Analysis of a Hybrid Finite Difference Scheme for the Black-Scholes Equation Governing Option Pricing

In this paper we present a hybrid finite difference scheme on a piecewise uniform mesh for a class of Black-Scholes equations governing option pricing which is path-dependent. In spatial discretization a hybrid finite difference scheme combining a central difference method with an upwind difference method on a piecewise uniform mesh is used. For the time discretiza- tion, we use an implicit difference method on a uniform mesh. Applying the discrete maximum principle and barrier function technique we prove that our scheme is second-order convergent in space for the arbitrary volatility and the arbitrary asset price. Numerical results support the theoretical results.