European option pricing models described by fractional operators with classical and generalized Mittag‐Leffler kernels

In this paper, we investigate novel solutions of fractional‐order option pricing models and their fundamental mathematical analyses. The main novelties of the paper are the analysis of the existence and uniqueness of European‐type option pricing models providing to give fundamental solutions to them and a discussion of the related analyses by considering both the classical and generalized Mittag‐Leffler kernels. In recent years, the generalizations of classical fractional operators have been attracting researchers' interest globally and they also have been needed to describe the dynamics of complex phenomena. In order to carry out the mentioned analyses, we take the Laplace transforms of either classical or generalized fractional operators into account. Moreover, we evaluate the option prices by giving the models' fractional versions and presenting their series solutions. Additionally, we make the error analysis to determine the efficiency and accuracy of the suggested method. As per the results obtained in the paper, it can be seen that the suggested generalized operators and the method constructed with these operators have a high impact on obtaining the numerical solutions to the option pricing problems of fractional order. This paper also points out a good initiative and tool for those who want to take these types of options into account either individually or institutionally.


INTRODUCTION
Recently, a massive volume of financial commodities have been utilized all around the world. One of the most significant instruments is a share option which permits the holder to buy or sell shares in a specified time. For instance, a European put option gives the right to the owner, but not the obligation, to sell a predetermined amount of an underlying asset at a predetermined price Q within a predetermined time T. While the pay-off function Θ for the European call option is given by the European put option is given by where Υ(T) is the asset price at time T. Similarly, there are a large number of different kinds of options such as American options which can be exercised at any time before T, barrier options which can be exercised only if the asset price reaches a specified level (barrier) and many other types such as bond options, exotic options, foreign exchange options.
The most important problem of financial derivatives is determining the present day value. In order to calculate this, one needs to compute the expected value of the option's pay-off functions at the exercise time. For that, the process of determining the underlying assets price can be modeled by considering a system of stochastic differential equations. In order to determine the price Υ t of a single security, the geometric Brownian motion with drift and volatility has been used in the Black-Scholes model [1]. The geometric Brownian motion is where W is a Wiener process. Not only a stochastic formulation technique, but also a partial differential equation (PDE) formulation has been widely used in order to determine value R(t, s). After necessary calculation we obtain the final value problem for European options with a number of underlying securities with share pricess = (s 1 , s 2 , … , s ) T as where is a matrix formed volatility. In this formula, every stochastic variable represents to one dimension in the PDE. In 1973, Black and Scholes [1] pointed out a model which can easily evaluate the prices of the options that is now well-known as the Black-Scholes model. This pricing model is one of the most effective mathematical equations in the mathematical finance literature. In Equation (4), we can obtain that R(0, t) = 0, R(Υ, t) ∼ Υ as Υ → ∞ , and we can find the following payoff functions: R c (Υ, T) = max(Υ − Q, 0) and R p (Υ, T) = max(Q − Υ, 0), where R c (Υ, T) and R p (Υ, T) show the value of vanilla call and put options, respectively. For transferring to the fractional version of the above equation, we make the following modifications: . This yields the equation: with initial condition: Equation (5) is called the Black-Scholes option pricing equation of fractional order. In Equation (5), we define k = 2 / 2 , where k shows the balance between the interest rates' and stock returns' variability.
Moreover, Cen and Le considered the generalized version of the fractional Black-Scholes equation (GFBSE) [2] by assigning = 0.06 and = 0.4(2 + sin ) in Equation (5): with the initial condition: Up to here, the emergence of the new fractional operators in the literature can be considered as a result of the reproduction of new problems that model different types of real-life events. Fractional derivative operators that can be stated as nonlocal have been developed to address these kinds of nonlinear differential equations such as the Riemann-Liouville fractional-order [3], Caputo-Fabrizio fractional-order derivative based on the exponential kernel [4], Atangana-Baleanu fractional-order derivative which is based on the generalized Mittag-Leffler (GML) function as nonlocal and nonsingular kernel [5]. One can see solid theoretical results and related essential applications of the mentioned operators in the literature [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. Moreover, in recent years the generalizations of classical fractional operators have been attracting effect all over the world and they are needed to describe the dynamics of complex phenomena. Abdeljawad et al. [26] pointed out very important related results for the generalized fractional operators with and without Mittag-Leffler kernel. Meanwhile, Abdeljawad [27,28] developed the corresponding fractional integrals of generalized ABC operator with arbitrary order by using the infinite binomial theorem, and studied their semi-group properties and their action on the ABC type fractional derivatives to prove the existence and uniqueness theorem for the ABC-fractional initial value problems.

