fractal and fractional
Article
Financial Applications on Fractional Lévy Stochastic Processes
Reem Abdullah Aljethi 1,2 and Adem Kılıçman 1, *
1
2
*
Department of Mathematics and Statistics, Institute for Mathematical Research, Universiti Putra Malaysia,
UPM Serdang, Seri Kembangan 43400, Selangor, Malaysia; gs52345@student.upm.edu.my
Department of Mathematics, Imam Mohammad Ibn Saud Islamic University, Riyadh 11564, Saudi Arabia
Correspondence: akilic@upm.edu.my
Abstract: In this present work, we perform a numerical analysis of the value of the European style
options as well as a sensitivity analysis for the option price with respect to some parameters of the
model when the underlying price process is driven by a fractional Lévy process. The option price is
given by a deterministic representation by means of a real valued function satisfying some fractional
PDE. The numerical scheme of the fractional PDE is obtained by means of a weighted and shifted
Grunwald approximation.
Keywords: European option pricing; Lévy process; fractional diffusion equations; sensitivity analysis
1. Introduction and Notations
The pricing of financial instruments is a problem of significant interest for risk management and hedging and/or in trading derivatives expecting more income in finance
industries. Financial instruments are linked to underlying assets whose dynamics are
modeled by an exponential of a Lévy process of the form
1
d(log St ) = (µ − σ2 )dt + σdLt ,
2
Citation: Aljethi, R.A.; Kılıçman, A.
Financial Applications on Fractional
Lévy Stochastic Processes. Fractal
Fract. 2022, 6, 278. https://doi.org/
10.3390/fractalfract6050278
Academic Editor: Ricardo Almeida
Received: 23 February 2022
Accepted: 13 May 2022
Published: 22 May 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
(1)
under the risk neutral measure Q, where µ > 0 is the average compounded growth of
the stock St , dLt is the increment of Lévy and σ is the volatility of the risky asset, induced
by Lévy jump. We refer to Raible (2000) [1], Schoutens [2] (2003) and Eberlein (2014) [3]
and the references therein for more details on non-fractional Lévy models in finance. It is
well known that the distribution of a Lévy process is characterized by its characteristic
function, which is given by the Lévy–Khintchine formula. We can characterize the Lévy
when a Lt dependent random variable with time is a Lévy process if and only if it has
independent stationary increments with an exponential characteristic function given by
the so-called Lévy–Khintchine representation. Indeed, if ( Lt )t≥0 is a Lévy process on R
with characteristic triplet (m, σ, ν), then E[ei̟Lt ] = etψ(̟ ) , t ∈ R. The function ψ is called
the characteristic exponent and can be expressed as
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Licensee MDPI, Basel, Switzerland.
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ψ(̟ ) = im̟ +
σ2
(i̟ )2 +
2
Z ∞
−∞
(ei̟κ − 1 − i̟I|κ |<1 )ν(dκ ),
(2)
where
m ∈ R, σ ≥ 0 and ν is the Lévy measure that controls the jumps such that
R
min
1, κ 2 ν(dκ ) is finite and I A is the indicator function defined by
R
distributed under the terms and
conditions of the Creative Commons
I A ( x ) :=
1
0
if x ∈ A
otherwise.
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Fractal Fract. 2022, 6, 278. https://doi.org/10.3390/fractalfract6050278
https://www.mdpi.com/journal/fractalfract
Fractal Fract. 2022, 6, 278
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An Example of a Lévy Process: Lévy Stable Processes (LSP)
α,β
Let Lt be a Lévy α-stable process with skew parameter β and stability index α. We
shall take β = −1 and assume that 1 < α < 2. Consequently, for each n > 0 and 0 < t < T,
h
i
πα
−1
α α
E exp nσLα,
=
exp
−
tn
σ
sec
< +∞.
t
2
Recall that the density function νLS ( x ) =
is given by
(
νLS ( x ) =
ν(dx )
dx
ad
| x | 1+ α
bd
x 1+ α
of the stable Lévy process with α ∈ (0, 2)
for x < 0,
for x > 0
where 1 = a + b and d > 0. Substituting the above density in Equation (2) yields the
characteristic exponent of a LSP with β = a − b
ψLS (̟ ) = i̟m −
απ
σα |̟ |α
× [1 − iβtan( )sign(̟ )].
