A Closed-Form Solution to American Options under General Diffusions

Abstract

This paper investigates American option pricing under general diffusion processes with nonconstant dividend yield. Specifically, the underlying asset price is assumed to have both the dividend yield and the volatility to be functions of time and the underlying asset price. Using homotopy analysis in Topology, the determination of the optimal early exercise boundary and that of the American option price are separated in the valuation procedure. Then, an exact and explicit solution for American options on a dividend-paying stock is derived as a Maclaurin series. Meanwhile, the corresponding optimal early exercise boundary is also obtained in a closed-form solution. As the solutions are in series expansion, an auxiliary parameter is introduced to control the convergence region and convergence rate. For practical use, the Pade technique is employed to further accelerate the convergence speed. Examples are given to demonstrate the validity, effectiveness and flexibility of the proposed approach.

Keywords: American Option, General Diffusion Process, Homotopy Analysis Method

JEL Classification: G120, 130