Accurate and Efficient Finite Difference Method for the Black–Scholes Model with No Far-Field Boundary Conditions

A fast and accurate explicit finite difference scheme for the Black–Scholes (BS) model with no far-field boundary conditions is proposed. The BS equation has been used to model the pricing of European options. The proposed numerical solution algorithm does not require far-field boundary conditions. Instead, the computational domain is progressively reduced one by one as the time iteration increases. A Saul’yev-type scheme for temporal discretization and non-uniform grids for the underlying asset variables are used. Because the scheme is stable, practically sufficiently large time steps can be applied. The main advantages of the proposed method are its speed, simplicity, and efficiency because it uses a stable explicit numerical scheme without using far-field boundary conditions. In particular, the proposed method is suitable for nonlinear boundary profiles such as power options because it does not require far-field boundary conditions. To validate the speed and efficiency of the proposed scheme, standard computational tests are performed. The computational test results confirmed the superior performance of the proposed method.