Asymptotic expansions of option price under regime-switching diffusions with a fast-varying switching process

摘要

This work aims to developing asymptotic expansions of solutions of a system of coupled differential equations with applications to option price under regime-switching diffusions. The main motivation stems from using switching diffusions to model stochastic volatility so as to obtain uniform asymptotic expansions of European-type options. By focusing on fast mean reversion, our effort is placed on finding the "effective volatility". 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, which will help us to gain insight on the behavior of the option price when the time approaches maturity. The asymptotic expansions obtained in this paper are interesting in their own right and can be used for other problems in control optimization of systems involving fast varying switching processes.