A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation

In this paper, the uniform cubic B-spline collocation method is implemented to find the numerical solution of the generalized Black-Scholes partial differential equation. We use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of a theta-method, theta is an element of [1/2, 1] (theta = 1 corresponds to the back-ward Euler method and theta = 1/2 corresponds to the Crank-Nicolson method), and a cubic B-spline collocation method on uniform meshes, respectively. The method corresponding to theta = 1 is shown to be unconditionally stable and first order accurate with respect to the time variable and second order accurate with respect to the space variable while the method corresponding to theta = 1/2 is shown to be unconditionally stable and second order accurate with respect to both the variables. Finally, the numerical examples demonstrate the stability and accuracy of the method. (C) 2011 Elsevier Ltd. All rights reserved.