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Analyzing the Black-Scholes equation with fractional coordinate derivatives using . . .
changes which could be a significant shortcom- Black-Scholes model stands out as a critical in-
ing. The application of the fractal structure in strument, enabling investors to make more in-
stochastic processes has led to the introduction formed decisions. It is also one of the most promi-
of both FC and FPDEs into stochastic models nent financial models for pricing options-contracts
1
in the theory of finance. The non-local prop- that grant the right to buy or sell an asset (such as
erty of fractional derivatives (FD) and the recog- a stock) at a predetermined price on a future date.
nition of fractal characteristics in financial mar- The Black-Scholes model not only facilitates the
kets have driven the introduction and rapid ad- pricing of options but also provides insights into
vancement of fractional calculus in finance. Com- the potential future value of such transactions.
pared to the classical Black-Scholes (B-S) equa- However, while it is a powerful tool for analyz-
tion, fractional Black-Scholes equations (FBSEs) ing and pricing options, its limitations must also
offer a more adaptable framework for modeling be acknowledged. Given its significance, a thor-
market behavior by accounting for long-range de- ough examination of this equation is essential for
pendence, heavy-tailed distributions, leptokurtic financial professionals, and we will delve into its
features, and multifractality. This enhanced flex- applications and implications in this context.
ibility enables more accurate modeling of extreme
events and complex market dynamics. Conse-
quently, FBSEs provide a more precise depiction 2. The equivalence of the TFB-S model
of price fluctuations in real-world financial mar-
Consider the following TFB-S model:
kets, serving as a more reliable foundation for
derivative pricing and risk management. 20 In, 2
2
β
the authors provide a space fractional order B-S ∂ υ(s, τ) + σ 2 s 2 ∂ υ(s, τ) + rs ∂υ(s, τ)
model for exotic options in markets with jumps. ∂τ β 2 ∂s 2 ∂s (3)
In addition in, 3 the time-fractional order B-S − rυ(s, τ) = 0,
model is used to price the European call option. where (s, τ) ∈ R × (0, T) and 0 < β < 1, with
+
In, 21 the B-S-Merton time-fraction model is pre-
sented, considering the relationship between the the boundary conditions:
fractal structure and the propagation process of
(
the options. υ(0, τ) = λ(τ),
(4)
With the increasing use of the B-S model in υ(∞, τ) = ζ(τ),
finance, researchers felt the need to provide an where
analytical solution for these equations. HPM, 22 υ(s, T) = η(s). (5)
HAM, 23 and wavelet-based hybrid methods 24
were presented. However, these methods were
usually difficult to calculate. Hence, according
In this equation, τ is the current time, s is
to the mentioned topics, some strategies are pro-
the stock price, υ(s, τ) is the value of an Amer-
posed to approximate the B-S model numerically.
ican option price or a European put option, T
The study of the numerical approximate solu-
is the maturity date of the contract, σ repre-
tions of these models expanded and some tech-
sents the volatility of the returns from the un-
niques were proposed for approximating the B-S derlying asset, and r is the risk-free interest rate.
model numerically. Ref. 25 compares the numer- β
∂ υ(s, τ)
ical solutions for the space fractional B-S model Furthermore, is the modified Riemann-
∂τ β
and presents the convergence conditions of each Liouville derivative which is defined in the fol-
1
of these models. In Ref. 26 by using the projection
lowing form:
method, a fast numerical method for discretiza-
β
tion is proposed. In, 27 the first-order and second- ∂ υ(s, τ)
=
order implicit finite difference methods were de- ∂τ β

1 d R T
scribed to solve the spatial fractional B-S model.  υ(s, α)(τ − α) −β dα, 0 < β < 1,

 τ

Several other approaches were presented in 28-30 to Γ(1 − β) dτ
solve the fractional order in the B-S model. 
∂υ(s, τ)

The intricate nature of financial markets has  β , β = 1.

∂τ
long driven the search for tools to predict and (6)
manage risk effectively. Among these tools, the
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