American Put Options with Regime-Switching Volatility

Abstract

In this paper we study the price of an American option with stochastic volatility. The nature of stochastic volatility in this paper is as follows: volatility can take two values and changes at the jump time of an independent Poisson process. Namely, the volatility of the underlying asset changes between two regimes, say 'high volatility regime' H and 'low volatility regime' L. We derive an analytic formula for the price of a perpetual put and prove that the price of the perpetual put is always higher in the high volatility regime than in the low volatility regime and the exercise boundary is lower in the high volatility regime than in the low volatility regime. We also show by numerical examples that the Richardson interpolation together with the randomization method by Carr (1998) provides a fast algorithm to approximate the price of an American put with finite expiry and compare our numerical results with the previous study by Bollen (1998).