2
2
An equivalent from of ψLS (̟ ) is given below
ψLS (̟ ) = i̟m −
σα
[(i̟ )α (1 − β) + (−i̟ )α (1 + β)]
4cos( απ
2 )
where the condition β = −1, yield a = 0 and b = 1 (see Cartea et al. [4] for more
details), hence
1
σα [(2)(i̟ )α ].
(3)
ψLS (̟ ) = i̟m −
)
4cos( απ
2
The price of an European call option may be expressed as the risk-neutral conditional expectation
V (St , t) = e−r(T −t) EQ [max(ST − K, 0)|Ft ],
and the value of an European put option may be expressed as the risk-neutral conditional expectation
V (St , t) = e−r(T −t) EQ [max(K − ST , 0)|Ft ],
where the price of the stock follows the dynamic (1) and Ft is the completed and right
continuous natural filtration associated to the Lévy stable process. Other extensions to
this result have been done in many directions; see Abouagwa and Li [5] (2019) and also
Wu et al. [6] (2021).
Our contribution in this paper is to perform a numerical analysis of the value of the
European style options as well as a sensitivity analysis for the option price with respect
to the model parameters. Numerical methods have been studied using a finite difference
scheme for non-fractional jump diffusion and exponential Lévy models for option pricing
by Cont and Voltchkova [7] (2005). For the sensitivity analysis, we shall focus mainly on the
very well-known Greeks (Delta, Gamma and Vega) and refer to a relatively recent paper by
Shokrollahi et al. [8] (2015).
The rest of the article is structured as follows: Section 2 is devoted to the stochastic model we are interested in as well as the deterministic representation of the option
price by a real valued function satisfying some fractional PDEs. Section 3 presents the
numerical scheme used to discretize the fractional PDE by means of the weighted and
shifted Grunwald approximation. Section 4 deals with a numerical implementation to
solve the fractional diffusion equation. Moreover, a subsection concerning the numerical
computations of Greek parameters giving some information about sensitivity analysis
is added.
Fractal Fract. 2022, 6, 278
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2. The Stochastic Model and Deterministic Representation
−1
The dynamic of the stock price driven by the α-stable Lévy process Lα,
with addit
tional impact term is given for each 0 ≤ t < T by the following equation:
−1
dγ St = St (r − q)dtγ + σdLα,
+ λSt dβγt , S0 initial price,
(4)
t
γ
where 0 < γ < 1 and 1 < α ≤ 2, β t is the number of shares of the stock at time t and the
γ
term λdβ t (λ ≥ 0) stands for the impact price of the investor’s trading strategy (see Chen
et al. (2014) [9] for more details and specific parameters).
Throughout this paper, we only consider trading strategies that can be written in the
following form
γ
−1
dβ t = η dtγ + ζ dLα,
,
(5)
t
γ
γ
with initial number of shares β 0 and some constants η, ζ. A self-financing strategy θt , β t t≥0
represented by the wealth process (Vt )t≥0 for trader is given by
γ
Vt = θt St0 + β t St = V0 +
Z t
0
θu dSu0 +
Z t
γ
0
β u dSu ,
where
dSt0 = rSt0 dt, S00 = 1.
According to the recent paper by Aljedhi and Kılıçman (2020) [10], the wealth process
(Vt )t≥0 can be written as Vt = u(St , t) where u( x, t) is the solution of the fractional partial
differential equation (FPDE)
∂γ u( x, t)
∂u( x, t)
∂α u( x, t)
+A
= ru( x, t),
+B
γ
∂t
∂x
∂x α
(6)
under the conditions u( x, 0) = h( x ), and ut ( x, 0) = f ( x ). Here, A and B are constants and
α
σα
απ
their expressions are A = r − q + λη + σ2 sec( απ
2 ) and B = 2 sec( 2 ).
Notice also that based on the martingale pricing approach under a risk neural probability measure, u(St , t) can be represented as
u(St , t) = EQ [φ(ST )|Ft ],
where
φ( x ) :=
European call option,
European put option,
( x − K )+
(K − x )+ .
