A Fast Computational Scheme for Solving the Temporal-Fractional Black–Scholes Partial Differential Equation

In this work, we propose a fast scheme based on higher order discretizations on graded meshes for resolving the temporal-fractional partial differential equation (PDE), which benefits the memory feature of fractional calculus. To avoid excessively increasing the number of discretization points, such...

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Bibliographic Details
Published inFractal and fractional Vol. 7; no. 4; p. 323
Main Authors Ghabaei, Rouhollah, Lotfi, Taher, Ullah, Malik Zaka, Shateyi, Stanford
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.04.2023
Subjects
Approximation
Black–Scholes PDE
Differential calculus
Differential equations
financial derivative
Finite difference method
Fractional calculus
high-order
non-uniform discretization
Numerical analysis
Partial differential equations
Smoothness
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Summary:In this work, we propose a fast scheme based on higher order discretizations on graded meshes for resolving the temporal-fractional partial differential equation (PDE), which benefits the memory feature of fractional calculus. To avoid excessively increasing the number of discretization points, such as the standard finite difference or meshfree methods, and, at the same time, to increase the efficiency of the solver, we employ discretizations on spatially non-uniform meshes with an attention on the non-smoothness area of the underlying asset. Therefore, the PDE problem is transformed to a linear system of algebraic equations. We perform numerical simulations to observe and check the behavior of the presented scheme in contrast to the existing methods.
ISSN:2504-3110
2504-3110
DOI:10.3390/fractalfract7040323