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Analyzing the Black-Scholes equation with fractional coordinate derivatives using . . .
and integrals, resulting in more precise model- than the fractional-order model. Specifically, the
ing methods. In traditional integer-order calcu- results show that fractional-order derivatives are
lus, market fluctuations are typically measured by superior to classical derivatives, offering greater
volatility, a key risk indicator in finance. Due to reliability and effectiveness in describing biologi-
the presence of long-range memory, heavytailed cal processes. Numerical simulations further re-
distributions, and leptokurtic characteristics in veal that the Caputo derivative provides more
financial markets, traditional methods may not precise results compared to several other applica-
fully capture the complexity of these dynamics. ble methods. To explore this further, we compare
The introduction of the fractional index param- the fractional derivative of Caputo with the frac-
eter in Fractional Black-Scholes Equations (FB- tional derivative of ψ-Hilfer’s in the Black-Scholes
SEs) allows for better modeling of long tails and equation. Additionally, the proposed method of-
non-normal features in stochastic processes. This fers advantages such as high computational speed,
approach improves the characterization of long- ease of implementation, and the assurance of ap-
term correlations and provides more accurate and proximate solutions due to its proven stability.
reliable methods for risk measurement. The so- Additionally, the proposed method offers advan-
lution techniques for FBSEs offer efficient and tages such as high computational speed, ease of
stable ways to analyze and predict market be- implementation, and the assurance of approxi-
havior. With the growing interest in FBSEs, mate solutions due to its proven stability. Future
coupled with ongoing advancements in computa- research will focus on applying this method to
tional methods and financial mathematics, fur- solve stochastic fractional differential equations,
ther improvements and innovations in this field as well as multidimensional stochastic fractional
are highly anticipated. This may involve lever- differential-integral equations. Additionally, at-
aging machine learning and neural network tech- tention will be given to exploring the similari-
niques to approximate solutions, as well as inte- ties, differences, and numerical treatments of frac-
grating hybrid and multifactor equations into the tional models for the Ebola virus, SARS, and
fractional-order Black-Scholes framework. We be- coronavirus.
lieve that as these equations continue to evolve,
they will play an increasingly vital role in in-
Acknowledgments
formed decision-making for financial professionals
and policymakers. None.
The main focus of this paper was on the TFB-
S model. In fact, the TFB-S model is a general- Funding
ized template proposed for the classical B-S model
in the realm of mathematical finance. Since the None.
analytical solution of this equation is difficult or
impossible, numerical treatments can be helpful Conflict of interest
and are sometimes the only choice.
At first, a Caputo derivative was obtained The authors declare that they have no conflict of
from the modified Riemann-Liouville derivative interest regarding the publication of this article.
using a variable transformation. Then, a descrip-
tion of how the issue is discretized was provided. Author contributions
Afterward, a numerical solution of an implicit
discrete design using the Crank-Nicolson scheme Conceptualization: Khosro Sayevand,
was demonstrated. We used the Fourier analysis Hossein Jafari
method to investigate the stability of the implicit Formal analysis: Khosro Sayevand, Iman Masti,
discrete design and showed that the proposed Maryam Mahdavi Parsa
method was unconditionally stable. The trunca- Investigation: Khosro Sayevand, Iman Masti,
tion error was checked. We also showed that the Maryam Mahdavi Parsa
proposed scheme for solving the TFB-S model was Methodology: Khosro Sayevand, Iman Masti,
convergent. This method was the second order in Hossein Jafari
space and 2 − β order in time, where 0 < β < 1 Validation: Khosro Sayevand, Iman Masti
was the order of the time-fractional derivative. Fi- Visualization: Khosro Sayevand
nally, by providing three examples and compar- Writing – original draft: Khosro Sayevand, Iman
ing their results, the efficiency and accuracy of Masti, Maryam Mahdavi Parsa
the method were evaluated. Our research indi- Writing – review & editing: Hossein Jafari, Iman
cates that the ordinary derivative is less accurate Masti
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