The first goal of this paper is to find numerical values of the price of European-style
option, based on implicit scheme for the (FPDE) (6) and the Euler–Maruyama scheme
for the (4). The second goal is to study the sensitivity of these European options while
parameters of the model change.
In our numerical analysis, we combine the method of Demirci et al. (2012) [11] and
the method adopted by Shen et al. (2012) [12] to solve our fractional partial differential
equation. In particular, in [11], they dealt with Caputo partial differential equations with
an initial condition of the form
D γ y ( t ) = g ( y ( t ), t ), y (0) = y0 .
It can be solved as a solution y∗ (ω ) to a non-fractional differential equation
tγ
y(t) = y∗
.
Γ ( γ + 1)
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Moreover, Jumarie (2010) [13] solved the equation,
dy = g(t)dtα , t ≥ 0, 0 < α ≤ 1, y(0) = y0 ,
where the integral of dtα has the form
Z t
0
g( x )dx α = α
Z t
0
(t − x )α−1 g( x )dx
(7)
and have used the expression
dα x = Γ(1 + α)dx.
Based on our specific structure of the strategy described in (5), the dynamic of the
stock price process (4) becomes
−1
−1
dγ St = St (r − q)dtγ + σdLα,
+ λSt (ηt dtγ + ζ t dLα,
).
(8)
t
t
It is well known that, using Itô’s formula to xt = log St , the (8) has a solution that can
be written as
Z T
(τ )γ
α,−1
x T = xt exp
,
(9)
(r − µλη − q) +
(σ + λζ t )dLu
Γ (1 + γ )
t
where τ = T − t. We shall use the Euler–Maruyama scheme to find some realizations of
the price process and then obtain some simulations of the underlying asset described in
Equation (9) by means of the following discrete scheme for 1 < α ≤ 2:
x n +1 = x n + υn +1
where x0 = log S0 , tn =
nT
N
1
(τn+1 )γ
−1
(r − q + λη − m) + υnα+1 (σ + λζ ) Lα,
n +1 ,
Γ (1 + γ )
for n = 0, . . . , N, τn+1 = T − tn+1 , υn+1 =
−1
α,−1
α,−1
Lα,
n +1 = L t n +1 − L t n .
(10)
σα
sec( απ
2 ) and
2
3. Numerical Discretization of the Fractional PDE
The price of the call or put European option associated to the underlying asset governed by the (9) is given by
u(St , t) = EQ [ h(ST )|Ft ],
where u( x, t) is the solution to the Fractional PDE (0 ≤ α, γ ≤ 2)
∂γ u( x, t)
∂u( x, t)
∂α u( x, t)
=
A
+ ru( x, t)
+
B
∂tγ
∂x
∂x α
(11)
with the conditions u( x, 0) = h( x ), and ut ( x, 0) = f ( x ). Here A and B are constants, and
their expressions are
A = r − q + λη +
σα
απ
sec(
),
2
2
B=
σα
απ
sec(
).
2
2
We shall use the finite difference method using the shifted Grunwald approximation
of the fractional derivative. To do this, we will be working with the notation uin = u( xmin +
ih, nτ ) with h as the step-size of the space variable and τ as the step-size of the time variable
i = 0, . . . , Nx and n = 1, . . . , Nt . We consider now (11) at ( xi , tn ), one gets:
∂un
∂α un
∂γ uin
= A i + B αi + ruin .
γ
∂t
∂x
∂x
(12)
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Before we move on to the method itself, we should first recall the definition for the
Caputo time-fractional derivative [12] as
1
∂γ u( x, tn+1 )
=
γ
∂t
Γ (1 − γ )
=
=
1
Γ (1 − γ )
n Z t k +1
∑
k =0 t k
n
u( x, tn+1 ) − u( x, tk )
∑
τ
k =0
n
τ −γ
Γ (2 − γ )
∂u
∂η ( x, η )
dη
(tn − η )γ
Z t
k +1
tk
dη
+ O(τ 2−γ )
(tn − η )γ
γ
∑ bk [u(x, tn+1−k ) − u(x, tn−k )] + O(τ2−γ )
k =0
γ
where the weights bk satisfy the following formula
γ
γ
bk = (k + 1)γ − k1−γ , b0 = 1,
k = 0, 1, · · · , n.
