doi:10.1155/2008/821243 Research Article A Maximum Principle Approach to Risk Indifference Pricing with Partial Information

In this paper we consider the problem of risk indierence pricing on an incomplete market, namely on a jump diusion market where the controller has limited access to market information. We use the maximum principle for stochastic dierential games to derive a formula for the risk indierence price p seller (G,E) of an European-type claim G.


Introduction
Suppose the value of a portfolio π t , S 0 t is given by X π x t x π t S t S 0 t , 1.1 where x is the initial capital, S t is a semimartingle price process of a risky asset, π t is the number of risky assets held at time t, and S 0 t is the amount invested in the risk-free asset at time t.Then, the cumulative cost at time t is given by If P t p-constant for all t, then the portfolio strategies π t , S 0 t is called self-financing.A contingent claim with expiration date T is a nonnegative F T -measurable random variable G 2 Journal of Applied Mathematics and Stochastic Analysis that represents the time T payoff from seller to buyer.Suppose that for a contingent claim G there exists a self-financing strategy such that X π x T G, that is, Then, p is the price of G in the complete market, that is, where Q is any martingale measure equivalent to P on the probability space Ω, F t , P .In an incomplete market, an exact replication of a contingent claim is not always possible.One of the approaches to solve the replicating problems in an incomplete market is the utility indifference pricing.See, for example, Grasselli and Hurd 1 for the case of stochastic volatility model, Hodges and Neuberger 2 for the financial model with constraints, and Takino 3 for model with incomplete information.The utility indifference price p of a claim G is the initial payment that makes the seller of the contract utility indifferent to the two following alternatives: either selling the contract with initial payment p and with the obligation to pay out G at time T or not selling the contract and hence receiving no initial payment.
Recently, several papers discuss risk measure pricing rather than utility pricing in incomplete markets.Some papers related to risk measure pricing are the following: Xu 4 propose risk measure pricing and hedging in incomplete markets; Barrieu and El Karoui 5 study a minimization problem for risk measures subject to dynamic hedging; Kl öppel and Schweizer 6 study the indifference pricing of a payoff with a minus dynamic convex risk measure.See also the references in these papers.
In our paper, we study a pricing formula based on the risk indifference principle in a jump-diffusion market.The same problem was studied by Øksendal and Sulem 7 with the restriction to Markov controls.So the problem is solved by using the Hamilton-Jacobi-Bellman equation.In our paper, the control process is required to be adapted to a given subfiltration of the filtration generated by the underlying Lévy processes.This makes the control problem non-Markovian.Within the non-Markovian setting, the dynamic programming cannot be used.Here we use the maximum principle approach to find the solution for our problem.
The paper is organized as follows.In Section 2, we will implement the option pricing method in an incomplete market.In Section 3, we present our problem in a jump-diffusion market.In Section 4, we use a maximum principle for a stochastic differential game to find the relation between the optimal controls of the stochastic differential game and of a corresponding stochastic control problem.Using this result, we derive the relationship between the two value functions of the two problems above, and then find the formulas for the risk indifference prices for the seller and the buyer.

Statement of the problem
Assume that a filtered probability space Ω, F, {F t } 0≤t≤T , P is given.Definition 2.1.A nonnegative random variable G on Ω, F t , P is called a European contingent claim.
From now on, we consider a European-type option whose payoff at time t is some nonnegative random variable G g S t .In the rest of the paper, we will identify a contingent claim with its payoff function g.
Let F be the space of all equivalence classes of real-valued random variables defined on Ω. Definition 2.2 see 8, 9 .A convex risk measure ρ : F → R ∪ {∞} is a mapping satisfying the following properties, for X, Y ∈ F: i convexity If an investor sells a liability to pay out the amount g S T at time T and receives an initial payment p for such a contract, then the minimal risk involved for the seller is where P is the set of self-financing strategies such that X π x t ≥ c, for some finite constant c and for 0 ≤ t ≤ T .
If the investor has not issued a claim and hence no initial payment is received , then the minimal risk for the investor is T .

2.3
Definition 2.3.The seller's risk indifference price, p p seller risk , of the claim G is the solution p of the equation Thus, p seller risk is the initial payment p that makes an investor risk indifferent between selling the contract with liability payoff G and not selling the contract.
In view of the general representation formula for convex risk measures see 10 , we will assume that the risk measure ρ, which we consider, is of the following type.
Theorem 2.4 representation theorem 8, 9 .A map ρ : F → R is a convex risk measure if and only if there exists a family L of measures Q P on F T and a convex "penalty" function By this representation, we see that choosing a risk measure ρ is equivalent to choosing the family L of measures and the penalty function ζ.
Using the representation 2.5 , we can write 2.2 and 2.3 as follows: for a given penalty function ζ.
Thus, the problem of finding the risk indifference price p p seller risk given by 2.4 has turned into two stochastic differential game problems 2.6 and 2.7 .In the complete information, Markovian setting this problem was solved in 7 where the authors use Hamilton-Jacobi-Bellman-Isaacs HJBI equations and PDEs to find the solution.In our paper, the corresponding partial information problem is considered by means of a maximum principle of differential games for SDEs.

