A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation

Abstract

In this paper, we evaluate and discuss different numerical methods to solve the Black–Scholes equation, including the θ-method, the mixed method, the Richardson method, the Du Fort and Frankel method, and the MADE (modified alternating directional explicit) method. These methods produce numerical drawbacks such as spurious oscillations and negative values in the solution when the volatility is much smaller than the interest rate. The MADE method sacrifices accuracy to obtain stability for the numerical solution of the Black–Scholes equation. In the present work, we improve the MADE scheme by using non-standard finite difference discretization techniques in which we use a non-local approximation for the reaction term (we call it the MMADE method). We will discuss the sufficient conditions to be positive of the new scheme. Also, we show that the proposed method is free of spurious oscillations even in the presence of discontinuous initial conditions. To demonstrate how efficient the new scheme is, some numerical experiments are performed at the end.