the formula (3.9) and the minimum entropy measure ˜Q used in the formula (3.22) coincide.
However note that the maximizer in the formula (3.22), denoted ̂Q, is not equal to the minimum
entropy measure ˜Q. As matter of fact, if ̂Q = ˜Q, then ν(C T ) = E Q [C T ]. That is, the price of the
claim C T is the expected (discounted) payoff of the claim. This is however untrue in general in an
incomplete market.
Theorem 3.8. Let ν(C T ) be the indifference price of the claim C T , let Q be the pricing measure
used in the formula (3.9), and let H(Q|P) be its associated relative entropy as given in (3.21).
(i) The minimal relative entropy measure ˜Q satisfies ˜Q = Q.
(ii) The value functions V 0 and V C T
are represented by
V 0 (x) = −e −γx−H(Q|P) = U
(x + 1 )
γ H(Q|P) , (3.23)
V C T
(x) = −e −γx−H(Q|P)+γν(C T ) = U
where U(x) = −e −γx is the utility function.
p i
(
x + 1 )
γ H(Q|P) − γν(C T ) , (3.24)
10

(iii) The indifference price ν(C T ) satisfies
(
ν(C T ) = sup E Q [C T ] 1 )
Q∈Q e
γ (H(Q|P) − H(Q|P) = E Q [C T ], (3.25)
where the nonlinear pricing measure E Q is given by
[ ]
1
E Q [C T ] = E Q γ log EQ [e γC T
|S T ] . (3.26)
3.5 Properties of Indifference Prices
Put
˜C T = 1 γ log EQ [e γC T
|S T ]
so that we have rewrite
ν(C T ) = E Q [C T ] = E Q [ ˜C T ].
Sometimes, we will explicitly indicate the dependence of the price ν(C T ) on the risk aversion
coefficient γ by writing ν(C T ) = ν(C T , γ).
Proposition 3.9. The price ν(C T , γ) is an increasing and continuous function of γ ∈ (0, ∞).
Moreover, if, for all claims C T , there holds
then γ = 1.
ν(C T , γ) = ν(C T , 1), (3.27)
Proof. Recall
[ ]
1
ν(C T , γ) = E Q γ log EQ [e γC T
|S T ]
from which the continuity of ν(C T , γ) in γ ∈ (0, ∞) follows.
To see that ν(C T , γ) is increasing in γ, let 0 < γ 1 < γ 2 . Then Holder’s inequality yields
It turns out that
E Q [e γ 1C T
|S T ] ≤
(
) γ
E Q [e γ 1
2C γ T
|S T ] 2
.
1
γ 1
log E Q [e γ 1C T
|S T ] ≤ 1 γ 2
log E Q [e γ 2C T
|S T ].
Taking expectation w.r.t. Q yields that ν(C T , γ 1 ) ≤ ν(C T , γ 2 ).
To prove the second part, we recall (3.15); namely,
ν(C T , γ) = q 1 γ log p 1e γc 1
+ p 2 e γc 2
p 1 + p 2
+ (1 − q) 1 γ log p 3e γc 3
+ p 4 e γc 4
p 3 + p 4
.
We also have
ν(C T , 1) = q log p 1e c 1
+ p 2 e c 2
p 1 + p 2
+ (1 − q) log p 3e c 3
+ p 4 e c 4
p 3 + p 4
.
11

Now the condition (3.27) implies that
q 1 γ log p 1e γc 1
+ p 2 e γc 2
+ (1 − q) 1 p 1 + p 2 γ log p 3e γc 3
+ p 4 e γc 4
p 3 + p 4
= q log p 1e c 1
+ p 2 e c 2
+ (1 − q) log p 3e c 3
+ p 4 e c 4
.
p 1 + p 2 p 3 + p 4
Taking a particular claim, for instance, a claim C T such that c 1 = C T (ω 1 ) > 0 and c i = C T (ω i ) = 0
for i = 2, 3, 4, we get from the last relation that, for all c 1 > 0,
Differentiating c 1 yields
It turns out that
Hence γ = 1.
1
γ log p 1e γc 1
+ p 2
p 1 + p 2
= log p 1e c 1
+ p 2
p 1 + p 2
.
e γc 1
p 1 e γc 1 + p2
=
e c 1
p 1 e c 1 + p2
.
e c 1
(p 1 e c 1
+ p 2 ) = e c 1
(p 1 e γc 1
+ p 2 ) ⇒ e γc 1
= e c 1
.
Proposition 3.10. There hold the following limiting relations:
Proof. Recall again the pricing formula
lim ν(C
γ→0 + T , γ) = E Q [C T ], (3.28)
[ ]
lim ν(C T , γ) = E Q ‖C T ‖ L ∞
γ→∞ Q{·|ST
. (3.29)
}
ν(C T , γ) = q 1 γ log p 1e γc 1
+ p 2 e γc 2
p 1 + p 2
+ (1 − q) 1 γ log p 3e γc 3
+ p 4 e γc 4
p 3 + p 4
. (3.30)
Taking the limit as γ → 0 + in (3.30) and observing the fact
p i
q i = q , i = 1, 2, q i = (1 − q) , i = 3, 4
p 1 + p 2 p 1 + p 2
p i
we obtain
(
lim ν(C p1 c 1
γ→0 + T , γ) = q
=
+ p )
2c 2
p 1 + p 2 p 1 + p 2
+ (1 − q)
4∑
q i c i = E Q [C T ].
i=1
(
p3 c 3
+ p )
4c 4
p 3 + p 4 p 3 + p 4
12

Next taking the limit as γ → ∞ in (3.30) by using the L’Hospital’s rule, we arrive at
lim ν(C c 1 p 1 e γc 1
+ c 2 p 2 e γc 2
T , γ) = q lim
γ→∞ γ→∞ p 1 e γc 1 + p2 e γc 2
+ (1 − q) lim
γ→∞
c 3 p 3 e γc 3
+ c 4 p 4 e γc 4
p 3 e γc 3 + p4 e γc 4
= q max{c [ 1 , c 2 } + (1 − q) max{c 3 , c 4 }
]
= E Q ‖C T ‖ L ∞
Q{·|ST
.
}
Proposition 3.11. The indifference price ν(C T , γ) satisfies
∂ν(C T , γ)
lim
= 1
γ→0 + ∂γ 2 EQ [Var Q (C T |S T )] . (3.31)
Thus,
ν(C T , γ) = E Q [C T ] + 1 2 EQ [Var Q (C T |S T )] + o(γ).
Proof. Recall the formula
[ ]
1
ν(C T , γ) = E Q γ log EQ [e γC T
|S T ] . (3.32)
From this we compute the partial derivative
∂ν(C T , γ)
∂γ
It turns out that, by using L’Hospital’s rule,
This gives that
∂ν(C T , γ)
lim
γ→0 + ∂γ
1
= lim
γ→0 + γ
[
= E Q − 1 γ 2 log EQ [e γC T
|S T ] + 1 E Q [C T e γC T
|S T ]
γ E Q [e γC T |ST ]
= 1 ( [ E
E Q Q [C T e γC ] )
T
|S T ]
γ E Q [e γC − ν(C T , γ) .
T |ST ]
(
E Q [ E Q [C T e γC T
|S T ]
E Q [e γC T |ST ]
] )
− ν(C T , γ)
[ E Q [C
= lim
γ→0 EQ T 2 eγC T
|S T ]E Q [e γC T
|S T ] − (E Q [C T e γC T
|S T ]) 2 ]
+ (E Q [e γC T |ST ]) 2
∂ν(C T , γ)
− lim
.
γ→0 + ∂γ
∂ν(C T , γ)
lim
= 1 [
γ→0 + ∂γ 2 EQ E Q (CT 2 |S T )] − (E Q (C T |S T )) 2]
= 1 2 EQ [Var Q (C T |S T )] .
]
13

