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In this study, we provide a numerical method to approximate the solution of the time fractional Black-Sholes equation by applying the multiquadric (MQ) quasi-interpolation scheme and the integrated radial basis function networks scheme. In the present approach, quadrature formula is used to discretize the temporal Caputo fractional derivative and the integrated form of the MQ quasi-interpolation scheme is used for approximation of the unknown function and its spatial derivatives. In order to show the accuracy and efficiency of the presented method, the L∞, L2 errors of several experiments are considered. Our numerical results are compared with the exact solutions as well as the results obtained from the other numerical schemes.

Content uploaded by Azim Aminataei

Author content

All content in this area was uploaded by Azim Aminataei on Jan 22, 2022

Content may be subject to copyright.

... (2021) [29] applied C-N format, and four kinds of Saul'yev asymmetric format to construct the alternating segmented C-N parallel scheme. Sarboland M. and Aminataei A. (2022) [30] applied the multiquadric quasi-interpolation scheme and the integrated radial basis function networks scheme to provide a numerical method to approximate the solution of the time fractional Black-Sholes equation. ...

Financial theory could introduce a fractional differential equation (FDE) that presents new theoretical research concepts, methods and practical implementations. Due to the memory factor of fractional derivatives, physical pathways with storage and inherited properties can be best represented by FDEs. For that purpose, reliable and effective techniques are required for solving FDEs. Our objective is to generalize the collocation method for solving time fractional Black–Scholes European option pricing model using the extended cubic B-spline. The key feature of the strategy is that it turns these type of problems into a system of algebraic equations which can be appropriate for computer programming. This is not only streamlines the problems but speed up the computations as well. The Fourier stability and convergence analysis of the scheme are examined. A proposed numerical scheme having second-order accuracy via spatial direction is also constructed. The numerical and graphical results indicate that the suggested approach for the European option prices agree well with the analytical solutions.

We study the fractional Black–Scholes model (FBSM) of option pricing in the fractal transmission system. In this work, we develop a full-discrete numerical scheme to investigate the dynamic behavior of FBSM. The proposed scheme implements a known L1 formula for the α-order fractional derivative and Fourier-spectral method for the discretization of spatial direction. Energy analysis indicates that the constructed discrete method is unconditionally stable. Error estimate indicates that the 2−α-order formula in time and the spectral approximation in space is convergent with order OΔt2−α+N1−m, where m is the regularity of u and Δt and N are step size of time and degree, respectively. Several numerical results are proposed to confirm the accuracy and stability of the numerical scheme. At last, the present method is used to investigate the dynamic behavior of FBSM as well as the impact of different parameters.

The current work aims to exploit two techniques namely: Residual Power Series method (RPSM) and collocation based meshfree method, for the solution of time-fractional Black-Scholes models with constant and variable coefficients. Firstly, using RPSM, we obtain exact solutions of the considered models and then numerical solution by meshfree method. Computer simulations are performed for three test problems of European options pricing. The simulations features excellent agreement with exact solutions. Accuracy and efficiency of the proposed numerical method is assessed via E_{2}, E_{∞} and E_{rms} error norms. Convergence of the proposed methods is also analyzed.

In this study, a new application of multivariate Padé approximation method has been used for solving European vanilla call option pricing problem. Padé polynomials have occurred for the fractional Black-Scholes equation, according to the relations of "smaller than", or "greater than", between stock price and exercise price of the option. Using these polynomials, we have applied the multivariate Padé approximation method to our fractional equation and we have calculated numerical solutions of fractional Black-Scholes equation for both of two situations. The obtained results show that the multivariate Padé approximation is a very quick and accurate method for fractional Black-Scholes equation. The fractional derivative is understood in the Caputo sense.

We introduce the mathematical modeling of American put option under the fractional Black–Scholes model, which leads to a free boundary problem. Then the free boundary (optimal exercise boundary) that is unknown, is found by the quasi-stationary method that cause American put option problem to be solvable. In continuation we use a finite difference method for derivatives with respect to stock price, Grünwal Letnikov approximation for derivatives with respect to time and reach a fractional finite difference problem. We show that the set up fractional finite difference problem is stable and convergent. We also show that the numerical results satisfy the physical conditions of American put option pricing under the FBS model.

