Time-fractional Black Scholes Equation Research Articles

Numerical investigation of the time-fractional Black-Scholes equation with barrier choice of regulating European option

The price variance of the associated fractal transmission mechanism was used to estimate the Black-Scholes fractional model of which a time-fractional derivative is $alpha$. In the current paper, the time-fractional Black-Scholes equation (TFBSE) that the temporal derivative is the Caputo fractional derivative is known by regulating the European option. At first, linear interpolation with a temporally $tau^{2-alpha}$ order accuracy is used for constructing the semi-discrete. Then, the spatial derivative terms are approximated with the help of the collocation approach centered on the Chebyshev polynomials of the third form (CPTF). Finally, The unconditional stability and convergence order are analyzed by applying the energy method. To show the precision of the numerical system, we solved two instances of the TFBSE. Numerical results and comparisons indicate the proposed approach is very reliable and efficient.

A compact difference scheme for time-fractional Black-Scholes equation with time-dependent parameters under the CEV model: American options

‎The Black-Scholes equation is one of the most important mathematical models in option pricing theory, but this model is far from market realities and cannot show memory effect in the financial market. This paper investigates an American option based on a time-fractional Black-Scholes equation under the constant elasticity of variance (CEV) model, which parameters of interest rate and dividend yield supposed as deterministic functions of time, and the price change of the underlying asset follows a fractal transmission system. This model does not have a closed-form solution; hence, we numerically price the American option by using a compact difference scheme. Also, we compare the time-fractional Black-Scholes equation under the CEV model with its generalized Black-Scholes model as α = 1 and β = 0. Moreover, we demonstrate that the introduced difference scheme is unconditionally stable and convergent using Fourier analysis. The numerical examples illustrate the efficiency and accuracy of the introduced difference scheme.

SPECTRALLY ACCURATE OPTION PRICING UNDER THE TIME-FRACTIONAL BLACK–SCHOLES MODEL

AbstractWe propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular$L1$finite difference approximation, which converges with order$\mathcal {O}((\Delta \tau )^{2-\alpha })$for functions which are twice continuously differentiable. However, when using the$L1$scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.

Application of the Generalized Laplace Homotopy Perturbation Method to the Time-Fractional Black–Scholes Equations Based on the Katugampola Fractional Derivative in Caputo Type

In the finance market, the Black–Scholes equation is used to model the price change of the underlying fractal transmission system. Moreover, the fractional differential equations recently are accepted by researchers that fractional differential equations are a powerful tool in studying fractal geometry and fractal dynamics. Fractional differential equations are used in modeling the various important situations or phenomena in the real world such as fluid flow, acoustics, electromagnetic, electrochemistry and material science. There is an important question in finance: “Can the fractional differential equation be applied in the financial market?”. The answer is “Yes”. Due to the self-similar property of the fractional derivative, it can reply to the long-range dependence better than the integer-order derivative. Thus, these advantages are beneficial to manage the fractal structure in the financial market. In this article, the classical Black–Scholes equation with two assets for the European call option is modified by replacing the order of ordinary derivative with the fractional derivative order in the Caputo type Katugampola fractional derivative sense. The analytic solution of time-fractional Black–Scholes European call option pricing equation with two assets is derived by using the generalized Laplace homotopy perturbation method. The used method is the combination of the homotopy perturbation method and generalized Laplace transform. The analytic solution of the time-fractional Black–Scholes equation is carried out in the form of a Mittag–Leffler function. Finally, the effects of the fractional-order in the Caputo type Katugampola fractional derivative to change of a European call option price are shown.

A space-time spectral method for time-fractional Black-Scholes equation

The purpose of this paper is to investigate a high order numerical method for solving time-fractional Black-Scholes equation in which the fractional operator is defined by the Caputo fractional derivative. The proposed space-time spectral method employs the Jacobi polynomials for the temporal discretisation and Fourier-like basis functions for the spatial discretisation. The stability and convergence of the numerical scheme are analyzed. Two numerical examples are considered to validate the accuracy and illustrate the practicability of the proposed method. The results agree with the theoretical analysis and this approach can be applied in dealing with option pricing models with smooth payoff functions.