A summary of numerical-approximate solutions in option pricing models
There exist a number of solutions which give us closed results to determine the price of a few options. However, since European-type options modeled Black-Scholes equation based on some underlying assets, one need to use numerical methods for valuation.
One of the most commonly used methods for valuation option prices is Monte Carlo simulation (MCS) method which simulates trajectories [50]. The major benefits are that the MCS method is easy to apply comparing with the other methods even with several underlying securities and that the implementation cost increases linearly with many securities. However, MCS methods are considerably slow to converge. In general, in order to decrease the variance in the estimates, MCS methods require simulating millions of trajectories. Another downside with MCS methods is that MCS is not applicable for Greeks. They might be obtained through additional MCS simulations presuming that the pay-off continuously differentiable.
In order to avoid high computational cost when pricing options, it can be addressed by raising hardware performance. In financial word, a number of people try to find a better computer clusters that provide more accurate and faster result via parallel programming. On the other hand, a large number of approximate-series solutions to option pricing models have been examined. Among them, Yavuz et al. [51][52][53][54] presented the solutions of fractional order BSE by using different types of fractional kernels and methods. Fall et al. [55] obtained an approximate solution to the mentioned problem by using the Caputo generalized fractional derivative. In [56] the authors have obtained the semi-analytical solution to the Ivancevic option pricing model of fractional order, which is an alternative of the standard Black-Scholes pricing equation and signifies a controlled Brownian motion related to the nonlinear Schrodinger equation. Additionally, one can see the related studies for the BSE of fractional order in [57][58][59][60][61][62].
The remaining parts of the study have been outlined as the following: in Section 2, we introduce the fractional-order derivatives, GML kernel, and their Laplace transformation which have been used in the paper. In Section 3, we present the mathematical investigation of the fractional model. In Section 4, we provide the existence and uniqueness of the solutions to fractional BSE. In Section 5, we describe the method by using the GML kernel for obtaining the approximate solutions of the models. Moreover, in this section, we give their corresponding solutions. In Section 6, we provide the error analysis of the approximate solutions we have obtained. In Section 7, we illustrate the main results by graphical representations and discuss the impact of the fractional order when we fix the other parameters , . We give the conclusions and perspectives in Section 8.

SOME PRELIMINARIES
In this section, we summarize the following fundamental definitions of fractional calculus which are used further in the present paper. In fractional calculus, most frequently used definition of integration of non-integer order comes from the extension of the well-known formula for the n-fold integration and is named Riemann-Louville operators. Moreover, in this section, we recall the definition of the fractional operators with singular kernel and their generalized versions which have been recently proposed in the literature by Abdeljawad et al. [26,28]. Before giving the fractional operator with their generalized form we begin with Mittag-Leffler function.

Definition 1
The Mittag-Leffler function with the parameters and is defined as following series where > 0, ∈ R, and q ∈ C. It is well known that the convergence of this series results from the assumptions > 0, and > 0.
Definition 5 [26,28]. The left and right ABC fractional derivatives which have been constructed with the GML function defined in Definition 2 are given as, respectively: and Definition 6 [26,28]. Assume that g( ) is defined on [a, b]. Then the generalized left and right Riemann-Liouville fractional integrals of the AB operator of order 0 < ≤ 1, > 0, Re(1 − ) > 0 are given as, respectively: , where a I {g( )} and I b {g( )} are the left and right Riemann-Liouville fractional integrals, respectively.

Definition 7
We assume f , g : [0, ∞) → R, then the convolution of these functions is given as and the following property holds where  represents the usual Laplace transform.
Definition 8 [5] The Laplace transform of the ABC fractional operator is given by Definition 9 [26]. The generalized Laplace transform of the ABC fractional operator is given by Lemma 1 [27]. We assume Re( ) > 0, , , , , s ∈ C, Re(s) > 0, and | s − | < 1. Then the Laplace transform of E , ( t ) is given as

MATHEMATICAL INVESTIGATION OF THE FRACTIONAL OPTION PRICING MODEL
In this section, we give the mathematical perspective of the fractional Black-Scholes model by considering the generalized ABC fractional operator. First, we present the fundamental instruments of the investigation: Consider the fractional BSE with the generalized ABC derivative given in Equation (5): subject to the initial condition Operating Equation (18) of order , , on Equation (25), we get Assigning ( , , ) = The function ( , ) has an upper bound if the kernel ( , , ) satisfies the Lipschitz condition. Exclusively, ( , ) has an upper bound, since we have Taking N = n 1 1 + n 2 2 + 3 , this intends that Accordingly, ( , , ) satisfies the Lipschitz condition. Therewith, the function ( , ) is bounded. We then give the following theorems: is a bounded function, then the operator Ψ( ( , )) given by satisfies the Lipschitz condition.
Proof. Postulate that ( , ) and ( , ) are bounded functions, then where Λ = N 1 N and Thus, the operator Ψ( ( , )) satisfies the Lipschitz condition. This proves the theorem. ▪ Theorem 2 Suppose that ( , ) is a bounded function, then the operator given by satisfies the condition Proof. Assume that ( , ) is a bounded function, then It implies that This gives the proof. ▪

