Abstract
This paper presents a numerical solution of the temporal-fractional Black–Scholes equation governing European options (TFBSE-EO) in the finite domain so that the temporal derivative is the Caputo fractional derivative. For this goal, we firstly use linear interpolation with the (2-alpha)-order in time. Then, the Chebyshev collocation method based on the second kind is used for approximating the spatial derivative terms. Applying the energy method, we prove unconditional stability and convergence order. The precision and efficiency of the presented scheme are illustrated in two examples.
Highlights
The Black–Scholes model (BSM) is a mathematical equation for pricing and options treaty
The TFBSE-EO can be considered as a generalization of the classical Black–Scholes model
Due to the “globality” characteristic of the model’s fractional-order derivative, the numerical solutions of this model are more complicated than the integer-order model
Summary
The Black–Scholes model (BSM) is a mathematical equation for pricing and options treaty. In 1973, pricing options have experienced a lot of consideration that the first time Black and Scholes [1] and Merton [2] proposed BSM for them This model is very favorite, it has some deficiencies like lacking the “volatility smile”[3] in the actual marketplaces. M,j = − M,j , a semi-discrete scheme of Eq (1) will be achieved by applying the Lemma 2 as FM + RM,. An explicit–implicit numerical method is proposed by Bhowmik In 2014 to solve the partial integrodifferential equation [28] which is the base of the option pricing hypothesis. In 2016, Zhang proposed a discrete implicit numerical scheme for pricing American options [30]. It is necessary to prove the following lemma to find the convergence order and denote the unconditional stability of the semi-discrete scheme. Theorem 1 The achieved semi-discrete scheme (6) by the linear interpolation of the Lemma 2 is unconditionally stable
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