AccScience Publishing / IJOCTA / Online First / DOI: 10.36922/ijocta.1661
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RESEARCH ARTICLE

Analyzing the Black-Scholes equation with fractional coordinate derivatives using an implicit discrete design

Maryam Mahdavi Parsa1 ,  Khosro Sayevand1* ,  Hossein Jafari2,3* ,  Iman Masti1
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1 Faculty of Mathematics and Statistics, Malayer University, Malayer, Iran
2 Department of Applied Mathematicas, University of Mazandaran, Babolsar, Iran
3 Department of Mathematical Sciences, University of South Africa, UNISA0003, South Africa
IJOCTA, 1661 https://doi.org/10.36922/ijocta.1661
Submitted: 6 August 2024 | Accepted: 21 January 2025 | Published: 20 March 2025
© 2025 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

An option is a financial contract or a derivative security entitling the owner to trade a certain quantity of a particular asset having a certain cost on or before a certain date. Therefore, in the last few years, not only mathematicians but also financial engineers have paid a great deal of attention to pricing options. Applying the fractal structure in the processes of stochasticity led to both fractional calculus (FC) and fractional partial differential equations (FPDEs) being associated with the stochastic models in financial theory. Thus, the beginning of the 20th century witnessed the use of stochastic processes to model the financial market. By studying the price behavior of assets, a model was presented, which is known as the Black-Scholes equation. The main focus of the present paper is the time-fractional Black-Scholes (TFB-S) model. The difficulty or impossibility of providing an analytical solution for the aforesaid equation has made numerical solutions more helpful or even the only option. In this work, using the Crank-Nicolson scheme, a numerical solution with an implicit discrete design is demonstrated. We use the Fourier analysis method to investigate the stability of the implicit discrete design and demonstrate that the proposed method is unconditionally stable. The truncation error is checked. We also show that the numerical scheme suggested to solve the TFBS model is convergent. This method is the second order in space and 2 – β order in time, where 0 < β < 1 is the order of the time-fractional derivative. Finally, the accuracy as well as the efficiency considered for the method are evaluated by providing three examples and comparing them with previous works. Finally, the method’s accuracy and efficiency are assessed through three examples, with results compared to previous studies. Additional advantages of the method include its high computational speed, ease of implementation, and the reliability of obtaining an approximate solution, supported by stability proof.

Keywords
Black-Scholes equation
Fractional derivatives
Crank-Nicolson scheme
Stability and convergence analysis
Funding
None.
Conflict of interest
The authors declare that they have no conflict of interest regarding the publication of this article.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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