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

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

The mathematical modeling in trade and finance issues is the key purpose in the computation of the value and considering option during preferences in contract. This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. Due to the outstanding memory effect present in the fractional derivatives, approximating financial options with regards to their hereditary characteristics can be well interpreted and stated. Motivated by the reason mentioned, relatively reliable and also efficient numerical approaches have to be found while facing with fractional differential equations. The main objective of the current paper is to obtain the approximation solution of the time fractional Black–Scholes model of order $$0<\alpha \le 1$$ governing European options based on the moving least-squares (MLS) method. In proposed method, firstly, the mentioned equation is discretized in the time sense based on finite difference scheme of order $${\mathcal {O}}(\delta t^{2-\alpha })$$ and then approximated by using MLS approach in the space variable. Furthermore, the stability and convergence of the proposed method are discussed in detail throughout the paper. Numerical evidences and comparisons demonstrate that the proposed method is very accurate and efficient.

A Computational Method Based on the Moving Least-Squares Approach for Pricing Double Barrier Options in a Time-Fractional Black–Scholes Model

Fractional Black–Scholes equation is a constructive financial equation. The model is used to determine the value of the option without a transaction cost. The analytical solutions of the fractional Black–Scholes equations have been addressed. The Caputo generalized fractional derivative has been used. The homotopy perturbation method has been developed for obtaining the analytical solutions of the fractional Black–Scholes equation (BSE) and the generalized fractionalBSE. The analytical solutions of the fractionalBSE and the generalized fractionalBSE have been represented graphically. The effect of the order ρ of the generalized fractional derivative in the diffusion processes has been analyzed.

Black–Scholes option pricing equations described by the Caputo generalized fractional derivative

A compact finite difference scheme for fractional Black-Scholes option pricing model

This work deals with the put option pricing problems based on the time-fractional Black-Scholes equation, where the fractional derivative is a so-called modified Riemann-Liouville fractional derivative. With the aid of symbolic calculation software, European and American put option pricing models that combine the time-fractional Black-Scholes equation with the conditions satisfied by the standard put options are numerically solved using the implicit
scheme of the finite difference method.

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Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method

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Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method