Numerical Methods for Differential and Integral EquationsView this Special Issue
On Comparative Analysis for the Black-Scholes Model in the Generalized Fractional Derivatives Sense via Jafari Transform
The Black-Scholes model is well known for determining the behavior of capital asset pricing models in the finance sector. The present article deals with the Black-Scholes model via the Caputo fractional derivative and Atangana-Baleanu fractional derivative operator in the Caputo sense, respectively. The Jafari transform is merged with the Adomian decomposition method and new iterative transform method. It is worth mentioning that the Jafari transform is the unification of several existing transforms. Besides that, the convergence and uniqueness results are carried out for the aforesaid model. In mathematical terms, the variety of equations and their solutions have been discovered and identified with various novel features of the projected model. To provide additional context for these ideas, numerous sorts of illustrations and tabulations are presented. The precision and efficacy of the proposed technique suggest its applicability for a variety of nonlinear evolutionary problems.
Recently, the subject of fractional calculus has garnered considerable prominence. Several well-known mathematicians have contributed to this field by proposing numerous fractional operators in various texts. The conclusions of contemporary calculus are often substantially more precise than those of ancient ones. It has presented the dynamic behavior of a variety of real-world situations that take place between two integers. Additionally, fractional operators have more dimensionality than integer differential operators such as Caputo, Liouville, Hadamard, Coimbra, Davison, Riesz, Riemann and Liouville, Weyl, and Jumarie, Caputo and Fabrizio , Atangana and Baleanu , and Scherer et al.  are some well-known fractional derivative formulations, see [4–9]. Furthermore, the Liouville-Caputo and AB operators are considered to be the best fractional filters in this field of research.
Several methods that assess fractional differential equations (FDEs) for their reliability and trustworthiness are actively considered in [10, 11]. Certain popular approximate-analytical approaches including Adomian decomposition method , new iterative transform method , homotopy perturbation method , Haar wavelet method , Padé approximation , reproducing kernel Hilbert space method , new Legendre wavelet decomposition method , Bäcklund transformation method , and Lie symmetry analysis .
Fischer Black and Myron Scholes developed a mathematical model for the capital asset pricing model in 1973. The revolutionary Black–Scholes model (BSM) is the foundation of modern financial theory which is remarkable to discuss contemporary economics without referencing the innovative BSM.
The purpose of this research is to obtain new solutions by employing both decomposition method and Jafari iterative transform method into a BSM. In banking and finance, the fractional formulation of BSM is represented by : subject to the playoff mapping where represents the option’s value at asset values at time and is the expiry period. The fundamental stock price is indicated by the letter . The risk-free mortgage to expiry is represented by the variable . The volatility of a financial commodity is represented by the constant . We also include the necessary hypotheses: a constant risk-free interest rate , no processing fees, the ability to purchase and sell an unlimited number of stocks, and no prohibitions on speculative trading. Finally, we include European options. Furthermore, it is remarkable that and as The BSM in (1) can then be represented as a parabolic diffusion equation. Introducing the subsequent transformations then, Equation (1) reduces to subject to initial conditions where signifies the equilibrium between the free relationship in inflation and stock volatility. In , Cen and Le introduced the generalized fractional BSM. The BSM is described this way: subject to initial conditions
The fractional BSM with one commodity has been explored extensively [21, 22]. The fractional BSM is an extended variant of the classical BSM that extends the model’s limitations. Meng and Wang  used the BSM to investigate fractional opportunity valuation. The covered call price for bank international trade in China was calculated using the fractional BSM. Their findings suggest that when it comes to evaluating the influence of the pricing system, the fractional BSM outperforms the classical BSM . Fall et al.  estimated the Black-Scholes option pricing equations via the homotopy perturbation method. Matadi and Zondi  contemplated the invariant solutions of BSM via theOrnstein-Uhlenbeck process. Kumar et al.  demonstrated numerical computation of fractional BSM arising in the financial market. Yavuz and Özdemir  proposed a diverse approach to the European option pricing model with a new fractional operator.
