Page 30 - IJOCTA-15-2

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.2, pp.225-244 (2025)
https://doi.org/10.36922/ijocta.1661

RESEARCH ARTICLE

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

1
1*
Maryam Mahdavi Parsa , Khosro Sayevand , Hossein Jafari 2,3 ∗ , and Iman Masti 1
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
mm.parsa.90@gmail.com, ksayehvand@malayeru.ac.ir, jafari.usern@gmail.com, iman.masty@gmail.com

ARTICLE INFO ABSTRACT

Article History: An option is a financial contract or a derivative security entitling the owner to
Received: August 6, 2024 trade a certain quantity of a particular asset having a certain cost on or before
Accepted: January 21, 2025 a certain date. Therefore, in the last few years, not only mathematicians but
Published Online: March 20, 2025 also financial engineers have paid a great deal of attention to pricing options.
Keywords: Applying the fractal structure in the processes of stochasticity led to both
fractional calculus (FC) and fractional partial differential equations (FPDEs)
Black-Scholes equation
being associated with the stochastic models in financial theory. Thus, the
Fractional derivatives
beginning of the 20th century witnessed the use of stochastic processes to
Crank-Nicolson scheme
model the financial market. By studying the price behavior of assets, a model
Stability and convergence analysis
was presented, which is known as the Black-Scholes equation. The main focus
AMS Classification: of the present paper is the time-fractional Black-Scholes (TFB-S) model. The
26A33; 65Z05; 65N06 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 TFB-
S 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.

1. Introduction attention to pricing options. An option is a fi-
nancial contract or a derivative security entitling
As versatile financial products, options are com- the owner to trade a certain quantity of a partic-
pletely frequent and significant in the financial ular asset having a certain cost (exercise price) on
market. Hence, knowing their price is essential for or before a certain date (maturity date). There-
financial and economic institutions. Therefore, in fore, these contracts include the buyer and the
the last few years, not only mathematicians but seller. Call options allow the holder to purchase
also financial engineers have paid a great deal of the asset at a specified time at a specified price.
*Corresponding Author
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