Stochastic Volatility Using Regime-Switching Diffusions

Abstract

This chapter aims to model stochastic volatility using regime-switching diffusions. Effort is devoted to developing asymptotic expansions of a system of coupled differential equations with applications to option pricing under regime-switching diffusions. By focusing on fast mean reversion, we aim at finding the “effective volatility." The main techniques used are singular perturbation methods. Under simple conditions, asymptotic expansions are developed with uniform asymptotic error bounds. The leading term in the asymptotic expansions satisfies a Black{Scholes equation in which the mean return rate and volatility are averaged out with respect to the stationary measure of the switching process. In addition, the full asymptotic series is developed. The asymptotic series helps us to gain insight on the behavior of the option price when the time approaches maturity. The asymptotic expansions obtained in this chapter are interesting in their own right and can be used for other problems in control optimization of systems involving fast-varying switching processes.