Novel Approaches for Getting the Solution of the...

Research ArticleNovel Approaches for Getting the Solution of the FractionalBlackndashScholes Equation Described by Mittag-LefflerFractional Derivative

Ndolane Sene 1 Babacar Sene2 Seydou Nourou Ndiaye3 and Awa Traore4

1Laboratoire Lmdan Departement de Mathematiques de la Decision Universite Cheikh Anta Diop de DakarFaculte des Sciences Economiques et Gestion BP 5683 Dakar Fann Senegal2Centre de Recherches Economiques Appliquees Directeur du Laboratoire de Finances pour le Developpement (LAFIDEV)Faculte des Sciences Economiques et Gestion Universite Cheikh Anta Diop de Dakar BP 5683 Dakar Fann Senegal3Centre de Recherches Economiques Appliquees Laboratoire de Finances pour le Development (LAFIDEV)Faculte des Sciences Economiques et Gestion Universite Cheikh Anta Diop de Dakar BP 5683 Dakar Fann Senegal4Centre de Recherches Economiques Appliquees Facultedes Sciences Economiques et GestionUniversite Cheikh Anta Diop de Dakar BP 5683 Dakar Fann Senegal

Correspondence should be addressed to Ndolane Sene ndolaneseneucadedusn

Received 6 May 2020 Accepted 27 May 2020 Published 25 July 2020

Guest Editor Qasem M Al-Mdallal

Copyright copy 2020 Ndolane Sene et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e value of an option plays an important role in finance In this paper we use the BlackndashScholes equation which is described bythe nonsingular fractional-order derivative to determine the value of an option We propose both a numerical scheme and ananalytical solution Recent studies in fractional calculus have included new fractional derivatives with exponential kernels andMittag-Leffler kernels ese derivatives have been found to be applicable in many real-world problems As fractional derivativeswithout nonsingular kernels we use a CaputondashFabrizio fractional derivative and a Mittag-Leffler fractional derivative Fur-thermore we use the AdamsndashBashforth numerical scheme and fractional integration to obtain the numerical scheme and theanalytical solution and we provide graphical representations to illustrate these methodse graphical representations prove thatthe AdamsndashBashforth approach is helpful in getting the approximate solution for the fractional BlackndashScholes equation Finallywe investigate the volatility of the proposedmodel and discuss the use of the model in financeWemainly notice in our results thatthe fractional-order derivative plays a regulator role in the diffusion process of the BlackndashScholes equation

1 Introduction

ere are many mathematical models [1ndash3] used in financeto predict the values of cost revenue and options In thispaper we address the application of fractional calculus ineconomics and finance Fractional derivatives occupy animportant place in fractional calculus so this paper inves-tigates the use of fractional derivatives for modeling financialand economic models e BlackndashScholes model is an im-portant tool used in finance to predict the value of an option[2] ere are many styles of options European options[3 4] American options [5 6] and Asian options epricing of options is a subject that has been very intensely

debated in economics So far the economic and financialliterature continues to take an interest in this subject Todaythe questions arise concerning alternative methods forpricing European options which are derivatives that canonly be exercised at maturity e literature is also interestedin the valuation of American options (there is not yet aconsensus model) American options are derivatives that canbe exercised at any time e methodology proposed in thispaper therefore makes it possible to price a Europeanoption It could be extended to American and Asian optionsand knock-in and knock-out barrier options Anotherpossible application is the determination of the default riskof bonds listed on the financial market e model could

HindawiDiscrete Dynamics in Nature and SocietyVolume 2020 Article ID 8047347 11 pageshttpsdoiorg10115520208047347

possibly be applied to determine the unknowns of theMertonndashBlackndashScholes-based model Note that predictingthe exact value of the American option is an ongoingproblem that is generally not common knowledge ForEuropean options the model proposed by Black and Scholes[2] has an exact analytical solution and a numerical schemeand that analytical solution uses the normal distributioneuse of the normal distribution in the formula for an option isnot always suitable thus our work proposes an analyticalsolution to this problem using recursive approximation thatavoids the need for a normal distribution in the formula forthe value of a European option

Some studies in the context of fractional calculus haveinvestigated the use of the Black and Scholes equation forEuropean options Much of this research has involved theuse of the CaputondashLiouville and the RiemannndashLiouvillederivatives In [4] Fall et al offered a new work on fractionalBlackndashScholes equations described by the generalizedfractional derivative In [4] the authors present a newprocedure to obtain the analytical solutions of the fractionalBlackndashScholes model which is called the homotopy per-turbation method Sawangston et al proposed an analyticalsolution for the fractional BlackndashScholes equation with theCaputondashLiouville derivative in [7] e research in [8] of-fered an analytical solution for the fractional BlackndashScholesdescribed by the conformable derivative For more nu-merical schemes proposed for the BlackndashScholes equationsee [1 6 9ndash13] For other numerical procedures in otherdifferential equations the readers can refer to [14ndash20]

Modeling the physical phenomena using the fractional-order derivative has many advantages First of all it permitsus to study the differential models with arbitrary noninteger-order derivatives Second it permits us to take into accountthe memory effect that is the next behavior of the dynamic isexplained by the past behavior of the dynamic As we willobserve in this paper the fractional-order derivative can playa regulator role in differential dynamics In this paper weinvestigate and introduce the fractional BlackndashScholesequation described by a new fractional-order derivative theMittag-Leffler fractional derivative [21] We prove this can beused in finance to evaluate the value of an option Note thatthe new model uses the AtanganandashBaleanu derivative andtakes into account the integer-order time derivative Wepropose a numerical scheme of the introduced model usingthe AdamsndashBashforth method and an analytical solutionusing the recursive procedure proposed by Liao in [22] Fi-nally we use the analytical solution to analyse the volatility ofthe option to a change in the price of the underlying security

is manuscript is structured as follows In Section 2 wedefine the fractional derivatives with Mittag-Leffler kernels InSection 3 we introduce and discuss the fractional Black-ndashScholes equation described by the Mittag-Leffler fractionalderivative In Section 4 we prove the existence and uniquenessof our introduced model In Section 4 we describe theAdamsndashBashforth method In Section 5 we propose the nu-merical discretization of the fractional BlackndashScholes equationIn Section 6 we propose an analytical solution for theBlackndashScholes equation described by the AtanganandashBaleanufractional derivative In Section 7 we analyse the volatility of

the fractional BlackndashScholes equation and in Section 8 wepresent graphical representations and a discussion Section 9includes our final remarks and conclusions

2 Fractional Operators withoutSingular Kernels

In this section we recall the definitions of fractional de-rivatives with nonsingular kernels Fractional calculus beganwith the RiemannndashLiouville fractional derivative and theCaputondashLiouville fractional derivative for definitions referto [23ndash25] ese two derivatives were proposed in responseto Leibnizrsquos question in 1695 Many other fractional de-rivatives were introduced in the literature after this date andhave been applied in physics mechanics and mathematicalmodeling [25ndash31] Some properties were lost in modelingreal-world phenomena specifically the physical aspectsRecently Caputo and Fabrizio proposed a new fractionalderivative with an exponential kernel [32] and gave it anassociated fractional integral e main advantage of thisfractional derivative is its lack of singularity and it can beused in modeling many physical phenomena [26 33] edefinitions are given in the following

Definition 1 (see [32]) e CaputondashFabrizio fractionalderivative of the function u R times [0 +infin[⟶ R of order αis defined in the form

DCFα u(y t)

M(α)

1 minus α1113946

t

0uprime(y(s) s)exp minus

α1 minus α

(t minus s)1113874 1113875ds

(1)

where tgt 0 and the order α isin (0 1) with the normalizationterm satisfying M(0) M(1) 1

Definition 2 (see [32]) e CaputondashFabrizio integral for agiven function u R times [0 +infin[⟶ R of order α isin (0 1] isdefined in the form

ICFα u(y t)

2(1 minus α)

(2 minus α)M(α)u(y t)

+2α

(2 minus α)M(α)1113946

t

0u(y(s) s)ds

(2)

for all tgt 0 and the order α isin (0 1) with the normalizationterm satisfying M(0) M(1) 1

e exponential form is a particular case of the Mittag-Leffler function Motivated by the fact that the Cauchyproblem with the CaputondashFabrizio derivative generates asolution with an exponential function Atangana andBaleanu proposed another fractional derivative with aMittag-Leffler kernel in 2016 [21] In other words we cantranslate the Cauchy equation using CaputondashFabrizio as thefirst-order equation with an integer-order derivative

Definition 3 (see [21]) e AtanganandashBaleanundashCaputoderivative for a function u R times [0 +infin[⟶ R of orderα isin (0 1] is defined in the form

2 Discrete Dynamics in Nature and Society

DABCα u(y t)

B(α)

1 minus α1113946

t

0uprime(y(s) s)Eα minus

α1 minus α

(t minus s)α

1113874 1113875ds

(3)for all tgt 0 where the function Γ( ) is the Euler gammafunction and Eα(middot) denotes the Mittag-Leffler function withthe normalization term satisfying B(0) B(1) 1