In this paper, we introduce a new family of infinitely smooth and “nearly” locally supported radial basis functions (RBFs), derived from the general solution of a heat equation arising from the American option pricing problem. These basis functions are expressed in terms of “the repeated integrals of the complementary error function” and provide highly efficient tools to solve the free boundary partial differential equation resulting from the related option pricing model. We introduce an integral operator with a function-dependent lower limit which is employed as a basic tool to prove the radial positive definiteness of the proposed basis functions and could be of independent interest in the RBF theory. We then show that using the introduced functions as expansion bases in the context of an RBF-based meshless collocation scheme, we could exactly impose the transparent boundary condition accompanying the heat equation. We prove that the condition numbers of the resulting collocation matrices are orders of magnitude less than those arising from other popular RBF families used in current literature. Some other properties of these bases such as their Fourier transforms as well as some useful representations in terms of positive Borel measures will also be discussed.

Our aim in this paper is to approximate the price of an American call option written on a dividend-paying stock close to expiry using an asymptotic analytic approach. We use the heat equation equivalent of the Black–Scholes partial differential equation defined on an unbounded spatial domain and decompose it into inner and outer problems. We extend the idea presented in [H. Han and X. Wu, A fast numerical method for the Black-Scholes equation of American options, SIAM Journal on Numerical Analysis 41 (6) (2003) 2081-2095.] in which a weakly singular memory-type transparent boundary condition (TBC) is obtained for the special case that the initial condition is equal to zero. We first derive this TBC in the general case and then focus on the outer problem in conjunction with an equivalent non-singular version of the TBC (dubbed ETBC) which is more tractable for analytical purposes. We then obtain the general solution of the outer problem in series form based on “the repeated integrals of the complementary error function” which also satisfies the introduced ETBC. As the next step, using the machinery of Poincaré asymptotic expansion and taking “time-to-expiry” as the expansion parameter, we find the general term of this series in closed form when the risk-free interest rate (r) is less than the dividend yield (δ). We also obtain the first five terms in the opposite case (r>δ) in a systematic manner. We also prove the convergence properties of the obtained series rigorously under some general conditions. Our numerical experiments based on the obtained asymptotic series, demonstrate the applicability and effectiveness of the results in valuation of a wide range of American option problems.

The moving least-squares (MLS) approximation is a powerful numerical scheme widely used in the meshfree literature to construct local multivariate polynomial basis functions for expanding the solution of a given differential or integral equation. For partial integro-differential equations arising from the valuation of multi-asset options written on correlated Lévy-driven assets, we propose here an MLS-based collocation scheme in conjunction with implicit-explicit (IMEX) temporal discretization to numerically solve the problem. We apply the method to price both European and American options and compute the option hedge parameters. In the case of American options, we use an operator splitting approach to solve the linear complementarity formulation of the problem. Our numerical experiments show the efficiency of the proposed scheme in comparison with some competing approaches, specially finite difference methods.

In this paper a time-fractional Black-Scholes equation is examined. We transform the initial value problem into an equivalent integral-differential equation with a weakly singular kernel and use an integral discretization scheme on an adapted mesh for the time discretization. A rigorous analysis about the convergence of the time discretization scheme is given by taking account of the possibly singular behavior of the exact solution and first-order convergence with respect to the time variable is proved. For overcoming the possibly nonphysical oscillation in the computed solution caused by the degeneracy of the Black-Scholes differential operator, we employ a central difference scheme on a piecewise uniform mesh for the spatial discretization. It is proved that the scheme is stable and second-order convergent with respect to the spatial variable. Numerical experiments support these theoretical results.

We present a numerical method to solve a time-space fractional Fokker-Planck equation with a space-time dependent force field F(x, t), and diffusion d(x, t). When the problem being modelled includes time dependent coefficients, the time fractional operator, that typically appears on the right hand side of the fractional equation, should not act on those coefficients and consequently the differential equation can not be simplified using the standard technique of transferring the time fractional operator to the left hand side of the equation. We take this into account when deriving the numerical method. Discussions on the unconditional stability and accuracy of the method are presented, including results that show the order of convergence is affected by the regularity of solutions. The numerical experiments confirm that the convergence of the method is second order in time and space for sufficiently regular solutions and they also illustrate how the order of convergence can depend on the regularity of the solutions. In this case, the rate of convergence can be improved by considering a non-uniform mesh.

When considering the price change of the underlying fractal transmission system, a fractional Black–Scholes(B-S) model with an -order time fractional derivative is derived. In this paper, we discuss the numerical simulation of this time fractional Black–Scholes model (TFBSM) governing European options. A discrete implicit numerical scheme with a spatially second-order accuracy and a temporally order accuracy is constructed. Then, the stability and convergence of the proposed numerical scheme are analyzed using Fourier analysis. Some numerical examples are chosen in order to demonstrate the accuracy and effectiveness of the proposed method. Finally, as an application, we use the TFBSM and the above numerical technique to price several different European options.