EXISTENCE AND UNIQUENESS OF THE SOLUTION
In this section, we analyze the existence and uniqueness of the solution of the problem which is given by Equation (25). Taking the unknown function ( , ) into consideration, we can generate the following iterative formula: where 0 = ( , 0). The difference between iterative terms can be given as We need to consider the following in order to proceed Taking the norm of both sides of Equation (37), we obtain By benefiting from the triangular inequality, we can conclude that Since the kernel satisfies the Lipschitz condition, then we get Taking Equations (36)-(41) into consideration, the following theorem is formed: (25) is said to have a solution if there exists 0 and the following inequality holds:

Theorem 3 Suppose that ( , ) is a bounded functions, then Equation
Proof. Assume that ( , ) is a bounded function. Using the fact that in Equation (41), and considering the recursive relation, we get Thus, Equation (38) exists and it is smooth. We then show that Equation (38) is a solution to Equation (25). Assume that ( , ) − ( , 0) = z ( , ) − Δ z ( , ), then we have Using the recursive scheme, we achieve as z → 0 reaches that ||Δ z ( , )|| → 0. This proves the theorem. ▪ We now keep going to show the uniqueness of the solution to Equation (25).

Theorem 4 Equation (25) is said to have a unique solution if
Proof. Assume that Equation (25) has two solutions, namely, 1 ( , ) and 2 ( , ), then we can write Taking norm of both sides of Equation (48), we have It follows that Therefore, If the condition given in Equation (51) holds, then we have which gives 1 ( , ) = 2 ( , ). Hence, Equation (25) has a unique solution. ▪

MAIN RESULTS
In this section, we give the solutions of the fractional order classical and generalized Black-Scholes option pricing models. In order to achieve that, we consider the fractional operators with both of the classical and GML kernels.

Fundamental solutions via the classical Mittag-Leffler kernel
In this subsection, we first define the suggested method by using the Laplace transformation to solve the Black-Scholes option pricing problems mentioned in Section 1. This method is combined with the classical homotopy method and Laplace transform. Now, we consider the FBSE which is given by Equation (5) and its initial condition in Equation (6): with initial condition given by where ABC 0 D , , { ( )} shows the generalized ABC operator. Other variables stated in Equation (53) are the same with those which were defined in Section 3. Using the LT, we define the { ( , )} = H( , s). We here construct the solution steps only according to the generalized ABC operator. For classical ABC operator one can obtain the method by using the similar steps as in this section. Then applying the homotopy to Equation (53) we derive the homotopies for the generalized ABC operator. Taking the LT of both sides of Equation (53), yields We assume that the solution is given by the following series then substituting Equation (56) into Equation (55) and applying the homotopy steps, we have Then the approximate solution is given as where By following the similar solution steps we can obtain the solution of the generalized fractional Black-Scholes equation which is given in Equations (7) and (8).
with initial condition given by where ABC 0 D , , {g( )} shows the GABC operator. Then applying the homotopy to Equation (60) we construct the homotopies for the ABC operator. Taking the LT of both sides of Equation (60), yields We regard that the solution of the GFBSE is given by the following series then substituting Equation (63) into Equation (62) and applying the homotopy steps, we have Then the approximate solution of the mentioned problem is given as where Now we achieve the solutions of the FBSE and GFBSE by considering the solution methods that have been constructed by using the fractional operators with both of the classical and GML kernels. First, we use the method for the FBSE with the classical Mittag-Leffler kernel operator: if we apply the Laplace transform of the ABC in Definition 9 to Equation (53) and considering its initial condition in Equation (54), we have the following relations: By equating the powers of , we get the approximate solution as the following By following similar steps above one can obtain the other parts of the series and then for the special case of the fractional parameter = 1, the exact solution of the problem is obtained as ( , ) = e (1 − e −k ) + e −k max(e − 1, 0). Second, we will also give the solution of the generalized FBSE by considering the classical ABC operator. Applying the LT of the ABC to Equation (60) and considering its initial condition in Equation (61), we have the following relations: Then by taking into account the powers of , we have the approximate solution as the following By following these steps one can obtain the other parts of the series and then for the special case of the fractional parameter = 1, the exact solution of the problem is obtained as ( , ) = max( − 25e −0.06 , 0)e 0.06 + x(1 − e 0.06 ).