Definition 4 (see [21]) e AtanganandashBaleanu integral for agiven function u R times [0 +infin[⟶ R of order α isin (0 1] isdefined as the form

IABα u(y t)

1 minus αB(α)

u(y t)

+α

B(α)

1Γ(α)

1113946t

0(t minus s)

αminus 1u(y(s) s)ds

(4)

for all tgt 0 where the function Γ( ) is the Euler gammafunction and Eα(middot) denotes the Mittag-Leffler function withthe normalization term satisfying B(0) B(1) 1

We were motivated to consider these two fractionalderivatives because of their successful application in mod-eling real-life phenomena In this paper we apply theAtanganandashBaleanu fractional derivative in modeling thevalue of options and investigate the fractional BlackndashScholesequation described by the AtanganandashBaleanundashCaputofractional derivative

3 BlackndashScholes in the Context of Mittag-Leffler Fractional Derivative

In this section we introduce the BlackndashScholes equation inthe context of the AtanganandashBaleanu fractional derivativeWe begin by recalling the classical model proposed by Blackand Scholes [2] Let the asset price be S at time the constantvolatility of an underlining asset be represented by theparameter σ and μ be the expected rate of return Myron andFischer stipulated that the stock price follows a Brownianmotion denoted by the parameter w thus we have thefollowing

dS μSdt + σSdw (5)

Note that equation (5) describes the asset price S asBrownian motion and represents a particular case of Itorsquoslemma ere are two types of derivations in the literaturefor pricing options in this paper we use Fisher and Myronrsquosderivation Black and Scholes expressed the value of theportfolio denoted by P in the following form

dP dV minuszV

zSdS (6)

where V represents the value of an option We havedecomposed the portfolio into the value of the option andthe asset price zVzS dS denotes the asset price obtained peryear Using Itorsquos lemma equation (6) can be represented inthe following form

dP zV

zτ+ μS

zV

zS+σ2

2S2zV2

zS2minus

zV

zSSμ1113890 1113891dt

+ σSzV

zSminus σ

zV

zSS1113890 1113891dw

(7)

Taking into account the interest rate r into the value ofthe portfolio we express equation (6) in the following form

dP rVdt minus rzV

zSSdt (8)

Combining equations (7) and (8) the differentialequation which calculates the value of a European option isgiven by the following equation

zV

zt+σ2

2S2z

2V

zS2+ rS

zV

zSminus rV 0 (9)

We made the following assumptions related to equation(9) it considers a European option the risk-less interest rater is constant and there are no transaction costs and weauthorize the possibility to buy and to sell any number ofstocks with no restriction to short selling at the last momente boundary conditions for equation (9) are defined asV(0 t) 0 V(S T) asymp S as S⟶infin and the terminalcondition is given by

V(S T) max(S minus E 0) (10)

where the parameter E denotes the strike price of the un-derlying stock and T represents the expiration time For theEuropean option we have the possibility to buy and to exercisethe option with no obligation at time T at means we can sellthe risky asset to a seller at a strike price E Equation (10) can beexplained as follows we exercise the option at time T when thecondition Elt S is held at is V(S T) max(S minus E 0)

S minus E in other words the buyer receives the playoff S minus E ebenefit is in selling the asset to the seller of the contract ratherthan on the financial market When the condition SltE is heldthe contract is not good for the buyer and the buyer can sell therisky asset for a larger price on the financial market

e use of equation (9) for the analytical solution or thenumerical scheme is not trivial but equation (9) is a dif-fusion equation and can be rewritten more simply We usethe following changes to the variables described by therelationships

S Eex

t T minus2τσ2

V Eu(x t)

(11)

From which it follows the classical Euler equation givenby

zu

zτ

z2u

zx2 +(k minus 1)zu

zxminus ku (12)

with the initial boundary condition defined by

Discrete Dynamics in Nature and Society 3

u(x 0) max ex

minus 1 0( 1113857 (13)

where k 2rσ2 denotes the balance between the free in-terest rate and the volatility of the stocks e differencebetween equation (9) and equation (12) shows the transitionbetween the finance model and the physical model Note thatit was provided in fractional calculus many models de-scribed by the classical derivative as in equation (9) cannotdescribe the real behavior of the modeled phenomena It hasbeen proved that many real-world problems follow frac-tional phenomena Equation (12) is a diffusion equation andthere are many diffusion processes in physics such assubdiffusion superdiffusion ballistic diffusion and hyper-diffusion which equation (12) does not take into accountis problem requires the introduction of a fractional de-rivative that takes into account all types of diffusion pro-cesses In this paper we replace the ordinary time derivativewith the fractional-order time derivative and we describethe fractional differential equation which we consider usingthe following fractional differential equation

DABCα u

z2u

zx2 +(k minus 1)zu

zxminus ku (14)


u(x 0) max ex

minus 1 0( 1113857 (15)

In the next section we try to prove our newmodel is welldefined admit a unique solution and use numerical andanalytical methods to approach it For the readers and moreunderstanding of the paper we summarize the description ofthe parameters used in this paper in Table 1

4 AdamsndashBashforth Numerical Approach

In this section we describe the procedure of discretizationused in this paper e method is called theAdamsndashBashforth numerical scheme and was introduced infractional calculus by Atangana in [34] AdamsndashBashforth isa useful method that involves the following fundamentaltheorem [34]

Theorem 1 (see [34]) 4e solution of the fractional differ-ential equation described by DABC

α v f(v t) with initialboundary condition v(0) satisfies the following relationship

v(t) minus v(0) 1 minus αB(α)

f(v t) +α

B(α)1113946

t

0(t minus s)

αminus 1f(v s)ds

(16)

e approximation of the function f uses a Lagrangepolynomial which is the main novelty of Atanganarsquos pro-posed numerical approximation e following relationshipdescribes the Lagrange polynomial

p(t) t minus tnminus1

tn minus tnminus1f vn tn( 1113857 +

t minus tnminus1

tnminus1 minus tn

f vnminus1 tnminus1( 1113857 (17)

Using equation (17) the discretized approximation ofequation (16) at time tn+1 and tn considering h tn minus tnminus1 asdescribed in [34] is obtained by the following equation

v tn+1( 1113857 minus v tn( 1113857 θ(α 1) + θ(α 2) (18)

where the function θ(α 1) is given by the expression de-scribed in the following equation

θ(α 1) f vn tn( 11138571 minus αB(α)

+α

B(α)Γ(α)h

2htα+1n+1α

minustα+1n+1

α + 11113888 1113889 minus

αB(α)Γ(α)h

htαnα

minustα+1

α + 11113888 11138891113896 1113897 (19)

and where the function θ(α 2) is given by the expressiondescribed in the following equation

θ(α 2) f vnminus1 tnminus1( 1113857α minus 1B(α)

minusα

B(α)Γ(α)h

htαn+1α

minustα+1n+1

1 + α+

tα+1

B(α)Γ(α)h1113888 11138891113896 1113897 (20)

e above discretization proposed by Atangana in [34] isvery useful in numerical solutions for fractional differentialequations described by certain fractional derivatives such asthe Caputo fractional derivative and the fractional derivativewith the Mittag-Leffler kernel e application of theAdamsndashBashforth scheme uses the discretization of theterms f(vn tn) and f(vnminus1 tnminus1) We can do the standarddiscretization procedures to discretize them Here we usethe central difference schemes for the second-order spacederivative and numerical approximation of the first-orderspace derivative Before moving on we look at the stability ofthe method used and see that stability is obtained when the

function f is Lipschitzian Using equations (19) and (20) thefollowing relationship is obtained

v tn+1( 1113857 minus v tn( 1113857 1 minus αB(α)

f vn tn( 1113857 minus f vnminus1 tnminus1( 11138571113858 1113859

+Ω(α 1) minusΩ(α 2)

(21)

where the function Ω(α 1) is given by the relationship

Ω(α 1) α

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds (22)


and the function Ω(α 2) is given by the relationship

Ω(α 2) α

B(α)1113946

tn

0tn minus s( 1113857


We will find a threshold for the functionΩ(α 1) minusΩ(α 2) by applying the Euclidean norm

Ω(α 1) minusΩ(α 2)leα

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds minus

αB(α)

1113946tn


αminus1f(v s)ds

leα

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds

+

αB(α)

1113946tn


αminus1f(v s)ds

leα

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds +

αB(α)

1113946tn


αminus1f(v s)ds

leαM

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1ds +αM

B(α)1113946

tn


αminus1ds

leαM

B(α)Γ(α + 1)tαn+1 + t

αn1113858 1113859

αMhα

B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(24)

Using the assumption tn nh we get the followingrelationship

Ω(α 1) minusΩ(α 2)leαMhα

B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(25)

Applying the norm to both sides of equation (21) weobtain the following relationships

v tn+1( 1113857 minus v tn( 1113857

le1 minus αB(α)



le1 minus αB(α)


+αMhα

B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(26)

From equation (26) it can be seen that when thefunction f is locally Lipschitz and h converges to zero weobtain the following relationship

v tn+1( 1113857 minus v tn( 1113857

⟶ 0 (27)

