Operator Splitting Methods for Pricing American Options with Stochastic Volatility

Stochastic volatility models lead to more realistic option prices than the Black-Scholes model which uses a constant volatility. Based on such models a two-dimensional parabolic partial differential equation can derived for option prices. Due to the early exercise possibility of American option contracts the arising pricing problems are free boundary problems. In this paper we consider the numerical pricing of American options when the volatility follows a stochastic process. We propose operator splitting methods for performing time stepping after a finite difference space discretization is done. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach we can use any efficient numerical method to solve the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank-Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting does not increase essentially the error. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.

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