Operator Splitting Methods for American Options with Stochastic Volatility

Abstract. Option pricing models with a stochastic volatility are more realistic than the Black-Scholes model which uses a constant volatility. The prices of options based on such models can be obtained by solving a parabolic partial differential equation. Particularly, we consider the model presented by Heston. The variables in these problems are the time, the underlying asset value, and the volatility. Due to the early exercise possibility of American options the arising problems are free boundary problems. This paper considers the numerical solution of this type of option pricing models. We study operator splitting methods for performing time stepping after a finite difference space discretization is done. The idea is to decouple the early exercise constraint for the prices at the grid points and the solution of the system of linear equations to separate fractional time steps. With this approach we can use any efficient numerical method to solve the systems of linear equations in the first fractional step and then make a simple update to satisfy the constraint in the second fractional step. This leads to more simple and efficient solution procedures. This paper studies the accuracy of the operator splitting methods without considering efficient solution procedures which is the topic of our future research. Numerical experiments compare the accuracy of our operator splitting methods with traditional unsplit implicit time discretizations. We demonstrate that the additional error due to the splitting does not essentially increase the time discretization error.