We can conclude the AdamsndashBashforth numericalscheme is unconditionally stable In the next section weapply the AdamsndashBashforth numerical scheme to the nu-merical approximation of the fractional BlackndashScholesequation described by the fractional derivative with Mittag-Leffler

5 Numerical Approach for FractionalBlackndashScholes Equation

In this section we describe the AdamsndashBashforth numericalscheme for the fractional BlackndashScholes equation repre-sented by the AtanganandashBaleanu fractional derivative Let usbegin the numerical approximation of the fractionalBlackndashScholes equation Let tn nh e AdamsndashBashforthnumerical scheme for the fractional BlackndashScholes equationdescribed by the AtanganandashBaleanu fractional derivativetakes the following form

u tn+1( 1113857 minus u tn( 1113857 θ(α 1) minus θ(α 2) (28)


Table 1 Parameters of the BlackndashScholes equation

Parameters Description of the parametersV and T e value of an option and the expiration time respectivelyσ e volatility of the underlining stockk e balance between the free interest rate and the volatility of the stockr e risk-less interest rateE and S e strike price of the underlying stock and the asset price respectively


θ(α 1) f un tn( 11138571 minus αB(α)

minusα

B(α)Γ(α)hα 2(n + 1)α

αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (29)


θ(α 2) f unminus1 tnminus1( 11138571 minus αB(α)

minusα

B(α)Γ(α)hα (n + 1)α

αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (30)

e next step consists of finding the discretization of thefunctions f(vn tn) and f(vnminus1 tnminus1) In the BlackndashScholesequation the function f is given by

f(u t) z2u

zx2 +(k minus 1)zu

zxminus ku (31)

Using the central difference approximation for thesecond-order derivative with respect to the space coordinateand the numerical approximation for the space derivativewe obtain the following discretization at the points un and tn

for the function f

f un tn( 1113857 un

j+1 minus 2unj + un

jminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δx

minus kunj + O(Δx)

(32)

Using the central difference approximation again for thesecond-order derivative with respect to the space coordinateand the numerical approximation for the space derivative

we obtain the following discretization at the points unminus1 andtnminus1 for the function f

f unminus1 tnminus1( 1113857 unminus1

j+1 minus 2unminus1j + unminus1

jminus1

Δx2 +(k minus 1)unminus1

j+1 minus unminus1jminus1

2Δx

minus kunminus1j + O(Δx)

(33)

Numerical discretization using the AdamsndashBashforthmethod for the BlackndashScholes equation is obtained bycombining equations (28)ndash(33) at is

un+1j u

nj + θ(α 1) minus θ(α 2) (34)

For the computation of our numerical schemes we makesome changes to the variables such that the terms dependingon n h and α are constants us we rewrite θ(α 1) andθ(α 2) respectively as in the following relationships

H(n h α 1) 1 minus αB(α)

minusα


αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (35)


minusα


αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (36)

Finally the numerical scheme using theAdamsndashBashforth method for the fractional BlackndashScholesequation is given by

un+1j u

nj + H(n h α 1)

unj+1 minus 2un

j + unjminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δxminus ku

nj1113890 1113891

minus H(n h α 2)unminus1


jminus1



2Δxminus ku

nminus1j1113890 1113891

(37)

To complete the numerical discretization given inequation (37) we recall the discretized form of the initialboundary condition described by the followingexpression

un0(x) max e

xminus 1 0( 1113857 (38)


6 Analytical Solution for FractionalBlackndashScholes Equation

In this section we use the fractional integrator to propose theanalytical solution of the fractional BlackndashScholes equationdescribed by the AtanganandashBaleanu fractional derivative emethod is described in the following theorem

Theorem 2 4e solution of the fractional differentialequation described by DABC

α v f(v τ) with initial boundarycondition v(0) satisfies the following relationship

vn+1(τ) minus vn+1(0) 1 minus αB(α)

f vn τ( 1113857 +α

B(α)1113946τ

0f s vn( 1113857ds

(39)

where n 0 1 2 Furthermore the solution is given by

v(x τ) v0(0) + v1(τ) + v2(τ) + (40)

Consider the fractional BlackndashScholes equation (14)Using 4eorem 2 with assumption u1(x 0) 0 we have thefollowing solution which is the first step

u1(x τ) u1(x 0) +1 minus αB(α)

k max ex 0( 1113857 minus max e

xminus 1 0( 1113857( 1113857 +

k max ex 0( ) minus max ex minus 1 0( )( )τα

B(α)Γ(1 + α)

1 minus αB(α)


xminus 1 0( 1113857( 1113857 +

αk max ex 0( ) minus max ex minus 1 0( )( )τα

B(α)Γ(1 + α)


xminus 1 0( 1113857( 11138571113858 1113859

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

(41)

Under assumption u2(x 0) 0 again using equation(39) the second step gives the following solution

u2(x τ) k2 max e

x 0( 1113857 minus max e

xminus 1 0( 1113857( 11138571113960 1113961(1 minus α)

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(α)

(1 minus α)τα

B(α)Γ(1 + α)+

ατ2α

B(α)Γ(1 + 2α)1113896 1113897

(42)

We adopt the procedure described in eorem 2 forthe rest of the steps e following expression gives theapproximate solution of the fractional BlackndashScholesequation in the context of the Mittag-Leffler fractionalderivative

u(x τ) u(x 0) + u1(x 0) + u2(x 0) + (43)

We recover the approximate solution of the classicalBlackndashScholes equation when α 1 In the context of theMittag-Leffler fractional derivative when we suppose α 1we get the following form

u(x τ) u(x 0) + u1(x 0) + u2(x 0) +

max ex

minus 1 0( 1113857 + k max ex 0( 1113857 minus max e

xminus 1 0( 1113857( 11138571113858 1113859τ

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961τ2 +

max ex

minus 1 0( 1113857 1 minus eminus kτ

1113872 1113873 + max ex 0( 1113857e

minus kτ

(44)

Due to space limitation all term are not written Finallywe describe the analytical solution for the classical Black-ndashScholes equation (12) in the following form

u(x τ) max ex 0( 1113857 1 minus e

minuskτ1113872 1113873 + max e

xminus 1 0( 1113857e

minuskτ

(45)

e method adopted here gives a solution which is ingood agreement with the classical solution of the Black-ndashScholes equation (12) Let us prove our solution (45) isanother representation for the traditional solution of theBlackndashScholes equation using the following

V(S E) SN d1( 1113857 minus Eeminuskτ

N d2( 1113857 (46)

where N designs the normal distribution functiond1 (log(SE) + (r + σ22)(T minus t))σ

T minus t

radic and

d2 d1 minus σT minus t

radic Note in the case ldquoin the moneyrdquo that is

Elt S we have high volatility in other words we have thefollowing equation

N d1( 1113857 N d2( 1113857 1 (47)

From which we rewrite equation (46) as the followingform

V(S E) S minus Eeminuskτ

(48)

Equation (48) represents the value of the option ldquoin themoneyrdquoWe recover this value with the solution proposed in


equation (45) using the following reasoning Note that in thecase ldquoin the moneyrdquo there exists the following relationship

max ex 0( 1113857 e

x

max ex

minus 1 0( 1113857 ex

minus 1(49)

Now we replace ex SE into equation (45) and usingthe assumption posed in equation (11) we obtain the fol-lowing relationship

u(x τ) S

E1 minus e

minuskτ1113872 1113873 minus

S

Eminus 11113874 1113875e

minus kτ

S

Eminus e

minus kτ

(50)

Multiplying the function u by E as in equation (11) weobtain

V(S E) S minus Eeminus kτ

(51)

We can see that the solution represented in equation (48)and the solution in equation (51) are the same us ouranalytical solution can be used in finance to determine thevalue of an option satisfying the BlackndashScholes equation

Note that in the money ldquocallrdquo is when the price of theunderlying asset is higher than the strike price It is in theinterest of the holder of the option to exercise it Heshe hasmade good anticipations (SgtE) in this case the value of theoption is given by the formula V max(S minus E 0) S minus EOut of the money ldquocallrdquo is when the price of the underlyingasset is lower than the strike price It is not in the interest ofthe option holder to exercise the option Heshe has madewrong expectations (SltE) in this case the value of theoption is given by V max(S minus E 0) 0 In the moneyldquoputrdquo is when the price of the underlying asset is lower thanthe strike price It is in the interest of the option holder toexercise the option Heshe has made good anticipations(SltE) in this case the value of the option is given byV max(E minus S 0) E minus S Out of the money ldquoputrdquo is whenthe price of the underlying asset is higher than the strikeprice It is not in the interest of the option holder to exercisethe option Heshe has made wrong anticipations (SgtE) inthis case the value of the option is given byV max(E minus S 0) 0

