Fractional Black-Scholes Equation Research Articles

Approximation of time fractional Black-Scholes equation via radial kernels and transformations

Approximation of time fractional Black-Scholes equation via radial kernels and transformations

A new iterative method based solution for fractional Black–Scholes option pricing equations (BSOPE)

In this manuscript, a new expansion technique namely residual power series method is used for finding the analytical solution of the Fractional Black–Scholes equation with an initial condition for European option pricing problem. The Black–Scholes formula is important for estimating European call and put option on a non-dividend paying stock in particular when it contains time-fractional derivatives. The fractional derivative is defined in Caputo sense. This technique is based on fractional power series expansion. The convergence analysis of the present method is also deliberated. Example problems are given to examine the efficacy of the proposed method. Obtained solutions are compared with exact solutions solved by other techniques which demonstrate that the present method is robust and easy to implement for other fractional problems arising in science and engineering.

Generalised class of Time Fractional Black Scholes equation and numerical analysis

It is well known now, that a Time Fractional Black Scholes Equation (TFBSE) with a time derivative of real order \begin{document}$ \alpha $\end{document} can be obtained to describe the price of an option, when for example the change in the underlying asset is assumed to follow a fractal transmission system. Fractional derivatives as they are called were introduced in option pricing in a bid to take advantage of their memory properties to capture both major jumps over small time periods and long range dependencies in markets. Recently new derivatives of Fractional Calculus with non local and/or non singular Kernel, have been introduced and have had substantial changes in modelling of some diffusion processes. Based on consistency and heuristic arguments, this work generalises previously obtained Time Fractional Black Scholes Equations to a new class of Time Fractional Black Scholes Equations. A numerical scheme solution is also derived. The stability of the numerical scheme is discussed, graphical simulations are produced to price a double barriers knock out call option.

European Vanilla Option Pricing Model of Fractional Order without Singular Kernel

Recently, fractional differential equations (FDEs) have attracted much more attention in modeling real-life problems. Since most FDEs do not have exact solutions, numerical solution methods are used commonly. Therefore, in this study, we have demonstrated a novel approximate-analytical solution method, which is called the Laplace homotopy analysis method (LHAM) using the Caputo–Fabrizio (CF) fractional derivative operator. The recommended method is obtained by combining Laplace transform (LT) and the homotopy analysis method (HAM). We have used the fractional operator suggested by Caputo and Fabrizio in 2015 based on the exponential kernel. We have considered the LHAM with this derivative in order to obtain the solutions of the fractional Black–Scholes equations (FBSEs) with the initial conditions. In addition to this, the convergence and stability analysis of the model have been constructed. According to the results of this study, it can be concluded that the LHAM in the sense of the CF fractional derivative is an effective and accurate method, which is computable in the series easily in a short time.

A different approach to the European option pricing model with new fractional operator

In this work, we have derived an approximate solution of the fractional Black-Scholes models using an iterative method. The fractional differentiation operator used in this paper is the well-known conformable derivative. Firstly, we redefine the fractional Black-Scholes equation, conformable fractional Adomian decomposition method (CFADM) and conformable fractional modified homotopy perturbation method (CFMHPM). Then, we have solved the fractional Black-Scholes (FBS) and generalized fractional Black-Scholes (GFBS) equations by using the proposed methods, which can analytically solve the fractional partial differential equations (FPDE). In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of these two option pricing problems by using in pricing the actual market data. Also, we have found out that the proposed models are very efficient and powerful techniques in finding approximate solutions of the fractional Black-Scholes models which are considered in conformable sense.

Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation

Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation

Block-pulse operational matrix method for solving fractional Black-Scholes equation

PurposeThe purpose of this paper is to evaluate a European option using the fractional version of the Black-Scholes model.Design/methodology/approachIn this paper, the authors employ the block-pulse operational matrix algorithm to approximate the solution of the fractional Black-Scholes equation with the initial condition for a European option pricing problem.FindingsThe fractional derivative will be described in the Caputo sense in this paper. The authors show the accuracy and computational efficiency of the proposed algorithm through some numerical examples.Originality/valueThis is the first paper that considers an alternative algorithm for pricing a European option using the fractional Black-Scholes model.

