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

AbstractThe 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.Keywords: Tempered time-fractional B–S modelnonuniform time stepsexponential transformationcompact difference schemeweak regularity2020 AMS Subject Classifications: 65M0665M5026A3391G20 AcknowledgmentsThe authors would like to thank anonymous reviewers, Dr. Can Li and Dr. Jinye Shen whose insightful comments and careful proof-checks helped to improve the current paper. X.-M. Gu also thanks Prof. Dongling Wang for helpful discussions during his visiting to Xiangtan University.Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 Due to two known functions ϕ(τ) and φ(τ), it is easy to compute the term 0CDτα,λz(x,τ) in the analytical (or numerical) manner.2 See some related experimental results in our arXiv preprint https://arxiv.org/abs/2303.10592v2.3 Evaluation of the M-L function with 2 parameters: https://www.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function.Additional informationFundingThis work was supported by the Applied Basic Research Program of Sichuan Province [grant number 2020YJ0007] and the Sichuan Science and Technology Program [grant number 2022ZYD0006 and grant number 2023NSFSC1326].