Abstract
We consider the problem of risk indifference pricing on an incomplete market, namely on a jump diffusion market where the controller has limited access to market information. We use the maximum principle for stochastic differential games to derive a formula for the risk indifference price of a European-type claim .
1. Introduction
Suppose the value of a portfolio is given by where is the initial capital, is a semimartingle price process of a risky asset, is the number of risky assets held at time , and is the amount invested in the risk-free asset at time . Then, the cumulative cost at time is given by If -constant for all , then the portfolio strategies is called self-financing. A contingent claim with expiration date is a nonnegative -measurable random variable that represents the time payoff from seller to buyer. Suppose that for a contingent claim there exists a self-financing strategy such that , that is,Then, is the price of in the complete market, that is,where is any martingale measure equivalent to on the probability space .
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; Klöppel 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.
2. Statement of the Problem
Assume that a filtered probability space is given.
Definition 2.1. A nonnegative random variable on is called a European contingent claim.
From now on, we consider a European-type option whose payoff at time is some nonnegative random variable . In the rest of the paper, we will identify a contingent claim with its payoff function .
Let be the space of all equivalence classes of real-valued random variables defined on .
Definition 2.2 (see [8, 9]). A convex risk measure is a mapping satisfying the following properties, for :(i) (convexity) (ii) (monotonicity) if , then
If an investor sells a liability to pay out the amount at time and receives an initial payment for such a contract, then the minimal risk involved for the seller iswhere is the set of self-financing strategies such that , for some finite constant c and for .
If the investor has not issued a claim (and hence no initial payment is received), then the minimal risk for the investor is
Definition 2.3. The seller's risk indifference price, , of the claim is the solution of the equation Thus, is the initial payment that makes an investor risk indifferent between selling the contract with liability payoff 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 , which we consider, is of the following type.Theorem 2.4 (representation theorem [8, 9]). A map is a convex risk measure if and only if there exists a family of measures on and a convex “penalty” function with such that By this representation, we see that choosing a risk measure is equivalent to choosing the family of measures and the penalty function .
Using the representation (2.5), we can write (2.2) and (2.3) as follows:for a given penalty function .
Thus, the problem of finding the risk indifference price 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.
3. The Setup Model
Suppose in a financial market, there are two investment possibilities:
(i) a bond with unit price , ;
(ii) a stock with price dynamics, for , Here is a Brownian motion and is a compensated Poisson random measure with Lévy measure . The processes , , and are -predictable processes such that , for a.s. , andfor all
Let be a given subfiltration. Denote by , , the fraction of wealth invested in based on the partial market information being available at time Thus, we impose on to be -predictable. Then, the total wealth with initial wealth is given by the SDE
In the sequel, we will call a portfolio admissible if is -predictable, permits a strong solution of (3.3), and satisfiesas well asThe class of admissible portfolios is denoted by .
Now, we define the measures parameterized by given -predictable processes such thatwhereWe assume thatThen, by the Itô formula, the solution of (3.7) is given byWe say that the control is admissible and write if is adapted to the subfiltration and satisfies (3.8) and
We set
We now define two sets of measures as follows:where
In particular, by the Girsanov theorem, all the measures with are equivalent martingale measures for the -conditioned market where(see, e.g., [11, Chapter 1]).
We assume that the penalty function has the formfor some convex functions , such thatfor all
Using the -notation, problem (2.6) can be written as follows:
Problem A. Find and such thatwhere
We will relate Problem A to the following stochastic control problem:Using the -notation, the problem gets the following form.
Problem B. Find and such thatwhere
Define the Hamiltonian for Problem A byand the Hamiltonian for Problem B byHere is the set of functions such that the integrals in (3.23) and (3.24) converge. We assume that and are differentiable with respect to , , and . The adjoint equations (corresponding to , , and ) in the unknown adapted processes , , are the backward stochastic differential equations (BSDEs)
Similarly, the adjoint equations (corresponding to and in the unknown processes , , are given byLemma 3.1. Let and suppose that is a solution of the corresponding adjoint
equations (3.28). For all ,
define If ,
then is a solution of the adjoint equations (3.25),
(3.26), and (3.27). Then, the following relation
holds:
Proof. Differentiating both sides of (3.29),
we getComparing this with (3.25) by
equating the , , coefficients, respectively, we
getSubstituting (3.35) and (3.36) into
(3.34), we getSince ,
(3.37) is satisfied, and hence is a solution of (3.25).
Proceeding as above with the processes and ,
we getWith the values , ,
and defined as above, relation (3.39) is satisfied
if .
Hence, , ,
and are solutions of (3.29), (3.30), and (3.31),
respectively.
Equations (3.23) and (3.24) give the following relation
between and :Hence,Lemma 3.2. Let , ,
and be as in Lemma 3.1. Suppose
that, for all ,
the function has a maximum point at .
Moreover, suppose that the function has a minimum point at .
Then, Proof. The first-order conditions for a
maximum point of the function iswhere is the gradient operator. The first-order
condition for a minimum point of the function isthat is,Choose .
Then, by (3.46) and (3.48), we havethat is,Substituting the values , ,
and as in Lemma 3.1 into (3.50), we
getThis gives,
4. Maximum Principle for Stochastic Differential Games
Problem A is related to what is known as stochastic games studied in [12]. Applying Theorem 2.1 in [12] 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 and suppose that the adjoint equations (3.25), (3.26), and (3.27) admit solutions , and , respectively. Moreover, suppose that, for all , the following partial information maximum principle holds: Suppose Then is an optimal control and
Theorem 4.2. Let , be, respectively, solutions of adjoint equations (3.28), and let , , be defined as in Lemma 3.1. Suppose is concave. Let be an optimal pair for Problem A, as given in Lemma 3.2. Then, is optimal for Problem B.Proof. By Theorem 4.1 for Problem B, solves Problem B under partial information ifthat is, if there exists such thatLet , be as in Lemma 3.2. Then,Hence, by Lemma 3.1, Therefore, if we choosewe see that (4.6) holds with , as claimed.
5. Risk Indifference Pricing
Let be as in Theorem 4.2 with the corresponding state process . Suppose that is the state process corresponding to an optimal control . Then, the value function , which is defined by (3.17) and (3.18), becomesWe have that, for all ,since is an equivalent martingale measure for -conditioned market. On the other hand, the first part of (5.1) does not depend on the parameter . Hence, (5.1) becomes
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.Lemma 5.1. The relationship between the value function for Problem B and the value function for Problem A is
We now apply Lemma 5.1 to find the risk indifference price , given as a solution of the equationBy Lemma 5.1, this becomeswhich has the solutionIn particular, choosing (i.e., all measures are probability measures), we get the following.Theorem 5.2. Suppose that the conditions of Theorem 4.2 hold. Then, the risk indifference price for the seller of claim , , is given by