Cubic Spline Method for a Generalized Black-Scholes Equation

Abstract EN

We develop a numerical method based on cubic polynomial spline approximations to solve a a generalized Black-Scholes equation.

We apply the implicit Euler method for the time discretization and a cubic polynomial spline method for the spatial discretization.

We show that the matrix associated with the discrete operator is an M-matrix, which ensures that the scheme is maximum-norm stable.

It is proved that the scheme is second-order convergent with respect to the spatial variable.

Numerical examples demonstrate the stability, convergence, and robustness of the scheme.