Chaos, Solitons and Fractals 125 (2019) 108–118

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals

Nonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier.com/locate/chaos

Black–Scholes option pricing equations describ e d by the Caputo

generalized fractional derivative

Aliou Niang Fall b , Seydou Nourou Ndiaye

b , Ndolane Sene

a , ∗

a Laboratoire Lmdan, Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Faculté des Sciences Economiques et Gestion,

Dakar Fann, BP 5683, Senegal b Centre de Recherche Economique Appliquées/Laboratoire Ingénierie Financiére et Economique (LIFE)/Faculté des Sciences Économiques et de Gestion,

Université Cheikh Anta Diop de Dakar, Dakar Fann, BP 5683, Senegal

a r t i c l e i n f o

Article history:

Received 12 February 2019

Revised 13 April 2019

Accepted 22 May 2019

Keywords:

Fractional Black–Scholes equation

European option pricing

Analytical solutions

a b s t r a c t

Fractional Black–Scholes equation is a constructive financial equation. The model is used to determine

the value of the option without a transaction cost. The analytical solutions of the fractional Black–

Scholes equations have been addressed. The Caputo generalized fractional derivative has been used. The

homotopy perturbation method has been developed for obtaining the analytical solutions of the frac-

tional Black–Scholes equation ( BSE ) and the generalized fractional BSE . The analytical solutions of the frac-

tional BSE and the generalized fractional BSE have been represented graphically. The effect of the order ρof the generalized fractional derivative in the diffusion processes has been analyzed.

© 2019 Elsevier Ltd. All rights reserved.

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1. Introduction

Fractional calculus attracts many mathematicians in the last

decade. It extends the integer-order differentiation and integra-

tion to non-integer order. Many models from physics [13,18,35] ,

mathematical modeling [13] , finance and economics [38] , mechan-

ics [12] can be modeled using the fractional derivatives.Fractional

calculus is important and introduced in many applications. The

fractional differential equations have played an important role in

the modeling of real-life problems in different fields (finance, eco-

nomics, mathematics). Many investigations and advancements re-

lated to the applications of fractional calculus to physical phenom-

ena and biological systems exist. We recall a few of them. In [20] ,

Iyiola et al. have addressed the fractional cancer tumor models,

have proposed the analytical solutions and the approximate solu-

tions using the homotopy analysis method and have discussed the

use of fractional order derivative in modeling the medical and bio-

logical models. In [21] , Iyiola et al. have proposed the analytical

solution of the nonlinear fractional 2D heat equation with non-

local integral term using the homotopy analysis method. In [22] ,

Iyiola et al. have addressed the exact and approximate solutions of

fractional diffusion equations with fractional reaction term using

the homotopy analysis method. In [31] , Saad et al. have proposed

∗ Corresponding author.

E-mail address: ndolane.sene@ucad.edu.sn (N. Sene).

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https://doi.org/10.1016/j.chaos.2019.05.024

0960-0779/© 2019 Elsevier Ltd. All rights reserved.

he homotopy analysis method for solving a cubic isothermal auto-

atalytic chemical system. For more relevant and recent investiga-

ions in applications of fractional calculus in the real world prob-

ems, see Al-Mdallal et al. in [7–10] , Abdeljawad et al. in [1,2] ,

garwal et al. in [5] , and Aman et al. in [11] . There exist several

ractional differential equations in fractional calculus: the fractional

iscoelastic equations [35,39] , the fractional burger equations [15] ,

he fractional Euler equations, the fractional diffusion equations

18,19] , the diffusion reaction equations [28] , and many others. In

ur paper, we are interested by the fractional BSE [26] , and the gen-

ralized fractional BSE [16] . In the financial market, the value of an

ption has several utilities.There exist many methods for deter-

ining the value of an option. The formula proposed by Fischer

lack and Myron Scholes in 1973 [14] is more useful. The scien-

ific community realizes the importance of the formula in 1997.

hey have received economic medal field in 1997. In Finance, the

lack–Scholes equation ( BSE ) is used to evaluate the value of an

uropean option and the value of an American option. Historically,

oness in 1964, Samuelson in 1965, Chen in 1970, propose a for-

ula to estimate the value of an option. Later, the results stated

y Boness, Samuelson, and Chen were extended by Fischer Black

nd Myron Scholes. In 1973, they proposed an explicit formula to

valuate the value of an option. The formula has physical concepts.

t’s a parabolic diffusion equation. The formula was extended in its

eneral form by Cen et al. in [16] . As we will notice later, the BSE

s an ordinary differential equation. There exist many contributions

elated to the approximate solutions and the numerical schemes

A.N. Fall, S.N. Ndiaye and N. Sene / Chaos, Solitons and Fractals 125 (2019) 108–118 109

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or the Black–Scholes model [6,29,30] . The analytical solution is

lso provided in the literature by Kumar, Yavuz, Sawangtong et al.

n [26,32,36] .

