Time-fractional Black Scholes Equation Research Articles

Lie symmetry, exact solutions and conservation laws of time fractional Black–Scholes equation derived by the fractional Brownian motion

Abstract The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black–Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black–Scholes equation.

An Efficient Numerical Method Based on Exponential B-splines for a Time-Fractional Black–Scholes Equation Governing European Options

An Efficient Numerical Method Based on Exponential B-splines for a Time-Fractional Black–Scholes Equation Governing European Options

Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations

In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments.

A Hybrid Spectral-Finite Difference Method for Numerical Pricing of Time-Fractional Black–Scholes Equation

A Hybrid Spectral-Finite Difference Method for Numerical Pricing of Time-Fractional Black–Scholes Equation

Compact Difference Schemes with Temporal Uniform/Non-Uniform Meshes for Time-Fractional Black–Scholes Equation

In this paper, we are interested in the effective numerical schemes of the time-fractional Black–Scholes equation. We convert the original equation into an equivalent integral-differential equation and then discretize the time-integral term in the equivalent form using the piecewise linear interpolation, while the compact difference formula is applied in the spatial direction. Thus, we derive a fully discrete compact difference scheme with second-order accuracy in time and fourth-order accuracy in space. Rigorous proofs of the corresponding stability and convergence are given. Furthermore, in order to deal effectively with the non-smooth solution, we extend the obtained results to the case of temporal non-uniform meshes and obtain a temporal non-uniform mesh-based compact difference scheme as well as the numerical theory. Finally, extensive numerical examples are included to demonstrate the effectiveness of the proposed compact difference schemes.

Numerical Method for American Option Pricing under the Time-Fractional Black–Scholes Model

The fractional Black–Scholes model has had limited applications in financial markets. Instead, the time-fractional Black–Scholes equation has attracted much research interest. However, it is difficult to obtain the analytic expression for American option pricing under the time-fractional Black–Scholes model. This paper will present an operator-splitting method to price the American options under the time-fractional Black–Scholes model. The fractional partial differential complementarity problem (FPDCP) that the American option price satisfied is split into two subproblems: a linear boundary value problem and an algebraic system. A high-order compact (HOC) scheme and a grid stretching (GS) method are considered for the linear boundary problem. Furthermore, numerical results show that the HOC scheme with a GS method gives an accurate numerical solution for American options under the time-fractional Black–Scholes model.

Two high-order compact difference schemes with temporal graded meshes for time-fractional Black-Scholes equation

<abstract><p>In this paper, two high-order compact difference schemes with graded meshes are proposed for solving the time-fractional Black-Scholes equation. We first eliminate the convection term in the equivalent form of the considered equation by using exponential transformation, then combine the sixth-order/eighth-order compact difference method with a temporal graded meshes-based trapezoidal formulation for the temporal integral term to obtain the fully discrete high-order compact difference schemes. The stability and convergence analysis of the two proposed schemes are studied by applying Fourier analysis. Finally, the effectiveness of the proposed schemes and the correctness of the theoretical results are verified by two numerical examples.</p></abstract>

A spectral collocation method based on fractional Pell functions for solving time–fractional Black–Scholes option pricing model

The fractional Black–Scholes equation has been widely studied by researchers in recent years. In this article, an efficient spectral collocation method based on fractional Pell functions is proposed for solving the time–fractional Black–Scholes equation. We introduce fractional Pell functions using the transformation x→xβ(β>0) on Pell polynomials, and we look for a solution of the model as a linear combination of these functions. Using operational matrices, we approximate the fractional derivative and other terms in a convenient form of the main equation. A system of algebraic equations is obtained by collocating resultant approximate equations. Convergence analysis of the numerical method has been investigated in Sobolev space. Finally, we have demonstrated the capability of the proposed method by considering numerical experiments in the form of tables and figures.

Lie symmetry analysis and exact solutions of time fractional Black–Scholes equation

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the constructive methods of exact solutions to time fractional Black–Scholes equation. By constructing one-parameter Lie symmetry transformations and their corresponding group generators, time fractional Black–Scholes equation is reduced to a fractional ordinary differential equation and some group-invariant solutions are obtained. Using the invariant subspace method, the analytical representations of two forms of exact solutions of time fractional Black–Scholes equation are given constructively, and the characteristics and differences of the two exact solutions are compared in the sense of geometric figures. In this paper, the form of the equation is generalized, and more group invariant solutions and analytical solutions in the form of separated variables are obtained.

