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The European option pricing problem with transaction costs is investigated for a risky asset price model with Lévy jump. By the aid of arbitrage pricing theory and the generalized Itô formula (which includes Poisson jump), the explicit solution to the risk asset price model is given. According to arbitrage-free principle, we first discretize the continuous-time model. Then, in each small time interval, the transaction costs are introduced. By using the

Recently, the stochastic differential equation theory has found more and more applications in many fields such as finance [

It should be pointed out that, in the frictional financial market, the models mentioned above have not taken the risks into account. For the pricing problem with transaction costs, we also need to consider the risk asset pricing model. Since the B-S [

In this paper, the risk asset pricing model with Lévy jump diffusion is considered and, following Leland’s [

The B-S model has been proposed in [

The following assumption is needed.

Suppose that the following conditions are satisfied:

risk-free rate

there is no dividend;

there are no transaction costs;

there is no arbitrage opportunity.

Consider the following asset model [

From (

We introduce the following lemmas that will be used to obtain the main results.

Let

Set

Defining

Let

One has

In an unstable financial market, investors use some financial instruments to hedge risks and price its value. The basic idea is to construct a portfolio to hedge risks. Similarly,

Assume that the asset price model is given by

Suppose that there is no arbitrage opportunity. We construct a portfolio

Then, we have

On the other hand, by using Itô formula, it can be obtained that

Substituting (

As an earlier derivation of option pricing is implemented under some “idealised” conditions, its applications are limited. In order to loosen these restrictions, the transaction costs are introduced in the derivation of option pricing. For example, the hedging strategy has been considered in [

In the interval

Our main results are given as follows.

For the risky asset price model (

By using Taylor’s formula, it follows from (

From Theorem

For

According to the definition of

In [

In this paper, the European option pricing problem with transaction costs has been studied. In order to make the pricing more practical, we have chosen the Lévy jump diffusion model instead of the standard geometric Brownian motion model. By using the

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by the National Natural Science Foundation of China under Grant 60974030, the Shanghai Rising-Star Program of China under Grant 13QA1400100, and the Fundamental Research Funds for the Central Universities of China.