The fractional calculus continues to attract many mathemati-

ians and econo-mists. Many of them conceived non-integer order

erivatives. We have the Riemann–Liouville derivative proposed by

iemann and Liouville,see the definition in [25] , the Caputo frac-

ional derivative proposed by Caputo, see the definition in [16] , the

aputo generalized fractional derivative proposed by Udita, see the

efinition in [25] , and the Hilfer fractional derivative proposed by

ilfer, see the definition in [25] . Some important investigations re-

arding the theory of generalized fractional derivatives in the Ca-

uto and Riemann senses exist, refer to [3,4,17,23,24] . In this pa-

er, we are interested by the formalistic fractional BSE and formal-

stic generalized fractional BSE both described by the Caputo gen-

ralized fractional derivative. We use the term ”formalistic” be-

ause we replace the integer order time derivative by the non-

nteger order time derivative. There exist several papers propos-

ng the approximate solution and the numerical solution of the

ractional BSE . We have the analytical solution proposed by Kumar

t al. using the homotopy perturbation method in [26] , the analyt-

cal solution proposed by Yavuz et al. using the homotopy analysis

ethod in [36] , a numerical scheme proposed by Akrami et al. in

6] , the approximate solution proposed by Özdemir et al. using the

ultivariante Padé method in [29] , an analytical solution for the

ractional BSE with two asset equations presented by Sawangtongn

t al. [32] , an implicit scheme proposed by Song et al. in [37] , the

pproximate solution proposed by Phaochoo et al. using the mesh-

es local Petrov Galerkin method in [30] . Cen et al. in [16] have

roposed an extension of the fractional BSE in [16] . In this paper,

e investigate the analytical solution of the fractional BSE and the

eneralized fractional BSE both described by the Caputo generalized

ractional derivative. Recently, the Laplace transform of the Caputo

eneralized fractional derivative (called the ρ-Laplace transform)

as introduced by Fahd and Thabet in [25] . In this paper, we com-

ine both the homotopy method and the ρ-Laplace transform. We

ainly focus — the effect of the order ρ in the diffusion pro-

esses. We analyze the impact generated by the order ρ in the

ractional BSE and the generalized fractional BSE . As proved with

he Riemann–Liouville fractional derivative, the Caputo fractional

erivative, the conformable fractional derivative, we will prove the

aputo generalized fractional derivative is an excellent compromise

o describe the diffusion process of an option.

The paper is structured as follows: in Section 2 , we recall the

ractional derivative operators which are necessary for this paper.

n Section 3 , we recall the constructive equations related to the

lack–Scholes models. In Section 4 , we recall the modified ho-

otopy perturbation method. In Section 5 , we propose the ana-

ytical solution of the fractional BSE described by the Caputo gen-

ralized fractional derivative and represent the obtained solution

raphically. In Section 6 , we propose the analytical solution of the

eneralized fractional BSE described by the Caputo generalized frac-

ional derivative and represent the obtained solution graphically. In

ection 7 , we give the concluding remarks.

. Generalized fractional derivatives

In this section, we introduce certain generalized fractional

erivatives. The generalized fractional derivatives are: the

iemann–Liouville generalized fractional derivative [25] and

he Caputo generalized fractional derivative [25] . We begin by

ecalling the generalized fractional integral operator, we will use it

o define the generalized fractional derivatives.

efinition 1 [25] . Consider the function defined by f : [ a, + ∞ [ −→ . The generalized integral of order α, ρ > 0 of the function f is

xpressed in the following form

( I α,ρ f ) (t) =

1

�(α)

∫ t

a

(t ρ − s ρ

ρ

)α−1

f (s ) ds

s 1 −ρ, (1)

here �( . . . ) is the Gamma function and t > a , and 0 < α < 1.

Observe that we recover the fractional integral when the order

= 1 . We have the following definition.

efinition 2 [25,33] . Consider the function defined by f :

a, + ∞ [ −→ R . The fractional integral of order α of the function f

s expressed in the following form

( I α f ) (t) =

(I α, 1 f

)(t) =

1

�(α)

∫ t

a ( t − s )

α−1 f (s ) ds, (2)

here �( . . . ) is the Gamma function and t > a , and 0 < α < 1.

Observe that we recover the classical integral when the order

= 1 . We use the generalized fractional integral to define the gen-

ralized fractional derivative in Riemann–Liouville sense. We have

he following definition.

efinition 3 [25] . Consider the function defined by f : [ a, + ∞ [ −→ . The generalized fractional derivative of order α, ρ > 0 of the

unction f in Riemann–Liouville sense is expressed in the follow-

ng form

a D

α,ρRL

f )(t) =

(I 1 −α,ρ f

)(t) =

1

�(1 − α)

(d

dt

)∫ t

a

(t ρ − s ρ

ρ

)−α

f (s ) ds

s 1 −ρ,

(3)

here �( . . . ) is the Gamma function and t > a , and 0 < α < 1.

