New Solutions of Time-and Space-Fractional Black–Scholes European Option Pricing Model via Fractional Extension of He-Aboodh Algorithm

. Te current study explores the space and time-fractional Black–Scholes European option pricing model that primarily occurs in the fnancial market. To tackle the complexities associated with solving models in a fractional environment, the Aboodh transform is hybridized with He’s algorithm. Tis facilitates in improving the efciency and applicability of the classical homotopy perturbation method (HPM) by ensuring the rapid convergence of the series form solution. Tree cases that are time-fractional scenario, space-fractional scenario, and time-space-fractional scenario are observed through graphs and tables. 2D graphical analysis is performed to depict the behaviour of a given option pricing model for varying time, stock price, and fractional parameters. Solutions of the European option pricing model at various fractional orders are also presented as 3D plots. Te results obtained through these graphs unfold the interchange between time-and space-fractional derivatives, presenting a comprehensive study of option pricing under fractional dynamics. Te competency of the proposed scheme is illustrated via solutions and errors throughout the fractional domain in tabular form. Te validity of the He-Aboodh results is exhibited by comparison with existing errors. Analysis shows that the proposed methodology (He-Aboodh algorithm) is a valuable scheme for solving time-space-fractional models arising in business and economics.


Introduction
One of the most crucial theories in contemporary fnance is the Black-Scholes model.Tis mathematical formula calculates the potential value of derivatives based on other fnancial instruments while accounting for the efects of time and other risk factors [1].It is frequently used in contracts for option pricing [2,3].Fischer Black and Myron Scholes developed the frst Black-Scholes model in 1973 [4].Later, Robert Merton published an article [5] to expand the model's applications.Te classical Black-Scholes equation was originated to determine the theoretical value of an option contract using current stock prices, the option's strike price, expected dividends, time of expiration, expected interest rates, and volatility.Some other modifcations of Black-Scholes models have been suggested that are the jump-difusion model [6], transaction cost models [7,8], stochastic interest model [9], and stochastic volatility model [10].After the fractal structures for the fnancial market [11] were discovered, the standard Brownian motion of the classical Black-Scholes equation was replaced by fractional Brownian motion to obtain the fractional Black-Scholes model.Some such models are the fractional Black-Scholes pricing model on arbitrage and replication [12], tempered fractional Black-Scholes equation for European double barrier option [13], pricing fnancial options model in fractal transmission system [14], fractional Black-Scholes model with stochastic volatility [15], pricing double barrier options in a time-fractional Black-Scholes model [16], fractional Black-Scholes equation under the constant elasticity of variance (CEV) model [17], fractional Black-Scholes model with European option [18], and two-dimensional fractional Black-Scholes equation [19].A time-space-fractional Black-Scholes European option pricing model [20] arising in the fnancial market is given as where (S, τ) ϵ (0, ∞) × (0, T), ρ, and η are the Caputo fractional derivatives with respect to τ and S, respectively.σ ( ≥ 0), r ( > 0), D, T, and S represent the volatility of the returns, risk-free rate, dividend rate, expiry time, and stock price, respectively.Te fnal and boundary conditions of ( 1) are Suppose τ � T − t and x � ln S. Ten, by defning B(x, t) � W(e x , T − t), the model in equation ( 1) can be rewritten in dimensionless form as where c 1 � σ 2 /2, c 2 � r − D − c 1 , c 3 � r, and g(x, t) are the source terms.Moreover, (I d , I u ) is the fnite domain, and the function B indicates the European option price.By introducing fractional derivatives in the above model, more complex phenomena such as the long-term memory efect can be observed.Several techniques have been introduced in the literature to solve ordinary and partial diferential equations.Galerkin method [21], implicit fnite diference scheme [22], Adomian decomposition method [23], Crank-Nicolson scheme [24], homotopy perturbation method [25], backward Euler method [26], diferential transform method [27], and stabilized meshless technique [28] are some of them.Many of these approaches have been utilized for