A new higher order compact finite difference method for generalised Black–Scholes partial differential equation: European call option

Abstract

This paper presents a high order numerical method based on a uniform mesh to obtain a highly accurate result for generalized Black–Scholes equation arising in the financial market. We first semi-discretize the underlying problem in the temporal direction by using Crank–Nicolson scheme and then discretize the resulting set of equations by using a high order compact finite difference method (HOCFDM). The stability of the scheme is analyzed. Convergence of the suggested method is proved. It is shown that the method is of order O ( Δ t 2 + h 4 ) . Some numerical experiments are carried out in order to illustrate the applicability and accuracy of the proposed method and to validate the theoretical results as well. It is shown that HOCFDM solution matches very well with the exact solution. Further, the rate of convergence predicted theoretically is the same as that obtained numerically. The computational time for our method in Matlab programming language is provided.