Observe that we recover the Riemann–Liouville fractional

erivative when the order ρ = 1 . We have the following definition.

efinition 4 [25] . Consider the function defined by f : [ a, + ∞ [ −→ . The fractional derivative of order α of the function f in

iemann–Liouville sense is expressed in the following form

( a D

αRL f ) (t) =

1

�(1 − α)

(d

dt

)∫ t

a ( t − s )

−α f ( s ) ds, (4)

here �( . . . ) is the Gamma function, t > a , and 0 < α < 1.

Observe that we recover the classical first-order derivative

hen α converges to 1 [25] . In the following definition, we recall

he Caputo generalized fractional derivative.

efinition 5 [25] . Consider the function defined by f : [ a, + ∞ [ −→ . The Caputo generalized fractional derivative ( GFD ) of order α,

> 0 of the function f is expressed in the following form

( a D

α,ρ f ) (t) =

1

�(1 − α)

∫ t

a

(t ρ − s ρ

ρ

)−α

f ′ ( s ) ds, (5)

here �(.) is the Gamma function and t > a , and 0 < α < 1.

We recall the ρ-Laplace transform of the Caputo GFD , recently

ntroduced in [25] . It plays an important role in our studies. The

-Laplace transform of the Caputo GFD is defined by

ρ{ ( a D

α,ρ f ) (t) } = s αL ρ{ f (t) } −n −1 ∑

k =0

s α−k −1 ( γ n f ) (0) , (6)

here the ρ-Laplace transform of the function f is expressed as

ollows

ρ{ f (t) } (s ) =

∫ ∞

0

e −s t ρ

ρ f (t ) dt

t 1 −ρ. (7)

et’s f (t) = t p , with p ≥ 0. We replace it into Eq. (7) , we obtain the

ollowing ρ-Laplace transform

ρ{ t p } (s ) = ρp ρ

�(1 +

p ρ

)s 1+ p ρ

. (8)

110 A.N. Fall, S.N. Ndiaye and N. Sene / Chaos, Solitons and Fractals 125 (2019) 108–118

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We will use Eq. (8) in many calculations. Eq. (8) plays a fundamen-

tal role in the paper. We finish this section by the definition of the

Mittag–Leffler function. We have the following definition.

Definition 6 [33] . The Mittag–Leffler function with two parameters

is defined as follows

E α,β ( z ) =

∞ ∑

k =0

z k

�(αk + β) , (9)

where α > 0, β ∈ R and z ∈ C .

The Mittag–Leffler function will be used to express the analyti-

cal solutions of the fractional BSE and the generalized fractional BSE

both described by the Caputo GFD .

3. Constructive equations

In finance, the formalistic fractional BSE described by the Caputo

fractional derivative [29] is defined by the following equation

D

αc V +

σ 2

2

S 2 ∂ 2 V

∂S 2 + rS

∂V

∂S − rV = 0 , (10)

with the playoff function for call defined by

(S, T ) = max ( S − E, 0 ) , (11)

where V ( S, t ) denotes the value of an option at asset price S at

time t. T denotes the expiration date. E denotes the stock price of

the underlying stock. The function r denotes the risk-free interest

rate to expiration. The constant σ denotes the volatility of an un-

derlining asset. Furthermore, we add the following assumptions:

constant risk-less interest rate r , without transaction costs, possi-

bility to buy and to sell any number of stocks, no restriction short

selling at last, and we have an European option. We observe also

(0 , t) = 0 and V ( S, T ) ≈ S as S → ∞ . The fractional BSE defined by

Eq. (10) can be rewritten as a parabolic diffusion equation. Let’s

the following transformations

S = Ee x t = T − 2 τ

σ 2 V = Eu (x, t) . (12)

Then Eq. (10) can be rewritten as the following form

τ D

αu =

∂ 2 u

∂x 2 + (k − 1)

∂u

∂x − ku, (13)

with initial boundary condition defined by

u (x, 0) = max ( e x − 1 , 0 ) , (14)

where k denotes the balance between the free interest rate and the

volatility of the stocks. In this paper, we replace the Caputo frac-

tional derivative by the generalized fractional derivative. We obtain

a formalistic Black–Scholes model. The objective will be to prove

the Caputo generalized fractional derivative is a good comprise to

describe the diffusion process of the value of an European option.

