Numerical analysis of time fractional Black–Scholes European option pricing model arising in financial market

• Published in: Computational & applied mathematics 2019-12, Vol.38 (4), p.1-24, Article 173
• Main Authors: Golbabai, Ahmad, Nikan, Omid, Nikazad, Touraj
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Applied physics; Boundary conditions; Computational mathematics; Computational Mathematics and Numerical Analysis; Differential equations; Finite difference method; Mathematical analysis; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematical models; Mathematics; Mathematics and Statistics; Meshless methods; Numerical analysis; Options trading; Pricing; Put & call options; Radial basis function
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-019-0957-7

The price variation of the correlated fractal transmission system is used to deduce the fractional Black–Scholes model that has an α -order time fractional derivative. The fractional Black–Scholes model is employed to price American or European call and put options on a stock paying on a non-dividend basis. Upon encountering fractional differential equations, the efficient and relatively reliable numerical schemes must be obtained for their solution due to fractional derivatives being non-local. The present paper is aimed at determining the numerical solution of the time fractional Black–Scholes model (TFBSM) with boundary conditions for a problem of European option pricing involved with the method of radial basis functions (RBFs), which is a truly meshfree scheme. The TFBSM is discretized in the temporal sense based on finite difference scheme of order O ( δ t 2 - α ) for 0 < α < 1 and approximated with the help of the RBF in the spatial derivative terms. In addition, the stability and convergence of the proposed method are theoretically proven. Numerical results illustrate the accuracy and efficiency of the presented technique which is examined in the present study.