7 The Volatility of the FractionalBlackndashScholes Equation

In this section we analyse the volatility of the fractionalBlackndashScholes equation using an analytical solution In themarket we can buy the call and put with different strikeprices and maturity e volatility analyses the liquidity ofthe cost of the call and the put in the market Many types ofvolatility can be generated by the BlackndashScholes equation Ingeneral volatility measures risk in the financial marketsVolatility is used to control both upward and downwardmovements It is calculated from log returns e delta ispart of the Greek letters of options It is the derivative of theprice of the call or put option in relation to the price of theunderlying asset It is used for trading arbitrage or hedging

operations on options It is an important indicator of marketrisk management e Basel Committee on InternationalBanking Regulation recommends that banks use delta forexposures to options In reality delta is a sensitivity factor

Here we recall the formula of the volatility delta of thefractional BlackndashScholes equation e delta measures thesensitivity of the option price to a change in the price of theunderlying security Under variable changes this isexpressed as follows

δ zu

zx (52)

Using the approximate solution (52) it is clear that thesensitivity of the option price to a change in the price of theunderlying security does not depend on the fractional-orderand is given by

δ max ex 0( 1113857 (53)

Given the conditions in equation (11) the volatility ofthe option price to a change in the price of the underlyingsecurity is given by

δ maxS

E 01113874 1113875 (54)

Earlier we stated that the volatility studied in this paperdoes not depend on the order of the fractional derivative orthe balance between the free interest rate and the volatility ofthe stocks k is remark can be explained simply from thefact that except for the first term u(x 0) we have thefollowing relationship

zui

zx 0 (55)

for all i 1 2 3 Figure 1 shows the volatility surface ofthe option price to a change in the price of the underlyingsecurity in Figure 1

e volatility surface gives the sensitivity of the optionprice when the asset price S and the strike price of theunderlying stock E both vary in time Let the asset price S befixed and let the strike price of the underlying stock E varyin time e behavior of the volatility can be seen in Fig-ure 2 In other words we can see that when SltE thevolatility decreases rapidly and converges to zero Let theunderlying stock E and the asset price S vary in time ebehavior of the volatility can be seen in Figure 2 us wecan see in Figure 2 that when Elt S the volatility quicklyincreases linearly and converges to infinity We note highvolatility

8 Graphical Representations and Discussion

In this section we illustrate our results graphically Spe-cifically we depict the behavior of the solutions of thefractional BlackndashScholes equation obtained with numericalschemes and the recursive method We begin with theapproximate solution generated by the recursive methodpreviously described in Section 7 In this context we con-sider the solution with three iterations given by equation(43) Figure 3 shows the behavior of the approximate


solution for the fractional BlackndashScholes equation with theAtanganandashBaleanu fractional derivative with order α 05and the balance between the free interest rate and thevolatility of the stocks k 2 e constant k is obtained withthe risk-free interest rate to expiration r 004 and thevolatility of the stocks σ 02 [35]

Figure 4 shows the behavior of the approximate solution forthe fractional BlackndashScholes equationwith theAtanganandashBaleanufractional derivativewith order α 1 and the balance between thefree interest rate and the volatility of the stocks k 2

Figure 5 depicts the analytical solution with the recursivemethod and the approximate solution obtained with theAdamsndashBashforth numerical scheme We suppose t 1 andα 05 and observed the numerical solution and the ap-proximate analytical solution is in good agreement

e main question now is how to find the optimal orderof α In our previous example we depicted the figures bychoosing the order α 05 or α 1 A complicated methodis required to find the optimal order α in fractional calculusFirst we must collect the data in the considered market

00510

2

2468

18

10121416

16

1820

14 12 151 08 06 04 02 02

Volat

ility

surfa

ce

S E

Figure 1 Volatility surface of the BlackndashScholes equation

0 05 1 15 2 25 3 35 4 45 502468

101214161820

S gt EE gt S

S or E

Volat

ility

line

Figure 2 Volatility when SltE and Elt S

04

10

35

20

43

30

3525

40

32

50

2515 2

60

151 105 050 0

u

xτ

Figure 3 Behavior of the BlackndashScholes equation with α 05


Second we create a figure generated by the obtained dataand finally we proceed by interpolation to find an order α

To support the numerical discretization we describe inTable 2 the different values of the options generated by ournumerical scheme and we compare them with values ob-tained with the homotopy perturbation method In otherwords the robustness of the used numerical scheme isauthenticated by comparing the numerical and analyticalresults in Table 2 In Table 2 we consider α 05 the risk-free interest rate to expiration α 05 the volatility of thestocks σ 02 the volatility of the stocks k 2[35] and thestrike price of the underlying stock E 10

We mainly observe that the results in Table 2 are in goodagreement with the results on the BlackndashScholes equationstudied in terms of the Caputo derivative in [35] In generalwe also notice that the order of the fractional derivative has asignificant impact on the value of the options In this in-vestigation the order α has a regulator impact and we notice

in Table 2 that the values obtained using the fractional orderare more beneficial rather than the values obtained with theclassical derivative in the financial market

9 Conclusion

In this paper we have discussed the numerical scheme andthe analytical solution for the fractional BlackndashScholesequation described by the AtanganandashBaleanu derivative Asobserved the analytical solution of the fractional Black-ndashScholes equation with the AtanganandashBaleanu fractionalderivative is not trivial We have used the AdamsndashBashforthnumerical scheme to approach the solution as it is usefuland straightforward for proposing the approximate solu-tions of the fractional BlackndashScholes equation We have alsoconsidered the liquidity of the cost of the call and the put inthe market namely the volatility e graphical represen-tations have proved the good agreements between the an-alytical solution and the numerical solutions for thefractional BlackndashScholes equation

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] M H Akrami and G H Erjaee ldquoNumerical solutions forfractional black-scholes option pricing equationrdquo GlobalAnalysis and Discrete Mathematics vol 1 pp 9ndash14 2015

[2] F Black andM Scholes ldquoe pricing of options and corporateliabilitiesrdquo Journal of Political Economy vol 81 no 3pp 637ndash654 1973

[3] S Kumar D Kumar and J Singh ldquoNumerical computationof fractional Black-Scholes equation arising in financialmarketrdquo Egyptian Journal of Basic and Applied Sciences vol 1no 3-4 pp 177ndash183 2014

[4] A N Fall S N Ndiaye and N Sene ldquoBlack-Scholes optionpricing equations described by the Caputo generalized frac-tional derivativerdquo Chaos Solitons amp Fractals vol 125pp 108ndash118 2019

0102030405060708090

Numerical approachAnalytical solution

0 05 1 15 2 25 3 35 4 45x

u

Figure 5 Analytical solution vs numerical approximation withα 05

Table 2 Option values when Elt S

x τ uNumerical uHomotopy uclassic α 1

01541 001 03240 03242 1146704677 003 08455 08456 1538102401 005 05745 05747 1176201697 002 04924 04925 11457

0

10

4

20

30

40

50

3

60

241 3532521510 050

u

x

τ



[5] W Chen K Du and X Qiu ldquoAnalytic properties of americanoption prices under a modified Black-Scholes equation withspatial fractional derivativerdquo Physica A Statistical Mechanicsand its Applications vol 491 pp 37ndash44 2017

[6] L Song and W Wang ldquoSolution of the fractional black-scholes option pricing model by finite difference methodrdquoAbstract and Applied Analysis vol 2013 Article ID 19428610 pages 2013

[7] P Sawangtong K Trachoo W Sawangtong andB Wiwattanapataphee ldquoe analytical solution for the black-scholes equation with two assets in the liouville-caputo fractionalderivative senserdquo Mathematics vol 6 no 8 p 129 2018

[8] M Yavuz and N Ozdemir ldquoA different approach to theEuropean option pricing model with new fractional operatorrdquoMathematical Modelling of Natural Phenomena vol 13 no 1p 12 2018

[9] Z Cen and A Le ldquoA robust and accurate finite differencemethod for a generalized Black-Scholes equationrdquo Journal ofComputational and Applied Mathematics vol 235 no 13pp 3728ndash3733 2011

[10] P Phaochoo A Luadsong and N Aschariyaphotha ldquoemeshless local Petrov-Galerkin based on moving kriginginterpolation for solving fractional Black-Scholes modelrdquoJournal of King Saud UniversitymdashScience vol 28 no 1pp 111ndash117 2016

[11] M Yavuz and N Ozdemir ldquoEuropean vanilla option pricingmodel of fractional order without singular kernelrdquo Fractaland Fractional vol 2 no 1 p 3 2018

[12] M Yavuz and N Ozdemir ldquoA quantitative approach tofractional option pricing problems with decomposition se-riesrdquo Konuralp Journal of Mathematics vol 6 no 1pp 102ndash109 2018

[13] M Yavuz N Ozdemir and Y Y Okur ldquoGeneralized dif-ferential transform method for fractional partial differentialequation from financerdquo in Proceedings of the InternationalConference on Fractional Differentiation and its Applicationspp 778ndash785 Novi Sad Serbia 2016

[14] AWakif ldquoA novel numerical procedure for simulating steadyMHD convective flows of radiative casson fluids over ahorizontal stretching sheet with irregular geometry under thecombined influence of temperature-dependent viscosity andthermal conductivityrdquoMathematical Problems in Engineeringvol 2020 Article ID 1675350 20 pages 2020