The fractional BSE represented by the Caputo GFD is defined by

τ D

α,ρu =

∂ 2 u

∂x 2 + (k − 1)

∂u

∂x − ku, (15)

with initial boundary condition defined by

u (x, 0) = max ( e x − 1 , 0 ) . (16)

The generalized fractional BSE was introduced in the literature

by Cen in [16] . The model under consideration is defined by

τ D

α,ρu = −0 . 08 ( 2 + sin x ) 2 x 2

∂ 2 u

∂x 2 − 0 . 06 x

∂u

∂x + 0 . 06 u, (17)

with initial boundary condition defined by ( −0 . 06 )

u (x, 0) = max x − 25 e , 0 . (18) e

. Homotopy perturbation method with ρ-Laplace transform

In this section, we recall the modified homotopy perturbation

ethod used for solving the fractional differential equations. Let’s

he fractional differential equation defined by

α,ρc u (x, t) + Lu (x, t) + Nu (x, t) = g(x, t) , (19)

ith initial boundary condition defined as u (x, 0) = f (x ) . We con-

truct the following homotopy [27,34]

α,ρc u (x, t) + p { Lu (x, t) + Nu (x, t) − g(x, t) } = 0 , (20)

here the parameter p ∈ [0, 1] is called the homotopy parameter.

denotes the linear operator which includes other integer or non-

nteger derivatives. N represents the nonlinear operator. The func-

ion g is the source term. Note that when p = 0 , we obtain the

ollowing generalized fractional differential equation:

α,ρc u (x, t) = 0 . (21)

e notice when p = 1 , we recover the initial generalized fractional

ifferential equation defined by

α,ρc u (x, t) + Lu (x, t) + Nu (x, t) = g(x, t) .

y the classical perturbation method, the homotopy parameter p is

sed to expand the solution in the following form

(x, t) = u 0 (x, t) + pu 1 (x, t) + p 2 u 2 (x, t) + p 3 u 3 (x, t) + . . . (22)

ubstituting Eq. (22) into Eq. (20) , the functions u 0 , u 1 , u 2 , u 3 , ...

ecome the solutions of the following fractional differential equa-

ions.

p 0 : D

α,ρc u 0 (x, t) = 0 ;

p 1 : D

α,ρc u 1 (x, t) = −Lu 0 (x, t) − h 1 (u 0 (x, t)) + g(x, t) ;

p 2 : D

α,ρc u 2 (x, t) = −Lu 1 (x, t) − h 2 (u 0 (x, t) , u 1 (x, t)) ;

p 3 : D

α,ρc u 3 (x, t) = −Lu 2 (x, t) − h 3 (u 0 (x, t) , u 1 (x, t) , u 2 (x, t)) ;

. . . : . . . (23)

here the functions h 1 , h 2 , h 3 , ... satisfy the following condition

h (u 0 (x, t) + pu 1 (x, t) + p 2 u 2 (x, t) + p 3 u 3 (x, t) + . . . ) = h 1 ((u 0 (x, t))

+ ph 2 (u 0 (x, t) , u 1 (x, t)) + p 2 h 3 (u 0 (x, t) , u 1 (x, t) , u 2 (x, t)) + . . .

e determine the functions u 0 , u 1 , u 2 , u 3 , ... by applying at each

tep of Eq. (23) , the ρ-Laplace transform. The boundary conditions

or Eq. (23) are given respectively by

1 (x, 0) = u 2 (x, 0) = u 3 (x, 0) = . . . = 0 . (24)

. Analytical solutions for the fractional Black–Scholes

quation

In this section, we address the analytical solutions of the frac-

ional BSE and the generalized fractional BSE both described by the

aputo GFD . We will focus on the effect of the order ρ in the diffu-

ion process. We use the modified Homotopy method. The BSE for

he value of an option described by the Caputo GFD (as defined in

q. (15) ) is given by

α,ρu =

∂ 2 u

∂x 2 + (k − 1)

∂u

∂x − ku, (25)

ith initial boundary condition (as defined in Eq. (16) ) defined by

(x, 0) = max ( e x − 1 , 0 ) . (26)

he fractional BSE is an fractional diffusion equation. It is well

nown, the fractional BSE (25) can be rewritten as a heat parabolic

quation.

A.N. Fall, S.N. Ndiaye and N. Sene / Chaos, Solitons and Fractals 125 (2019) 108–118 111

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.1. Homotopy perturbation method combined with ρ-Laplace

ransform

In this section, we propose the analytical solution of the BSE de-

cribed by the Caputo GFD defined by Eq. (25) . We use the modi-

ed homotopy method described in previous section. The novelty

f this section is the use of the Laplace transform of the Caputo GFD

ecently introduced by Fahd et al. in [25] . The Laplace transform of

he Caputo GFD is called ρ-Laplace transform.

