On the price of risk under regime switching exponential levy model: a case of CGMY Process

In this work, we study the pricing of an option when the price dynamics of the underlying asset are governed by a regime switching exponential levy process. We suppose that the market parameters, for instance, the market interest rate, appreciation rate and volatility of the underlying risky asset are time dependent and are modulated by a continuous time, finite state hidden Markov chain. We firs study the problem in a general framework, that is, a general regime switching exponential levy models is incomplete and hence the equivalent martingale measure is not unique. We adopt the regime switching Esscher transform in order to determine an equivalent martingale pricing measure. This regime switching Esscher transform is taken conditional on the information about the Markov chain and as a result, we obtain two families of equivalent martingale measures. We study two families of equivalent martingale measure for a general regime-switching Levy model and this allows us to assess the rice of risk. We use the minimization of maximum entropy between an equivalent martingale measure and the real world measure over the different states in order to select appropriate Esscher parameters. We reformulate the problem to particular cases and study the regime switching Black-Scholes, regime switching jump diffusion and finally the regime switching CGMY model. Numerical experiments are conducted and their results reveal that the impact of pricing the regime switching risk o the price of an option is significant.