Stability and Error Analysis of Operator Splitting Methods for American Options Under the Black–Scholes Model

Abstract

The operator splitting method has shown to be an effective approach for solving the linear complementarity problem for pricing American options. It has been successfully applied to various Black–Scholes models, and it is implementation friendly because the differential equation and the complementarity conditions are decoupled and easily solved on its own part. However, despite its popularity, no stability and error analysis is available for these operator splitting methods. The challenge mainly arises from the special splitting associated with the slack function and the complementarity constraints. In this paper, we establish stability results for the operator splitting schemes based on the backward Euler and BDF2 methods, as well as an error estimate for the scheme based on the backward Euler method. We also provide numerical experiments to demonstrate the convergence behaviors of the two operator splitting methods.