In first iteration p 0 , we solve the fractional differential equation

efined by

α,ρu 0 (x, t) = 0 , (27)

ith initial boundary condition defined by u 0 (x, 0) = u (x, 0) =ax ( e x − 1 , 0 ) . Applying the ρ-Laplace transform to both sides of

q. (27) , we have

αu 0 (x, s ) − s α−1 u 0 (x, 0) = 0

s αu 0 (x, s ) = s α−1 u 0 (x, 0)

s αu 0 (x, s ) = s α−1 max ( e x − 1 , 0 )

u 0 (x, s ) =

max ( e x − 1 , 0 )

s , (28)

pplying the inverse of ρ-Laplace transform to both sides of Eq.

28) , we obtain the analytical solution of the fractional differential

quation defined by Eq. (27)

0 (x, t) = max ( e x − 1 , 0 ) . (29)

n second iteration p 1 , we reduce the fractional differential equa-

ion defined by

α,ρu 1 =

∂ 2 u 0

∂x 2 + (k − 1)

∂u 0

∂x − ku 0

= max ( e x , 0 ) + (k − 1) max ( e x , 0 ) − k max ( e x − 1 , 0 )

= k max ( e x , 0 ) − k max ( e x − 1 , 0 ) , (30)

ith boundary condition defined by u 1 (x, 0) = 0 . Applying the ρ-

aplace transform to both sides of Eq. (30) , we have

αu 1 (x, s ) − s α−1 u 1 (x, 0) =

k max ( e x , 0 )

s − k max ( e x − 1 , 0 )

s

s αu 1 (x, s ) =

k max ( e x , 0 )

s − k max ( e x − 1 , 0 )

s

u 1 (x, s ) =

k max ( e x , 0 )

s 1+ α − k max ( e x − 1 , 0 )

s 1+ α . (31)

pplying the inverse of ρ-Laplace transform to both sides of Eq.

31) and using identity (8), we obtain the analytical solution of the

ractional differential equation defined by Eq. (30)

1 (x, t) =

[k max ( e x , 0 )

�(1 + α) − k max ( e x − 1 , 0 )

�(1 + α)

](t ρ

ρ

)α

. (32)

n third iteration p 2 , we reduce the fractional differential equation

efined by

α,ρu 2 =

∂ 2 u 1

∂x 2 + (k − 1)

∂u 1

∂x − ku 1

= −ku 1

=

[−k 2 max ( e x , 0 )

�(1 + α) +

k 2 max ( e x − 1 , 0 )

�(1 + α)

](t ρ

ρ

)α

, (33)

ith boundary condition defined by u 2 (x, 0) = 0 . Applying the ρ-

aplace transform to both sides of Eq. (33) , we have

αu 2 (x, s ) − s α−1 u 2 (x, 0) = − k 2 max ( e x , 0 )

s 1+ α +

k 2 max ( e x − 1 , 0 )

s 1+ α

s αu 2 (x, s ) = − k 2 max ( e x , 0 )

s 1+ α +

k 2 max ( e x − 1 , 0 )

s 1+ α

u 2 (x, s ) = − k 2 max ( e x , 0 )

s 1+2 α+

k 2 max ( e x − 1 , 0 )

s 1+2 α. (34)

pplying the inverse of ρ-Laplace transform to both sides of Eq.

34) and using identity (8), we obtain the analytical solution of the

ractional differential equation defined by Eq. (33)

2 (x, t) = −[

k 2 max ( e x , 0 )

�(1 + 2 α) +

k 2 max ( e x − 1 , 0 )

�(1 + 2 α)

](t ρ

ρ

)2 α

. (35)

e repeat the same reasoning for the other iterations p 3 , p 4 , ...

ote, we have to solve for all n > 2 the following fractional differ-

ntial equation

α,ρu n =

∂ 2 u n −1

∂x 2 + (k − 1)

∂u n −1

∂x − ku n −1 .

inally, the approximate solution of the fractional BSE described by

he Caputo GFD is given by

(x, t) = u 0 (x, t) + pu 1 (x, t) + p 2 u 2 (x, t) + p 3 u 3 (x, t) + . . . (36)

hen p converges to 1, we have the following analytical solution

(x, t) = max ( e x − 1 , 0 )

{1 − k

�(1 + α)

(t ρ

ρ

)α

− k 2

�(1 + 2 α)

(t ρ

ρ

)2 α

+ . . .

}

+ max ( e x , 0 )

{1 − 1 − k

�(1 + α)

(t ρ

ρ

)α

− k 2

�(1 + 2 α)

(t ρ

ρ

)2 α

+ . . .

