Option Pricing With Transaction Costs Under Regime Switching

The trend of general market level is a key factor which can govern the price movements of individual risky assets, so it is of practical importance to allow the the market parameters to respond to the movements of general market level. The Markov-modulated regime switching model can provide a better way to describe and explain the market environment. It offers one possible way to model the situation where the market parameters depend on a market mode which switches among a finite number of states and reflects that the interest rates, exchange rates, volatility and expected returns are related to the macro-economic environment, business cycles and other economic factors. In this paper, we mainly investigate the option pricing when the underlying risky assets are governed by a Markov-modulated geometric Brownian motion or a fractional Brownian motion process. That is to say, assume that in a given market economy state, the underlying assets are characterized by the standard geometric Brown motion or fractional Brown motion model. When the market economy status changes, the process of asset price switches among a number of states in the corresponding model. The main results are as follows:(1) The pricing of European option is studied when the volatility of the underlying asset depends on a hidden Markov process. It is assumed that the stock price is driven by a geometric Brown motion process and a fixed proportion of transaction costs is payed for the trade. Applying the risk-hedging principle, the system of coupled nonlinear parabolic partial differential equations is derived. By variable substitution, the equations are transformed into simpler form. The explicit difference scheme is constructed for the transformed. Then stability and consistency are both proved theoretically. Finally, the effectiveness and accuracy are tested through a numerical example.(2) The pricing of lookback option with transaction costs is studied when the price dynamics of underlying asset are governed by a Markov-modulated fractional Brownian motion process. Adopting the method of Leland, a system of N-coupled partial difference equations for the lookback put option prices over different economic states is obtained. It is very difficult for us to get the analytic solutions of the system of partial differential equations, and the strike price at the expiration date for the lookback option is uncertain, so we need to reduce the dimension of the resulting model through variable substitution, then construct Crank-Nicolson scheme to get thenumerical solution of the equations. Finally, we discuss the convergence of the numerical scheme and analyze the influence of market economic conditions, values of transaction costs and Hurst parameters on values of lookback put option. From the numerical results, it is observed that the difference in option values with and without regime switching is substantial.(3) We study the more realistic problem that the interest rates and the volatility of the underlying risky assets both depend on a hidden Markov process. In the trade,the transaction costs is not fixed, but decreases with the increase of the assets.To solve this problem, the system of coupled partial difference equations and corresponded boundary conditions is obtained through replication strategy. By variable substitution,the four-dimensional equations are transformed into three-dimensional. Then the differential term for the transformed is approximated by means of constructing explicit discrete format and the stability and consistency of the scheme are analyzed.Finally, we employ the Matlab software to discuss the impact of market parameter values on the lookback put option by a numerical example.There are totally 9 ?gures, 13 tables,87 references in this paper.