Stability of central finite difference schemes on non-uniform grids for the Black–Scholes equation

Abstract

This paper deals with stability in the numerical solution of the Black–Scholes partial differential equation. We investigate the semi-discretization on non-uniform grids with central, second-order finite difference schemes. Our stability analysis concerns the important question of whether, for the obtained semi-discrete matrices A, the norm ‖ e t A ‖ of the matrix exponential of tA ( t ⩾ 0 ) can be bounded suitably. Even though the considered semi-discretization of the Black–Scholes equation is widely known in the literature, a rigorous stability analysis for non-uniform grids appears to be lacking. In most cases, the matrices A are non-normal and an analysis based solely on the eigenvalues of A does not provide adequate estimates. In the present paper, we prove rigorous useful upper bounds on ‖ e t A ‖ for general non-uniform grids. Here we consider scaled spectral norms as well as the maximum norm. Our theoretical estimates are illustrated by ample numerical experiments, and practical conclusions about the stability of the schemes on non-uniform grids are derived. The results in this paper can directly be used in obtaining stability results also for time discretization schemes and are significant to a variety of applications beyond the Black–Scholes model.