A second order numerical method for the time-fractional Black–Scholes European option pricing model

In this paper, we design an efficient and accurate numerical method for solving the time-fractional Black–Scholes equation governing European options. The time-fractional Black–Scholes equation is transformed into an equivalent integro-differential equation. The numerical method for the integro-differential equation is developed by using a numerical integration scheme for time discretization and central difference formulas for space discretization. The stability and convergence of the method are analyzed. The numerical method is proved to be second-order accurate in both space and time. Richardson extrapolation is also introduced to obtain a modified version of the method that exhibits faster convergence for the time-fractional Black–Scholes equation with non-smooth initial data. Finally, two numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.

High-order compact finite difference schemes for the time-fractional Black-Scholes model governing European options

Based on the price fluctuations of the underlying fractal transmission system, the α-order time-fractional Black-Scholes model is obtained. In this paper, we introduce two compact finite difference schemes for the solution of the time-fractional Black-Scholes equation governing European option pricing. In proposed schemes, in order to gain sixth-order and eighth-order accuracy in space, we first use exponential transformation to eliminate the convection term of the Black-Scholes equation. Then, the time-fractional derivative discretizes by a 3−αth order numerical formula (called the L1–2 formula here) which is constructed by a linear interpolating polynomial on the first subinterval and the quadratic interpolating polynomials on the other subintervals. We investigate stability and convergence of proposed schemes by Fourier method. Finally, some numerical examples perform to demonstrate the theoretical order of accuracy and illustrate the effectiveness of proposed methods. We also discuss the influence of different parameters on the option price in the time-fractional Black-Scholes model.

A posteriori grid method for a time-fractional Black-Scholes equation

<abstract><p>In this paper, a posteriori grid method for solving a time-fractional Black-Scholes equation governing European options is studied. The possible singularity of the exact solution complicates the construction of the discretization scheme for the time-fractional Black-Scholes equation. The $ L1 $ method on an arbitrary grid is used to discretize the time-fractional derivative and the central difference method on a piecewise uniform grid is used to discretize the spatial derivatives. Stability properties and a posteriori error analysis for the discrete scheme are studied. Then, an adapted a posteriori grid is constructed by using a grid generation algorithm based on a posteriori error analysis. Numerical experiments show that the $ L1 $ method on an adapted a posteriori grid is more accurate than the method on the uniform grid.</p></abstract>

Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

In this study, we present an accurate and efficient nonuniform finite difference method for the three-dimensional (3D) time-fractional Black–Scholes (BS) equation. The operator splitting scheme is used to efficiently solve the 3D time-fractional BS equation. We use a nonuniform grid for pricing 3D options. We compute the three-asset cash-or-nothing European call option and investigate the effects of the fractional-orderαin the time-fractional BS model. Numerical experiments demonstrate the efficiency and fastness of the proposed scheme.

Localized kernel‐based meshless method for pricing financial options underlying fractal transmission system

The variation in the option pricing of the fractal transmission system is modelled by the time fractional Black–Scholes equation (TFBSE). This paper proposes an efficient local meshless method for the numerical simulation of the TFBSE. At the first step, a difference formula of L1 type is employed to get a semi‐discrete algorithm in the temporal variable with a accuracy of order 2 − α in the case of smooth solutions, where 0 < α ≤ 1 is the fractional‐order derivative. At the second step, a localized radial basis function finite difference is adopted to derive a full‐discrete scheme. Moreover, the unconditional stability and convergence of the proposed method are analyzed based on energy norm. The exact expressions for the weights of the first and second derivatives are used by imposing a multiquadric function generated by finite difference. The proposed technique produces linear systems with tridiagonal and diagonal matrices. Numerical experiments highlight the performance of the method.

Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs

Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs

Group formalism of Lie transformations, conservation laws, exact and numerical solutions of non-linear time-fractional Black–Scholes equation

The topic of the present paper concentrates on a systematic investigation of Lie group analysis of non-linear time-fractional Black–Scholes equation including numerical approximations. Lie point symmetries of the equation are found via Riemann–Liouville derivative with four different volatility models. Also the invariant solutions as a kind of exact solutions are constructed by a useful explained method. Conservation laws for the intended equation are derived by using a modified version of Noether’s theorem. A numerical simulation by Chebyshev pseudo-spectral (CPS) method is computed for the equation. Approximate solutions together with their absolute errors are found and the graphs of solutions are plotted by the action of order of derivative α.

Spectrally accurate option pricing under the time fractional Black-Scholes model

We 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. doi:10.1017/S1446181121000286

Novel ANN Method for Solving Ordinary and Time-Fractional Black–Scholes Equation

The main aim of this study is to introduce a 2-layered artificial neural network (ANN) for solving the Black–Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of ordinary differential equations (ODE). Subsequently, each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine-tuning for speeding up the process and domain mapping to confront the infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of the Black–Scholes model are reported.

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.