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

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