Portfolio Strategy of Financial Market with Regime Switching Driven by Geometric Lévy Process

The problem of a portfolio strategy for financial market with regime switching driven by geometric L´evy 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 L´evy 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 Itˆo 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.


Introduction
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 [1].Extension of Markowitz's work to the multiperiod model has given by Li and Ng [2] which derived the analytical optimal portfolio policy.These previous researches were assuming that the underlying market has only one state or mode.But the real market might have more than one state and could switch among them.Then, portfolio policies under regime switching have been widely discussed.In a financial market model, the key process  that models the evolution of stock price should be a Brownian motion.Indeed, this can be intuitively justified on the basis of the central limit theorem if one perceives the movement of stocks.The analysis of Øksendal [3] was mainly based on the generalized Black-Scholes model which has two assets () and () as () = ()() and () = ()()+()()(), where () is a Brownian motion.In that case, Øksendal formulated optimal selling decision making as an optimal stopping problem and derived a closed-form solution.The underlying problem may be treated as a free boundary value problem, which was extended to incorporate possible regime switching by Guo and Zhang [4] and Pemy et al. [5] with the switching represented by a two-state Markov chain.The rate of return (t) in the above Black-Scholes models in [4,5] is a Markov chain which is different from the general one.As an application, Wu and Li [6,7] have given the strategy of multiperiod mean-variance portfolio selection with regime switching and a stochastic cash flow which depends on the states of a stochastic market following a discrete-time Markov chain.Being put in the Markov jump, Black-Scholes model with regime switching is much closer to the real market.
In recent years, Lévy process as a more general process than Brownian motion has been applied in financial portfolio optimization.Kallsen [8] gave an optimal portfolio strategy of securities market under exponential Lévy process.More specific than exponential Lévy process, a financial market model with stock price following the geometric Lévy process was discussed by Applebaum [9] in which a Lévy process () and geometric Lévy motion () =  () were introduced.Taking  to be a Lévy process could force our stock prices clearly not moving continuously, and a more realistic approach is that the stock price is allowed to have small jumps in small time intervals.Some applications of financial market driven by Lévy process are taken on life insurance.Vandaele and Vanmaele [10] show the real riskminimizing hedging strategy for unit-linked life insurance in financial market driven by a Lévy process while Weng [11] has analyzed the constant proportion portfolio insurance by assuming that the risky asset price follows a regime switching exponential Lévy process and obtained the analytical forms of the shortfall probability, expected shortfall and expected gain.Optimizing proportional reinsurance and investment policies in a multidimensional Lévy-driven insurance model is discussed by Bäuerle and Blatter [12].Moreover, under a generally method, Yuen and Yin [13] have considered the optimal dividend problem for the insurance risk process in a general Lévy process which shows that if the Lévy density is a completely monotone function, then the optimal dividend strategy is a barrier strategy.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 singlestock 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.

Problem Formulation
Assume that (Ω, F, ) is a complete probability space and {F  :  ≥ 0} is a nondecreasing family of -algebra subfields of F. {() :  ≥ 0} denotes a Markov chain in (Ω, F, ) as the regime of financial market, for example, the bull market or bear market of a stock market.Let  = {1, 2, . . ., } be the regime space of this Markov chain, and let Γ = (  ) × be the transition rate matrix which is satisfying where Δ > 0 is the increment of time, In this paper, we consider a financial market model driven by geometric Lévy process.The market consists of one risk-free asset denoted by  and  risky assets denoted by  1 ,  2 , . . .,   .The price process of these assets obeys the following dynamic equations in which the price process of the risky assets follows the geometric Lévy process; that is, where () is the price of  with the interest rate (, ()) and   () is the price of   with the expect rate of return   (, ()) and the volatility   (, ()), which follow the regime switching of financial market. 1 (),  2 (), . . .,   () are independent from each other.  () is the Brownian motion which is independent from {() :  ≥ 0}.Ñ (⋅, ⋅) is defined as below where   (, ) and   () indicate the number of jumps and average number of jumps within time  and jump range  of price process   (), respectively.That is where E is the expectation operator.Moreover, we assume that   (, ), (), and   () ( = 1, 2, . . ., ) are independent of each other.
Remark 1.The finance market model ( 2) is an extension of the B-S market model 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.
For finance market model (2), we introduce the concept of self-financing portfolio as follows.
Definition 2. A self-financing portfolio (, ) = (,  1 ,  2 , . . .,   ) for the financial market model ( 2) is a series of predictable processes that is, for each  > 0, and the corresponding wealth process {()} ≥0 , defined by is an Itô process satisfying Problem Formulation.In this note, we will propose a portfolio strategy for the financial market model ( 2) which is determined by a partial differential equation (PDE) of parabolic type by using Itô formula.The solvability of the PDE is researched by making use of variables transformation.Furthermore, the relationship between the solution of the PDE and the wealth process will be discussed.

Main Results and Proofs
In this section, we will give the following fundamental results.
To obtain the main result, we give the solution of (2) and the characteristic of the derivation (8) of the wealth process.
Thus, the rate of return and the volatility in ( 20) and ( 12) should be coincided, and hence We can easily get   ≥ 0 from (11), which together with the first equation of ( 21) and the independence of   ( = 1, 2, . . ., ) yields ( 16).
Next, we will show that (, ) is a self-financing strategy; that is, (12) Those together with (16) yield that (20) implies (12).The proof of Theorem 4 is completed.
Remark 5.In order to determine the portfolio strategy (, ) and obtain the final value (), from Theorem 4, we should find the solution of the PDF (17) with the final data (18).This is the key problem in the rest of this section.We have the following result in terms of method of variables transformation.Theorem 6.Let (, ()) in (2) be a constant .The function (, ),  ≤ ,  > 0 given by the following formula: is a solution of the general Black-Scholes equation (17) with the final data (18).
Proof.We are going to do some equivalent transformations of general B-S equation (17), in order to get an appropriate equivalent equation with analytic solutions.The procedure will be divided into four steps. Step