A posteriori grid method for a time-fractional Black-Scholes equation

Zhongdi Cen; Jian Huang; Aimin Xu; Zhongdi Cen; Jian Huang; Aimin Xu

doi:10.3934/math.20221148

AIMS Mathematics

2022, Volume 7, Issue 12: 20962-20978. doi: 10.3934/math.20221148

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A posteriori grid method for a time-fractional Black-Scholes equation

Institute of Mathematics, Zhejiang Wanli University, Ningbo 315100, China

Received: 26 July 2022 Revised: 10 December 2022 Accepted: 19 December 2022 Published: 27 December 2022
MSC : 65M06, 65M12, 65M15

In this paper, a posteriori grid method for solving a time-fractional Black-Scholes equation governing European options is studied. The possible singularity of the exact solution complicates the construction of the discretization scheme for the time-fractional Black-Scholes equation. The $ L1 $ method on an arbitrary grid is used to discretize the time-fractional derivative and the central difference method on a piecewise uniform grid is used to discretize the spatial derivatives. Stability properties and a posteriori error analysis for the discrete scheme are studied. Then, an adapted a posteriori grid is constructed by using a grid generation algorithm based on a posteriori error analysis. Numerical experiments show that the $ L1 $ method on an adapted a posteriori grid is more accurate than the method on the uniform grid.
- option valuation,
- Black-Scholes equation,
- fractional differential equation,
- a posteriori grid,
- a posteriori error analysis
Citation: Zhongdi Cen, Jian Huang, Aimin Xu. A posteriori grid method for a time-fractional Black-Scholes equation[J]. AIMS Mathematics, 2022, 7(12): 20962-20978. doi: 10.3934/math.20221148

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Abstract

In this paper, a posteriori grid method for solving a time-fractional Black-Scholes equation governing European options is studied. The possible singularity of the exact solution complicates the construction of the discretization scheme for the time-fractional Black-Scholes equation. The $ L1 $ method on an arbitrary grid is used to discretize the time-fractional derivative and the central difference method on a piecewise uniform grid is used to discretize the spatial derivatives. Stability properties and a posteriori error analysis for the discrete scheme are studied. Then, an adapted a posteriori grid is constructed by using a grid generation algorithm based on a posteriori error analysis. Numerical experiments show that the $ L1 $ method on an adapted a posteriori grid is more accurate than the method on the uniform grid.

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