}

= max ( e x − 1 , 0 ) E α

(−k

(t ρ

ρ

)α)

+ max ( e x , 0 )

{1 − E α

(−k

(t ρ

ρ

)α)}

.

he explicit analytical solution of the fractional BSE described by

he Caputo GFD is given by

(x, t) = max ( e x − 1 , 0 ) E α

(−k

(t ρ

ρ

)α)

+ max ( e x , 0 )

{1 − E α

(−k

(t ρ

ρ

)α)}

. (37)

Observe that we recover the analytical solution of the frac-

ional BSE described by the Caputo GFD when ρ = 1 . We have the

ollowing analytical solution

(x, t) = max ( e x − 1 , 0 ) E α( −kt α) + max ( e x , 0 ) { 1 − E α( −kt α) } . (38)

Observe that we recover the analytical solution of the classi-

al BSE [26] when α = ρ = 1 . We have the following approximate

olution

(x, t) = max ( e x − 1 , 0 ) e −kt + max ( e x , 0 ) {

1 − e −kt }. (39)

.2. Interpretation and utility of the approximate solutions

Let the fractional BSE Eq. (25) defined with the risk-free inter-

st rate r = 0 . 01 and the stock’s volatility σ = 0 . 03 . The balance

etween the free-interest rate and the stock’s volatility is given by

he following expression k =

2 r σ 2 . The classical values of the options

α = ρ = 1 ) are simulated in Fig. 1 . In Fig. 2 , we have depicted the

alues of the options for α = 1 and ρ = 0 . 65 . The values of the

ptions for α = ρ = 0 . 65 are depicted in Fig. 3 . The differences ex-

sting between Figs. 1–3 can be seen in Figs. 4 and 5 . We note the

112 A.N. Fall, S.N. Ndiaye and N. Sene / Chaos, Solitons and Fractals 125 (2019) 108–118

5

405 34.5

t

43.5 23

50

S

2.52 1

V

1.51

0.5

100

00

150

Fig. 1. Values of the options for α = ρ = 1 .

05

50

4 54.5

V

3 4

100

S

3.53

t

2 2.5

150

21.51 1

0.50 0

Fig. 2. Values of the options for α = 1 and ρ = 0 . 65 .

6

B

e

f

(

6

t

b

analytical solution of the fractional BSE and the analytical solution

of the classical BSE are in good agreement.

In Table 1 are assigned the values of the European option when

the Caputo GFD is used. For clarity, the used conversion to obtain

the value in Table 1 are

S = Ee x , t = T − 2 τ

σ 2 , V = Ev (x, τ ) (40)

We observe the order ρ has an acceleration effect in the diffu-

sion process when ρ < 1, see Fig. 4 . The order ρ has an retardation

effect in the diffusion process when ρ > 1, see Fig. 5 . Thus, it im-

pacts the price of the European option. We observe a slight delay

in the cost of the European option when ρ > 1.

. Analytical solutions for the generalized fractional

lack–Scholes equation

In this section, we address the analytical solution of the gen-

ralized fractional BSE proposed by Cen in [16] . The generalized

ractional differential equation under consideration is given by Eq.

17) with initial boundary condition defined by Eq. (18) .

.1. Homotopy perturbation method combined with ρ-Laplace

ransform

In this section, we solve the generalized fractional BSE described

y the Caputo GFD defined by Eq. (17) . As in Section 5 , we combine

A.N. Fall, S.N. Ndiaye and N. Sene / Chaos, Solitons and Fractals 125 (2019) 108–118 113

05

50

V

100

4 00.5

150

13 1.5

t

2

S

2 2.533.51 44.50 5

Fig. 3. Values of the options for α = ρ = 0 . 65 .

0 0.5 1 1.5 2 2.5 3 3.5 4

S

0

10

20

30

40

50

60

V

=1; =0.85

=1; =1

Fig. 4. Values of the options for α = 1 and ρ < 1, t = 0 . 5 years.

b

f

t

D

w

m

s

s

A

(

e

u

oth the homotopy perturbation method and the ρ-Laplace trans-

orm.

The first iteration p 0 , we solve the fractional differential equa-

ion defined by

α,ρu 0 (x, t) = 0 , (41)

ith boundary condition defined by u 0 (x, 0) = u (x, 0) =ax

(x − 25 e −0 . 06 , 0

). Applying the ρ-Laplace transform to both

ides of Eq. (41) , we have

αu 0 (x, s ) − s α−1 u 0 (x, 0) = 0

s αu 0 (x, s ) = s α−1 u 0 (x, 0)

s αu 0 (x, s ) = s α−1 max (x − 25 e −0 . 06 , 0

)u 0 (x, s ) =

max (x − 25 e −0 . 06 , 0

)s

. (42)

pplying the inverse of ρ-Laplace transform to both sides of Eq.

