A robust nonuniform B-spline collocation method for solving the generalized Black-Scholes equation

In this paper, we present a robust and accurate numerical method for solving the generalized Black-Scholes equation governing option pricing. We use a horizontal method of lines to discretize the temporal variable and the spatial variable by means of an implicit finite difference method and a cubic B-spline collocation method, respectively. The method is shown to be stable and second-order convergent with respect to both variables. It approximates not only the option value but also some of its important 'Greeks' (Delta and Gamma), at the same time without any extra effort. Furthermore, the present paper efficiently treats the singularity of the nonsmooth pay-off function by condensing the mesh near the singularity. Numerical examples demonstrate the stability, convergence and robustness of the method.