Amidst Gorge Adomian’s massive boost in 1980, the Adomian decomposition method introduced a well-noted terminology. It has been intensively implemented for a diverse set of nonlinear PDEs, for instance, Fisher’s model  and Zakharov–Kuznetsov equation . The ADM was determined to be significantly related to a variety of integral transforms, including Laplace, Swai, Mohand, Aboodh, and Elzaki. Very recently, Jafari  propounded a well-known integral transform which is known as the Jafari transform. The dominant feature of this transformation is that it has the ability to recapture several existing transformations, see Remark 8.
In 2006, Yavuz and Özdemir  expounded a new iterative transform method (NITM) that is intensively employed by numerous researchers due to its frequent applicability in fractional ODES and PDEs. The recursive technique tends to the exact solution if it exists via successive approximations. A small proportion of estimates can be employed for analytical reasons with a reasonable level of accuracy for particular problems. The NITM does not need some restricted hypothesis for handling nonlinearity factors. For example, in , the authors employed NITM to find the numerical solution of the fifth- and sixth-order nonlinear boundary value problems, Rashid et al.  applied NITM to obtain the solution of the fractional Fornberg-Whitham equation, and Jafari  constructed the Laplace decomposition algorithm via NITM.
Owing to the aforementioned trend, to obtain the explicit solution of the time-fractional BSM, we employ the Jafari transform decomposition technique (JDM) and Jafari iterative transform method (JITM). The Jafari transform merged the Adomian decomposition method and a new iterative method in an efficient manner to develop novel algorithmic approaches. The Jafari transform is the refinement of several existing transforms, see Remark 8. Both projected schemes yield analytical solutions in a convergent series form. Mathematical characterizations of the BSM are illustrated via the AB fractional derivative operator in the Caputo sense. Simulation and tabulation studies depict a clear picture of the proposed approaches. The analytical solution, especially for fractional PDEs, is a useful mechanism for analyzing the behaviors of solutions that are challenging to numerically solve. The analytical solution can be used to investigate macroeconomic behaviors.
This article’s entire content is divided into seven parts, which are described in the following order: Section 2 summarises and presents the core concepts and terminology of the singular power law fractional derivative and nonsingular Atangana-Baleanu fractional derivative in the Caputo sense. In Section 3, two novel algorithms are developed via the new integral transform. In Section 4, convergence and uniqueness analyses are discussed and presented for the proposed model. Section 5 is the main part of the proposed work where we present a debate on the results and their interpretation. Finally, concluding remarks are presented in Section 7.
In this section, we present some essential concepts, notions, and definitions concerning fractional derivative operators depending on power and Mittag-Leffler as a kernel, along with the detailed consequences of the Jafari transform.
Definition 1 (see ). The Caputo fractional derivative is described as follows:
Definition 2 (see ). The Atangana-Baleanu fractional derivative operator in the Caputo form is stated as follows: where and signifies a normalization function as
Definition 3 (see ). The fractional integral of the operator is described as follows:
Definition 4 (see ). Consider an integrable mapping defined on a set then
Definition 5 (see ). Suppose the mappings such that The Jafari transform of the mapping presented by is described as
Theorem 6 (see ) (convolution property). For Jafari transform, the subsequent holds true:
Definition 7. The Jafari transform of the CFD operator is stated as follows:
Remark 8. Definition 7 leads to the following conclusions: (1)Taking and then we acquire the Laplace transform (2)Taking and then we acquire the -Laplace transform (3)Taking and then we acquire the Sumudu transform (4)Taking and then we acquire the Aboodh transform (5)Taking and then we acquire the Pourreza transform [39, 40](6)Taking and then we acquire the Elzaki transform (7)Taking and then we acquire the natural transform (8)Taking and then we acquire the Mohand transform (9)Taking and then we acquire the Sawi transform (10)Taking and then we get the Kamal transform .(11)Taking and then we acquire the transform [46, 47]
Definition 9 (see ). The Jafari transform of the ABC fractional derivative operator is described as
Remark 10. Definition 9 leads to the following conclusions: (1)Taking and then we acquire the Laplace transform of ABC fractional derivative operator [2, 49](2)Taking and then we acquire the Elzaki transform of ABC fractional derivative operator (3)Taking then we get the Sumudu transform of ABC fractional derivative operator (4)Taking and then we get the Shehu transform of ABC fractional derivative operator 
Definition 11 (see ). The Mittag-Leffler function for a single parameter is described as
3. New Semianalytical Approach for Nonlinear PDEs
Consider the generic fractional form of PDE: with ICs where symbolizes the Caputo and ABC fractional derivative of order while and denotes the linear and nonlinear factors, respectively. Also, represents the source term.