Pricing European Options under Fractional Black–Scholes Model with a Weak Payoff Function

The purpose of this paper is to obtain an explicit solutions of the fractional Black–Scholes model with a weak payoff function. To do this, we derive fractional Black–Scholes equation by creating a self-financing portfolio strategy under Leland’s strategy. Then, we use the Mellin transform method for solving this equation and obtain the price of a European option as a particular case of the proposed solution. A sensitivity analysis is carried out through numerical experiments which shows the differences between Black–Scholes model and the fractional Black–Scholes model. Moreover, an empirical analysis shows that the fractional Black–Scholes model with Hurst exponent greater than one-half is more precise to predict the real market prices than the classical Black–Sholes model.

Fractional Black–Scholes equation

In this paper, it has been shown that the combined use of exponential operators and special functions provides a powerful tool to solve certain class of generalized space fractional Laguerre heat equation. It is shown that exponential operators are powerful and effective method for solving certain singular integral equations and space fractional Black–Scholes equation.

Fractional Black-Scholes model with regularized Prabhakar derivative

We introduce a fractional type Black-Scholes model in European options including the regularized Prabhakar derivative. We apply the reconstruction of variational iteration method to get the approximate analytical solutions for some models of generalized fractional Black-Scholes equations in terms of the generalized Mittag-Leffler functions.

A universal difference method for time-space fractional Black-Scholes equation

The fractional Black-Scholes (B-S) equation is an important mathematical model in finance engineering, and the study of its numerical methods has very significant practical applications. This paper constructs a new kind of universal difference method to solve the time-space fractional B-S equation. The universal difference method is analyzed to be stable, convergent, and uniquely solvable. Furthermore, it is proved that with numerical experiments the universal difference method is valid and efficient for solving the time-space fractional B-S equation. At the same time, numerical experiments indicate that the time-space fractional B-S equation is more consistent with the actual financial market.

The numerical simulation of the tempered fractional Black–Scholes equation for European double barrier option

In recent years, the Finite Moment Log Stable(FMLS), KoBoL and CGMY models, which follow a jump process or a Lévy process, have become the most popular modeling frameworks in the financial field because they can capture some of the important characteristics in the dynamic process of stock price changes, such as large movements or jumps over small time steps. In this paper, we consider the numerical simulation of these three models. We construct a discrete implicit numerical scheme with second order accuracy, and provide a stability and convergence analysis of the numerical scheme. Furthermore, a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) is used to reduce the storage space from O(M2) to O(M) and the computational cost from O(M3) to O(Mlog M) per iteration, where M is the number of space grid points. Some numerical examples are chosen in order to demonstrate the accuracy and efficiency of the proposed method and technique. Finally, as an application, we use the above numerical technique to price a European double-knock-out barrier option, and then the characteristics of the three fractional Black–Scholes (B–S) models are analyzed through comparison with the classical B–S model.

The meshless local Petrov–Galerkin based on moving kriging interpolation for solving fractional Black–Scholes model

In this paper, the fractional Black–Scholes equation in financial problem is solved by using the numerical techniques for the option price of a European call or European put under the Black–Scholes model. The MLPG and implicit finite difference method are used for discretizing the governing equation in option price and time variable, respectively. In MLPG method, the shape function is constructed by a moving kriging approximation. The Dirac delta function is chosen to be the test function. The numerical examples for varieties of variables are also included.

