An improved approximate method for solving two-dimensional time-fractional-order Black-Scholes model: a finite difference approach

Din Prathumwan; Thipsuda Khonwai; Narisara Phoochalong; Inthira Chaiya; Kamonchat Trachoo; Din Prathumwan; Thipsuda Khonwai; Narisara Phoochalong; Inthira Chaiya; Kamonchat Trachoo

doi:10.3934/math.2024836

AIMS Mathematics

2024, Volume 9, Issue 7: 17205-17233. doi: 10.3934/math.2024836

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Research article

An improved approximate method for solving two-dimensional time-fractional-order Black-Scholes model: a finite difference approach

1.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
2.
Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham 44150, Thailand

Received: 30 January 2024 Revised: 30 April 2024 Accepted: 06 May 2024 Published: 17 May 2024
MSC : 26A33, 65M06, 91B26, 91G20, 91G60

In this paper, we considered the two-dimensional fractional-order Black-Scholes model in the Liouville-Caputo sense. The Black-Scholes model was an important tool in the financial market, used for determining option prices in the European-style market. However, finding a closed-form analytical solution for the fractional-order partial differential equation was challenging. To address this, we introduced an improved finite difference method for approximating the solution of the two-dimensional fractional-order Black-Scholes model in the Liouville-Caputo sense, based on the Crank-Nicolson finite difference method. This method combined the concepts of the finite difference method for solving the multidimensional Black-Scholes model and the finite difference method for solving the fractional-order heat equation. We analyzed the conditional stability and the order of convergence. Furthermore, numerical examples were provided to illustrate the determination of option prices.
- Crank-Nicolson,
- finite difference method,
- time fracional-order Black-Scholes model,
- numerical simulation,
- multi-dimensional model
Citation: Din Prathumwan, Thipsuda Khonwai, Narisara Phoochalong, Inthira Chaiya, Kamonchat Trachoo. An improved approximate method for solving two-dimensional time-fractional-order Black-Scholes model: a finite difference approach[J]. AIMS Mathematics, 2024, 9(7): 17205-17233. doi: 10.3934/math.2024836

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Abstract

In this paper, we considered the two-dimensional fractional-order Black-Scholes model in the Liouville-Caputo sense. The Black-Scholes model was an important tool in the financial market, used for determining option prices in the European-style market. However, finding a closed-form analytical solution for the fractional-order partial differential equation was challenging. To address this, we introduced an improved finite difference method for approximating the solution of the two-dimensional fractional-order Black-Scholes model in the Liouville-Caputo sense, based on the Crank-Nicolson finite difference method. This method combined the concepts of the finite difference method for solving the multidimensional Black-Scholes model and the finite difference method for solving the fractional-order heat equation. We analyzed the conditional stability and the order of convergence. Furthermore, numerical examples were provided to illustrate the determination of option prices.

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