On Comparative Analysis for the Black-Scholes Model in the Generalized Fractional Derivatives Sense via Jafari Transform

Department of Mathematics, Government College University, Faisalabad 38000, Pakistan Department of Mathematics, Imam Mohammad Ibn Saud Islamic University, Riyadh 12211, Saudi Arabia Department of Mathematics, Lahore College Women University, 54000 Lahore, Pakistan Gofa Camp, Near Gofa Industrial College and German Adebabay, Nifas Silk-Lafto, 26649 Addis Ababa, Ethiopia Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia


Introduction
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 [1], Atangana and Baleanu [2], and Scherer et al. [3] are some well-known fractional derivative formulations, see [4][5][6][7][8][9]. Furthermore, the Liouville-Caputo and AB operators are considered to be the best fractional filters in this field of research.
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 [16]: subject to the playoff mapping where UðS, ςÞ represents the option's value at S asset values at time ς and T is the expiry period. The fundamental stock price is indicated by the letter E. The risk-free mortgage to expiry is represented by the variable ζ. The volatility of a financial commodity is represented by the constant ϖ. We also include the necessary hypotheses: a constant risk-free interest rate u, 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 Uð0, ςÞ = 0 and UðS, T Þ ≈ S as S ↦ ∞: 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 η signifies the equilibrium between the free relationship in inflation and stock volatility. In [20], Cen and Le introduced the generalized fractional BSM. The BSM is described this way: D ϑ ς Q w 1 , ς ð Þ= −0:08 2 + sin w 1 ð Þ 2 w 2 1 ∂ 2 Q w 1 , ς ð Þ ∂w 2 1 − 0:06w 1 ∂Q w 1 , ς ð Þ ∂w 1 + 0:06Q w 1 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 [23] 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 [24]. Fall et al. [14] estimated the Black-Scholes option pricing equations via the homotopy perturbation method. Matadi and Zondi [25] contemplated the invariant solutions of BSM via theOrnstein-Uhlenbeck process. Kumar et al. [26] dem-onstrated numerical computation of fractional BSM arising in the financial market. Yavuz and Özdemir [27] 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 [28] and Zakharov-Kuznetsov equation [29]. The ADM was determined to be significantly related to a variety of integral transforms, including Laplace, Swai, Mohand, Aboodh, and Elzaki. Very recently, Jafari [30] 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 [27] 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 [28], the authors employed NITM to find the numerical solution of the fifthand sixth-order nonlinear boundary value problems, Rashid et al. [29] applied NITM to obtain the solution of the fractional Fornberg-Whitham equation, and Jafari [30] 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 2 Journal of Function Spaces interpretation. Finally, concluding remarks are presented in Section 7.

Preliminaries
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.

New Semianalytical Approach for Nonlinear PDEs
Consider the generic fractional form of PDE: with ICs where D ϑ ς = ∂ ϑ Qðw 1 , ςÞ/∂ς ϑ symbolizes the Caputo and ABC fractional derivative of order ϑ ∈ ð0, 1 while L and ℕ denotes the linear and nonlinear factors, respectively. Also, Fðw 1 , ςÞ 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: The inverse Jafari transform of (20) and (21), respectively, yields Therefore, the Jafari decomposition method was utilized to derive the solution of (17) by satisfying the assumption that Qðw 1 , ςÞ has a solution of this equation which can be expressed as Thus, the nonlinear term ℕðw 1 , ςÞ can be evaluated by the Adomian decomposition method prescribed as wherẽ Inserting (23) and (24) into (26) and (27), respectively, we have Consequently, the recursive technique for (26) and (18) are established as 4 Journal of Function Spaces 3.2. Construction of Jafari Iterative Transform Method. With the aid of Jafari transform to (17) along with the IC (18), we obtain First, we apply the differentiation rule of Jafari transform for CFD, and then, we consider for ABC fractional derivative operator, respectively, we get By the virtue of the inverse Jafari transform of (30) and (31), respectively, this yields Using the fact of an iterative process, we find Also, the operator L is linear; therefore, and the nonlinearity ℕ dealt by (see [27]) where (39), and (36) into (32) and (33), respectively, we attain Finally, we derive the following iterative process for CFD:

Journal of Function Spaces
Again, the iterative process for ABC fractional derivative operator is presented as follows: Finally, (37), (39), and (40) produce the q-term solution in series representation, stated as

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 [53].
Journal of Function Spaces Since 0 < ε < 1, 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.

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.