Proposition 3.12. The indifference price is consistent with the no-arbitrage principle, that is, for
γ > 0,
inf E Q [C T ] ≤ ν(C T , γ) ≤ sup E Q [C T ], (3.33)
Q∈Q e Q∈Q e
where Q e is the set of equivalent martingales.
Proof. Since ν(C T , γ) is increasing in γ, we find
lim ≤ ν(C
γ→0 + T , γ) ≤ lim ≤ ν(C T , γ).
γ→∞
By Proposition 3.10, we get
E Q [C T ] ≤ ν(C T , γ) ≤ E Q [ ‖C T ‖ L ∞
Q{·|ST }
]
.
Taking infimum gives us that
inf E Q [C T ] ≤ E Q [C T ] ≤ ν(C T , γ).
Q∈Q e
This is the left side of (3.33). To see the right side, it is suffices to observe the relations (we assume
c 1 < c 2 and c 3 < c 4 )
[ ]
E Q ‖C T ‖ L ∞
Q{·|ST
= q max{c
}
1 , c 2 } + (1 − q) max{c 3 , c 4 } = E ¯Q[C T ],
where ¯Q is the martingale measure with elementary probabilities
¯Q(ω 1 ) = 0, ¯Q(ω2 ) = q, ¯Q(ω3 ) = 0, ¯Q(ω4 ) = 1 − q.
Hence,
ν(C T , γ) ≤ E Q [ ‖C T ‖ L ∞
Q{·|ST }
]
= E ¯Q[C T ] ≤ sup
Q∈Q e
E Q [C T ].
Proposition 3.13. The indifference price ν(C T ) is an increasing and convex function of the payoff,
that is,
(i) if C 1 T ≤ C2 T , then ν(C1 T ) ≤ ν(C2 T ),
(ii) for α ∈ (0, 1), ν(αC 1 T + (1 − α)C2 T ) ≤ αν(C1 T ) + (1 − α)ν(C2 T ).
Proof. Recall the pricing formula
[ ]
1
ν(C T ) = E Q γ log EQ [e γC T
|S T ] .
From this formula, it is trivial to find that ν(C T ) is an increasing function of the payoff C T .
14

We next show the convexity of ν(C T ). Applying Holder’s inequality, we derive that
The convexity of ν(C T ) follows.
ν(αCT 1 + (1 − α)CT 2 )
[ ]
1
= E Q γ log EQ [e γ(αC1 T +(1−α)C2 T ) |S T ]
[ ]
1
≤ E Q γ log(EQ [e γC1 T |ST ]) α (E Q [e γC2 T |ST ]) 1−α
[ ]
[ ]
1 1
= αE Q γ log(EQ [e γC1 T |ST ] + (1 − α)E Q γ log EQ [e γC2 T |ST ] .
Proposition 3.14. The indifference price ν(C T ) satisfies the properties:
(i) ν(αC T ) ≤ αν(C T ) for α ∈ (0, 1),
(ii) ν(αC T ) ≥ αν(C T ) for α ≥ 1.
Proof. We have
[ ]
1
ν(αC T ) = E Q γ log EQ [e γαC T
|S T ] .
If α ∈ (0, 1), then setting ¯γ = γα < γ, we get
[ ]
ν(αC T ) = ν(αC T , γ) = αE 1¯γ Q log EQ [e¯γC T
|S T ] = αν(C T ), ¯γ).
Since ν(αC T , γ) is increasing in γ, we derive from the last relation that
ν(αC T ) = αν(C T ), ¯γ) ≤ αν(C T ), γ) = αν(C T ).
If α > 1, then ¯γ > γ. Hence it follows again from the increasingness of ν(αC T , γ) that
ν(αC T ) = αν(C T ), ¯γ) ≥ αν(C T ), γ) = αν(C T ).
Proposition 3.15. The indifference pricing operator is additively invariant w.r.t. hedgeable risks;
that is, if C T = C 1 T + C2 T , with C1 T = C1 (S T ) and C 2 T = C2 (S T , Y T ), then
ν(C T ) = E Q [C 1 (S T )] + ν(C 2 (S T , Y T )) = ν(C 1 (S T )) + ν(C 2 (S T , Y T )). (3.34)
Proof. Recall the pricing formula
[ ]
1
ν(C T ) = E Q (C T ) = E Q γ log EQ [e γC T
|S T ] .
15

Now ifC T = CT 1 + C2 T , with C1 T = C1 (S T ) and CT 2 = C2 (S T , Y T ), then we get
[ ]
1
ν(C T ) = E Q γ log EQ [e γ(C1 T (S T )+CT 2 (S T ,Y T )) |S T ]
[ ]
1
= E Q [CT 1 (S T )] + E Q γ log EQ [e γC2 T (S T ,Y T ) |S T ]
= E Q [CT 1 (S T )] + ν(CT 2 (S T , Y T )).
Remark 3.16. (i) If C T = C 1 (S T ) + C 2 (S T , Y T ), where Y T is functionally dependent on S T , then
CT 2 = C2 (S T , Y T ) is F S T
T
-measurable. It turns out that
and, from (3.34), that
C 2 (S T , Y T ) = ˜C 2 (S T , Y T ) = 1 γ log EQ [e γC2 T |ST ]
ν(C T ) = E Q [C 1 (S T )] + ν(C 2 (S T , Y T )) = E Q [C 1 (S T )] + E Q [C 2 (S T , Y T )].
Therefore, the indifference price simplifies to the no-arbitrage price and the pricing measure is
coincides with the risk-neutral measure.
(ii) If C T = C 1 (S T ) + C 2 (Y T ), where S T and Y T are independent under P, then
˜C 2 (Y T ) = 1 γ log EQ [e γC2 T (Y T ) |S T ] = 1 γ log EP [e γC2 T (Y T ) ]
and the formula (3.34) becomes
ν(C T ) = E Q [C 1 (S T )] + 1 γ log EP [e γC2 T (Y T ) ].
Namely, the indifference price ν(C T ) is the sum of the arbitrage-free price of the first claim and the
traditional certainty equivalent price of the second claim.
Definition 3.17. A mapping ρ : F T → R is said to be a convex risk measure if it satisfies the
following conditions, for all C T , CT 1 , C2 T ∈ F T ,
(a) Convexity: ρ(αCT 1 + (1 − α)C2 T ) ≤ αρ(C1 T ) + (1 − α)ρ(C2 T
) for all α ∈ (0, 1),
(b) Monotonicity: If C 1 T ≤ C2 T , then ρ(C1 T ) ≥ ρ(C2 T ),
(c) Translation invariance: If m ∈ R, then ρ(C T + m) = ρ(C T ) − m.
Define a mapping on F T by
[ ]
1
ρ(C T ) = ν(−C T ) = E Q γ log EQ [e −γC T
|S T ] . (3.35)
From the above properties of the indifference price, we immediately get the following result.
Proposition 3.18. The mapping ρ defined by (3.35) is a convex risk measure.
16

