A new iterative method based solution for fractional Black–Scholes option pricing equations (BSOPE)

Abstract

In this manuscript, a new expansion technique namely residual power series method is used for finding the analytical solution of the Fractional Black-Scholes equation with an initial condition for European option pricing problem. The Black-Scholes formula is important for estimating European call and put option on a non-dividend paying stock in particular when it contains time-fractional derivatives. The fractional derivative is defined in Caputo sense. This technique is based on fractional power series expansion. The convergence analysis of the present method is also deliberated. Example problems are given to examine the efficacy of the proposed method. Obtained solutions are compared with exact solutions solved by other techniques which demonstrate that the present method is robust and easy to implement for other fractional problems arising in science and engineering.

On the solution of time‐fractional dynamical model of Brusselator reaction‐diffusion system arising in chemical reactions

“…The most iterating feature of these models is their global characteristics, which do not exist in the classical‐order models. FC has occupied considerable attention in various fields such as solid mechanics, fluid dynamics, ecology, financial mathematics, biological diseases, and other areas of science and engineering (see previous studies). Sometimes it is tough to find the solution of the FDEs, so one may also need efficient computational methods for the solution of FDEs.…”

Exact solutions to the family of Fisher's reaction‐diffusion equation using Elzaki homotopy transformation perturbation method

“…However, some methods can perform better on some models than others due to the nonlinearity of the model and the radius of convergence of such method. Bildik, Rajarama, Chakraverty, and so on buttressed this based on their convergence analysis on some iterative methods …”

“…NPDEs have been widely studied by numerous researchers over the years and has become ubiquitous in nature, they can be classified into integrable and nonintegrable.…”

“…Exact solution rarely exists generally for nonlinear differential equations (ordinary and partial) and as a result of this, efforts have been made over the years by mathematicians in constructing semianalytical and numerical means, schemes or algorithms for solving them, namely, Taylor collocation method, Euler collocation methods, wavelet collocation methods [44, 52, 53], iterative differential quadrature method, variational iteration method, homotopy perturbation method (HPM), perturbation iteration method, Adomian decomposition method, new iterative method, homotopy analysis method, reduced differential transform method, and residual power series method …”

“…Very recently, Cherif and Djelloul coupled Elzaki integral transform and variational iteration method on PDEs of fractional order; Dhunde and Waghmare coupled the double Laplace transform with the new iterative method on NPDEs; Jena and Chakraverty coupled the Elzaki transform with the homotopy perturbation on time‐fractional Navier‐Stokes equation, Again in Reference , they implemented the residual power series approach on the fractional Black‐Scholes option pricing equations, Alderremy et al proposed a novel decomposition method involving the Elzaki transform and Adomian decomposition method on nonlinear fractional differential equations, Singh and Sharma implemented EHTPM and HPTM method on nonlinear fractional PDEs as a comparative study of these two methods. Series solutions were obtained using all of these methods of which converges rapidly to the exact solutions of the problems/models being solved.…”