The Black-Scholes (B-S) equation has been recently extended as a kind of tempered time-fractional B-S equations, which becomes 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 exact solution, we combine the compact difference operator with L1-type approximation under 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 the tempered Caputo fractional derivative is approximated by sum-of-exponentials, which leads to a fast unconditionally 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) 方程最近已被扩展为一类调和时间分数阶B-S方程，这成为期权定价中一个有趣的数学模型。本研究提供一种快速数值方法来逼近调和时间分数阶B-S模型的解。为在空间上实现高阶精度并克服精确解的弱初始奇异性，我们将紧致差分算子与非均匀时间步长下的L1型近似相结合，构建了数值格式。所提出差分格式的收敛性被证明是无条件稳定的。此外，通过指数和近似逼近调和Caputo分数阶导数中的核函数，我们得到一种快速无条件稳定的紧致差分方法，从而降低了计算成本。最后，数值结果验证了所提方法的有效性。