When considering the price change of the underlying fractal transmission system, a fractional Black–Scholes(B-S) model with an α-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 2−α 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.
In the paper by Jian Wang et al. Chaos, Solitons and Fractals 157 (2022) 111895, the fractional Black-Scholes PDE and the corresponding Greeks are given. We find that Greeks do not conform to the ...fractional Black-Scholes PDE proposed by Jian Wang et al. 1.
This work deals with the construction and analysis of a high‐order computational scheme for a time‐fractional Black‐Scholes model that governs the European option pricing. The time‐fractional ...derivative is considered in the sense of Caputo and the L1 − 2 formula is employed to approximate the Caputo temporal‐fractional derivative of order α, where α ∈ (0, 1). A compact difference scheme is designed for discretization of space variable. The convergence of the method is discussed in detail. It is shown that the proposed method has fourth order accuracy in space and
(3−α)−th order accuracy in time. One numerical example with the known exact solution is considered to demonstrate the applicability and accuracy of present numerical scheme. Moreover, the suggested numerical scheme is employed for pricing three European option problems, namely European call option, European double barrier knock‐out option and European put option. The effect of fractional order derivative on option price profile is investigated. Furthermore, the effects of three relevant parameters, namely volatility, strike price and interest rate on the price of European double barrier knock‐out option are investigated. The computational time for the proposed method is provided.
The Ivancevic option pricing model is an alternative of the standard Black–Scholes pricing equation, which signifies a controlled Brownian motion related to the nonlinear Schrodinger equation. Even ...though many researchers have studied the applicability and practicality of this model, but the analytical approach of this model is rarely found in the literature. In this paper, a novel semi-analytical technique called fractional reduced differential transform method has been applied to solve the Schrodinger type option pricing model, which is characterized by the time-fractional derivative. Two problems are explained to validate and prove the effectiveness of the proposed technique. Obtained results are compared with the solution of other existing methods for a particular case. This comparison shows that the attained results are in good agreement with the existing solutions.
•The time-fractional order Ivancevic option pricing model has been developed.•This model has been studied analytically by using FRDTM.•Convergence analysis at α=1,0.2,0.5 with increasing value of the number of terms has been included.
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.
•Tempered fractional Black–Scholes equation.•European double barrier option.•Numerical simulation.•Stability and convergence.•Fast bi-conjugate gradient stabilized method.
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.
In this paper, the Black–Scholes equation of the option pricing theory in order to minimize the risk through the stocks is studied. The solutions are obtained in terms of exceptional Laguerre ...polynomials. Moreover, higher-order supesymmetric representations are studied with a special case of third order. The Darboux transformation of the heat equation linked to the Black–Scholes system is given and a new potential is shown.
•Black–Scholes system adapted in position dependent mass.•Transformed system solutions are obtained through exceptional orthogonal polynomials.•Higher-order SUSY transformations are used to obtain a model for Black–Scholes system.•Heat equation is obtained from Black–Scholes system and Darboux transformation is applied.
The main purpose of this work is to present a new numerical method based on Hahn hybrid functions (HHFs) for solving of Black–Scholes option pricing distributed order time‐fractional partial ...differential equation. To this end, HHFs are introduced and their fractional integral operator with some properties of HHFs is calculated. In the next, with the help of fractional integral operator of HHFs, Gauss–Legendre quadrature formula and collocation method, distributed order time‐fractional Black–Scholes model is reduced to a system of algebraic equations. Furthermore, convergence analysis of the mentioned scheme is discussed. Finally, some test problems have been included to confirm the validity and efficiency of the mentioned numerical scheme. Moreover, Black–Scholes equations are studied through a bibliometric viewpoint.
•A time-fractional Black-Scholes model with smooth payoff function (TFBSM-APF) is considered.•A high-order numerical scheme is constructed for solving TFBSM-APF.•Convergence and stability analyses of ...the proposed schemes are presented.•Numerical examples are given.•The theoretical analysis and numerical approach can be extended to solve other similar time-fractional models.
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.
In this study, we derive a new exact solution for pricing European options in a two-state regime-switching economy. Two coupled Black–Scholes partial differential equations (PDEs) under the regime ...switching are solved using the Fourier Transform method. A key feature of the newly-derived solution is its simplicity in the form of a single integral with a real integrand, which leads to great computational efficiency in comparison with other closed-form solutions previously presented in the literature. Numerical examples are provided to demonstrate some interesting results obtained from our pricing formula.
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