[15] A Wakif A Chamkha T umma I L Animasaun andR Sehaqui ldquoermal radiation and surface roughness effectson the thermo-magneto-hydrodynamic stability of alumina-copper oxide hybrid nanofluids utilizing the generalizedBuongiornorsquos nanofluid modelrdquo Journal of 4ermal Analysisand Calorimetry 2020 In press

[16] A Wakif Z Boulahia S R Mishra M M Rashidi andR Sehaqui ldquoInfluence of a uniform transverse magnetic fieldon the thermo-hydrodynamic stability in water-basednanofluids with metallic nanoparticles using the generalizedBuongiornorsquos mathematical modelrdquo 4e European PhysicalJournal Plus vol 133 p 181 2018

[17] M K Nayak A Wakif I L Animasaun and M S H AlaouildquoNumerical differential quadrature examination of steadymixed convection nanofluid flows over an isothermal thinneedle conveying metallic and metallic oxide nanomaterials acomparative investigationrdquo Arabian Journal for Science andEngineering 2020 In press

[18] A Zaib U Khan A Wakif and M Zaydan ldquoNumericalentropic analysis of mixedMHD convective flows from a non-isothermal vertical flat plate for radiative tangent hyperbolic

blood biofuids conveying magnetite ferroparticles dualsimilarity solutionsrdquo Arabian Journal for Science and Engi-neering 2020 In press

[19] A Wakif M Qasim M I Afridi S Saleem and M M Al-Qarni ldquoNumerical examination of the entropic energy har-vesting in a magnetohydrodynamic dissipative flow of Stokesrsquosecond problem utilization of the gear-generalized diferentialquadrature methodrdquo Journal of Non-Equilibrium 4ermo-dynamics vol 44 no 4 pp 385ndash403 2019

[20] T umma A Wakif and I L Animasaun ldquoGeneralizeddifferential quadrature analysis of unsteady three-dimen-sional MHD radiating dissipative Casson fluid conveying tinyparticlesrdquo Heat Transfer vol 49 no 5 pp 2595ndash2626 2020

[21] A Atangana and D Baleanu ldquoNew fractional derivatives withnonlocal and non-singular kernel theory and application toheat transfer modelrdquo 4ermal Science vol 20 no 2pp 763ndash769 2016

[22] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147no 2 pp 499ndash513 2004

[23] K M Owolabi and A Atangana ldquoRobustness of fractionaldifference schemes via the Caputo subdiffusion-reaction equa-tions chaosrdquo Solitons amp Fractals vol 111 pp 119ndash127 2018

[24] A A Kilbas H M Srivastava and J J Trujillo 4eory andApplications of Fractional Differential Equations p 204North-Holland Mathematics Studies Elsevier AmsterdamNetherlands 2006

[25] N Sene ldquoSecond-grade fluid model with Caputo-Liouvillegeneralized fractional derivativerdquo Chaos Solitons amp Fractalsvol 133 Article ID 109631 2020

[26] K A Abro and J F Gomez-Aguilar ldquoA comparison of heatand mass transfer on a Walterrsquos-B fluid via Caputo-Fabrizioversus Atangana-Baleanu fractional derivatives using the Fox-H functionrdquo 4e European Physical Journal Plus vol 134no 3 pp 1ndash10 2019

[27] A Atangana and T Mekkaoui ldquoTrinition the complex numberwith two imaginary parts fractal chaos and fractional calcu-lusrdquo Chaos Solitons amp Fractals vol 128 pp 366ndash381 2019

[28] N Sene ldquoStokesrsquo first problem for heated flat plate withAtangana-Baleanu fractional derivativerdquo Chaos Solitons ampFractals vol 117 pp 68ndash75 2018

[29] N Sene ldquoSIR epidemic model with Mittag-Leffler fractionalderivativerdquo Chaos Solitons amp Fractals vol 137 Article ID109833 2020

[30] Q Al-Mdallal K A Abro and I Khan ldquoAnalytical solutionsof fractional walterrsquos B fluid with applicationsrdquo Complexityvol 2018 Article ID 8131329 10 pages 2018

[31] T Abdeljawad M A Hajji Q Al-Mdallal and F JaradldquoAnalysis of some generalized ABC-Fractional logisticmodelsrdquo Alexandria Engineering Journal 2020 In press

[32] M Caputo and M Fabrizio ldquoA new definition of fractionalderivative without singular kernelrdquo Progress in FractionalDifferentiation and Applications vol 1 no 2 pp 1ndash15 2015

[33] T Abdeljawad R Amin K Shah Q Al-Mdallal and F JaradldquoEfficient sustainable algorithm for numerical solutions of systemsof fractional order differential equations by Haar wavelet collo-cation methodrdquo Alexandria Engineering Journal 2020 In press

[34] A Atangana and K M Owolabi ldquoNew numerical approachfor fractional differential equationsrdquoMathematical Modellingof Natural Phenomena vol 13 no 1 p 3 2018

[35] N Ozdemir and M Yavuz ldquoNumerical solution of fractionalblack-scholes equation by using the multivariate pade ap-proximationrdquo Acta Physica Polonica A vol 132 pp 1050ndash1053 2017


possibly be applied to determine the unknowns of theMertonndashBlackndashScholes-based model Note that predictingthe exact value of the American option is an ongoingproblem that is generally not common knowledge ForEuropean options the model proposed by Black and Scholes[2] has an exact analytical solution and a numerical schemeand that analytical solution uses the normal distributioneuse of the normal distribution in the formula for an option isnot always suitable thus our work proposes an analyticalsolution to this problem using recursive approximation thatavoids the need for a normal distribution in the formula forthe value of a European option

Some studies in the context of fractional calculus haveinvestigated the use of the Black and Scholes equation forEuropean options Much of this research has involved theuse of the CaputondashLiouville and the RiemannndashLiouvillederivatives In [4] Fall et al offered a new work on fractionalBlackndashScholes equations described by the generalizedfractional derivative In [4] the authors present a newprocedure to obtain the analytical solutions of the fractionalBlackndashScholes model which is called the homotopy per-turbation method Sawangston et al proposed an analyticalsolution for the fractional BlackndashScholes equation with theCaputondashLiouville derivative in [7] e research in [8] of-fered an analytical solution for the fractional BlackndashScholesdescribed by the conformable derivative For more nu-merical schemes proposed for the BlackndashScholes equationsee [1 6 9ndash13] For other numerical procedures in otherdifferential equations the readers can refer to [14ndash20]

Modeling the physical phenomena using the fractional-order derivative has many advantages First of all it permitsus to study the differential models with arbitrary noninteger-order derivatives Second it permits us to take into accountthe memory effect that is the next behavior of the dynamic isexplained by the past behavior of the dynamic As we willobserve in this paper the fractional-order derivative can playa regulator role in differential dynamics In this paper weinvestigate and introduce the fractional BlackndashScholesequation described by a new fractional-order derivative theMittag-Leffler fractional derivative [21] We prove this can beused in finance to evaluate the value of an option Note thatthe new model uses the AtanganandashBaleanu derivative andtakes into account the integer-order time derivative Wepropose a numerical scheme of the introduced model usingthe AdamsndashBashforth method and an analytical solutionusing the recursive procedure proposed by Liao in [22] Fi-nally we use the analytical solution to analyse the volatility ofthe option to a change in the price of the underlying security

is manuscript is structured as follows In Section 2 wedefine the fractional derivatives with Mittag-Leffler kernels InSection 3 we introduce and discuss the fractional Black-ndashScholes equation described by the Mittag-Leffler fractionalderivative In Section 4 we prove the existence and uniquenessof our introduced model In Section 4 we describe theAdamsndashBashforth method In Section 5 we propose the nu-merical discretization of the fractional BlackndashScholes equationIn Section 6 we propose an analytical solution for theBlackndashScholes equation described by the AtanganandashBaleanufractional derivative In Section 7 we analyse the volatility of

the fractional BlackndashScholes equation and in Section 8 wepresent graphical representations and a discussion Section 9includes our final remarks and conclusions

2 Fractional Operators withoutSingular Kernels

In this section we recall the definitions of fractional de-rivatives with nonsingular kernels Fractional calculus beganwith the RiemannndashLiouville fractional derivative and theCaputondashLiouville fractional derivative for definitions referto [23ndash25] ese two derivatives were proposed in responseto Leibnizrsquos question in 1695 Many other fractional de-rivatives were introduced in the literature after this date andhave been applied in physics mechanics and mathematicalmodeling [25ndash31] Some properties were lost in modelingreal-world phenomena specifically the physical aspectsRecently Caputo and Fabrizio proposed a new fractionalderivative with an exponential kernel [32] and gave it anassociated fractional integral e main advantage of thisfractional derivative is its lack of singularity and it can beused in modeling many physical phenomena [26 33] edefinitions are given in the following

Definition 1 (see [32]) e CaputondashFabrizio fractionalderivative of the function u R times [0 +infin[⟶ R of order αis defined in the form

DCFα u(y t)

M(α)

