This paper presents the numerical solution of the Black–Scholes partial differential equation (PDE) for the evaluation of European call and put options. The proposed method is based on the finite difference and Legendre wavelets aproximation scheme. We derive a matrix structure for the Legendre wavelets integral operator which has been widely used so far. Moreover, in order to use the payoff function, another operational matrix is derived. By the proposed combined method, the solving Black–Scholes PDE problem reduces to those of solving a Sylvester equation. The proposed algorithms show that in compared to literature methods, the proposed method is easy to be implemented and have high execution speed. Furthermore, we prove that the obtained Sylvester equation has a unique solution. In addition, the effect of the finite difference space step size to the computational accuracy is studied. For having suitable solution, the numerical solutions show that there is no need to select very small step size. Also only a small number of basis functions in the Legendre wavelets series is needed. The numerical results demonstrate efficiency and capability of the proposed method.

本文介绍了用于评估欧洲看涨期权和看跌期权的Black-Scholes偏微分方程（PDE）的数值解。所提出的方法基于有限差分和勒让德小波逼近方案。我们推导了迄今为止广泛使用的勒让德小波积分算子的矩阵结构。此外，为了使用收益函数，导出了另一个运算矩阵。通过提出的组合方法，求解Black-Scholes PDE问题简化为求解Sylvester方程的问题。所提出的算法表明，与文献方法相比，该方法易于实现，执行速度快。此外，我们证明所获得的Sylvester方程具有唯一解。此外，研究了有限差分空间步长对计算精度的影响。为了获得合适的解决方案，数值解决方案表明不需要选择非常小的步长。此外，在勒让德小波系列中只需要少量的基函数。数值结果证明了该方法的有效性和有效性。