A fractional reduced differential transform method for solving time fractional Black Scholes American option pricing equation

In this paper, fractional reduced differential transform method (FRDTM) is operated to solve time fractional Black-Scholes American option pricing equation paying no dividends.The Black-Scholes model plays a significant role in the evaluation of European or American call and put options. The advantage of the proposed method to other existing methods is that it finds the solution without discretization or transformation. While using this method, no recommended assumptions are needed and hence the computational work reduces to a greater extent. Numerical experiments prove that the proposed method is efficient and valid for obtaining the solution of time fractional Black-Scholes equation governing American options. This method proves to be powerful for solving general fractional order partial differential equations (PDEs) existing in the field of Science, Engineering and other related fields.

Recovery of the time-dependent implied volatility of time fractional Black–Scholes equation using linearization technique

Abstract This paper tries to examine the recovery of the time-dependent implied volatility coefficient from market prices of options for the time fractional Black–Scholes equation (TFBSM) with double barriers option. We apply the linearization technique and transform the direct problem into an inverse source problem. Resultantly, we get a Volterra integral equation for the unknown linear functional, which is then solved by the regularization method. We use L 1 {L_{1}} -forward difference implicit approximation for the forward problem. Numerical results using L 1 {L_{1}} -forward difference implicit approximation ( L 1 {L_{1}} -FDIA) for the inverse problem are also discussed briefly.

The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown.

A novel numerical scheme for a time fractional Black–Scholes equation

This paper consists of two parts. On one hand, the regularity of the solution of the time-fractional Black–Scholes equation is investigated. On the other hand, to overcome the difficulty of initial layer, a modified L1 time discretization is presented based on a change of variable. And the spatial discretization is done by using the Chebyshev Galerkin method. Optimal error estimates of the fully-discrete scheme are obtained. Finally, several numerical results are given to confirm the theoretical results.

Homotopy analysis method and its applications in the valuation of European call options with time-fractional Black-Scholes equation

• HAM and its applications in the valuation of ECO with TFBSE in a Caputo sense are presented. • By means of the marked point process, the classical Black-Scholes equation turns to a time-fractional Black-Scholes equation. • The valuation formula for the price of a European call option with fractional order is obtained via HAM. • Illustrative examples were presented to measure the performance of HAM in terms of accuracy, effectiveness and suitability. • The physical behaviour of the option prices has been shown in terms of plots for different values of fractional order. This paper presents the applications of Homotopy Analysis Method (HAM) in the valuation of a European Call Option (ECO) with Time-Fractional Black-Scholes Equation (TFBSE). The fractional derivative is considered in the sense of Caputo. Also, it is assumed that the stock price pays no dividend and follows the geometric Brownian motion. Based on HAM, a series solution for TFBSE has been obtained successfully. The valuation formula for the price of ECO with fractional order is also obtained. The accuracy, effectiveness and suitability of HAM were tested on two illustrative examples in the context of the Crank Nicolson Method (CRN), Binomial Model (BM) and the Black-Scholes Model (BSM). The comparative study of the results obtained via HAM, CRN, BM and BSM is presented. Furthermore, the physical behaviour of the option prices obtained via HAM has been shown in terms of plots for diverse fractional order. Moreover, HAM is found to be accurate, effective and suitable for the solution of TFBSE. Hence, it can be concluded that HAM converges faster to the analytical solution and is a good alternative tool to determine the price of ECO with fractional order.

A compact quadratic spline collocation method for the time-fractional Black–Scholes model

A compact quadratic spline collocation (QSC) method for the time-fractional Black–Scholes model governing European option pricing is presented. Firstly, after eliminating the convection term by an exponential transformation, the time-fractional Black–Scholes equation is transformed to a time-fractional sub-diffusion equation. Then applying $$L1 - 2$$ formula for the Caputo time-fractional derivative and using a collocation method based on quadratic B-spline basic functions for the space discretization, we establish a higher accuracy numerical scheme which yields $$3-\alpha $$ order convergence in time and fourth-order convergence in space. Furthermore, the uniqueness of the numerical solution and the convergence of the algorithm are investigated. Finally, numerical experiments are carried out to verify the theoretical order of accuracy and demonstrate the effectiveness of the new technique. Moreover, we also study the effect of different parameters on option price in time-fractional Black–Scholes model.