Fundamental solutions via the GML kernel
In this subsection, we aim to obtain a fundamental solution to both of the FBSE and GFBSE by considering the generalized fractional operator with Mittag-Leffler kernel. For this purpose, first, we use the method for the FBSE via the fractional operator with the GML kernel. We start to the solution by applying the LT of the generalized ABC in Definition 9 to Equation (53) and considering its initial condition in Equation (54), we have the following relations: By expanding the series according to the powers of , we point out the following decomposition where E , (q) is the Mittag-Leffler function with three parameters which has been given in Definition 2.
Second, we will also have the solution of the generalized FBSE by considering the generalized ABC operator. The later step is to apply the LT of the generalized ABC to Equations (60) and (61), we obtain the following results: Then, by taking cognizance of the powers of , we reveal the approximate solution as where E , (q) is the GML function with three parameters.

ERROR ANALYSIS OF THE METHOD
In this section, we point out the error norms of the suggested method by considering L 2 and L ∞ error norms. By the aid of a computer package program, we have calculated the error norms which are given. In numerical computations, it is well known that these L 2 and L ∞ error norms have been used to test the accuracy of the numerical results to the exact solution. The formulas of these error norms are given as: The L 2 error norm for the BSE of fractional order which has been constructed with the ABC operator can be defined as [63] and L ∞ error norm can be defined as The error rates we have obtained are quite meaningful and they point out the method we have used is very accurate and effective even we have used only first few components of the series. Therefore using this mentioned method is extremely suggested for obtaining the approximate solution of the fractional PDE.
In Table 1 and Figure 1, we have depicted the series solution results and comparison of the solution with the exact solution which we have obtained in Section 5.1. In addition, we have shown the absolute error values that are very lower even they have been provided by only first four components of the series solution.

GRAPHICAL REPRESENTATIONS AND DISCUSSION
In this section, we figure out the solutions we have obtained in Section 3 and we discuss the results that these figures reveal out. In Figures 2 and 3, we have depicted the numerical and exact solutions of the model with respect to the ABC operator which is defined by the classical Mittag-Leffler kernel. In Figure 4, one can see the exact solution which has been obtained by considering the classical Mittag-Leffler kernel. In Figure 5, numerical solutions have been shown according to different values of the fractional parameter. It can be concluded from this figure that as the fractional order increases, option values approach to the exact solution. Figures 6 and 7 represent the exact solution and the numerical solution of the fractional order option pricing problem which includes the GML kernel, respectively. One can understand from Figure 7 that the numerical solutions are totally agreement to the exact solution. Figure 8 depicts the option prices varying according to fractional parameter values for Equation (5). It points out that the fractional parameter has an important effect on the option prices. In addition to these results, we can represent the effects of the volatility values on the option prices with Figure 9.

CONCLUSION
In this paper, the existence and uniqueness have been investigated for the fractional option pricing models which have been described by a specific-type fractional derivative operator with the both classical and GML kernel. Moreover, novel solutions of the mentioned problems have been achieved numerically, the successive approximation method has been also discussed and the computational simulations have been depicted graphically. We also have pointed out the error analysis of the method and according to the error values we can conclude that the method we have used is totally agree with the exact solution. With the aid of the results of this paper, some effective studies can be achieved by considering different types of fractional operators. These results can be regarded as the main novelties of the paper. These results have also shown that both of the classical and GML kernels are quite compatible to reveal the option prices. It is clear that when , , → 1, the ABC operator which is defined by the GML kernel turns to the integer order ordinary differential equation. Moreover, although the generalized type fractional derivatives have singular kernels for 0 < < 1, the one parameter ML function has a nonsingular kernel. In recent years, the generalizations of classical fractional operators have been attracting researchers' interest globally and they are needed to describe the dynamics of complex phenomena. In order to carry out the mentioned items, we have taken the Laplace transforms of the either classical or generalized fractional operators into account. Moreover, we have evaluated the option prices by giving the models' fractional versions and presenting their series solutions. In addition to all results, this paper has pointed out a good initiative and tool for those who want to take these types of options into account either individually or institutionally. For future studies, different types of options such as American options, barrier options, interest options, bond options, exotic options, foreign exchange options, and so on, or various applications of the mentioned method constructed with the GML kernel can be considered.