the numerical solution of fractional diferential equations including the Navier-Stokes equation [29,30], Schrödinger equation [31], COVID-19 model [32], Kundu-Mukherjee-Naskar equation [33], and Black-Scholes model [34].Chen et al. [35] employed a Laguerre neural network for generalized Black-Scholes models.Chebyshev collocation method is applied by Mesgarani et al. [36] to analyze time-fractional Black-Scholes models.Te Crank-Nicolson scheme is employed by Roul and Goura [37] to solve generalized Black-Scholes with the European call option.An et al. [38] proposed a space-time spectral method for the solution of Black-Scholes equations.Time-fractional Black-Scholes European option pricing equations are solved through the residual power series method by Dubey et al. [39].Roul and Goura [40] introduced a fnite diference scheme for the fractional Black-Scholes equation.Te homotopy analysis method is utilized by Fadugba [41] for European call options with the time-fractional Black-Scholes model.
Te homotopy perturbation method (HPM) provides a semianalytical algorithm for solving both linear and nonlinear ordinary/partial diferential equations [42].It is also applied to diferential system of equations [43].In order to solve diferential equations in fractional form more accurately, many modifcations of HPM have been introduced.Baleanu and Jassim [44] extended the modifed fractional homotopy perturbation technique on Helmholtz and coupled Helmholtz equations.Qayyum et al. [45,46] utilized the He-Laplace method to solve generalized third-and ffth-order timefractional KdV models.Fractional Navier-Stokes equations are investigated by Jena and Chakraverty [47] through homotopy perturbation Elzaki transform.Another modifcation is the He-Aboodh algorithm [48] which combines Aboodh transform and HPM.Manimegalai et al. [49] studied strongly nonlinear oscillators by applying the Aboodh transform and the homotopy perturbation method.An iterative scheme and Aboodh transform are employed by Gbenga and Mahmudov [50] to analyze the fractional spatial difusion of a biological population model.Compared to classical HPM, He-Aboodh has broader applicability and improved convergence and accuracy.It eliminates integral terms which give an efcient procedure when dealing with fractional derivatives.Tus, in this paper, we have adapted the He-Aboodh algorithm for the solution and analysis of time-space-fractional Black-Scholes model (3).Te scenarios involving time-fractional, spacefractional, and time-space-fractional derivatives are taken in the Caputo sense.Te results obtained from this study indicate improvement in the predictive accuracy of option pricing particularly the systems involving noninteger-order derivatives.It also enhanced the comprehension of risks associated with option pricing in fnancial markets.
Te format of this research article is as follows: Section 2 contains some basic defnitions of Aboodh transform, Caputo fractional derivatives, and their Aboodh transform.A general methodology of the He-Aboodh algorithm is in Section 3 whereas Section 4 is centered on the application and solutions of the time-space-fractional Black-Scholes model.Results and discussion is given in Section 5, and the conclusion of the paper is in Section 6.
Defnition 2 (see [52]).Te inverse Aboodh transform A − 1 of the function K(x, s) is given as ( Defnition 3 (see [53]).Te Caputo time-and spacefractional derivatives CD ρ t and CD η x of a function B(x, t) is, respectively, defned by ρ and η are fractional parameters.Defnition 4 (see [54]).Te Aboodh transform A of Caputo time-and space-fractional derivatives is, respectively, given as Defnition 5 (see [55]).Te double Aboodh transform w.r.t.t and x on Caputo time-and space-fractional derivatives (CD ρ t & CD η x ) can be described as Journal of Mathematics

General Methodology of He-Aboodh Algorithm for Space-Time Fractional Models
Consider a general nonlinear time-space-fractional model with conditions where D ρ t and D η x represent the time-and space-fractional derivatives of unknown function B, respectively.L and N are the linear and nonlinear operators of B.
3.1.Time-Fractional Scenario.Te process will begin by applying the Aboodh transform with respect to time and taking into account the space derivative in integer order Application of Aboodh transform on Caputo fractional derivative gives Te general homotopy is Hom: where B 0 represents the initial guess and 0 ≤ λ ≤ 1. Expansion of B(x, t) in power series w.r.t.λ leads to Substituting equation ( 14) in equation ( 13) and then comparing identical coefcients of λ gives the following equations: At 1 st order, 4

