Risk-Based Indifference Pricing in Jump Diffusion Markets with Regime-Switching

Abstract

This paper is concerned with risk indifference pricing of a European type contingent claim in an incomplete market, where the evolution of the price of the underlying stock is modeled by a regime-switching jump diffusion. The rationale of using such a model is that it can naturally capture the inherent randomness of a prototypical stock market by incorporating both small and big jumps of the prices as well as the qualitative changes of the market. While the model provides a realistic description of the real market, it does introduces substantial difficulty in the analysis. In particular, in contrast with the classical Black-Scholes model, there are infinitely many equivalent martingale measures and hence the price is not unique in our incomplete market. In particular, there exists a big gap between the commonly used sub- and super-hedging prices.\\ We approach this problem using the framework of risk-indifference pricing. By transforming the pricing problem to an equivalent stochastic game problem, we solve this problem via the associated Hamilton-Jacobi-Bellman-Issac equations. Consequently we obtain a new interval which is smaller than the interval from super- and sub-hedging.