This paper presents a numerical solution of the temporal-fractional Black–Scholes equation governing European options (TFBSE-EO) in the finite domain so that the temporal derivative is the Caputo fractional derivative. For this goal, we firstly use linear interpolation with the \((2-\alpha)\)-order in time. Then, the Chebyshev collocation method based on the second kind is used for approximating the spatial derivative terms. Applying the energy method, we prove unconditional stability and convergence order. The precision and efficiency of the presented scheme are illustrated in two examples.

中文翻译：

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切比雪夫搭配方法对时间分数Black-Scholes解的影响

本文提出了在有限域中控制欧洲期权的时分分数Black-Scholes方程（TFBSE-EO）的数值解，因此时导数是Caputo分数导数。为了这个目标，我们首先使用线性插值，其时间顺序为\（（2- \ alpha）\）。然后，基于第二种类型的切比雪夫搭配方法被用于近似空间导数项。应用能量法，证明了无条件稳定性和收敛阶。在两个示例中说明了所提出的方案的精度和效率。