Spectral Solutions for Fractional Black–Scholes Equations

is paper presents a numerical method to solve accurately the fractional Black–Scholes model of pricing evolution. A fully spectral collocation technique for the two independent variables is derived.e shifted fractional Jacobi–Gauss–Radau and shifted fractional Jacobi–Gauss–Lobatto collocation techniques are utilized. Firstly, the independent variables are interpolated at the shifted fractional Jacobi nodes, and the solution of the model is approximated by means of a sequence of shifted fractional Jacobi orthogonal functions. en, the residuals at the shifted fractional Jacobi quadrature locations are estimated. As a result, an algebraic system of equations is obtained that can be solved using any appropriate approach.e accuracy of the proposedmethod is demonstrated using two numerical examples. It is observed that the new technique is more accurate, ecient, and feasible than other approaches reported in the literature. Indeed, the results show the exponential convergence of the method, both for smooth and nonsmooth solutions.

e Black-Scholes model (BSM) is a mathematical description of pricing evolution [28,29]. Speci cally, the model estimates nancial instruments via time variation. ese instruments (such as stocks or futures) track a lognormal distribution of prices. Based on this hypothesis and taking into account other variables, the BSM derives the price of call options. Despite there being several models that describe pricing evolution, the BSM has been one of the most signi cant and prevalent in the last decades, and its generalization to fractional order was proposed [30,31].
In [32], the Laplace homotopy analysis approach was employed to solve the fractional BSM (FBSM) in the sense of the Caputo-Fabrizio derivative. In [33], the authors employed the Sumudu and Laplace transforms to address various economic models with di erent fractional operators. In [34,35], the Adomian and fractional Adomian decomposition were adopted to solve the FBSM. In [36], the timefractional BSM was solved using the generalized di erential transform technique. Moreover, in [37], the existence and uniqueness solution of the European-type option pricing model was discussed, while in [38], the multivariate Padé approximation was utilized to solve the Caputo fractional European vanilla call option pricing problem.
In most circumstances, it is impossible to provide explicit analytical solutions to space and/or time-fractional differential equations. Hence, developing effective numerical techniques is a critical necessity. Many high-accuracy numerical approaches have been devised to tackle diverse issues in various applications. Because of their high-order precision, fractional differential and integral equations [39] have seen tremendous development in recent years. But, contrasting with the effort put into evaluating finite difference and element schemes, little research has been dedicated to designing and assessing global spectral schemes [40]. In cases with periodic boundary conditions, spectral techniques based on the Fourier expansion have been used. When compared to other schemes, spectral techniques take the main place due to their robustness and exponential rates of convergence [41][42][43][44]. Despite some drawbacks, such as the inability to represent physical processes in spectral space and the difficulty in parallelizing on distributed memory computers, they reveal excellent accuracy, high speed of convergence, and simplicity in solving many types of differential equations [45]. e spectral collocation method is particularly valuable because it can estimate the solution of a wide range of equations. Moreover, its exponential rate of convergence is extremely useful in delivering very exact solutions. e collocation approach has grown in prominence in recent decades for dealing with specific difficulties posed by fractional derivatives.
Herein, the shifted fractional Jacobi-Gauss-Radau collocation (SFJ-GR-C) and shifted fractional Jacobi-Gauss-Lobatto collocation (SFJ-GL-C) techniques are utilized to solve the FBSM. We interpolate the independent variables using the shifted fractional-order Jacobi nodes, and we approximate the solution of the model by a sequence of shifted fractional-order Jacobi orthogonal functions. After that, the residuals at the shifted fractional-order Jacobi quadrature locations are estimated. is yields an algebraic system of equations that can be solved using a suitable approach. e accuracy of the new method is demonstrated using numerical examples. e paper is organized into four sections: Section 2 introduces the new spectral collocation technique. Section 3 applies the method to solve numerical examples. Section 4 summarizes the main conclusions.

