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.