Page 31 - IJOCTA-15-2

M. M. Parsa, K. Sayevand, H. Jafari, I. Masti / IJOCTA, Vol.15, No.2, pp.225-244 (2025)
On the other hand, the holder can benefit from is one of the most famous applications of Itˆo’s
call options to purchase the meant asset within a lemma. For more details see. 6
specified period of time at a specified price. Each Fractional calculus extends the traditional
call option has an ascending buyer and a descend- concepts of derivatives and integrals to noninteger
ing seller, while the put options have a descending orders, offering a powerful mathematical frame-
buyer and an ascending seller. For more details work for modeling complex systems with memory
see. 1,2 and hereditary properties. Differential equations
Moreover, the division of trading options ac- involving fractional derivatives 7–13 have become
cording to the time of application is as follows essential tools for exploring fractal dynamics and
(see): 3 geometry. Recently, fractional differential equa-
• European options: The options that are appli- tions (FDEs) have garnered significant attention
cable at the exact time specified in the contract due to their broad applicability in mathemati-
are called European options. cal modeling across diverse fields such as physics,
• American options: The options that apply at engineering, biological systems, and viscoelastic
maturity and at any time before maturity are materials. The fractional-order differential opera-
called American options. tor is now regarded as a more versatile extension
Although both American and European op- of the conventional integer-order operator. The
tions can be exercised at any time before being growing interest in this area has spurred extensive
expired, the potential value attached to the Amer- research, leading to numerous advancements. For
ican option is higher than that of the European instance, fractional integral inequalities in frac-
option at the same exercise price. tional calculus play a crucial role in developing
Modeling the financial market with stochastic innovative models and techniques in prominent
processes started at the beginning of the last cen- areas of computer science, including artificial in-
telligence, machine learning, and data science. To
tury. In several monographs, the standard Black-
Scholes (B-S) model has been investigated as a further advance fields like big data, artificial intel-
prototype example. In the standard B-S model, ligence, and machine learning, an understanding
it is assumed that s follows a geometric Gaussian of both fractional dynamics and fractional-order
14,15
process: 4 reasoning is indispensable. Non-integer-order
integral inequalities have emerged as powerful and
ds
− µ∆τ − σdW = 0, (1) versatile tools in both applied 16 and pure 17 math-
s
ematics. The application of the Caputo derivative
where s is the price of the underlying asset, σ
with a variable fractional order to time depen-
is the volatility of the process, µ is the drift rate
dent models of Ordinary Differential Equations
or the expected return on the asset, and dW is a
(ODEs) aims to improve the simulation accuracy
standard Brownian motion. As another example,
of dynamic systems exhibiting complex and non-
using the property of the Ito integral, the price of 18
linear temporal behaviors. This approach offers
a European vanilla option υ(s, τ) can be governed a deeper understanding and enhanced predictive
by the B-S model as follows: 5
capabilities for non-constant real-world phenom-
ena, driving the advancement of innovative scien-
2
∂υ 1 2 2 ∂ υ ∂υ tific and engineering solutions. Additionally, the
+ σ s + rs − rυ = 0,
∂τ 2 ∂s 2 ∂s (2) study in 19 investigates the increased convergence
s ∈ [0, ∞), τ ∈ [0, T), rate of the Caputo-based Newton solver for solv-
where r is the risk-free rate. Itˆo’s lemma is ing one-dimensional nonlinear equations, further
used to find the derivative of a time-dependent demonstrating the utility of fractional calculus in
function of a stochastic process. Under the sto- addressing challenging mathematical problems.
chastic setting that deals with random variables, With the discovery of the fractal structure for
Itˆo’s lemma plays a role analogous to the chain the stochastic process and financial field, FC and
rule in ordinary differential calculus. The Black- FPDEs have been introduced more and more into
Scholes formula is often used in the finance sector the stochastic models and financial theory. In
to evaluate option prices. Although the deriva- real financial data, we have jumped in a short pe-
tion of the Black-Scholes formula does not use sto- riod of time. On the other hand, the B-S model
chastic calculus, it is essential to understand the is applied to Gaussian shocks. Therefore, the
significance of the Black-Scholes equation which proposed B-S model will probably minimize the
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