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) 方程近期被扩展为一种回火时间分数阶B-S方程，成为期权定价中一个有趣的数学模型。本研究提出一种快速数值方法，用于逼近回火时间分数阶B-S模型的解。为在空间上实现高阶精度并克服解的初始弱奇异性，我们将紧致算子与基于非均匀时间步长的回火L1近似相结合，构建数值格式。所提出的差分格式被证明是无条件稳定的。此外，通过将回火Caputo分数阶导数中的核函数近似为指数和，我们得到一种快速无条件稳定的紧致差分方法，显著降低了计算成本。最后，数值结果验证了所提方法的有效性。