MELLIN TRANSFORM METHOD FOR SOLVING THE BLACK-SCHOLES EQUATION

In this paper, we obtain an analytical solution of the Black-Scholes equation for the European and the American put options by using the Mellin integral transform method. The analytical solution thus obtained of the Black- Scholes equation for the European put option is compared with the existing already numerical solutions. By using a join Mellin-Laplace integral transforms method, a solution of a boundary value problem for a fractional Black-Scholes equation is also derived. For some prescribed values of the parameters, com- putational values of the analytical results obtained are compared with some known numerical solutions.

Numerical computation of fractional Black–Scholes equation arising in financial market

The aim of present paper is to present a numerical algorithm for time-fractional Black–Scholes equation with boundary condition for a European option problem by using homotopy perturbation method and homotopy analysis method. The fractional derivative is described in the Caputo sense. The methods give an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The methods show improvements over existing analytical techniques. Two examples are given and show that the homotopy perturbation method and homotopy analysis method are very effective and convenient overcomes the difficulty of traditional methods. The numerical results show that the approaches are easy to implement and accurate when applied to time-fractional Black–Scholes equation.

A finite difference method for pricing European and American options under a geometric Lévy process

In this paper we develop a numerical approach to a fractional-order differential Linear Complementarity Problem (LCP) arising in pricing European and American options under a geometric Lévy process. The LCP is first approximated by a nonlinear penalty fractional Black-Scholes (fBS) equation. We then proposea finite difference scheme for the penalty fBS equation. We show that both the continuous and the discretized fBS equations are uniquely solvable and establish the convergence of the numerical solution to the viscosity solution of the penalty fBS equation by proving the consistency, stability and monotonicity of the numerical scheme. We also show that the discretization has the 2nd-order truncation error in both the spatial and time mesh sizes. Numerical results are presented to demonstrate the accuracy and usefulness of the numerical method for pricing both European and American options under the geometric Lévy process.

Game theoretic analysis of incomplete markets: emergence of probabilities, nonlinear and fractional Black–Scholes equations

Expanding the ideas of the author's paper 'Nonexpansive maps and option pricing theory' (Kibernetica 34:6 (1998), 713-724) we develop a pure game-theoretic approach to option pricing, by-passing stochastic modeling. Risk neutral probabilities emerge automatically from the robust control evaluation. This approach seems to be especially appealing for incomplete markets encompassing extensive, so to say untamed, randomness, when the coexistence of infinite number of risk neutral measures precludes one from unified pricing of derivative securities. Our method is robust enough to be able to accommodate various markets rules and settings including path dependent payoffs, American options and transaction costs. On the other hand, it leads to rather simple numerical algorithms. Continuous time limit is described by nonlinear and/or fractional Black-Scholes type equations.

Fractional Variational Iteration Method and Its Application to Fractional Partial Differential Equation

We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations in fluid mechanics and in financial models. The fractional derivatives are described in Riemann-Liouville sense. To show the efficiency of the considered method, some examples that include the fractional Klein-Gordon equation, fractional Burgers equation, and fractional Black-Scholes equation are investigated.

An efficient wavelet based approximation method to time fractional Black-Scholes European option pricing problem arising in financial market

In this paper, a wavelet based hybrid method is employed to provide the quick and accurate solutions of fractional Black-Scholes equation with boundary condition for a European option pricing (EOP) problem. The fractional Black-Scholes is used as a model for valuing European or American call and put options on a non-dividend paying stock. To the best of our knowledge, until now there is no rigorous Legendre wavelet solutions have been reported for the fractional Black-Scholes

Homotopy Perturbation Method for Fractional Black-Scholes European Option Pricing Equations Using Sumudu Transform

The homotopy perturbation method, Sumudu transform, and He’s polynomials are combined to obtain the solution of fractional Black-Scholes equation. The fractional derivative is considered in Caputo sense. Further, the same equation is solved by homotopy Laplace transform perturbation method. The results obtained by the two methods are in agreement. The approximate analytical solution of Black-Scholes is calculated in the form of a convergence power series with easily computable components. Some illustrative examples are presented to explain the efficiency and simplicity of the proposed method.