Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation

Abstract

In this study, a new application of multivariate Padé approximation method has been used for solving European vanilla call option pricing problem. Padé polynomials have occurred for the fractional Black-Scholes equation, according to the relations of "smaller than", or "greater than", between stock price and exercise price of the option. Using these polynomials, we have applied the multivariate Padé approximation method to our fractional equation and we have calculated numerical solutions of fractional Black-Scholes equation for both of two situations. The obtained results show that the multivariate Padé approximation is a very quick and accurate method for fractional Black-Scholes equation. The fractional derivative is understood in the Caputo sense.

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

“…For m = 0 , the 0th RPS solution of w(x, t) may be written as Using Eqs. (15) and (14) can be modified as So mth RPS solution can be evaluated after obtaining all b k (x) , k = 1, 2, … , m.…”

“…Adomian Decomposition Method (ADM) with conformal derivative and Modified Homotopy Perturbation Method (MHPM) [14], Multivariate Padé Approximation [15] and ADM [16]. Recently fractional order European Vanilla option pricing model has also been studied by Yavuz and Özdemir [17].…”

European Vanilla Option Pricing Model of Fractional Order without Singular Kernel

“…For example, these techniques have been used to evaluate the performance of an electrical resistance inductance and capacitance (RLC) circuit using a new fractional operator with a local and nonlocal kernel [1], to price fractional European vanilla-type options [2,3], to analyze a new model of H1N1 spread [4], to model the population growth [5], to apply the homotopy analysis method [6], the Adomian decomposition method [7,8], the homotopy perturbation method [9,10], He's variational iteration method in conformable derivative sense [11], the generalized differential transform method [12], the finite difference method [13] and the multivariate Padé approximation method [14]. Moreover, these proposed fractional techniques have been used to obtain the solution of the optimal control problem [15], the constrained optimization problem [16], the portfolio optimization problem [17], the diffusion-wave problem [18], etc.…”

On the Solutions of Fractional Cauchy Problem Featuring Conformable Derivative

“…In recent years, mathematical modelling with fractional PDEs in some special areas such like physics, mathematics, engineering, finance, biology and medicine [1-8] have been studied. Analytical solutions for fractional differential equations (FDEs) are not very popular in the literature, but the methods applied to avoid effective engineering analyzes, and therefore, numerical methods are frequently applied [9][10][11][12][13].…”