1 minus α1113946

t

0uprime(y(s) s)exp minus

α1 minus α

(t minus s)1113874 1113875ds

(1)

where tgt 0 and the order α isin (0 1) with the normalizationterm satisfying M(0) M(1) 1

Definition 2 (see [32]) e CaputondashFabrizio integral for agiven function u R times [0 +infin[⟶ R of order α isin (0 1] isdefined in the form

ICFα u(y t)

2(1 minus α)

(2 minus α)M(α)u(y t)

+2α

(2 minus α)M(α)1113946

t

0u(y(s) s)ds

(2)

for all tgt 0 and the order α isin (0 1) with the normalizationterm satisfying M(0) M(1) 1

e exponential form is a particular case of the Mittag-Leffler function Motivated by the fact that the Cauchyproblem with the CaputondashFabrizio derivative generates asolution with an exponential function Atangana andBaleanu proposed another fractional derivative with aMittag-Leffler kernel in 2016 [21] In other words we cantranslate the Cauchy equation using CaputondashFabrizio as thefirst-order equation with an integer-order derivative

Definition 3 (see [21]) e AtanganandashBaleanundashCaputoderivative for a function u R times [0 +infin[⟶ R of orderα isin (0 1] is defined in the form


DABCα u(y t)

B(α)

1 minus α1113946

t


α1 minus α

(t minus s)α

1113874 1113875ds



IABα u(y t)

1 minus αB(α)

u(y t)

+α

B(α)

1Γ(α)

1113946t

0(t minus s)


(4)







dP dV minuszV

zSdS (6)


dP zV

zτ+ μS

zV

zS+σ2

2S2zV2

zS2minus

zV

zSSμ1113890 1113891dt

+ σSzV

zSminus σ

zV

zSS1113890 1113891dw

(7)


dP rVdt minus rzV

zSSdt (8)


zV

zt+σ2

2S2z

2V

zS2+ rS

zV

zSminus rV 0 (9)






S Eex

t T minus2τσ2

V Eu(x t)

(11)


zu

zτ

z2u

zx2 +(k minus 1)zu

zxminus ku (12)



u(x 0) max ex

minus 1 0( 1113857 (13)


DABCα u

z2u

zx2 +(k minus 1)zu

zxminus ku (14)


u(x 0) max ex

minus 1 0( 1113857 (15)







f(v t) +α

B(α)1113946

t

0(t minus s)

αminus 1f(v s)ds

(16)




t minus tnminus1

tnminus1 minus tn






+α

B(α)Γ(α)h

2htα+1n+1α

minustα+1n+1

α + 11113888 1113889 minus

αB(α)Γ(α)h

htαnα

minustα+1

α + 11113888 11138891113896 1113897 (19)



minusα

B(α)Γ(α)h

htαn+1α

minustα+1n+1

1 + α+

tα+1

B(α)Γ(α)h1113888 11138891113896 1113897 (20)






(21)


Ω(α 1) α

B(α)1113946

tn+1

0tn+1 minus s( 1113857




Ω(α 2) α

B(α)1113946

tn





B(α)1113946

tn+1

0tn+1 minus s( 1113857


αB(α)

1113946tn


αminus1f(v s)ds

leα

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds

+

αB(α)

1113946tn


αminus1f(v s)ds

leα

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds +

αB(α)

1113946tn


αminus1f(v s)ds

leαM

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1ds +αM

B(α)1113946

tn


αminus1ds

leαM

B(α)Γ(α + 1)tαn+1 + t

αn1113858 1113859

αMhα

B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(24)



B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(25)


v tn+1( 1113857 minus v tn( 1113857

le1 minus αB(α)



le1 minus αB(α)


+αMhα

B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(26)


v tn+1( 1113857 minus v tn( 1113857

⟶ 0 (27)










minusα


αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (29)



minusα


αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (30)


f(u t) z2u

zx2 +(k minus 1)zu

zxminus ku (31)


for the function f

f un tn( 1113857 un

j+1 minus 2unj + un

jminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δx

minus kunj + O(Δx)

(32)





jminus1



2Δx


(33)


un+1j u

nj + θ(α 1) minus θ(α 2) (34)



minusα


αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (35)


minusα


αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (36)


un+1j u

nj + H(n h α 1)

unj+1 minus 2un

j + unjminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δxminus ku

nj1113890 1113891



jminus1



2Δxminus ku

nminus1j1113890 1113891

(37)


un0(x) max e

xminus 1 0( 1113857 (38)







f vn τ( 1113857 +α

B(α)1113946τ

0f s vn( 1113857ds

(39)


v(x τ) v0(0) + v1(τ) + v2(τ) + (40)




xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)

1 minus αB(α)


xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)


xminus 1 0( 1113857( 11138571113858 1113859

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

(41)


u2(x τ) k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(1 minus α)

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(α)

(1 minus α)τα

B(α)Γ(1 + α)+

ατ2α

B(α)Γ(1 + 2α)1113896 1113897

(42)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) + (43)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) +

max ex


xminus 1 0( 1113857( 11138571113858 1113859τ

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961τ2 +

max ex


1113872 1113873 + max ex 0( 1113857e

minus kτ

(44)



minuskτ1113872 1113873 + max e

xminus 1 0( 1113857e

minuskτ

(45)



N d2( 1113857 (46)


T minus t

radic and




N d1( 1113857 N d2( 1113857 1 (47)



(48)




max ex 0( 1113857 e

x

max ex

minus 1 0( 1113857 ex

minus 1(49)


u(x τ) S

E1 minus e


S

Eminus 11113874 1113875e

minus kτ

S

Eminus e

minus kτ

(50)



(51)







δ zu

zx (52)


δ max ex 0( 1113857 (53)


δ maxS

E 01113874 1113875 (54)


zui

zx 0 (55)










00510

2

2468

18

10121416

16

1820

14 12 151 08 06 04 02 02

Volat

ility

surfa

ce

S E


0 05 1 15 2 25 3 35 4 45 502468

101214161820

S gt EE gt S

S or E

Volat

ility

line


04

10

35

20

43

30

3525

40

32

50

2515 2

60

151 105 050 0

u

xτ







9 Conclusion


Data Availability




References





0102030405060708090


0 05 1 15 2 25 3 35 4 45x

u




01541 001 03240 03242 1146704677 003 08455 08456 1538102401 005 05745 05747 1176201697 002 04924 04925 11457

0

10

4

20

30

40

50

3

60

241 3532521510 050

u

x

τ




































DABCα u(y t)

B(α)

1 minus α1113946

t


α1 minus α

(t minus s)α

1113874 1113875ds



IABα u(y t)

1 minus αB(α)

u(y t)

+α

B(α)

1Γ(α)

1113946t

0(t minus s)


(4)







dP dV minuszV

zSdS (6)


dP zV

zτ+ μS

zV

zS+σ2

2S2zV2

zS2minus

zV

zSSμ1113890 1113891dt

+ σSzV

zSminus σ

zV

zSS1113890 1113891dw

(7)


dP rVdt minus rzV

zSSdt (8)


zV

zt+σ2

2S2z

2V

zS2+ rS

zV

zSminus rV 0 (9)






S Eex

t T minus2τσ2

V Eu(x t)

(11)


zu

zτ

z2u

zx2 +(k minus 1)zu

zxminus ku (12)



u(x 0) max ex

minus 1 0( 1113857 (13)


DABCα u

z2u

zx2 +(k minus 1)zu

zxminus ku (14)


u(x 0) max ex

minus 1 0( 1113857 (15)







f(v t) +α

B(α)1113946

t

0(t minus s)

αminus 1f(v s)ds

(16)




t minus tnminus1

tnminus1 minus tn






+α

B(α)Γ(α)h

2htα+1n+1α

minustα+1n+1

α + 11113888 1113889 minus

αB(α)Γ(α)h

htαnα

minustα+1

α + 11113888 11138891113896 1113897 (19)



minusα

B(α)Γ(α)h

htαn+1α

minustα+1n+1

1 + α+

tα+1

B(α)Γ(α)h1113888 11138891113896 1113897 (20)






(21)


Ω(α 1) α

B(α)1113946

tn+1

0tn+1 minus s( 1113857




Ω(α 2) α

B(α)1113946

tn





B(α)1113946

tn+1

0tn+1 minus s( 1113857


αB(α)

1113946tn


αminus1f(v s)ds

leα

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds

+

αB(α)

1113946tn


αminus1f(v s)ds

leα

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds +

αB(α)

1113946tn


αminus1f(v s)ds

leαM

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1ds +αM

B(α)1113946

tn


αminus1ds

leαM

B(α)Γ(α + 1)tαn+1 + t

αn1113858 1113859

αMhα

B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(24)



B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(25)


v tn+1( 1113857 minus v tn( 1113857

le1 minus αB(α)



le1 minus αB(α)


+αMhα

B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(26)


v tn+1( 1113857 minus v tn( 1113857

⟶ 0 (27)










minusα


αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (29)



minusα


αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (30)


f(u t) z2u

zx2 +(k minus 1)zu

zxminus ku (31)


for the function f

f un tn( 1113857 un

j+1 minus 2unj + un

jminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δx

minus kunj + O(Δx)

