A nonstandard finite difference scheme for a nonlinear Black–Scholes equation

In this paper we construct a numerical method based on a nonstandard finite difference scheme to solve numerically a nonlinear Black–Scholes equation modeling illiquid markets. In particular, the proposed method uses an exact difference scheme in the linear convection–reaction term and the spatial derivative is approximated using a nonstandard finite difference scheme. It is shown that the proposed numerical scheme preserves the positivity as well as stability and consistence. In order to illustrate the accuracy of the method, the numerical results are compared with other methods.