For the Riemann–Liouville space derivative with the weighted and shifted Grunwald
approximation [14,15], we use
1
∂α u( x, t)
= α
∂x α
h
m
∑ ωkα u(x, tm−k ) + O(h),
k =0
where the Grünwald–Letnikov weights ωkα = (−1)k
α
, k ≥ 0 are the coefficients of the
k
∞
power series of the generating function (1 − z)k = ∑ ωkα zk . These coefficients satisfy the
k =0
recursive formula
ωkα =
1−
α+1
ωkα−1 ,
k
ω0α = 1,
k = 1, · · · , m.
Hence, the numerical scheme of (12) is given by
τ −γ
Γ (2 − γ )
n −1
γ
∑ bn − k − 1
k =0
A n
B
ui+1 − uin−1 + α
uik+1 − uik =
2h
h
i +1
∑ ωkα uin−k+1 + ruin .
(13)
k =1
where
γ
γ
bk = (k + 1)γ − k1−γ , b0 = 1,
α+1
ωkα−1 , w0α = 1.
ωkα = 1 −
k
By developing the left side of the Equation (13), one gets
n −1
τ −γ
γ
uin − ∑ (bn−k−1 − bn−k )uik − bn−1 u0i
Γ (2 − γ )
k =1
!
=
A n
B
u
− uin−1 + α
2h i+1
h
i +1
∑ ωkα uin−k+1 + ruin .
(14)
k =1
Rearranging and substituting uin in Equation (14) by its numerical approximation Uin ,
one obtains for i ∈ {1, . . . , Nx − 1} and n ∈ {1, . . . , Nt − 1} :
i +1
n −1
γ
Uin − µ1 Uin+1 − Uin−1 − µ2 ∑ ωkα Uin−k+1 = µ3 ∑ (bn−k−1 − bn−k )Uik + µ0 Ui0 ,
k =1
U0n = u( x0 , tn ),
(15)
k =1
n
UN
= u( x Nx , tn ),
x
(16)
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where
µ0 =
bn − 1 τ γ Γ ( 2 − γ )
τ γ AΓ(2 − γ)
, µ1 =
,
1 − rΓ(2 − γ)
2h(1 − rΓ(2 − γ))
and
µ2 =
τ γ BΓ(2 − γ)
hα (1 − rΓ(2 − γ))
, µ3 =
1
.
1 − rτ γ Γ(2 − γ)
In order to develop the numerical result, we must write Equation (15) in matrix form
= U n−1 , which is completed with boundary conditions (16). We note that, unlike in
integer-order PDEs, the coefficient matrix M is a square matrix of size Nx , and will not be
tri-diagonal, which will slow down our numerical algorithm quite significantly. This is due
to the non-local nature of the time-fractional and space-fractional derivatives. The structure
of M is described in the following way, for i = 1, . . . , Nx − 2 and j = 1, . . . , Nx − 2
MU n
Mi,j
−µ2 ωiα− j+1
−µ1 − µ2 ω2α
=
1 − µ2 ω1α
α
− µ 1 − µ 2 ω0
0
if
if
if
if
if
j
j
j
j
j
< i−1
= i−1
=i
= i+1
> i+1
To complete this matrix with the boundary conditions, we have that M0,0 = 1, M0,j = 0
for j = 1, . . . , Nx − 1, M Nx −1,Nx −1 = 1, M Nx −1,j = 0 for j = 1, . . . , Nx − 1 and the second
member is given by
Uin−1 = µ3
n −1
γ
∑ (bn−k−1 − bn−k )Uik + µ0 Ui0 .
k =1
Now, all we have to deal with are the boundary conditions. For European-type options,
the behavior of numerical scheme is similar regardless of performing call and put options.
However, in the context of our Lévy stable model, this is not the case. Hence, we describe
each case separately.
•
•
For a call option, the payoff is defined by Π T = max(ST − K, 0). Since we must truncate
the spatial domain in order to make it workable from a numerical point of view, we will
use the following boundary conditions u( xmin , t) = 0 and u( xmax , t) = e xmax − Ke−r(T −t)
where xmin = − log(4K ) and xmax = log(4K ).
For a put option, the payoff is defined by Π T = max(K − ST , 0) and we will use
the following conditions u( xmax , t) = 0 and u( xmin , t) = Ke−r(T −t) − e xmin where
xmin = − log(4K ) and xmax = log(4K ).
4. Numerical Simulations and Discussions
In this section, we present some numerical simulations in order to validate the stochastic model with associated partial differential equations and illustrate the utility of the novel
model. In addition, we compute the price of call and put options in order see the difference
with the classical Black and Schole’s price. Finally, we perform the sensitivity analysis,
usually applied by financial engineers, that provide useful information for the investors.
To this end, we use the parameter settings reported in Table 1 for the one-year option.
Table 1. Parameters’ value.
S0
Strike K
r
q
λ
η
m
σ
β
α
100
80
0.02
1
10
0.01
0.02
0.15
−1
1.76
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Under the risk neutral equivalent probability measure and thanks to the martingale
property, the value of an option at the present time can be represented as
u(St , t) = EQ [(K − ST )+ |Ft )]
(17)
that is the conditional expected value of the discounted payoff under a chosen risk neutral
measure with respect to the all the information available in the market up to time t. This
representation allows us to use Monte Carlo techniques to estimate values of the option
price by simulating a large number of sample values of ST . Notice that this method uses
the risk-neutral valuation approach, which means that one has to take m = r.
In order to compare the numerical solutions using others numerical computations,
we use the Euler–Maruyama method for stochastic differential equation associated with
our Lévy stable process. However, the numerical solution of the FPDE is obtained using
the finite difference method cited in (15), which is reported in the Figure 1. Moreover, the
numerical solution of our SDE is obtained by the Euler–Maryuma scheme.
Figure 1. Comparison between numerical solutions (SDE and FPDE). Upon the comparison, using
the same parameters, of the left hand side of Equation (17) obtained by a FPDE approach and the
right hand side of Equation (17) computed using Monte Carlo simulation, we observe that both
solutions are close to each other.
In order to validate our Lévy model, we firstly solve the SDE and FPDE for call options.
Secondly, we present the solutions for put option. In Tables 2 and 3, we present the call and
put prices obtained by SDE and FPDE under the same parameters.
Table 2. Call price obtained with Euler scheme and finite difference. Upon comparison,
the two call prices are very closed to each other, with small error Emax which is defined by
Emax = max |u(xi , tn ) − Uin |.
Strike K
Call Price with SDE
Call Price with FPDE
Emax
80
43.9820
43.9789
0.0040
100
40.9454
40.8985
0.0068
110
39.8865
39.8783
0.0082
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Table 3. Put price obtained with Euler scheme and finite difference. Upon comparison, the two put
prices are very close to each other, with small error Emax which defined by Emax = max |u(xi , tn ) − Uin |.
Strike K
Put Price with SDE
Put Price with FPDE
Emax
80
26.0659
26.1043
0.0384
100
5.5968
5.5481
0.0487
110
2.7987
2.7683
0.0304
In what follows, in Figure 2, we plot the price of the call obtained by solving the
fractional PDE (12).
Figure 2. This figure shows the evolution of the call option price with respect to the time to maturity
using the FPDE representation. In comparison with the Black–Scholes call price when the underlying
is driven by the Brownian motion, the two prices have the same shape.
Secondly, we are interested in the calculus of the put price by using the FPDE and
SDE. We use the same parameters setting in Table 1 completed by the following conditions
1
.
u( xmax , 0) = 0 and u( xmin , 0) = Ke−rT − 4K
In Figure 3, we plot the price of the put obtained by solving the fractional PDE (12).
Fractal Fract. 2022, 6, 278
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Figure 3. This figure shows the evolution of the call put option price with respect to the time to
maturity using the FPDE representation. In comparison with the Black–Scholes put price when the
underlying is driven by the Brownian motion, the two prices have the same shape.
Approximation of Lévy Model Greeks
In the Black and Scholes model, the derivation and analytic expressions for the Greeks
for put and call prices can be done. We refer to De Olivera and Mordecki (2014) [16] for
the computation of Greeks using the Fourier transform approach. However, due to the
complexity of our model, we chose to use finite differences to approximate the derivatives.
