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Mathematical Problems in Engineering
Volume 2014, Article ID 484362, 7 pages
Research Article

Cubic Spline Method for a Generalized Black-Scholes Equation

Institute of Mathematics, Zhejiang Wanli University, Ningbo, Zhejiang 315100, China

Received 10 January 2014; Revised 6 February 2014; Accepted 6 February 2014; Published 6 March 2014

Academic Editor: Kim Meow Liew

Copyright © 2014 Jian Huang and Zhongdi Cen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.