The Black-Scholes (B-S) equation has been recently extended as a kind of tempered time-fractional B-S equations, which become an interesting mathematical model in option pricing. In this study, we provide a fast numerical method to approximate the solution of the tempered time-fractional B-S model. To achieve high-order accuracy in space and overcome the weak initial singularity of the solution, we combine the compact operator with a tempered L1 approximation with nonuniform time steps to yield the numerical scheme. The convergence of the proposed difference scheme is proved to be unconditionally stable. Moreover, the kernel function in tempered Caputo fractional derivative is approximated by sum-of-exponentials, which leads to a fast unconditional stable compact difference method that reduces the computational cost. Finally, numerical results demonstrate the effectiveness of the proposed methods.

翻译：Black-Scholes（B-S）方程最近被扩展为温和时分 Black-Scholes 方程的一种，成为期权定价中的一个有趣的数学模型。在本研究中，我们提供了一种快速数值方法来近似温和时分 B-S 模型的解。为了在空间方面获得高阶精度并克服解的弱初始奇异性，我们将紧致算子与温和 L1 近似和非均匀时间步长相结合，以得到数值格式。证明了所提出的差分格式的收敛性是无条件稳定的。此外，用指数和来逼近温和 Caputo 分数导数中的核函数，可以得到快速无条件稳定的紧致差分方法，减少了计算成本。最后，数值结果证明了所提出的方法的有效性。