(32)





jminus1



2Δx


(33)


un+1j u

nj + θ(α 1) minus θ(α 2) (34)



minusα


αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (35)


minusα


αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (36)


un+1j u

nj + H(n h α 1)

unj+1 minus 2un

j + unjminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δxminus ku

nj1113890 1113891



jminus1



2Δxminus ku

nminus1j1113890 1113891

(37)


un0(x) max e

xminus 1 0( 1113857 (38)







f vn τ( 1113857 +α

B(α)1113946τ

0f s vn( 1113857ds

(39)


v(x τ) v0(0) + v1(τ) + v2(τ) + (40)




xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)

1 minus αB(α)


xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)


xminus 1 0( 1113857( 11138571113858 1113859

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

(41)


u2(x τ) k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(1 minus α)

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(α)

(1 minus α)τα

B(α)Γ(1 + α)+

ατ2α

B(α)Γ(1 + 2α)1113896 1113897

(42)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) + (43)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) +

max ex


xminus 1 0( 1113857( 11138571113858 1113859τ

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961τ2 +

max ex


1113872 1113873 + max ex 0( 1113857e

minus kτ

(44)



minuskτ1113872 1113873 + max e

xminus 1 0( 1113857e

minuskτ

(45)



N d2( 1113857 (46)


T minus t

radic and




N d1( 1113857 N d2( 1113857 1 (47)



(48)




max ex 0( 1113857 e

x

max ex

minus 1 0( 1113857 ex

minus 1(49)


u(x τ) S

E1 minus e


S

Eminus 11113874 1113875e

minus kτ

S

Eminus e

minus kτ

(50)



(51)







δ zu

zx (52)


δ max ex 0( 1113857 (53)


δ maxS

E 01113874 1113875 (54)


zui

zx 0 (55)










00510

2

2468

18

10121416

16

1820

14 12 151 08 06 04 02 02

Volat

ility

surfa

ce

S E


0 05 1 15 2 25 3 35 4 45 502468

101214161820

S gt EE gt S

S or E

Volat

ility

line


04

10

35

20

43

30

3525

40

32

50

2515 2

60

151 105 050 0

u

xτ







9 Conclusion


Data Availability




References





0102030405060708090


0 05 1 15 2 25 3 35 4 45x

u




01541 001 03240 03242 1146704677 003 08455 08456 1538102401 005 05745 05747 1176201697 002 04924 04925 11457

0

10

4

20

30

40

50

3

60

241 3532521510 050

u

x

τ




































u(x 0) max ex

minus 1 0( 1113857 (13)


DABCα u

z2u

zx2 +(k minus 1)zu

zxminus ku (14)


u(x 0) max ex

minus 1 0( 1113857 (15)







f(v t) +α

B(α)1113946

t

0(t minus s)

αminus 1f(v s)ds

(16)




t minus tnminus1

tnminus1 minus tn






+α

B(α)Γ(α)h

2htα+1n+1α

minustα+1n+1

α + 11113888 1113889 minus

αB(α)Γ(α)h

htαnα

minustα+1

α + 11113888 11138891113896 1113897 (19)



minusα

B(α)Γ(α)h

htαn+1α

minustα+1n+1

1 + α+

tα+1

B(α)Γ(α)h1113888 11138891113896 1113897 (20)






(21)


Ω(α 1) α

B(α)1113946

tn+1

0tn+1 minus s( 1113857




Ω(α 2) α

B(α)1113946

tn





B(α)1113946

tn+1

0tn+1 minus s( 1113857


αB(α)

1113946tn


αminus1f(v s)ds

leα

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds

+

αB(α)

1113946tn


αminus1f(v s)ds

leα

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds +

αB(α)

1113946tn


αminus1f(v s)ds

leαM

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1ds +αM

B(α)1113946

tn


αminus1ds

leαM

B(α)Γ(α + 1)tαn+1 + t

αn1113858 1113859

αMhα

B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(24)



B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(25)


v tn+1( 1113857 minus v tn( 1113857

le1 minus αB(α)



le1 minus αB(α)


+αMhα

B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(26)


v tn+1( 1113857 minus v tn( 1113857

⟶ 0 (27)










minusα


αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (29)



minusα


αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (30)


f(u t) z2u

zx2 +(k minus 1)zu

zxminus ku (31)


for the function f

f un tn( 1113857 un

j+1 minus 2unj + un

jminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δx

minus kunj + O(Δx)

(32)





jminus1



2Δx


(33)


un+1j u

nj + θ(α 1) minus θ(α 2) (34)



minusα


αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (35)


minusα


αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (36)


un+1j u

nj + H(n h α 1)

unj+1 minus 2un

j + unjminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δxminus ku

nj1113890 1113891



jminus1



2Δxminus ku

nminus1j1113890 1113891

(37)


un0(x) max e

xminus 1 0( 1113857 (38)







f vn τ( 1113857 +α

B(α)1113946τ

0f s vn( 1113857ds

(39)


v(x τ) v0(0) + v1(τ) + v2(τ) + (40)




xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)

1 minus αB(α)


xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)


xminus 1 0( 1113857( 11138571113858 1113859

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

(41)


u2(x τ) k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(1 minus α)

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(α)

(1 minus α)τα

B(α)Γ(1 + α)+

ατ2α

B(α)Γ(1 + 2α)1113896 1113897

(42)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) + (43)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) +

max ex


xminus 1 0( 1113857( 11138571113858 1113859τ

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961τ2 +

max ex


1113872 1113873 + max ex 0( 1113857e

minus kτ

(44)



minuskτ1113872 1113873 + max e

xminus 1 0( 1113857e

minuskτ

(45)



N d2( 1113857 (46)


T minus t

radic and




N d1( 1113857 N d2( 1113857 1 (47)



(48)




max ex 0( 1113857 e

x

max ex

minus 1 0( 1113857 ex

minus 1(49)


u(x τ) S

E1 minus e


S

Eminus 11113874 1113875e

minus kτ

S

Eminus e

minus kτ

(50)



(51)







δ zu

zx (52)


δ max ex 0( 1113857 (53)


δ maxS

E 01113874 1113875 (54)


zui

zx 0 (55)










00510

2

2468

18

10121416

16

1820

14 12 151 08 06 04 02 02

Volat

ility

surfa

ce

S E


0 05 1 15 2 25 3 35 4 45 502468

101214161820

S gt EE gt S

S or E

Volat

ility

line


04

10

35

20

43

30

3525

40

32

50

2515 2

60

151 105 050 0

u

xτ







9 Conclusion


Data Availability




References





0102030405060708090


0 05 1 15 2 25 3 35 4 45x

u




01541 001 03240 03242 1146704677 003 08455 08456 1538102401 005 05745 05747 1176201697 002 04924 04925 11457

0

10

4

20

30

40

50

3

60

241 3532521510 050

u

x

τ





































Ω(α 2) α

B(α)1113946

tn





B(α)1113946

tn+1

0tn+1 minus s( 1113857


αB(α)

1113946tn


αminus1f(v s)ds

leα

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds

+

αB(α)

1113946tn


αminus1f(v s)ds

leα

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1f(v s)ds +

αB(α)

1113946tn


αminus1f(v s)ds

leαM

B(α)1113946

tn+1

0tn+1 minus s( 1113857

αminus1ds +αM

B(α)1113946

tn


αminus1ds

leαM

B(α)Γ(α + 1)tαn+1 + t

αn1113858 1113859

αMhα

B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(24)



B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(25)


v tn+1( 1113857 minus v tn( 1113857

le1 minus αB(α)



le1 minus αB(α)


+αMhα

B(α)Γ(α + 1)(n + 1)

α+(n)

α1113858 1113859

(26)


v tn+1( 1113857 minus v tn( 1113857

⟶ 0 (27)










minusα


αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (29)



minusα


αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (30)


f(u t) z2u

zx2 +(k minus 1)zu

zxminus ku (31)


for the function f

f un tn( 1113857 un

j+1 minus 2unj + un

jminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δx

minus kunj + O(Δx)

(32)





jminus1



2Δx


(33)


un+1j u

nj + θ(α 1) minus θ(α 2) (34)



minusα


αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (35)


minusα


αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (36)


un+1j u

nj + H(n h α 1)

unj+1 minus 2un

j + unjminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δxminus ku

nj1113890 1113891



jminus1



2Δxminus ku

nminus1j1113890 1113891

(37)


un0(x) max e

xminus 1 0( 1113857 (38)







f vn τ( 1113857 +α

B(α)1113946τ

0f s vn( 1113857ds

(39)


v(x τ) v0(0) + v1(τ) + v2(τ) + (40)




xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)

1 minus αB(α)


xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)


xminus 1 0( 1113857( 11138571113858 1113859

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

(41)


u2(x τ) k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(1 minus α)

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(α)