Then, we use the finite differences to find the Greeks (Delta “∆”, Vega “Vega”, Gamma
“Γ”, Rho “Rho”). When this method is used to calculate Delta and Vega Greeks (first order
derivative), however, the computation time is increased because the option price must
be calculated more than once, multiple times in the case of the Gamma Greek (secondorder derivative).
In order to approximate the first-order derivative of Greek sensitivity of the option
price with respect to a parameter or to variable, we use the first-order central differences:
∆ Levy =
and
Vega Levy =
∂u(St , r, σ, t)
u(S + dS, r, σ, t) − u(S − dS, r, σ, t)
:=
,
∂S
2dS
u(S, r, σ + dσ, t) − u(S − dS, r, σ − dσ, t)
∂u(St , r, σ, t)
:=
,
∂σ
2dσ
(18)
(19)
and
u(S, r + dr, σ, t) − u(S, r − dr, σ, t)
∂u(St , r, σ, t)
:=
.
(20)
∂r
2dr
For the second-order derivative of Greeks, such as Gamma, we use the second order
central differences for a single variable:
Rho Levy =
Γ Levy =
∂2 u(St , r, σ, t)
u(S + dS, r, σ, t) − 2u(S, r, σ, t) + u(S − dS, r, σ, t)
:=
.
∂S2
(dS)2
(21)
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To illustrate the Greek sensitivity analysis for put and call option prices, we develop
MATLAB codes. It is informative to present visual simulations of the Greeks (Delta, Vega,
Gamma and Rho) in 3D. Indeed, the Figure 4 presents a plot of the Greeks for the considered
Lévy model.
In Figure 5, we plot the Greeks for the put option in 3D. We show that the Delta is
constant with value equal to 1.51 for put option’s price for Lévy model, but it oscillates for
Black–Scholes. We have also the same remarks for Gamma, Vega and Rho.
Figure 4. Lévy model Greeks for the call option: By analyzing the Greeks plots, we observe that the
call option’s price have much higher Delta values than out of the call option’s price of Black–Scholes
model, and this value oscillates around 2.5, which ranges between 2.49 and 2.51. Gamma reaches
its maximum when the underlying price is a little bit smaller, exactly equal to the strike of the call
option, and the chart shows that Gamma is significantly constant for the Lévy model. We display also
Vega as a function of the asset price and time to maturity for a call options with strike at 80 and we
highlight the fact that Vega is much higher for the call option’s price than for the call option’s price
for Black–Scholes. At last, Rho reaches its maximum when the underlying price is a little smaller, not
exactly equal to the strike of the call option’s price, and it is significantly higher with hyper-volatility
than for the call option’s price of Black–Scholes.
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Figure 5. Lévy model Greeks for the put option: in the figures above, we plotted the Greeks for the
put option in 3D. We observe that the Delta is constant with value equal to 1.51 for put option’s price
for Levy model, but it oscillates for Black–Scholes. We can see that the put–call parity is maintained
for the Lévy model: Vega(Call) = Vega(Put) and Gamma(Call) = Gamma(Put). We have a negative
Rho which ranges between −1000 and 0, and the figure displays its fluctuation with respect to the
underlying asset.
5. Conclusions
Through this work, the fractional diffusion equation in terms of the Lévy process is
solved. We also discussed the Euler–Maruyama equation for solving the price of fractional
financial derivatives of European options price.
Lévy processes have proven to be a suitable method that strikes the right balance
between the properties of the mathematical approach and the value of option evolution.
Furthermore, we made a comparison between European call and put option prices
based on the Lévy diffusion equation and provided numerical solutions of the fractional
partial differential equation that were derived by the fractional diffusion model.
Author Contributions: Both authors (R.A.A. and A.K.) contributed equally to this work. All authors
have read and agreed publishing this manuscript.
Funding: The second author would like to acknowledge that this research is partially supported by
Ministry of Higher Education under the Fundamental Research Grants Scheme(FRGS) with project
number FRGS/2/2014/SG04/UPM/01/1 and having vot number 5524674.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments: The authors would like to thank the reviewers for their fruitful comments that
have improved the presentation of the paper.
Conflicts of Interest: The authors declare there is no conflict of interest.
Fractal Fract. 2022, 6, 278
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