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In general, at k th order, A solution can be obtained by taking the inverse Aboodh transform.
At λ 1 , Te approximate series solution of equation ( 11) is For η � 1, we may obtain the residual function by inserting equation (19) in equation ( 9) 3.2.Space-Fractional Scenario.We will initiate the process by considering Aboodh transform w.r.t.space.
Aboodh transform of space derivative gives Homotopy of equation is Hom:

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Te same process as in Case 6 results in the following equation: At λ p , Solutions can be found by using the inverse Aboodh transform.
At λ 1 , Adding these terms gives the approximate series solution.Te residual function is produced by substituting the obtained approximate solution in equation ( 9) at ρ � 1.
3.3.Time-Space-Fractional Scenario.For both time-and space-fractional model, we will take double Aboodh transform w.r.t.time and space.
Aboodh transform of time-space derivative gives 6

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Homotopy equation of equation ( 30) is Hom: Mapping the same process as in Case 6 gives the following equation: At λ p , Inverse Aboodh transform and then the sum of obtained terms lead towards an approximate series solution.Te residual function is given by

Application and Solution of Time-Space-Fractional Black-Scholes European Option Pricing Model
Example 1 (see [20]).

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Substituting equation (14) in equation ( 37) and comparing alike coefcients of λ leads to the following equation: Application of Aboodh transform inverse generates the following equation: At λ 2 , Aboodh transform inverse gives the following equation: Hence, the continuation of the process gives the ffthorder approximate solution in series form as shown in the following equation: 4.1.2.Case 2: Space Fractional.Take ρ � 1 in equation (35) and then apply Aboodh transform w.r.t.space (A x ).Te acquired homotopy equation is Hom: 8

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where A is the dummy variable which is introduced to convert the boundary condition into the initial condition.A similar procedure as Case 9 leads to the following equation: At λ 2 , Taking the inverse Aboodh transform and adding the terms gives the ffth-order series solution.Te optimal value of dummy variable A can be found by using the right-side boundary condition of interval.
4.1.3.Case 3: Time-Space Fractional.Aboodh transform with respect to both time (A t ) and space (A x ) gives the following equation: HPM procedure leads to homotopy. Hom: Substituting equation ( 14) and comparing coefcients of λ gives the following equation: At λ 2 , Applying the double inverse Aboodh transform and then the summation of obtained terms gives a ffth-order approximate series solution.
Journal of Mathematics where g(x, t) � ((2t the exact solution of equation ( 50) is shown in the following equation [56]: Hom: Equation ( 14) leads to the following equation: At λ 1 , Fifth-order approximate solution is obtained from the following equation: 4.2.2.Case 2: Space Fractional.By following the same steps as in Section 3, we get the following equation: Utilizing equation (14) gives the following equation: At λ 1 , At λ 3 , 10

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Taking the inverse Aboodh transform and adding the terms gives the ffth-order series solution.

Case 3: Time-Space
Fractional.Te procedure given in Section 3 leads to the following equation: Substituting equation ( 14) and comparing similar coefcients of λ gives the following equation: At λ 3 , Applying the double inverse Aboodh transform and then the summation of obtained terms gives a ffth-order approximate series solution.