42) , we obtain the analytical solution of the fractional differential

quation defined by Eq. (41)

0 (x, t) = max (x − 25 e −0 . 06 , 0

). (43)

114 A.N. Fall, S.N. Ndiaye and N. Sene / Chaos, Solitons and Fractals 125 (2019) 108–118

0 0.5 1 1.5 2 2.5 3 3.5 4

S

0

10

20

30

40

50

60

V

=1; =5.025

=1; =1

Fig. 5. Values of the options for α = 1 and ρ > 1, t = 0 . 5 years.

Table 1

Values of European option.

α 1 1 1 1 1 1 1

ρ 0.65 1 0.65 1 0.95 1 1.5

σ 0.03 0.03 0.2 0.2 0.1 0.1 0.1

r 0.01 0.01 0.05 0.05 0.08 0.08 0.08

T 1 1 1 1 1 1 1

τ 0.5 0.5 0.5 0.5 0.5 0.5 0.5

E 100 100 100 100 100 100 100

S 100 100 100 100 100 100 100

u 0.8868 0.6708 0.9138 0.7135 0.9998 0.9997 0.9770

V 88.68 67.08 91.38 71.35 99.98 99.97 97.70

t

D

w

L

s

A

(

f

u

N

e

D

T

b

The second iteration p 1 , we solve the fractional differential

equation defined by

D

α,ρu 1 = −0 . 08 ( 2 + sin x ) 2 x 2

∂ 2 u 0

∂x 2 − 0 . 06 x

∂u 0

∂x + 0 . 06 u 0

= −0 . 0 6 x max ( 1 , 0 ) + 0 . 0 6 max (x − 25 e −0 . 06 , 0

)= −0 . 06 x + 0 . 06 max

(x − 25 e −0 . 06 , 0

), (44)

with boundary condition defined by u 1 (x, 0) = 0 . Applying the ρ-

Laplace transform to both sides of Eq. (44) , we have

s αu 1 (x, s ) − s α−1 u 1 (x, 0) = − 0 . 06 x

s +

0 . 06 max (x − 25 e −0 . 06 , 0

)s

s αu 1 (x, s ) = − 0 . 06 x

s +

0 . 06 max (x − 25 e −0 . 06 , 0

)s

u 1 (x, s ) =

0 . 06 x

s 1+ α −0 . 06 max

(x − 25 e −0 . 06 , 0

)s 1+ α . (45)

Applying the inverse of ρ-Laplace transform to both sides of Eq.

(45) and identity (8), we obtain the analytical solution of the frac-

tional differential equation defined by Eq. (44)

u 1 (x, t) =

[

− 0 . 06 x

�(1 + α) +

0 . 06 max (x − 25 e −0 . 06 , 0

)�(1 + α)

] (t ρ

ρ

)α

. (46)

u

In third iteration p 2 , we solve the fractional differential equa-

ion defined by

α,ρu 2 = −0 . 08 ( 2 + sin x ) 2 x 2

∂ 2 u 1 ∂x 2

− 0 . 06 x ∂u 1 ∂x

+ 0 . 06 u 1

= 0 . 06 u 1

=

[

− 0 . 06 2 x

�(1 + α) +

0 . 06 2 max (x − 25 e −0 . 06 , 0

)�(1 + α)

] (t ρ

ρ

)α

, (47)

ith boundary condition defined by u 2 (x, 0) = 0 . Applying the ρ-

aplace transform to both sides of Eq. (47) , we have

αu 2 (x, s ) − s α−1 u 2 (x, 0) =

−0 . 06 2 (x − max (x − 25 e −0 . 06

))

s

s αu 2 (x, s ) =

−0 . 06 2 (x − max (x − 25 e −0 . 06

))

s

u 2 (x, s ) =

−0 . 06 2 (x − max (x − 25 e −0 . 06

))

s 1+2 α. (48)

pplying the inverse of ρ-Laplace transform to both sides of Eq.

48) and using identity (8), we obtain the analytical solution of the

ractional differential equation defined by Eq. (47)

2 (x, t) =

[

− 0 . 06

2 x

�(1 + 2 α) +

0 . 06

2 max (x − 25 e −0 . 06 , 0

)�(1 + 2 α)

] (t ρ

ρ

)2 α

.

(49)

We repeat the same procedure in the other iterations p 3 , p 4 , ...

ote, we have to solve for all n > 2 the following fractional differ-

ntial equation

α,ρu n =

∂ 2 u n −1

∂x 2 + (k − 1)

∂u n −1

∂x − ku n −1 .

he analytical solution of the generalized fractional BSE described

y the Caputo GFD is given by

(x, t) = u 0 (x, t) + pu 1 (x, t) + p 2 u 2 (x, t) + p 3 u 3 (x, t) + . . . (50)

A.N. Fall, S.N. Ndiaye and N. Sene / Chaos, Solitons and Fractals 125 (2019) 108–118 115

Fig. 6. Values of the options for α = ρ = 1 and various value of ρ .

