The Valuation of American Options for a Class of Diffusion Processes

We present an integral equation approach for the valuation of American-style derivatives when the underlying asset price follows a general diffusion process and the interest rate is stochastic. Our contribution is fourfold. First, we show that the exercise region is determined by a single exercise boundary under very general conditions on the interest rate and the dividend yield. Second, based on this result, we derive a recursive integral equation for the exercise boundary and provide a parametric representation of the American option price. Third, we apply the results to models with stochastic volatility or stochastic interest rate, and to American bond options in one-factor models. For the cases studied, explicit parametric valuation formulas are obtained. Finally, we extend results on American capped options to general diffusion prices. Numerical schemes based on approximations of the optimal stopping time (such as approximations based on a lower bound, or on a combination of lower and upper bounds) are shown to be valid in this context.