4 Risk Monitoring Strategies
In a complete market, the payoff of every claim can be replicated by a self-financing portfolio and
thus any risk associated with the claim is eliminated. However, in an incomplete market, not all
risk associated with (some) claim can not be eliminated completely. Therefore, managing risk
generated by derivative contracts is an important issue for an incomplete market.
For this one-period model, if a claim C T is F S T
T
-measurable, then its indifference price coincides
with the arbitrage-free price; it is also replicable and there holds the formula
with
C T = ν(C T ) + ∂ν(C T )
∂S 0
(S T − S 0 ) (4.1)
ν(C T ) = E Q [C T ],
∂ν(C T )
∂S 0
= ∂EQ [C T ]
∂S 0
. (4.2)
For a general claim C T = c(S T , Y T ), (4.1) does not hold in general.
Let ν(C T ) be the indifference price of a claim C T . Recall that the value function of C T is
V C T
(x) = sup
α
E P [−e −γ(X T −C T ) ] = e −γx sup E P [−e −γα(S T −S 0 )+γC T
]. (4.3)
α
Let α C T ,∗ denote the optimal solution to the optimal problem (4.3).
Recall also the relative entropy of the measure Q w.r.t. P:
H(Q|P) =
4∑
i=1
q i log q i
p i
.
Proposition 4.1. The optimal number of shares, α C T ,∗ , in the the optimal investment problem
(4.3) satisfies
α C T ,∗ = α 0,∗ + ∂ν(C T )
, (4.4)
∂S 0
where
α 0,∗ = − 1 ∂H(Q|P)
γ ∂S 0
represents the number of shares held optimally in the absence of the claim. Both optimal controls
α C T ,∗ and α 0,∗ are wealth independent.
Proof. We have computed α C T ,∗ :
α C T ,∗ =
Setting c 1 = c 2 = c 3 = c 4 = 0, we get
Recall (S u = S 0 ξ u , S d = S 0 ξ d )
α 0,∗ =
1
γS 0 (ξ u − ξ d ) log (ξu − 1)(p 1 e γc 1
+ p 2 e γc 2
)
(1 − ξ d )(p 3 e γc 3 + p4 e γc 4 )
. (4.5)
1
γS 0 (ξ u − ξ d ) log (ξu − 1)(p 1 + p 2 )
(1 − ξ d )(p 3 + p 4 ) . (4.6)
q = 1 − ξd
ξ u − ξ d = S 0 − S d
S u − S d , thus 1 − q = ξu − 1
ξ u − ξ d .
17

We can rewrite
and
Note that we have
Since (noting q = q 1 + q 2 )
α C T ,∗ =
q i
p i
=
1
γ(S u − S d ) log (1 − q)(p 1e γc 1
+ p 2 e γc 2
)
q(p 3 e γc 3 + p4 e γc . (4.7)
4 )
α 0,∗ 1
=
γ(S u − S d ) log (1 − q)(p 1 + p 2 )
. (4.8)
q(p 3 + p 4 )
∂q
∂S 0
=
q
p 1 + p 2
, i = 1, 2,
we can express H(Q|P) as a function of q as follows:
Consequently, we get
H(Q|P) =
∂H(Q|P)
∂S 0
2∑
i=1
= q log
q i log q i
p i
+
1
S u − S d
. (4.9)
q i
p i
=
4∑
i=3
q
p 3 + p 4
, i = 3, 4,
q i log q i
p i
q
p 1 + p 2
+ (1 − q) log 1 − q
p 3 + p 4
.
= ∂H(Q|P) ∂q
∂q ∂S
(
0
= log
q
− log 1 − q ) ∂q
p 1 + p 2 p 3 + p 4 ∂S 0
= − log (1 − q)(p 1 + p 2 )
q(p 3 + p 4 )
Combining (6.45), (4.9) and (4.10), we get α 0,∗ = − 1 γ
To prove (4.4), recall we have proved that
It follows that
·
∂H(Q|P)
∂S 0
.
∂q
∂S 0
. (4.10)
ν(C T ) = q 1 γ log p 1e γc 1
+ p 2 e γc 2
p 1 + p 2
+ (1 − q) 1 γ log p 3e γc 3
+ p 4 e γc 4
p 3 + p 4
.
∂ν(C T )
= ∂ν(C T ) ∂q
∂S 0 ∂q ∂S 0
= 1 γ log (p 3 + p 4 )(p 1 e γc 1
+ p 2 e γc 2
)
(p 1 + p 2 )(p 3 e γc 3 + p4 e γc 4 ) ·
Combining (4.10) and (4.11) yields that, using (4.7),
α 0,∗ + ∂ν(C T )
= 1 ∂S 0 γ log (1 − q)(p 1e γc 1
+ p 2 e γc 2
) 1
q(p 3 e γc 3 + p4 e γc ·
4 ) S u − S d
= α C T ,∗ .
∂q
∂S 0
. (4.11)
18

Consider the agent’s optimal wealths X C T ,∗ with the claim and X 0,∗ without the claim. In the
first case, the agent starts with the initial wealth x + ν(C T ) and buys α C T ,∗ shares of stock; while
in the latter case, the agent starts with the wealth x and follows the strategy α 0,∗ . Namely,
X C T ,∗ = x + ν(C T ) + α C T ,∗ (S T − S 0 ) (4.12)
X 0,∗ = x + α 0,∗ (S T − S 0 ). (4.13)
Definition 4.2. The residual optimal wealth process is defined as
By definition, we get
L t = X C T ,∗
t − X 0,∗
t , t = 0, T. (4.14)
L 0 = X C T ,∗
0 − X 0,∗
0 = x + ν(C T ) − x = ν(C T ), (4.15)
L T = X C T ,∗
T
− X 0,∗
T
= ν(C T ) + (α C T ,∗ − α 0,∗ )(S T − S 0 ). (4.16)
In a complete model, the residual optimal wealth is exactly the value of perfectly replicating
portfolio, and is therefore a martingale under the unique risk-neutral measure, and generates the
claim’s payoff at the expiration time T . However, in an incomplete model, the situations are
different. The residual terminal optimal wealth L T reproduces the claim only partially. In addition,
it is FT S -measurable and remains to be a martingale under all martingale measures. Its most
important property lies in its coincidence with the conditional certainty equivalent.
Proposition 4.3. The residual optimal wealth process satisfies the following properties:
(i) L 0 = ν(C T ),
(ii) L T = ν(C T ) + ∂ν(C T )
∂S 0
(S T − S 0 ),
(iii) L T equals the conditional certainty equivalent: L T = ˜C T ,
(iv) L t is a martingale under all equivalent martingale measures: E Q [L T ] = L 0 = ν(C T ) for all
Q ∈ Q e .
Proof. (i) and (ii) follows from (4.15) and (ii) follows from (4.16) together with Proposition 4.1.
To see (iii), recall that
From (ii) we get
ν(C T ) = E Q [ ˜C T ], ˜CT = 1 γ log EQ [e γC T
|S T ].
L T = E Q [ ˜C T ] + ∂EQ [ ˜C T ]
∂S 0
(S T − S 0 ).
On the other hand, since ˜C T is FT S -measurable, it is replicable and decomposed as the form
˜C T = E Q [ ˜C T ] + ∂EQ [ ˜C T ]
∂S 0
(S T − S 0 ). (4.17)
Consequently, L T = ˜C T .
(iv) follows from (i) and (ii) as for each Q ∈ Q e , E Q [S T ] = S 0 .
19

Definition 4.4. The indifference price process ν t (C T ), t = 0, T , is defined as
ν 0 (C T ) = ν(C T ), ν T (C T ) = C T . (4.18)
Definition 4.5. The residual risk process R t , t = 0, T , is defined as the difference between the
indifference price and the residual optimal wealth, that is,
R t = ν t (C T ) − L t .
In other words,
R 0 = ν(C T ) − L 0 , R T = C T − L T . (4.19)
If perfect replication is viable, then the residual risk R = 0 throughout. In general, it represents
the component of the claim that is not replicable.
Proposition 4.6. The residual risk process satisfies the following properties:
(i) R 0 = 0 and R T = C T − ˜C T ,
(ii) its conditional certainty equivalent is zero: ˜RT = 0,
(iii) its indifference price is zero: ν(R T ) = 0,
(iv) it is a supermartingale under the pricing measure Q: E Q [R T ] ≤ R 0 = 0,
(v) its expected certainty equivalent under the historical measure P is zero: log E P [e γR T
] = 0.
Proof. (i) follows from Proposition 4.3(i) and (iii).
(ii) We compute, as ˜C T is F S T -measurable,
˜R T = 1 γ log EQ [e γ(C T − ˜C T ) |S T ]
= 1 γ log e−γ ˜C T
E Q [e γ(C T ) |S T ]
= 1 γ log EQ [e γ(C T ) |S T ] − ˜C T
= ˜C T − ˜C T = 0.
(iii) ν(R T ) = E Q [ ˜R T ] = 0.
(iv) E Q [R T ] = E Q [C T − ˜C T ] = E Q [C T ] − E Q [ ˜C T ] = E Q [C T ] − ν(C T ) ≤ 0.
(v) By (ii) and (3.8) we obtain
0 = ˜R T = 1 γ log EQ [e γR T
|S T ] = 1 γ log EP [e γR T
|S T ].
Hence
and the result of (v) follows.
E P [e γR T
|S T ] = 1 ⇒ E P [e γR T
] = 1
By the Doob-Meyer decomposition theorem, the supermartingale R t can be decomposed.
20

