The main aim of this paper is to construct a new computational approach for the numerical solution of generalized Black–Scholes equation. In this approach, the temporal variable is discretized using Crank–Nicolson scheme and spatial variable is discretized using sextic B-spline collocation method. Convergence analysis of the method is discussed. The proposed method is proved to be stable and has second-order convergence with respect to time variable and sixth order convergence with respect to space variable. To illustrate the applicability and efficiency of the proposed method, we consider some benchmark problems describing European call options. Numerical results verify the orders of convergence predicted by the analysis. Numerical results reveal that the present method provides better results as compared to some existing numerical methods based on B-spline collocation approach.

本文的主要目的是为广义Black-Scholes方程的数值解建立一种新的计算方法。在这种方法中，使用Crank–Nicolson方案离散时间变量，并使用六性B样条搭配方法离散时间变量。讨论了该方法的收敛性分析。所提出的方法被证明是稳定的，并且对于时间变量具有二阶收敛性，对于空间变量具有六阶收敛性。为了说明该方法的适用性和效率，我们考虑了一些描述欧洲看涨期权的基准问题。数值结果验证了分析预测的收敛阶数。