The problem of a portfolio strategy for financial market with regime switching driven by geometric Lévy process is investigated in this paper. The considered financial market includes one bond and multiple stocks which has few researches up to now. A new and general Black-Scholes (B-S) model is set up, in which the interest rate of the bond, the rate of return, and the volatility of the stocks vary as the market states switching and the stock prices are driven by geometric Lévy process. For the general B-S model of the financial market, a portfolio strategy which is determined by a partial differential equation (PDE) of parabolic type is given by using Itô formula. The PDE is an extension of existing result. The solvability of the PDE is researched by making use of variables transformation. An application of the solvability of the PDE on the European options with the final data is given finally.

To make a portfolio strategy is to search for a best allocation of wealth among different assets in markets. Taking the European options, for instance, how to distribute the appropriate proportions of each option to maximize total returns at expire time is the core of portfolio strategy problem. There are two points mentioned among the relevant literatures for portfolio selection problems: setting up a market model that approximates to the real financial market and the way of solving it.

Portfolio strategy researches are based on portfolio selection analysis given by Markowitz [

In recent years, Lévy process as a more general process than Brownian motion has been applied in financial portfolio optimization. Kallsen [

Among all the above literatures, those portfolios are always based on one risk-free asset and only one risky asset which may limit the chosen stocks. However, in a real financial market, there always exists more than one risky asset in a portfolio. That is why we are going to extend the single-stock financial market model to a multistock financial market model driven by geometric Lévy process which is more closer to the real market than proposed portfolios cited above. In this paper, we set up a general Black-Scholes model with geometric Lévy process. For the general Black-Scholes model of the financial market, a portfolio strategy which is determined by a partial differential equation (PDE) of parabolic type is given by using Itô formula. The solvability of the PDE is researched by making use of variables transformation. An application of the solvability of the PDE on the European options with the final data is given finally. The contributions of this paper are as follows. (i) The B-S market model is extended into general form in which the interest rate of the bond, the rate of return, and the volatility of the stock vary as the market states switching and the stock prices are driven by geometric Lévy process. (ii) The PDE determining the portfolio strategy and its solvability are extensions of the existing results.

Assume that

In this paper, we consider a financial market model driven by geometric Lévy process. The market consists of one risk-free asset denoted by

The finance market model (

For finance market model (

A self-financing portfolio

In this section, we will give the following fundamental results. For the sake of simplification, we write

To obtain the main result, we give the solution of (

The exact solutions of

To solve the second equation in (

Consider the price model (

Substituting (

Conversely, from (

This completes the proof of the above proposition.

Now we give the following fundamental results.

Consider the model (

Moreover, if

For the converse part, we assume that

We proof the direct part of Theorem

For

On the other hand, since our strategy is self-financing, the formula (

Thus, the rate of return and the volatility in (

We can easily get

From the first equation of (

Substituting (

Conversely, assume that

Firstly, we will show that a process

This proves (

Next, we will show that

By applying the Itô formula to the process

Furthermore, by (

Those together with (

In order to determine the portfolio strategy

Let

We are going to do some equivalent transformations of general B-S equation (

Recalling the relationship between

In this way we proved Theorem

As an application, we consider the European call option. In Theorem

For the European call option, the solution to the general Black-Scholes value problem (

For a European call option, we infer that

In this way, we have proved Corollary

The above result is about the European call option. A similar representation to those from the above corollary in the European put option case can be obtained by taking

In this paper, we have considered a financial market model with regime switching driven by geometric Lévy process. This financial market model is based on the multiple risky assets

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

This work is supported by the National Natural Science Foundation of China (Grants no. 61075105 and no. 71371046).