A fast compact difference scheme with unequal time-steps for the tempered time-fractional Black-Scholes model

• Published in: International journal of computer mathematics Vol. 101; no. 9-10; pp. 989 - 1011
• Main Authors: Zhou, Jinfeng, Gu, Xian-Ming, Zhao, Yong-Liang, Li, Hu
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02.10.2024; Taylor & Francis Ltd
• Subjects: Approximation; compact difference scheme; Exact solutions; exponential transformation; Finite differences; Kernel functions; Mathematical analysis; Mathematical models; nonuniform time steps; Numerical methods; Operators (mathematics); Tempered time-fractional B-S model; weak regularity
• ISSN: 0020-7160; 1029-0265
• DOI: 10.1080/00207160.2023.2254412

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.