Fully Spectral Collocation Technique
We start by mapping the variable τ as τ � t − T. Additionally, the Riemann-Liouville is transformed to the Caputo fractional derivative D μ Y(x, τ) (see [31,46]): where and D μ Z(x, τ) is a Caputo fractional derivative.
Similarly, we have [48,49] e time Caputo fractional derivative, on the contrary, is obtained as [48,49]  Mathematical Problems in Engineering Otherwise, the initial boundary can be obtained by Eq. (3) is constrained to be zero at (N − 1) × (M) points: where Δ x where n � 1, . . . , N − 1, and m � 1, . . . , M, and Mathematical Problems in Engineering Finally, a solvable linear algebraic system is provided. It should be mentioned that the convergence spectral rate of Jacobi polynomials has been thoroughly studied in the literature [50,51], while the convergence spectral rate of fractional Jacobi functions has been addressed in [48,49,52], to cite a few.

Numerical Results
In this section, we demonstrate the resilience and accuracy of our strategy to solve two problems. Example 1. We solve the FBSM [47]. with choosing H(x, t) such that Table 1 summarizes the L ∞ errors between the exact and approximate solutions achieved by our method and those reported in [47], taking α � 0.25, r 1 � 0.01875, r 2 � 0.05, and μ � 0.6. We verify that the new strategy is superior and that good estimates are obtained with a small number of collocation points. Figures 1 and 2 present the 3D charts of the numerical solution and the absolute error, respectively, when α 1 � β 1 � α 2 � β 2 � 0, λ 1 � 1, λ 2 � 1, and N � M � 20. Figure 3 depicts the corresponding exact and numerical solutions along x. Figures 4 and 5 represent the absolute errors along x and t, respectively, while Figure 6 shows the maximum absolute error (M E ) convergence for the following cases:

Example 2.
We solve the FBSM [46]. with choosing H(x, t) such that Y(x, t) � t μ + e x + x + 1. Table 2 summarizes the L ∞ errors between the exact and approximate solutions achieved by our method and those reported in [46], taking α � 0.1, r � 0.06, and 0 < μ < 1. We verify that the new strategy is superior and that good estimates are obtained with a small number of collocation points. Figures 7 and 8 present the 3D charts of the numerical solution and the absolute error, respectively, when α 1 � −α 2 � −0.5, β 1 � β 2 � 0, λ 1 � 0.9, λ 2 � 1, and N � M � 14. Figure 9 depicts the corresponding exact and numerical solutions along x. Figures 10 and 11 represent the absolute errors along x and t, respectively, while Figure 12 shows the M E convergence for the following cases: Table 1: e L ∞ errors between the exact and approximate solutions achieved by our method and those reported in [47] for Example 1, taking α � 0.25, r 1 � 0.01875, r 2 � 0.05, and μ � 0.6.  Figures 8, 10, and 11 show that the results are fairly precise, as the absolute error approaches zero. We can also see in Figures 7 and 9 that the approximate solution matches precisely the exact solution. In addition, Figure 12 clearly reveals that the method yields exponential convergence of the error.
To sum up, we emphasize that the majority of numerical techniques for the problem at hand are based on orthogonal polynomials. e use of classical polynomials in problems with nonsmooth solutions leads to low accuracy or even failure to converge. Using the fractional, rather than the classical, Jacobi functions mitigates the problem. With the proposed method, all numerical computations could be completed with good precision and a low number of degrees of freedom.
Moreover, our technique outperformed other existing approaches. Finally, we can conclude that the fully spectral collocation approach is a useful, efficient, and acceptable strategy for dealing with problems that have singular solutions.  Table 2: e L ∞ errors between the exact and approximate solutions achieved by our method and those reported in [46] for Example 2, taking α � 0.1, r � 0.06, and 0 < μ < 1.

Conclusion
e FBSM was treated using a fully spectral collocation technique for the two independent variables x and t. It was verified that the new technique is superior in terms of accuracy and efficiency to other methods, for both smooth and nonsmooth solutions. To deal with the FBSM, we devised an approach that yields an algebraic system from which an approximated solution can be computed. e simulation results revealed that the proposed approach is effective for the goal at hand. Furthermore, because of its ease of use, our technique is relevant to a wide range of fractional problems. In the future, we can concentrate on the usage of the spectral Galerkin and the tau approaches for solving more complicated pricing models, such as the tempered FBSM. Finally, we should mention that the maximum absolute error for a given boundary value problem with a smooth solution is exponentially convergent. For nonsmoothness in time (or in space), the method's order of convergence degrades. is, however, can be mitigated by employing the fractional-order Jacobi functions described herein.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.