Proposition 4.7. The residual risk R t admits the Doob-Meyer decomposition
R t = R m t
+ R d t
where
and
R m 0 = 0 and R m T = R T − E Q [R T ] (4.20)
R d 0 = 0 and R d T = E Q [R T ] (4.21)
Moreover, R m t is F (S,Y )
T
-measurable, and R d t is decreasing and adapted to the trivial filtration F (S,Y )
0 .
The next result is the payoff decomposition result which is central to the study of risks of
associated with the indifference valuation method since it provides with an analog of the arbitragefree
payoff decomposition (4.1).
Theorem 4.8. Let ˜C T and R T be the conditional certainty equivalent and the residual risk associated
with the claim C T , respectively. Let (R m t , R d t ) be the Doob-Meyer decomposition of the residual
risk R t . Define a process M ˜C
t , t = 0, T , by
M ˜C
0 = ν(C T ), M ˜C
T = ν(C T ) + ∂ν(C T )
∂S 0
(S T − S 0 ). (4.22)
(i) The Claim C T admits a unique payoff decomposition under the measure Q:
C T = ˜C T + R T
= ν(C T ) + ∂ν(C T )
∂S 0
(S T − S 0 ) + R T (4.23)
= M ˜C
T + R m T + R d T .
(ii) The indifference price process ν t , defined in Definition 4.4, is an F (S,Y )
T
-supermartingale under
the measure Q. It admits the unique decomposition
where
ν t (C T ) = M t + R d t , (4.24)
M t = M ˜C
t + R m t .
The components M t and R d t represent the associated martingale and respectively, the nonincreasing
parts of the price process ν t .
Proof. (i) Since nu(C T ) = E Q [ ˜C T ] and using Proposition 4.6(i), (4.17) and the definition of M ˜C
T
,
we immediately get
C T = ˜C T + R T
= ν(C T ) + ∂ν(C T )
(S T − S 0 ) + R T
∂S 0
= M ˜C
T + R m T + R d T .
21

(ii) That (4.24) holds trivially when t = 0 since, by definition, ν 0 (C T ) = ν(C T ) and
M 0 + R d 0 = M ˜C
0 + R 0 = ν(C T )
for M ˜C
0 = ν(C T ) and R 0 = 0.
When t = T , by definition of M ˜C
T
, we find that the relation ν T (C T ) = M T + RT d
relation (4.23).
is precisely the
Proposition 4.9. The expected residual risk satisfies
E Q [R T ] = − 1 2 γEQ [Var(C T |S T )] + o(γ),
E Q [R T ] = − 1 2 γEQ [Var(R T |S T )] + o(γ).
Proof. By Propositions 4.6 and 3.11, we get
E Q [R T ] = E Q [C T ] − E Q [ ˜C T ] = E Q [C T ] − ν(C T )
= − 1 2 γEQ [Var(C T |S T )] + o(γ).
Since R T = C T − ˜C T and since ˜C T is F S T
T
-measurable, it follows that
[ ]
Var(R T |S T ) = E Q (R T − E Q [R T |S T ]) 2 |S T
= E Q [ (C T − ˜C T − E Q [C T − ˜C T |S T ]) 2 |S T
]
= E Q [ (C T − E Q [C T |S T ]) 2 |S T
]
= Var(C T |S T ).
The second equality now follows from the last relation and the first equality.
22

5 Relative Indifference Prices
The indifference price ν(C T ) is nonlinear. It is not homogeneous: for α ≠ 0, 1,
It is not additive: for two claims CT i , i = 1, 2,
More generally, for claims CT i , 2 ≤ i ≤ n,
( n∑
ν
i=1
ν(αC T ) ≠ αν(C T ).
ν(C 1 T + C 2 T ) ≠ ν(C 1 T ) + ν(C 2 T ).
C i T
)
≠
n∑
ν(CT i ).
The nonhomogeneity reflects the fact that the agents are not willing to expose herself to double
risk when she sells or buys two units of a claim. The nonadditivity interprets the nondiversity of
risks in this nonlinear valuation mechanism.
In a complete market, the valuation function is both homogeneous and additive (i.e., linear).
The introduction of the concept of relative indifference price remedies the nonlinear indifference
valuation method in some sense. It helps twofold.
Definition 5.1. Let CT 1 = C1 (S T , Y T ) and CT 2 = C2 (S T , Y T ) be two claims that have indifference
prices ν(CT 1 ) and ν(C2 T ), respectively. Let V C1 T , V C2 T and V C1 T +C2 T be the value functions
corresponding to CT 1 , C2 T and C1 T + C2 T , respectively.
The relative indifference price ν(CT 2 |C1 T ) and ν(C1 T |C2 T
) are defined, respectively, as the amounts
satisfying, for all wealth levels x ∈ R,
and
i=1
V C1 T (x) = V
C 1 T +C2 T (x + ν(C
2
T |C 1 T )) (5.1)
V C2 T (x) = V
C 1 T +C2 T (x + ν(C
1
T |C 2 T )). (5.2)
Proposition 5.2. Let the claims CT 1 = C1 (S T , Y T ) and CT 2 = C2 (S T , Y T ) have indifference prices
ν(CT 1 ) and ν(C2 T ), and relative indifference prices ν(C1 T |C2 T ) and ν(C2 T |C1 T
), respectively. Then the
indifference price of the claim CT 1 + C2 T satisfies
Proof. By Theorem 3.8,
It turns out that
ν(C 1 T + C 2 T ) = ν(C 1 T ) + ν(C 2 T |C 1 T ), (5.3)
ν(C 1 T + C 2 T ) = ν(C 2 T ) + ν(C 1 T |C 2 T ).
V C1 T (x) = −e
−γx−H(Q|P)+γν(C 1 T ) ,
V C1 T +C2 T (x) = −e
−γx−H(Q|P)+γν(C 1 T +C2 T ) .
V C1 T +C2 T (x + ν(C
2
T |C 1 T )) = −e −γ(x+ν(C2 T |C1 T ))−H(Q|P)+γν(C1 T +C2 T )
= V C1 T (x)e
−γ[ν(C 1 T )+ν(C2 T |C1 T ))−ν(C1 T +C2 T )] .
23