The setup model
Suppose in a financial market, there are two investment possibilities: i a bond with unit price S 0 t 1, t ∈ 0, T ; ii a stock with price dynamics, for t ∈ 0, T ,

3.1
Here B t is a Brownian motion and N dt, dz N dt, dz − ν dz dt is a compensated Poisson random measure with Lévy measure ν.The processes α t , σ t , and γ t, z are F t -predictable processes such that γ t, z > −1, for a.s.t, z, and for all T ≥ 0. Let E t ⊆ F t be a given subfiltration.Denote by π t , t ≥ 0, the fraction of wealth invested in S t based on the partial market information E t ⊆ F t being available at time t.Thus, we impose on π t to be E t -predictable.Then, the total wealth X π t with initial wealth x is given by the SDE

3.3
In the sequel, we will call a portfolio π ∈ P admissible if π is E t -predictable, permits a strong solution of 3.3 , and satisfies as well as The class of admissible portfolios is denoted by Π.Now, we define the measures Q θ parameterized by given F t -predictable processes θ θ 0 t , θ 1 t, z such that where We assume that

3.8
Then, by the It ô formula, the solution of 3.7 is given by

3.9
We say that the control θ θ 0 , θ 1 is admissible and write θ ∈ Θ if θ is adapted to the subfiltration E t and satisfies 3.8 and We set

3.11
We now define two sets L, M of measures as follows:

3.13
In particular, by the Girsanov theorem, all the measures Q θ ∈ M with E K θ T 1 are equivalent martingale measures for the E t -conditioned market S 0 t , S 1 t , where see, e.g., 11, Chapter 1 .
We assume that the penalty function ζ has the form Using the Y t -notation, problem 2.6 can be written as follows: where λ t, θ 0 t, y , θ 1 t, y, z , y, z ν dz .

3.19
We will relate Problem A to the following stochastic control problem:

3.20
Using the Y t -notation, the problem gets the following form.

Problem B.
Find Ψ E G t, y and θ ∈ M such that where

3.24
Here R is the set of functions r : 0, T × R → R such that the integrals in 3.23 and 3.24 converge.We assume that H and H are differentiable with respect to k, s, and x.The adjoint equations corresponding to θ, π, and Y t in the unknown adapted processes p t , q t , r t, z are the backward stochastic differential equations BSDEs 25 3.26

3.27
Similarly, the adjoint equations corresponding to θ and Y t in the unknown processes p t , q t , r t, z are given by

3.32
Proof.Differentiating both sides of 3.29 , we get

3.33
Comparing this with 3.25 by equating the dt, dB t , N dt, dz coefficients, respectively, we get 3.34

3.37
Since θ ∈ M, 3.37 is satisfied, and hence p 1 t is a solution of 3.25 .

Journal of Applied Mathematics and Stochastic Analysis
Proceeding as above with the processes p 2 t and p 3 t , we get γ t, z r 3 t, z ν dz 0, 3.39

3.40
With the values p 3 t , q 3 t , and r 3 t, z defined as above, relation 3.39 is satisfied if θ ∈ M. Hence, p 1 t , p 2 t , and p 3 t are solutions of 3.29 , 3.30 , and 3.31 , respectively.Equations 3.23 and 3.24 give the following relation between H and H:

3.42
Lemma 3.2.Let p 1 t , p 2 t , and p 3 t be as in Lemma 3.1.Suppose that, for all π ∈ R, the function Suppose that Y Y θ π ,π is the state process corresponding to an optimal control θ π , π .Then, the value function Φ E G , which is defined by 3.17 and 3.18 , becomes Λ θ * u, Y * u du − h K θ * T , S T K θ * T g S T − K θ * T X π T .

5.1
We have that, for all π ∈ Π,

5.3
We have proved the following result for the relation between the value function for Problem A and the value function for Problem B in the partial information case that is the same as in Øksendal and Sulem 7 for the full information case.
Let θ ∈ Θ and suppose that p t p 1 t , p 2 t is a solution of the corresponding adjoint equations 3.28 .For all π ∈ R, define If θ ∈ M, then p t p 1 t , p 2 t , and p 3 t is a solution of the adjoint equations 3.25 , 3.26 , and 3.27 .Then, the following relation holds: Problem A is related to what is known as stochastic games studied in 12 .Applying in 12 , Theorem 2.1 to our setting we get the following jump-diffusion version of the maximum principle of Ferris and Mangasarian type 13 .Theorem 4.1 maximum principle for stochastic differential games 12 .Let θ, π ∈ Θ × Π and suppose that the adjoint equations 3.25 , 3.26 , and 3.27 admit solutions p 1 t , q 1 t , r 1 t, z , p 2 t , q 2 t , r 2 t, z , and p 3 t , q 3 t , r 3 t, z , respectively.Moreover, suppose that, for all t ∈ 0, T , the following partial information maximum principle holds: * , π * : θ, π is an optimal control and 4.3 Theorem 4.2.Let p 1 t , p 2 t be, respectively, solutions of adjoint equations 3.28 , and let p 1 t , p 2 t , p 3 t be defined as in Lemma 3.1.Suppose θ → H t, Y t , θ, p t ; q t , r t, • is concave.Let θ π , π be an optimal pair for Problem A, as given in Lemma 3.2.Then,E ∇ θ H t, Y t , θ, p t , q t , r t, z − C t M θ θ θ | E E ∇ θ H t, Y t , θ, p t , q t ,r t, z − 2S t π t K θ t Mθ θ θ π t | E t .* , π * θ, π be as in Theorem 4.2 with the corresponding state process Y * Y θ * ,π * .
kx, 5.2 since 1/k Q θ is an equivalent martingale measure for E t -conditioned market.On the other hand, the first part of 5.1 does not depend on the parameter π.Hence, 5.1 becomes Lemma 5.1.The relationship between the value function Ψ E G t, y for Problem B and the value function Φ E G t, y for Problem A is In particular, choosing k 1 i.e., all measures Q ∈ L are probability measures , we get the following.Suppose that the conditions of Theorem 4.2 hold.Then, the risk indifference price for the seller of claim G, p seller risk G, E , is given by