(1 minus α)τα

B(α)Γ(1 + α)+

ατ2α

B(α)Γ(1 + 2α)1113896 1113897

(42)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) + (43)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) +

max ex


xminus 1 0( 1113857( 11138571113858 1113859τ

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961τ2 +

max ex


1113872 1113873 + max ex 0( 1113857e

minus kτ

(44)



minuskτ1113872 1113873 + max e

xminus 1 0( 1113857e

minuskτ

(45)



N d2( 1113857 (46)


T minus t

radic and




N d1( 1113857 N d2( 1113857 1 (47)



(48)




max ex 0( 1113857 e

x

max ex

minus 1 0( 1113857 ex

minus 1(49)


u(x τ) S

E1 minus e


S

Eminus 11113874 1113875e

minus kτ

S

Eminus e

minus kτ

(50)



(51)







δ zu

zx (52)


δ max ex 0( 1113857 (53)


δ maxS

E 01113874 1113875 (54)


zui

zx 0 (55)










00510

2

2468

18

10121416

16

1820

14 12 151 08 06 04 02 02

Volat

ility

surfa

ce

S E


0 05 1 15 2 25 3 35 4 45 502468

101214161820

S gt EE gt S

S or E

Volat

ility

line


04

10

35

20

43

30

3525

40

32

50

2515 2

60

151 105 050 0

u

xτ







9 Conclusion


Data Availability




References





0102030405060708090


0 05 1 15 2 25 3 35 4 45x

u




01541 001 03240 03242 1146704677 003 08455 08456 1538102401 005 05745 05747 1176201697 002 04924 04925 11457

0

10

4

20

30

40

50

3

60

241 3532521510 050

u

x

τ





































minusα


αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (29)



minusα


αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (30)


f(u t) z2u

zx2 +(k minus 1)zu

zxminus ku (31)


for the function f

f un tn( 1113857 un

j+1 minus 2unj + un

jminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δx

minus kunj + O(Δx)

(32)





jminus1



2Δx


(33)


un+1j u

nj + θ(α 1) minus θ(α 2) (34)



minusα


αminus

(n + 1)α+1

α + 11113890 1113891 minus

αB(α)Γ(α)

hα (n)α

αminus

(n)α+1

α + 11113890 11138911113896 1113897 (35)


minusα


αminus

(n + 1)α+1

α + 1+

nα+1

B(α)Γ(α)h1113890 11138911113896 1113897 (36)


un+1j u

nj + H(n h α 1)

unj+1 minus 2un

j + unjminus1

Δx2 +(k minus 1)un

j+1 minus unjminus1

2Δxminus ku

nj1113890 1113891



jminus1



2Δxminus ku

nminus1j1113890 1113891

(37)


un0(x) max e

xminus 1 0( 1113857 (38)







f vn τ( 1113857 +α

B(α)1113946τ

0f s vn( 1113857ds

(39)


v(x τ) v0(0) + v1(τ) + v2(τ) + (40)




xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)

1 minus αB(α)


xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)


xminus 1 0( 1113857( 11138571113858 1113859

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

(41)


u2(x τ) k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(1 minus α)

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(α)

(1 minus α)τα

B(α)Γ(1 + α)+

ατ2α

B(α)Γ(1 + 2α)1113896 1113897

(42)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) + (43)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) +

max ex


xminus 1 0( 1113857( 11138571113858 1113859τ

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961τ2 +

max ex


1113872 1113873 + max ex 0( 1113857e

minus kτ

(44)



minuskτ1113872 1113873 + max e

xminus 1 0( 1113857e

minuskτ

(45)



N d2( 1113857 (46)


T minus t

radic and




N d1( 1113857 N d2( 1113857 1 (47)



(48)




max ex 0( 1113857 e

x

max ex

minus 1 0( 1113857 ex

minus 1(49)


u(x τ) S

E1 minus e


S

Eminus 11113874 1113875e

minus kτ

S

Eminus e

minus kτ

(50)



(51)







δ zu

zx (52)


δ max ex 0( 1113857 (53)


δ maxS

E 01113874 1113875 (54)


zui

zx 0 (55)










00510

2

2468

18

10121416

16

1820

14 12 151 08 06 04 02 02

Volat

ility

surfa

ce

S E


0 05 1 15 2 25 3 35 4 45 502468

101214161820

S gt EE gt S

S or E

Volat

ility

line


04

10

35

20

43

30

3525

40

32

50

2515 2

60

151 105 050 0

u

xτ







9 Conclusion


Data Availability




References





0102030405060708090


0 05 1 15 2 25 3 35 4 45x

u




01541 001 03240 03242 1146704677 003 08455 08456 1538102401 005 05745 05747 1176201697 002 04924 04925 11457

0

10

4

20

30

40

50

3

60

241 3532521510 050

u

x

τ









































f vn τ( 1113857 +α

B(α)1113946τ

0f s vn( 1113857ds

(39)


v(x τ) v0(0) + v1(τ) + v2(τ) + (40)




xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)

1 minus αB(α)


xminus 1 0( 1113857( 1113857 +


B(α)Γ(1 + α)


xminus 1 0( 1113857( 11138571113858 1113859

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

(41)


u2(x τ) k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(1 minus α)

1 minus αB(α)

+ατα

B(α)Γ(1 + α)1113896 1113897

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961(α)

(1 minus α)τα

B(α)Γ(1 + α)+

ατ2α

B(α)Γ(1 + 2α)1113896 1113897

(42)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) + (43)


u(x τ) u(x 0) + u1(x 0) + u2(x 0) +

max ex


xminus 1 0( 1113857( 11138571113858 1113859τ

minus k2 max e


xminus 1 0( 1113857( 11138571113960 1113961τ2 +

max ex


1113872 1113873 + max ex 0( 1113857e

minus kτ

(44)



minuskτ1113872 1113873 + max e

xminus 1 0( 1113857e

minuskτ

(45)



N d2( 1113857 (46)


T minus t

radic and




N d1( 1113857 N d2( 1113857 1 (47)



(48)




max ex 0( 1113857 e

x

max ex

minus 1 0( 1113857 ex

minus 1(49)


u(x τ) S

E1 minus e


S

Eminus 11113874 1113875e

minus kτ

S

Eminus e

minus kτ

(50)



(51)







δ zu

zx (52)


δ max ex 0( 1113857 (53)


δ maxS

E 01113874 1113875 (54)


zui

zx 0 (55)










00510

2

2468

18

10121416

16

1820

14 12 151 08 06 04 02 02

Volat

ility

surfa

ce

S E


0 05 1 15 2 25 3 35 4 45 502468

101214161820

S gt EE gt S

S or E

Volat

ility

line


04

10

35

20

43

30

3525

40

32

50

2515 2

60

151 105 050 0

u

xτ







9 Conclusion


Data Availability




References





0102030405060708090


0 05 1 15 2 25 3 35 4 45x

u




01541 001 03240 03242 1146704677 003 08455 08456 1538102401 005 05745 05747 1176201697 002 04924 04925 11457

0

10

4

20

30

40

50

3

60

241 3532521510 050

u

x

τ





































max ex 0( 1113857 e

x

max ex

minus 1 0( 1113857 ex

minus 1(49)


u(x τ) S

E1 minus e


S

Eminus 11113874 1113875e

minus kτ

S

Eminus e

minus kτ

(50)



(51)







δ zu

zx (52)


δ max ex 0( 1113857 (53)


δ maxS

E 01113874 1113875 (54)


zui

zx 0 (55)










00510

2

2468

18

10121416

16

1820

14 12 151 08 06 04 02 02

Volat

ility

surfa

ce

S E


0 05 1 15 2 25 3 35 4 45 502468

101214161820

S gt EE gt S

S or E

Volat

ility

line


04

10

35

20

43

30

3525

40

32

50

2515 2

60

151 105 050 0

u

xτ







9 Conclusion


Data Availability




References





0102030405060708090


0 05 1 15 2 25 3 35 4 45x

u




01541 001 03240 03242 1146704677 003 08455 08456 1538102401 005 05745 05747 1176201697 002 04924 04925 11457

0

10

4

20

30

40

50

3

60

241 3532521510 050

u

x

τ








































00510

2

2468

18

10121416

16

1820

14 12 151 08 06 04 02 02

Volat

ility

surfa

ce

S E


0 05 1 15 2 25 3 35 4 45 502468

101214161820

S gt EE gt S

S or E

Volat

ility

line


04

10

35

20

43

30

3525

40

32

50

2515 2

60

151 105 050 0

u

xτ







9 Conclusion


Data Availability




References





0102030405060708090


0 05 1 15 2 25 3 35 4 45x

u




01541 001 03240 03242 1146704677 003 08455 08456 1538102401 005 05745 05747 1176201697 002 04924 04925 11457

0

10

4

20

30

40

50

3

60

241 3532521510 050

u

x

τ








































9 Conclusion


Data Availability




References





0102030405060708090


0 05 1 15 2 25 3 35 4 45x

u




01541 001 03240 03242 1146704677 003 08455 08456 1538102401 005 05745 05747 1176201697 002 04924 04925 11457

0

10

4

20

30

40

50

3

60

241 3532521510 050

u

x

τ





































































Date post:	31-Jul-2020
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