So the defining equation (5.1) for the relative indifference price ν(CT 1 |C2 T
) is reduced to the equation:
This is equation (5.3).
ν(C 1 T ) + ν(C 2 T |C 1 T )) − ν(C 1 T + C 2 T ) = 0.
Corollary 5.3. The indifference prices ν(CT 1 ) and ν(C2 T
) and their relative indifference prices
ν(CT 1 |C2 T ) and ν(C2 T |C1 T
) satisfy the equation
ν(C 1 T ) − ν(C 2 T ) = ν(C 1 T |C 2 T ) − ν(C 2 T |C 1 T ).
Corollary 5.4. The indifference price of the claim C T := CT 1 + C2 T
ν(CT 1 |C2 T ) and ν(C2 T |C1 T
) satisfy the equation
is given by indifference prices
ν(C 1 T + C 2 T ) = 1 2 (ν(C1 T ) + ν(C 2 T )) + 1 2 (ν(C1 T |C 2 T ) + ν(C 2 T |C 1 T )).
Moreover, the above relation can be rewritten as
ν(C 1 T + C 2 T ) − (ν(C 1 T ) + ν(C 2 T )) = 1 2 (ν(C1 T |C 2 T ) + ν(C 2 T |C 1 T ))
− 1 2 (ν(C1 T ) + ν(C 2 T )).
Remark 5.5. (i) Assume C T = C 1 T + C2 T with C1 T = C1 (S T , Y T ) and C 2 T = C2 (S T ). Then by
Proposition 3.15, the price is additive:
ν(C T ) = ν(C 1 T ) + ν(C 2 T ).
Note that ν(C 2 T ) = EQ [C 2 (S T )]. Proposition 5.2 then implies that
ν(C 2 T |C 1 T ) = ν(C 2 T ) = E Q [C 2 (S T )].
That is, if a claim CT
2 = C2 (S T ) is F S T
T
-measurable, then its relative indifference price w.r.t.
another F S T ,Y T
T
-measurable claim CT 2 = C2 (S T , Y T ) is exactly its arbitrage-free price E Q [C 2 (S T )].
If, in addition, Y T functionally depends on S T so that CT 2 is F S T
T
-measurable, then we deduce
that (as CT 1 + C2 T is F S T
T
-measurable)
It turns out that
ν(C 1 T + C 2 T ) = E Q [C 1 T ] + E Q [C 2 ]. (5.4)
ν(C 1 T |C 2 T ) = ν(C 1 T ), ν(C 2 T |C 1 T ) = ν(C 2 T ).
(ii) Assume C T = C 1 T (S T ) + C 2 T (Y T ) with S T and Y T being independent under the historical
measure P. Then it has been shown that
ν(C T ) = ν(C 1 T + C 2 T ) = ν(C 1 T ) + ν(C 2 T ),
where
ν(C 1 T ) = E Q [C 1 T ], ν(C 2 T ) = 1 γ log EP [e γC2 (Y T ) ].
Proposition 5.2 then implies that
ν(C 1 T |C 2 T ) = ν(C 1 T ), ν(C 2 T |C 1 T ) = ν(C 2 T ).
24

6 Wealth, Preferences and Numeraires
This section extends indifference price to the case of nonzero rate of interest.
6.1 Indifference Prices in Spot and Forward Units
Consider the one-period model of a riskless bond B and two risky assets S and Y , of which only
the asset S is traded. The dynamics of the risky assets remain unchanged. However, the rate of
interest for the period is now nonzero. Thus,
B 0 = 1,
B T = 1 + r
with the no-arbitrage conditions:
ξ d < 1 + r < ξ u .
Since we have now nonzero rate of interest, the indifference price formula (3.9) does not apply.
In order to produce meaningful prices, one must be consistent with the units in which quantities
that are used in price specifications are expressed. We thus consider the valuation problem in spot
and forward units and will force the price to be independent of the unit price. We first consider
the case of spot units.
Let a portfolio ϕ be composed of α shares of stock and β amount of bond. We use X 0 to denote
its initial value and XT s its spot terminal wealth. Thus,
X 0 = x = β + αS 0 ,
( )
XT s ST
= x + α
1 + r − S 0 .
Note that the stock is discounted from time T to 0.
The utility of the investor remains exponential and its constant absolute risk aversion coefficient
is γ s . It is important to notice that for the utility to be well defined, the coefficient γ s needs to be
expressed in the reciprocal of the spot unit. The value function of a claim C T is defined by
[
V s,C T
(x) = sup E P −e −γs
α
X s T − C T
1+r
]
. (6.1)
[Note that the payoff of the claim, C T , is discounted from time T to 0.] The indifference price of
the claim C T (in spot units) is defined as follows.
Definition 6.1. The indifference price of the claim C T (in spot units) is defined as the amount
ν s (C T ) for which the two spot value functions V s,0 and V s,C T
coincide. More precisely, ν s (C T ) is
the unique solution of the equation
for all wealth levels x ∈ R.
V s,0 (x) = V s,C T
(x + ν s (C T )) (6.2)
Theorem 6.2. Let Q s be the measure such that
[ ]
ST
E Qs = S 0 , (6.3)
1 + r
Q s [Y T |S T ] = P[Y T |S T ]. (6.4)
25

The spot indifference price of a claim C T = C(S T , Y T ) is given by
[ ]
ν s CT
(C T ) = E Qs 1 + r
[ ( )]
1
= E Qs γ s log EQs e γs C T
1+r ∣ ST . (6.5)
[Note: (6.3) says that under Q s , the discounted stock price ˜S = S/(1 + r) is a martingale.]
Proof. The proof follows the lines of that of Theorem 3.5. First we work out the value function
(6.1). Recall c i = C T (ω i ), i = 1, 2, 3, 4. We have
[
]
V C T
(x) = sup E P −e −γs XT s − C T
1+r
α
[ ]
= sup E P −e −γs (x−αS 0 + 1
1+r (αS T −C T ))
α
[
= e −γsx sup e γs αS 0
E P −e − γs
1+r (αS T −C T )
α
4∑
= e −γsx sup e γs αS 0
α
= e −γsx sup
α
≡ e −γsx sup g(α).
α
i=1
]
−p i e − γs
1+r (αS T (ω i )−c i )
(
{−e γs αS 0 (1− ξu
1+r ) p 1 e γs c 1
1+r + p2 e γs c 2
1+r
(
−e γs αS 0 (1− ξd
1+r ) p 3 e γs c 3
1+r + p4 e γs c 4
1+r
)
)}
We have
g ′ (α) = −γ s S 0 (1 −
− γ s S 0 (1 −
(
)
ξu αS 0 (1− ξu
1 + r )eγs 1+r ) p 1 e γs c 1
1+r + p2 e γs c 2
1+r
(
ξd αS 0 (1− ξd
1 + r )eγs 1+r ) p 3 e γs c 3
1+r + p4 e γs c 4
1+r
)
.
The unique solution to the equation g ′ (α) = 0 is
α ∗ =
=
1 + r
γ s S 0 (ξ u − ξ d ) log (ξu − (1 + r))(p 1 e γs c 1
1+r + p2 e γs c 2
1+r )
(1 + r − ξ d )(p 3 e γs c 3
1+r + p4 e γs c 4
1+r )
1 + r
γ s S 0 (ξ u − ξ d ) log (1 − qs )(p 1 e γs c 1
1+r + p2 e γs c 2
1+r )
q s (p 3 e γs c 3
1+r + p4 e γs c 4
, (6.6)
1+r )
where
q s = 1 + r − ξd
ξ u − ξ d , 1 − qs = ξu − (1 + r)
ξ u − ξ d . (6.7)
26

Noticing
γ s α ∗ S 0 (1 −
ξu
1 + r ) = −(1 − qs ) log (1 − qs )(p 1 e γs c 1
= log
1+r + p2 e γs c 2
1+r )
q s (p 3 e γs c 3
1+r + p4 e γs c 4
(
(q s ) 1−qs p 3 e γs c 3
1+r + p4 e γs c 4
1+r
γ s α ∗ S 0 (1 −
ξd
1 + r ) = qs log (1 − qs )(p 1 e γs c 1
= log
1+r )
) 1−q s
(
)
(1 − q s ) 1−qs p 1 e γs c 1
1+r + p2 e γs c 1−q s ,
2
1+r
1+r + p2 e γs c 2
1+r )
q s (p 3 e γs c 3
1+r + p4 e γs c 4
1+r )
(
(1 − q s ) qs p 1 e γs c 1
1+r + p2 e γs c 2
1+r
) q s
(
)
(q s ) qs p 3 e γs c 3
1+r + p4 e γs c q s ,
4
1+r
we get
− g(α ∗ )
=
and the value function
In particular,
(
1
(q s ) qs (1 − q s p 1 e γs c 1
1+r + p2 e γs c 2
1+r
) 1−qs
V s,C T
(x) = e −γsx g(α ∗ )
(
e −γs x
= −
(q s ) qs (1 − q s ) 1−qs
)
p 1 e γs c 1
1+r + p2 e γs c q s (
2
1+r
e −γs x
) q s (
)
p 3 e γs c 3
1+r + p4 e γs c 1−q s
4
1+r
p 3 e γs c 3
1+r + p4 e γs c 4
1+r
) 1−q s
. (6.8)
V s,0 (x) = −
(q s ) qs (1 − q s (p 1 + p 2 ) qs (p 3 + p 4 ) 1−qs . (6.9)
) 1−qs
It turns out that the equation (6.15) is reduced to
(
)
e −γs ν s (C T )
p 1 e γs c 1
1+r + p2 e γs c q s (
2
1+r p 3 e γs c 3
1+r + p4 e γs c 4
1+r
= (p 1 + p 2 ) qs (p 3 + p 4 ) 1−qs .
) 1−q s
Therefore,
⎛
⎞
ν s (C T ) = q s ⎝ 1 γ s log p 1e γs c 1
1+r + p2 e γs c 2
1+r
⎠
p 1 + p 2
⎛
⎞
+ (1 − q s ) ⎝ 1 γ s log p 3e γs c 3
1+r + p4 e γs c 4
1+r
⎠ . (6.10)
p 3 + p 4
27