On implied volatility recovery of a time-fractional Black-Scholes equation for double barrier options

ABSTRACT This paper investigates a time-fractional Black-Scholes equation with an implied volatility which is assumed to be associated with underlying price. Two aspects are considered. One is for the forward problem, i.e. a robust -CDIA (-central difference implicit approximation) scheme is utilized to solve the initial boundary value problem effectively. Remarkably, convergence analysis for the forward problem solver is also given. The other is for the inverse problem, i.e. recovery of the implied volatility via additional data. By linearization which is similar as [Bouchouev et al. Quant. Finance, 2 (2002) 257–263.], a Fredholm integral equation of the first kind for the unknown volatility is obtained. Based upon the integral equation, a uniqueness of the recovery is shown under some assumptions and an efficient numerical reconstruction algorithm is proposed to reconstruct the coefficient. Numerical results for both direct problem and inverse problem are presented to illustrate the validity and effectiveness of the proposed schemes.

Lie symmetry analysis and conservation laws for the time fractional Black–Scholes equation

In this paper, the Lie symmetry algebra admitted by the time fractional Black–Scholes equation is obtained by using the Lie group method. The constructed symmetry generators are investigated to construct a family of exact solutions and conservation laws for the studied equation. At the same time, the family of solutions is extended by using the invariant subspace method.

A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European options

This paper is concerned with the design of a high order numerical approach based on a uniform mesh for efficient numerical solution of time-fractional Black-Scholes equation, governing European options. The time-fractional derivative is defined in the Caputo sense. A collocation method based on quintic B-spline basis functions is used for space discretization and time-stepping is done using a backward Euler method. The stability and convergence of the method are analyzed. The method is shown to be unconditionally stable and fourth order accurate in space and (2−β) order accurate in time, where β is the order of the time fractional derivative (0<β<1). Two numerical examples with the known exact solutions are considered to validate theoretical results and demonstrate the accuracy of the method. Moreover, this scheme is used to price three different European options governed by a time-fractional Black-Scholes model; (i) European double barrier knock-out call option (ii) European call option and (iii) European put option. The effect of the order of fractional derivative on the option price is studied.

Numerical Simulation of Time-fractional Black-Scholes Equation using Fractional Variational Iteration Method

Numerical Simulation of Time-fractional Black-Scholes Equation using Fractional Variational Iteration Method

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.

Numerical approximation of a time-fractional Black–Scholes equation

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.

A New Version of Black–Scholes Equation Presented by Time-Fractional Derivative

In this article, a new time-fractional-order Black–Scholes equation has been derived. In this derivation, the asset price satisfies in a fractional-order stochastic differential equation. Here, the effect of trend memory in financial pricing is considered. Finally, a new approximate analytical method has been used to solve our new proposed time-fractional Black–Scholes equation

Numerical solution of time-fractional Black–Scholes equation

In this paper we present a numerical method for a time-fractional Black–Scholes equation, which is used for modeling the fractional structure of the financial market. The method is based on—first, discretization in time and then the weighted finite difference spatial approximation. Some properties of the spatial discretization are studied. The main difficulty (that originates from the non-local structure of the differential operator) of the algorithm is the impossibility to advance layer by layer in time. Numerical experiments are discussed.

Analytically pricing double barrier options based on a time-fractional Black–Scholes equation

This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. In this scenario, the option price is governed by a modified Black–Scholes equation with a time-fractional derivative. In comparison with standard derivatives of integer order, the fractional-order derivatives are characterized by their “globalness”, i.e., the rate of change of a function near a point is affected by the property of the function defined in the entire domain of definition rather than just near the point itself. The existence of the time-fractional derivative, in conjunction with the presence of two barriers of the double-barrier options, has added an additional degree of difficulty not only when a purely numerical solution is sought but also when an analytical method is attempted. Albeit difficult, we have managed to find an explicit closed-form analytical solution for double-barrier options, which has been taken to price the single barrier options and European path-independent options under the same framework as a special case of the current solution. In addition, not only have we provided a theoretical proof for the convergence of the newly-found analytic solution in series form, which is a vital step to show the closeness of our solution, we have also proposed an efficient numerical evaluation technique to facilitate the implementation of our formula so that it can be easily used in trading practice.