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We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset are governed by a jump diffusion equation. We obtain the Radon-Nikodym derivative in the minimal martingale measure and a partial integrodifferential equation (PIDE) of European call option. In a special case, we get the exact solution for European call option by Fourier transformation methods. Finally, we employ the pricing kernel to calculate the optimal portfolio selection by martingale methods.

Option pricing problem is one of the predominant concerns in the financial market. Since the advent of the justly celebrated Black-Scholes option pricing formula in [

Different from the Black-Scholes framework, we use jump diffusion to describe the price dynamics of underlying assets and let the market of our model be incomplete; that is, it is not possible to replicate the payoff of every contingent claim by a portfolio, and there are several equivalent martingale measures. How to choose a consistent pricing measure from the set of equivalent martingale measures becomes an important problem. That means we need to find some criteria to determine one from the set of equivalent martingale measures in some economically or mathematically motivated fashion. Föllmer and Leukert (2000), Kallsen (1999), Cvitanić et al. (2001), and Bielecki and Jeanblanc (2008) in [

General equilibrium framework is also a popular method to deal with the option pricing in an incomplete market. Pan (2002), Liu and Pan (2003), and Liu et al. (2005) in [

Anther contribution of our paper is finding the optimal portfolio selection and terminal wealth by martingale methods. The popular methods to solve the optimal portfolio problem are stochastic control methods, which derive some complex partial differential equations (PDE). For example, see [

The rest of the paper is organized as follows. In Section

In this paper, we consider the financial market with the following two basic assets:

In this model, the jump process

Now, we want to find a unique optimal strategy hedging of contingent claims under risk-minimization criterion. Based on [

With the Doob-Meyer decomposition, the discounted risky asset price process,

We introduce the notions of minimal martingale measure in this section. Föllmer and Schweizer (1991) [

A local martingale measure

(i) The minimal martingale measure

(ii)

Using Theorem

The Radon-Nikodym derivative in the minimal martingale measure

The theory of the Girsanov transformation shows that the predictable process of bounded variation can also be computed in terms of

Throughout this paper, we make use of the notations that

From (

In the minimal martingale measure

From the Girsanov theorem, the Brown motion in the minimal martingale measure

Then (

In the minimal martingale measure

Substituting

Using (

In the minimal martingale measure

By the fact that the discounted price of the European call option is a martingale under

In the minimal martingale measure

The total derivative of the discounted option price is

We make the drift term be zero, since the discounted price of the European put option is a martingale. Then we obtain the equation in Theorem

It is difficult to get the solution of European call option

(i) In physical measure

(ii) In the minimal martingale measure

Then the stock price process (

The stock price process (

Using Fourier transformation methods, it is easy to obtain the solution of (

The pricing formula of European call option under Assumption

First, we denote

Denote

Hence,

Although the pricing formula (

Then, we try to find the relationship of central moments between the physical measure and risk-neutral measure which can help us to study the negative variance risk premium, the implied volatility smirk, and the prediction of realized skewness. The typical literature is Bakshi et al. (2003) in [

However, to the best of our knowledge, except Zhang et al. (2012) in [

The first moment, and the second, third, and fourth central moments of the continuously compounded return within

The first moment and the second, third, and fourth central moments of the continuously compounded return in the physical measure are given by

See Zhang et al. (2012) in [

The first moment and the second, third, and fourth central moments of the continuously compounded return within

To compute central moments in the physical measure, we have

Under Assumption

The investor maximizes his/her expected utility,

We suppose

Consider

We define the pricing kernel as follows:

In order to solve Problem

Consider

Solving Problem

If

The Lagrangian function of Problem

Hence,

With risk-minimization criterion, we employ the minimal martingale measure to solve the pricing problem in an incomplete market. Then we obtain the Radon-Nikodym derivative in the minimal martingale measure and a PIDE with respect to the European option. In a special case, we get the exact solution of European call option by Fourier transformation methods. Finally, we employ the pricing kernel to calculate the optimal portfolio selection by martingale methods.

This work is supported by the Fundamental Research Funds for the Central Universities (JBK130401).