Journal of Function Spaces
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 Qðw 1 , ςÞ 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

Journal of Function Spaces
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 Qðw 1 , ςÞ can be written by an infinite series of the form: The approximate solution for Example 1 is expressed as: The integer-order solution of Example 1 can be obtained with the aid of Taylor series expansion and setting ϑ = 1 as follows: Example 2 (see [20]). Assume the time-fractional onedimensional BSM (6) subject to IC (7).
Case 1. Firstly, we solve the (6) in the sense of Caputo fractional derivative operator incorporating the Jafari decomposition method.
Applying Jafari transform to both sides of (6), we have Using the differentiation rule of Jafari transform, we have In view of (7), we get

Journal of Function Spaces
Employing the inverse Jafari transform on both sides yields By the Jafari decomposition method, we find Here, we surmise that the unknown function Qðw 1 , ςÞ can be written by an infinite series of the form: The approximate solution for Example 2 is expressed as Case 2. Now, we solve the (6) by using ABC fractional derivative operator incorporating the Jafari decomposition method.
Considering (64) and using the differentian rule of Jafari transform, we have In view of (7), we get Employing the inverse Jafari transform on both sides yields: By the Jafari decomposition method, we find: Here, we surmise that the unknown function Qðw 1 , ςÞ can be written by an infinite series of the form:

Journal of Function Spaces
The approximate solution for Example 2 is expressed as: The integer-order solution of Example 2 can be obtained with the aid of Taylor series expansion and setting ϑ = 1 as follows:
Case 1. First, we surmise the Caputo fractional derivative operator for (4) with the new iterative transform method via the Jafari transform.
Implementing the Jafari transform to (4) with IC (5), we have: Correspondingly, we get  Table 3: The comparsion analysis among HPM [14], JDM CFD , and JDM ABC of Example 1 for approximated results of Qðw 1 , ςÞ and absolute error E abs = ∥E exact − E approx ∥ at ϑ = 1 with varying values of w 1 and ς: The series form solution is Consequently, we have Case 2. Now, we consider the ABC fractional derivative operator for (4) with the new iterative transform method via the Jafari transform.
Implementing the Jafari transform to (4) with IC (5), we have Correspondingly, we get Employing the suggested analytical method, we obtain The series form solution is Consequently, we have Example 4 (see [20]). Assume the time-fractional onedimensional BSM (6) subject to IC (7). Implementing the Jafari transform to (6) with IC (7), we have Correspondingly, we get: Employing the suggested analytical method, we obtain Table 6: The comparsion analysis among HPM [14], JDM CFD , and JDM ABC of Example 2 for approximated results of Qðw 1 , ςÞ and absolute error E abs = ∥E exact − E approx ∥ at ϑ = 1 with varying w 1 and ς: Consequently, we have Case 2. Now, we consider the ABC fractional derivative operator for (6) with the new iterative transform method via the Jafari transform.
Implementing the Jafari transform to (6) with IC (7), we have Correspondingly, we get   Tables 1 and 2. Table 3 exibits a numerical comparative analysis of the Homotopy perturbation approach and the Jafari decomposition method in perspective of absolute error for (4) taking into consideration both fractional derivative operators.
The findings of a simulation study for the BSM considered in Examples 2 and 4 are shown in Tables 4 and 5. The comparative analysis of the homotopy perturbation with the projected methods is represented in Table 6. This comparison clearly illustrates that the synthesised trajectories are more powerful and effective than the existing ones. Figure 1 depicts the behavior of the Jafari decomposition method's outcome from Qðw1, ςÞ for Example 1. show the absolute error and response of acquired data for (6) with various standard and Brownian motions of 0:6,0:65,0:7,0:8,0:9, and 1, respectively. Thus, we conclude that the order of the profile pictures increases as the value of the time-dependent variable increases. It is worth mentioning that the fractional order has an acceleration effect on the diffusion processes.

Conclusions
This article has investigated the more general integral transform with the Adomian decomposition method and new iterative transform method. The Caputo and ABC fractional derivative operators have been implemented to deal with the BSM. Several distinct solutions have been proposed with the assumption of fractional orders. Various representations have been used to understand these solutions, which have clarified the significant properties of the fractional models in consideration. Without any restrictive assumptions, discretization, or linearization, the proposed methodology locates solutions. Elegance and originality have been invoked to describe our trajectory. Contrasting proposed findings to those acquired in earlier scholarly articles demonstrate the specificality of our solutions in the European option pricing model. The strategy's powerful and successful implementation is explored and validated in order to demonstrate its applicability to additional nonlinear evolution equations that arise in banking and finance issues.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no competing interests.