Now define the spot pricing measure Q s by setting Q s (ω i ) = q s i ,
i = 1, · · · , 4, where
p i
qi s = q s , i = 1, 2, qi s = (1 − q s ) , i = 3, 4.
p 1 + p 2 p 3 + p 4
Introduce the conditional certainty equivalent in spot units by
˜C T s = 1 [
]
γ s log EQs e γs C T
1+r
∣ S T . (6.11)
Then we can rewrite the formula (6.10) as
This is (6.5).
p i
ν s (C T ) = E Qs [ ˜Cs T
]
. (6.12)
Next we express the indifference price in terms of forward units. Consider the forward terminal
wealth
( ( ))
X f T = ST
Xs T (1 + r) = x + α
1 + r − S 0 (1 + r)
= x(1 + r) + α(S T − S 0 (1 + r))
= f + α(F T − F 0 ), (6.13)
where f = x(1 + r) is the forward value of the current value x, and F 0 = S 0 (1 + r) and F T = S T is
the forward stock price process. The forward value function is defined as
[ ]
V f,C T
(f) = sup E P −e −γf (X f T −C T )
. (6.14)
α
Definition 6.3. The indifference price of the claim C T (in forward units) is defined as the amount
ν f (C T ) for which the two spot value functions V f,0 and V f,C T
coincide. More precisely, ν f (C T ) is
the unique solution of the equation
for all wealth levels x ∈ R.
Theorem 6.4. Let Q f be the measure such that
Then
V f,0 (x) = V f,C T
(x + ν f (C T )) (6.15)
E Qf [F T ] = F 0 , (6.16)
Q f [Y T |f T ] = P[Y T |F T ]. (6.17)
Q f = Q s .
The spot indifference price of a claim C T = C(S T , Y T ) in the forward units is given by
ν f (C T ) = E Qf [C T ]
= E Qf [ 1
γ f log EQf ( e γf C T
∣ ∣ F T
) ] . (6.18)
28

Proof. Since the interest rate is constant, the equations (6.16) and (6.17) are equivalent to the
equations (6.3) and (6.4), respectively. It turns out that Q f = Q s .
The value function V f,C T
can be expressed [ as
]
V f,C T
(x) = sup E P −e −γf (X f T −C T )
α
= sup
α
E P [ −e −γf (x(1+r)+α(S T −S 0 (1+r))−C T
]
[
= sup E P −˜γ x+α
−e S
α
= V s,C T
(x),
T
1+r −S 0 − C T
1+r
where γ s = ˜γ = γ f (1 + r). The results thus follow from Theorem 6.2.
In the rest of the section, we set Q = Q s = Q f .
Proposition 6.5. The indifference prices, in terms of spot and forward units, are consistent with
the no-arbitrage condition; that is,
if and only if the spot and forward risk aversion coefficients satisfy
ν f (C T ) = (1 + r)ν s (C T ) (6.19)
γ f =
]
γs
1 + r . (6.20)
Proof. (i) (6.20) ⇒ (6.19). We have from (6.5) and (6.18), we get
[ ( )] 1
ν f (C T ) = (1 + r)E Q γ s log EQ e γs C T ∣
1+r ∣S T = (1 + r)ν s (C T ).
(ii) (6.19) ⇒ (6.20). From (6.19) it follows that, for all claims C T ,
[
1 1
(
1 + r EQ γ f log ∣ ) ] [ ( )]
EQ e γf C T 1
F T = E Q γ s log EQ e γs C T ∣
1+r ∣S T .
Alternatively, setting ĈT = γs
1+r C T and ˆγ = γf (1+r)
γ
, we get
s
In other words, we have, for all claims ĈT ,
[
E 1ˆγ (
Q log ∣ ) ]
EQ eˆγĈT ∣S [ ( T = E Q log E Q ∣ )]
∣S eĈT T .
ν(ˆγ, ĈT ) = ν(1, ĈT ).
It follows from Proposition 3.9, we get ˆγ = 1; hence (6.20) holds.
Proposition 6.6. The value function V s,C T
and V f,C T
are given by
(
V s,C T
= −e −γs (x−ν s (C T ))−H(Q|P) = U s x − ν s (C T )) + 1 )
γ s H(Q|P) ,
(
V f,C T
= −e −γs (x−ν f (C T ))−H(Q|P) = U f x − ν f (C T )) + 1 )
γ f H(Q|P) ,
where U s (x) = −e −γsx and U f (x) = −e −γf x are the spot and forward utility functions.
29

6.2 Indifference Prices and State-Dependent Preferences
Consider a random risk aversion coefficient which is an F S T -measurable variable γ T = γ T (S T ) taking
values:
γ u = γ(S 0 ξ u ), γ d = γ(S 0 ξ d )
when the events
A = {S T = S 0 ξ u } = {ω 1 , ω 2 } and A c = {S T = S 0 ξ d } = {ω 3 , ω 4 }
occur. The risk tolerance is the reciprocal of the risk aversion:
δ T := 1
γ T
.
Assume we choose to work with the spot unit so that
( )
X T = XT s ST
= x + α
1 + r − S 0 .
Theorem 6.7. Assume that the risk aversion coefficient γ T is of the form γ T = γ(S T ). Let Q be
the measure satisfying
Q[Y T |S T ] = P[Y T |S T ]. (6.21)
Let C T = C(S T , Y T ) be a claim to be priced under exponential utility with risk aversion coefficient
γ T . Then the indifference price of C T is given by
[ ( )]
1
ν(C T , γ T ) = E Q log E Q e γ T C T ∣
1+r ∣S T . (6.22)
γ T
Proof. We start by computing the value functions V C T
(x, γ T ) and V 0 (x, γ T ). We have
[
]
V C T
(x, γ T ) = sup E P −e −γ T x+α S T
1+r −S 0 − C T
1+r
. (6.23)
α
Put
( )
( )
ξ
β u = γ u u
1 + r − 1 , β d = γ d 1 − ξd
. (6.24)
1 + r
By computing the expectation in (6.23), we can express
V C T
(x, γ T ) = sup g(α),
α
where
Solving the equation
g(α) = −e αS 0β u ( e −γu (x− c 1
1+r ) p1 + e −γu (x− c 2
1+r ) p2
)
− e αS 0β d ( e −γd (x− c 3
1+r ) p3 + e −γd (x− c 4
1+r ) p4
)
.
g ′ (α) = 0
30

gets the maximum point
α C T ,∗ =
(
1 β u e −γu x
S 0 (β u + β d ) log β d e −γd x
e γu c 1
1+r p1 + e γu c 2
1+r p2
)
(
e γd c 3
1+r p3 + e γd c 4
1+r p4
) . (6.25)
Further calculations give that
and
e −αC T ,∗ S 0 β u =
e −αC T ,∗ S 0 β d =
( β
u
β d ) −
( β
u
β d ) −
β u
β u +β d e βu (γ u −γ d )
β u +β d
β d
β u +β d e βd (γ u −γ d )
β u +β d
It then turns out after some manipulations that
V C T
(x, γ T ) = sup g(α C T ,∗ )
α
⎛
( )
= − ⎝ β
u −
β u ( )
β u +β d β
u −
β d +
β d
x
(e γu c 1
1+r p1 + e γu c 2
1+r p2
e γd c 3
1+r p3 + e γd c 4
x
(e γu c 1
1+r p1 + e γu c 2
1+r p2
⎞
β d
β u +β d
e γd c 3
1+r p3 + e γd c 4
⎠ e − βu γ d +β d γ u )
β u +β d
) −
β u
β u +β d
1+r p4
) −
β d
β u +β d .
1+r p4
(
)
× e γu c 1
1+r p1 + e γu c β d ( )
2
1+r
β u +β d p2 e γd c 3
1+r p3 + e γd c β u
4
1+r
β u +β d p4 . (6.26)
Taking C T = 0 gives the value function
⎛
( )
V 0 (x; γ T ) = − ⎝ β
u −
β u ( )
β u +β d β
u −
β d +
β d
β d
βu
⎞
β d
β u +β d
x
⎠ e − βu γ d +β d γ u )
β u +β d
× (p 1 + p 2 ) β u +β d (p 3 + p 4 ) β u +β d . (6.27)
Using (6.26) and (6.27), and definition of the indifference price ν ( C T , γ T ), we get
( β u γ d + β d γ u ) −1
(
) β u
c 1
ν(C T , γ T ) =
β u + β d β u + β d log eγu 1+r p1 + e γu c 2
1+r p2
p 1 + p 2
)
c
+ βd
3
β u + β d log eγd 1+r p3 + e γd c 4
1+r p4
. (6.28)
p 1 + p 2
Manipulations also show that
γ u β d
β u γ d + β d γ u = 1 + r − ξd
ξ u − ξ d
= S 0(1 + r) − S d
S u − S d = q, (6.29)
γ d β u
β u γ d + β d γ u = ξu − (1 + r)
ξ u − ξ d = Su − S 0 (1 + r)
S u − S d = 1 − q, (6.30)
x
31

where S u = S 0 ξ u and S d = S 0 ξ d .
Substituting (6.29) and (6.30) into (6.28), we obtain
ν(C T , γ T ) = q 1
c 1
γ u log eγu 1+r p1 + e γu c 2
1+r p2
p 1 + p 2
Similar to the argument used in Theorem 3.5, we can find that
+ (1 − q) 1
c 3
γ d log eγd 1+r p3 + e γd c 4
1+r p4
. (6.31)
p 1 + p 2
e γu c 1
1+r p1 + e γu c 2 [
]
1+r p2
= E Q e γ T C T ∣
1+r ∣S T = S u , (6.32)
p 1 + p 2
e γd c 3
1+r p3 + e γd c 4 [
]
1+r p4
= E Q e γ T C T
1+r ∣ ST = S d . (6.33)
p 3 + p 4
Substituting (6.32) and (6.33) into (6.31), we can rewrite (6.31) as
ν(C T , γ T ) = q 1 [
]
γ u log EQ e γ T C T
1+r ∣ ST = S u
This is (6.22).
+ (1 − q) 1 [
]
γ d log EQ e γ T C T
1+r ∣ ST = S d
[ ( )]
1
= E Q log E Q e γ T C T
1+r ∣ ST .
γ T
Theorem 6.8. Assume that the risk aversion coefficient γ T is of the form γ T = γ(S T ). Let Q be
the measure satisfying
Q[Y T |S T ] = P[Y T |S T ]. (6.34)
Let C T = C(S T , Y T ) be a claim to be priced under exponential utility with risk aversion coefficient
γ T . Then the value functions V 0 (x, γ T ) and V C T
(x, γ T ) have the representations
(
V 0 (x, γ T ) = − exp −
x
)
E Q [δ T ] − H(Q∗ |P) , (6.35)
(
V C T
(x, γ T ) = − exp − x − ν(C )
T ; γ T )
E Q − H(Q ∗ |P) , (6.36)
[δ T ]
where
( [ ])
1
ν(C T ; γ T ) = E Q log E Q e γ T C T
1+r ∣ ST ,
γ T
(6.37)
dQ ∗
dQ (ω) = δ T (ω)
E Q [δ T ] . (6.38)
32

Proof. We start with the representations (6.26) and (6.27) for V C T
(x, γ T ) and V 0 (x, γ T ), respectively.
Observe that
Observe also that
β u γ d + β d γ u ( β u + β d ) −1
β u + β d =
β u γ d + β d γ u )
(
= q 1
γ u + (1 − q) 1 ) −1
γ d
=
1
E Q [δ T ] .
β u
β d = γu (1 − q)
γ d ,
q
β u
β u + β d =
1 − q
γ d E Q [δ T ] ,
β d
β u + β d =
q
γ u E Q [δ T ] .
Let
where
q ∗ =
q = 1 + r − ξd
ξ u − ξ d
Define a measure Q ∗ with elementary probabilities
p i
q
γ u E Q [δ T ] ,
= S 0(1 + r) − S d
S u − S d .
qi ∗ = q ∗ , i = 1, 2, qi ∗ = (1 − q ∗ ) , i = 3, 4.
p 1 + p 2 p 3 + p 4
Substituting the above into (6.27), we obtain the representation (6.35):
( )
V 0 (x; γ T ) = −e − x p1 + p q ∗ ( )
E Q [δ T ] 2 p3 + p 1−q ∗
4
q ∗ 1 − q
(
∗
= − exp −
x
)
E Q [δ T ] − H(Q∗ |P) .
We find that the measure Q ∗ satisfies the properties:
(i) E Q [γ T (S T − S 0 (1 + r))] = 0,
(ii) E Q∗ [Y T |S T ] = E P [Y T |S T ].
p i
Alternatively, Q ∗ can also be constructed by its Radon-Nikodym density w.r.t.
measure Q:
the pricing
dQ ∗
dQ (ω i) = q∗
q = 1
γ u E Q , i = 1, 2,
[δ T ]
dQ ∗
dQ (ω i) = 1 − q∗
1 − q = 1
γ d E Q , i = 3, 4.
[δ T ]
33

Using the formulas (6.26) and (6.27) we get
(
)
V C T
(x; γ T )
V 0 (x; γ T ) = e γu c 1
1+r p1 + e γu c β d ( )
2
1+r
β u +β d p2 e γd c 3
1+r p3 + e γd c β u
4
1+r p4
p 1 + p 2 p 3 + p 4
= exp
( ν(CT ; γ T )
E Q [δ T ]
This, together with (6.35), implies (6.36).
)
.
Theorem 6.9. The writer’s optimal investment policy α C T ,∗ given in (6.25) has the decomposition:
where
Proof. Recall α C T ,∗ given in (6.25):
(
α C T ,∗ 1 β u e −γu x
=
S 0 (β u + β d ) log β d e −γd x
We can thus decompose α C T ,∗ as follows:
where
α 0,∗ =
α 1,∗ =
α 2,∗ =
β u +β d
α C T ,∗ = α 0,∗ + α 1,∗ + α 2,∗ , (6.39)
α 0,∗ = − ∂H(Q∗ |P)
∂S 0
E Q [δ T ], (6.40)
α 1,∗ = ∂ log EQ [δ T ]
x, (6.41)
∂S 0
( )
α 2,∗ = E Q ∂ ν(CT ; γ T )
[δ T ]
∂S 0 E Q . (6.42)
[δ T ]
e γu c 1
1+r p1 + e γu c 2
1+r p2
)
(
e γd c 3
1+r p3 + e γd c 4
1+r p4
) . (6.43)
α C T ,∗ = α 0,∗ + α 1,∗ + α 2,∗ , (6.44)
1
S 0 (β u + β d ) log βu (p 1 + p 2 )
β d (p 1 + p 2 ) , (6.45)
1
S 0 (β u + β d ) log e−γu x
e = − 1
−γd x S 0 (β u + β d (γu − γ d )x, (6.46)
(
)
1 e γu c 1
1+r p1 + e γu c 2
1+r p2 (p 3 + p 4 )
S 0 (β u + β d ) log (
) . (6.47)
e γd c 3
1+r p3 + e γd c 4
1+r p4 (p 1 + p 2 )
It remains to show that (6.45)-(6.47) coincide with (6.40)-(6.42), respectively. Towards this we
34

ecall
Also note the following computations:
q = 1 + r − ξd
ξ u − ξ d = S 0(1 + r) − S d
S u − S d ,
∂q
=
1 + r
∂S 0 S u − S d ,
β d
β u + β d = q
γ u E Q [δ T ] = q∗ ,
β u
β u + β d = 1 − q
γ d E Q [δ T ] = 1 − q∗ .
log βu (p 1 + p 2 )
β d (p 1 + p 2 ) = log (1 − q∗ )(p 1 + p 2 )
q ∗ , (6.48)
(p 1 + p 2 )
q ∗
H(Q ∗ |P) = q ∗ log + (1 − q ∗ ) log 1 − q∗
, (6.49)
p 1 + p 2 p 3 + p
( )
4
∂
∂q
H(Q ∗ ∗
|P) = − log (1 − q∗ )(p 1 + p 2 )
∂S 0 ∂S 0 q ∗ , (6.50)
(p 3 + p 4 )
∂q ∗
= ∂q 1
∂S 0 ∂S 0 γ u γ d (E Q [δ T ]) 2 , (6.51)
1
S 0 (β u + β d ) = ∂q 1
∂S 0 γ u γ d E Q [δ T ] = ∂q∗ E Q [δ T ].
∂S 0
(6.52)
Substituting (6.52), (6.48), (6.50) into (6.45) yields
We compute the partial derivative
It turns out from (6.46) that
α 0,∗ = − ∂H(Q∗ |P)
∂S 0
E Q [δ T ].
∂ log E Q [δ T ]
= ∂ log EQ [δ T ] ∂q
∂S 0 ∂q ∂S
( 0
1 1
=
E Q [δ T ] γ u − 1 ) ∂q
γ d ∂S 0
1
= −
S 0 (β u + β d ) (γu − γ d ).
α ∗,1 = ∂ log EQ [δ T ]
∂S 0
x.
Finally after tedious computations from (6.47), we have arrive at
α ∗,2 = (E Q [δ T ]) ∂q∗
∂S 0
{
= E Q ∂
[δ T ]
∂S 0
(
= E Q ∂ ν(CT ; γ T )
[δ T ]
∂S 0 E Q [δ T ]
{ [ (
∂
∂q ∗ E Q∗ log E Q∗ e γ T C T
1+r
[ (
∣
E Q∗ log E Q∗ 1+r
)
.
e γ T C T
∣S T
)]}
)]}
∣ ST
35

7 Indifference Prices and General Numeraires
The wealth
( )
ST
X T = x + α
1 + r − S 0
is expressed in the spot units. If we choose the stock S as the numeraire, the wealth will be
expressed as the number of shares of stock and not in the dollar amount. In other words, the
current wealth x is expressed as
X0 S = x =: x S
S T
and the terminal wealth X S T
is written as
X S T = x
S T
+ α
V S,C T
(x) = sup E
[− P exp
α
( 1
1 + r − S )
0
= x S + α
S T
The value function of a claim C T is then defined as
(
−γ S (S T )
where γ S (S T ) is the risk aversion associated with this unit.
(
X S T −
( 1
1 + r − S )
0
. (7.1)
S T
))]
C T
, (7.2)
S T (1 + r)
Definition 7.1. The indifference price of a claim C T is defined as the number of shares of stock,
ν S (C T ), such that the two value functions V S,C T
and V S,0 coincide. Namely, ν S (C T ) is the solution
to the equation
V S,0 (x S ) = V S,C T
(x S + ν S (C T )) (7.3)
for all initial number of shares x S ∈ R.
Theorem 7.2. Let Q S be a measure under which the discounted, by S T , riskless bond process B t
S t
(t = 0, T ) is a martingale, and at the same time, the conditional distribution of the nontraded asset
given the traded one is preserved w.r.t. the historical measure, i.e.,
Q S [Y T |S T ] = P[Y T |S T ]. (7.4)
Let C T = c(S T , Y T ) be a claim to be priced under exponential preferences with state-dependence risk
aversion coefficient γ S (S T ). Then the indifference price of C T is given by
( [ (
) ∣ ])
1
ν S (C T ) = E QS γ S (S T ) log EQS exp γ S C T ∣∣∣
(S T )
S T (7.5)
S T (1 + r)
Proof. The elementary probabilities of Q S are given by
where
qi S = q S p i
, i = 1, 2, qi S = (1 − q S ) , i = 3, 4,
p 1 + p 2 p 3 + p 4
q S =
( 1
ξ d − 1 ) ξ u ξ d
1 + r ξ u − ξ d .
p i
36

We can also use the Radon-Nikodym density w.r.t. Q to specify Q S :
Indeed, we have
Set
[ ] 1 + r
E QS S T
dQ S
dQ =
= E Q [
Then we can rewrite the value function V S,C T
V S,C T
(x) = sup E
[− P exp
α
S T
(1 + r)S 0
.
S T
· 1 + r
(1 + r)S 0
λ T = γT (S T )
S T
.
(−λ T
(
x + α
]
= 1 .
S T S 0
defined by (7.2) as
(
ST
1 + r − S 0
)
−
C ))]
T
,
1 + r
Similar to the proof of Theorem 6.8, we find
(
)
V S,C T
(x S ) = V C T
(x; λ T ) = − exp − x − ν(C T ; λ T )
E Q [λ −1
T ] − H(˜Q|P) ,
where
d˜Q
dQ =
λ−1 T
Q[λ −1
T ].
Since the wealth argument x in V C T
(x; λ T ) and in the price
( [ ])
1
ν(C T ; λ T ) = E Q log E Q e λ T C T
1+r ∣ ST
λ T
are expressed in the spot units, it turns out that for all claims C T ,
which implies that
ν(C T ; γ T ) = ν(C T ; λ T )
γ T = λ T ⇒ γ S (S T ) = γ T S T .
Now recall, that for any payoff G which is dependent only on S T ,
[ ] [ ]
E Q G(ST )
G(ST )
= S 0 E QS .
1 + r
S T
Therefore,
It turns out that
[ ( )]
1
ν(C T ; γ T ) = E Q log E Q e γ T C T
1+r ∣ ST
γ T
E QS [ 1
γ T S T
log E QS (
= (1 + r)S 0 E QS [ 1
γ T S T
log E QS (
e γ C
S(S T ) T
S T (1+r)
)]
∣ ST
e γ C
S(S T ) T
S T (1+r)
)]
∣ ST .
= ν(C T ; γ T )
(1 + r)S 0
= ν S (C T ).
37

References
[1] V. Henderson and D. Hobson, Utility Indifference Pricing: An Overview, in “Indifference
Pricing: Theory and Applications” Edited by R. Carmona, Princeton University Press, 2009,
pp. 44-74.
[2] J. Kallsen, J. Muhle-Karbey, and R. Vierthauer, Asymptotic power utility-based pricing and
hedging. (Preprint.)
[3] S. Malamud, E. Trubowitz and M. V. Wüthrich, Indifference pricing for power utilities.
(Preprint.)
[4] M. Monoyios, Utility indifference pricing with market incompletness. (Preprint.)
[5] M. Musiela and T. Zariphopoulou, The single period binomial model, in “Indifference Pricing:
Theory and Applications” Edited by R. Carmona, Princeton University Press, 2009, pp. 1-42.
38