Fig. 7. Values of the options for ρ = 1 and α = 0 . 65 .

116 A.N. Fall, S.N. Ndiaye and N. Sene / Chaos, Solitons and Fractals 125 (2019) 108–118

Fig. 8. Values of the options for α = ρ = 0 . 65 .

Fig. 9. Values of the options for α = ρ = 0 . 65 .

A.N. Fall, S.N. Ndiaye and N. Sene / Chaos, Solitons and Fractals 125 (2019) 108–118 117

Fig. 10. Values of the options for α = ρ = 0 . 65 .

W

u

F

t

u

a

f

u

e

α

u

6

T

d

0

w

w

ρ

W

c

d

a

o

o

w

7

t

s

r

c

p

ρ

c

s

f

D

i

hen p converges to 1, we have the following analytical solution

(x, t) = max (x − 25 e −0 . 06 , 0

)+

{x − max

(x − 25 e −0 . 06

)}×{

1 − 1 − 0 . 06

�(1 + α)

(t ρ

ρ

)α

− 0 . 06 2

�(1 + 2 α)

(t ρ

ρ

)2 α

+ . . .

}= max

(x − 25 e −0 . 06 , 0

)+

{x − max

(x − 25 e −0 . 06 , 0

)}{

1 − E α

(0 . 06

(t ρ

ρ

)α)}

.

inally, the analytical solution of the generalized BSE described by

he Caputo GFD is in the following form

(x, t) = max (x − 25 e −0 . 06 , 0

)+

{x − max

(x − 25 e −0 . 06 , 0

)}×{

1 − E α

(0 . 06

(t ρ

ρ

)α)}

. (51)

Observe that we recover the analytical solution of the gener-

lized BSE described by the Caputo GFD when ρ = 1 . We have the

ollowing analytical solution

(x, t) = max (x − 25 e −0 . 06 , 0

)+

{x − max

(x − 25 e −0 . 06 , 0

)}×{ 1 − E α( 0 . 06 t α) } . (52)

Observe that we recover the analytical solution of the gen-

ralized BSE described by the integer order time derivative when

= ρ = 1 . We have the following analytical solution

(x, t) = max (x − 25 e −0 . 06 , 0

)+

{x − max

(x − 25 e −0 . 06 , 0

)}×{

1 − e 0 . 06 t }. (53)

.2. Interpretation and utility of the approximate solutions

We analyze the effect of the order ρ in the diffusion process.

he values of the options when the orders α = 1 and ρ = 0 . 65 are

epicted in Fig. 6 . The values of the options when the orders α = . 65 and ρ = 1 are depicted in Fig. 7 . The values of the options

hen α = ρ = 0 . 65 are depicted in Fig. 8 .

In Fig. 9 , we represent the values of the options graphically

hen we fix α = 1 and the time t = 0 . 5 and the values of the order

< 1. The order of the profiles follows the decrease of the order ρ .

e notice the order ρ has a retardation effect in the diffusion pro-

esses. Thus, we note decay in the cost of the option. In Fig. 10 , we

epicted the values of the options graphically when we fix α = 1

nd the time t = 0 . 5 and the values of the order ρ > 1. The order

f the profiles follows the increase of the order ρ . We conclude the

rder ρ has an acceleration effect in the diffusion processes. Thus

e note an increase in the cost of the option.

. Conclusion

In this paper, we have discussed the analytical solution of

he BSE and the analytical solution of the generalized BSE both de-

cribed by the Caputo GFD . The order ρ of the Caputo GFD has a

etardation in the diffusion process when ρ < 1, thus we note de-

ay in the cost of the European option. The order ρ of the Ca-

uto GFD has an acceleration effect in the diffusion process when

> 1, thus an increase in the cost of the European option. Note the

lassical BSE is recovered when the order α = ρ = 1 . The numerical

chemes of the BSE will be the subject of future investigations in a

orthcoming paper.

eclaration of Competing Interest

The authors declare that there is no conflict of interests regard-

ng the publication of this paper.

118 A.N. Fall, S.N. Ndiaye and N. Sene / Chaos, Solitons and Fractals 125 (2019) 108–118

[

[

[

[

[

[

[

CRediT authorship contribution statement

Aliou Niang Fall: Conceptualization, Investigation, Formal anal-

ysis. Seydou Nourou Ndiaye: Investigation. Ndolane Sene: Con-

ceptualization, Formal analysis, Methodology, Investigation, Vali-

dation, Visualization, Writing - original draft, Writing - review &

editing.

Acknowledgments

The first and second authors enjoy their Ph.D. The last author

has supervised the investigations.

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