An Efficient Numerical Model for the Black–Scholes Equations

In this paper, a novel numerical model for the Black–Scholes equations is developed. To address some potential issues that may arise when solving this equation using the conventional model, the original Black–Scholes equation is reformulated as a convection–diffusion equation. The Crank–Nicolson scheme is utilized to discretize the diffusion and source terms in time, and the convection term is dealt explicitly using the Adams–Bashforth method. Spatial derivatives for the diffusion term are approximated with the central difference scheme. However, the spatial discretization of the convection term can be chosen adaptively according to the problem setting and the parameter values. By employing various discretization schemes for the convection term, a series of models are constructed for solving the Black–Scholes equations. We demonstrated that the discrete matrix with the new models has a better diagonal dominance property than that of the conventional model, which is beneficial for the solution. Subsequently, the new model is validated with several single-asset and multiasset benchmark problems, and the numerical results are compared with analytic solutions. It is observed that the choice of the discretization scheme for the convection term has little impact on the computational efficiency but can be significant on the accuracy and robustness. We find that the new model shows clear advantages over conventional methods, particularly in stable market conditions.