3.1. Configuration of Jafari Decomposition Method
Taking into account the Jafari transform to (17), we acquire
Firstly, the differentiation rule of Jafari transform with respect to CFD was applied; then, we apply the ABC fractional derivative operator as follows:
Therefore, the Jafari decomposition method was utilized to derive the solution of (17) by satisfying the assumption that has a solution of this equation which can be expressed as
Thus, the nonlinear term can be evaluated by the Adomian decomposition method prescribed as where
3.2. Construction of Jafari Iterative Transform Method
First, we apply the differentiation rule of Jafari transform for CFD, and then, we consider for ABC fractional derivative operator, respectively, we get
Using the fact of an iterative process, we find
Also, the operator is linear; therefore, and the nonlinearity dealt by (see ) where
Finally, we derive the following iterative process for CFD:
Again, the iterative process for ABC fractional derivative operator is presented as follows:
4. Convergence and Uniqueness Analyses of BSM via ABC Fractional Derivative Operator
The subsequent subsections will highlight how sufficient requirements guarantee the emergence of a unique solution. Our anticipated existence of solutions in the case of JDM is followed by .
Theorem 12 (uniqueness theorem). Equation (24) has a unique solution whenever where
Proof. Assuming all continuous functions on the Banach space are denoted by Also, suppose that have the norm Now, we define a function such that where and Here, suppose that and are also Lipschitzian with and where and are Lipschitz constant, respectively, and are distinct functional values. Since the mapping is contraction. Consequently, by the Banach contraction fixed point theorem, (17) has a unique. This gives the desired result.
Theorem 13 (convergence analysis). The general form solution of (17) will be convergent.
Proof. Assume that be the th partial sum, i.e., Here, we prove that a Cauchy sequence in Banach space
We acquire by considering a new form of Adomian polynomials. Now, Consider then, where Now, from triangular inequality we have since , we have then, Therefore, (since is bounded). Furthermore, as then Thus, is a Cauchy sequence in Consequently, the series is convergent and this yields the immediate consequence.
5. Physical Interpretation of Time-Fractional Black-Scholes Models
In this section, we compute the approximate analytical solution of BSM via the CFD and ABC fractional derivative operators by using the Jafari decomposition method.
5.1. Jafari Decomposition Method
Case 1. Firstly, we solve the (4) by using Caputo fractional derivative operator incorporating the Jafari decomposition method.
Applying Jafari transform to both sides of (4), we have
Using the differentiation rule of Jafari transform, we have
In view of (5), we get
Employing the inverse Jafari transform on both sides yields
With the help of Jafari decomposition method, we find
Here, we surmise that the unknown function can be written by an infinite series of the form
The approximate solution for Example 1 is expressed as
Case 2. Now, we solve(4) by using the ABC fractional derivative operator incorporating the Jafari decomposition method.
Considering (50) and using the differentiation rule of Jafari transform, we have
In view of (5), we get
Employing the inverse Jafari transform on both sides of the above equation yields
By the Jafari decomposition method, we find:
Here, we surmise that the unknown function can be written by an infinite series of the form:
The approximate solution for Example 1 is expressed as: