Abstract
In this paper, a high-order compact finite difference method is proposed for a class of temporal fractional subdiffusion equation. A numerical scheme for the equation has been derived to obtain 2-α in time and fourth-order in space. We improve the results by constructing a compact scheme of second-order in time while keeping fourth-order in space. Based on the L2-1σ approximation formula and a fourth-order compact finite difference approximation, the stability of the constructed scheme and its convergence of second-order in time and fourth-order in space are rigorously proved using a discrete energy analysis method. Applications using two model problems demonstrate the theoretical results.
Highlights
The Black-Scholes model, proposed in 1973 by Black and Scholes [1] and Merton [2], gives a theoretical estimate of the price of European-style options
In [7], H.Zhang et al use some numerical technique to price a European doubleknock-out barrier option, and the characteristics of the three fractional Black-Scholes models are analysed through comparison with the classical Black-Scholes model
Some computationally efficient numerical methods have been proposed for solving fractional differential equation, for example, which include finite difference methods, finite element methods, finite volume methods, spectral methods, and meshless methods [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]
Summary
The Black-Scholes model, proposed in 1973 by Black and Scholes [1] and Merton [2], gives a theoretical estimate of the price of European-style options. In [3], Wyss priced a European call option by a time-fractional Black-Scholes model. In [4], Liang et al derive a biparameter fractional Black-Merton-Scholes equation and obtain the explicit option pricing formulas for the European call option and put option, individually. An explicit closed-form analytical solution for barrier options under a generalized time-fractional Black-Scholes model by using eigenfunction expansion method together with the Laplace transform is derived in [5]. In [7], H.Zhang et al use some numerical technique to price a European doubleknock-out barrier option, and the characteristics of the three fractional Black-Scholes models are analysed through comparison with the classical Black-Scholes model. The rest of the paper is organized as follows: in Section 2, an efficient implicit numerical scheme with second-order accuracy in time and fourth-order accuracy in space is constructed.
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