Results and Discussion
In the current work, Black-Scholes European option pricing model in space-and time-fractional environment is solved through a hybrid technique, the He-Aboodh transform, in which the Aboodh transform and homotopy perturbation method are integrated together.For simulation purposes, Wolfram Mathematica 13.3 is utilized on TinkPad that has a display size of 14.00-inch, resolution 3840 × 2160 pixels, processor core i9, and 16 GB RAM.Tree cases that are timefractional (Case 6), space-fractional (Case 7), and timespace-fractional (Case 8) are analyzed via 2D and 3D graphs.Absolute errors obtained through the proposed methodology are compared with the multiquadric-radial basis function (MQ-RBF) method errors in Tables 1 and  2. It can be deduced that the He-Aboodh algorithm gives more precise results than the existing technique.For different values of fractional parameter ρ, Tables 3 and 4 exhibit the solutions and errors at varying time t whereas, in Tables 5  and 6, solutions and errors are calculated throughout the whole domain of stock price x.From these tables, the convergence of the scheme for the whole fractional domain is observed.Te consistency of obtained solutions can also be seen from them.It is noted that He-Aboodh transform is a reliable technique for solving space-and time-fractional models as the obtained solutions are nearer to their exact solutions.
In Example 1 for Case 9, Figure 1 displays the solution pattern of the Black-Scholes model at fractional parameter ρ � 0.24 and 0.86.By considering volatility σ � 0.37, risk-free Journal of Mathematics rate r � 0.16, and dividend rate D � 0.05, both plot shows that initially increase in time does not have any impact on option price.However, when the time becomes large and the stock price x increases in its domain, the European option pricing also displays an increase in its value.As compared to plot (a), the value of European option pricing B is greater in plot (b) for larger fractional parameter value.Tis shows the efects of long-range memory on the dynamics of Black-Scholes model.Figure 2 demonstrates that an increase in stock price and time causes the European option price to rise.For Case 10, Figure 3 with fractional parameter η � 0.24 and 0.86 illustrates that option price increases as time and stock price escalates.Moreover, the profle of European option pricing also expands at a greater value of time (see Figure 4).In the case of time-space fractional, plot (a) of Figure 5 at ρ � 0.24 and η � 0.5 exhibits that option price keeps increasing as time and stock price rises.Te peak of the European option pricing model is at stock price x � 0.8; after that, it started to get lower.Solution at ρ � 0.84 and η � 0.7 in plot (b) also presents the surge in European option pricing model.Additionally, enlarging the values of fractional parameters ρ and η in their domains indicates a drop in Black-Scholes   6.It can be seen from these fgures that noninteger-order derivatives allow the characterization of fnancial models more accurately than integer-order derivatives.
By taking the values of volatility σ � 0.4, risk-free rate r � 0.21, and dividend rate D � 0.1 in the time-fractional case of Example 2, Figure 7 illustrates that as stock price and fractional parameter expand, the profle of Black-Scholes option pricing model depicts an increasing behaviour.In Case 13 (see Figure 8), it is shown through the arrow that as time increases the option price indicates an elevation.On the other hand, a decline in the option pricing profle is observed for larger values of fractional parameter η. 3D Figure 9 in Case 14 demonstrates that in the beginning, the option price was at higher value, but as the stock price and time goes up, it began to decrease.Moreover, the value of fractional parameters ρ and η is smaller in plot (a) as compared to plot (b).For smaller values of ρ and η, the option price is greater.Tis indicates the complex and memory-dependent behaviour of option pricing models that are efectively captured through fractional calculus.

Conclusion
Fractional analysis of Black-Scholes European option pricing model in both space and time is the primary focus of this research article.Te proposed methodology, He-Aboodh algorithm, is utilized to obtain approximate solutions of the given model.Diferential property of Aboodh transform on Caputo time-fractional, space-fractional, and time-spacefractional derivatives is applied to efciently tackle the complexities arising from noninteger-order dynamics.Solution and errors at diferential values of time, stock price, and fractional parameters are displayed in tabular form which shows that the application of the proposed methodology improves the predictive accuracy of option pricing models especially when dealing with memory-dependent procedures.
Additionally, the error comparison of the multiquadric-radial basis function and He-Aboodh algorithm at fractional parameters ρ � 0.2 and 0.7 leads to the conclusion that the proposed methodology gives better results in terms of accuracy.By increasing the rates of time, stock price, and fractional parameters, change in the profle of the given Black-Scholes model is demonstrated with the help of twodimensional fgures.Te ups and downs in the value of European option pricing are illustrated through threedimensional fgures.Solutions at diferent values of fractional parameters for all three cases indicate that for Example 1, with an increase in time and stock price, the value of the option price also increases.On the other hand, the option price is initially at a higher value in Example 2, but it started to decline when stock price and time increased.Tese results provide a benefcial understanding of the interaction between time-and dynamics in pricing models.Tus, it can be concluded that the He-Aboodh algorithm is an efcient technique that can be extended to solve nonlinear complex time-and space-fractional Black-Scholes models arising in the fnancial market.