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 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) 方程近期被推广为一种变时间分数阶B-S方程，成为期权定价中一类有趣的数学模型。本研究提出一种快速数值方法来逼近变时间分数阶B-S模型的解。为在空间上获得高阶精度并克服解的初始弱奇异性，我们结合紧致算子与采用非均匀时间步长的L1逼近，从而构建出数值格式。所提出的差分格式被证明具有无条件稳定性收敛。此外，通过指数和逼近变时间Caputo分数阶导数中的核函数，得到一种降低计算成本的快速无条件稳定紧致差分方法。最后，数值结果验证了所提方法的有效性。