An Implicit Martingale Restriction in a Closed-Form Higher Order Moments Option Pricing Formula Based on Multipoint Padé Approximants

The purpose of this paper is to develop a new non-parametric method to price options based on normalized Multipoint Padé Approximants. Following the seminal paper of Padé (1892), we propose to approximate the risk-neutral distribution by a rational function of polynomials that can accommodate the asymmetric and leptokurtic characteristics of the implied state price densities. After recalling the general framework of Padé Approximants we present an analytical formula where we use a power series expansion of the risk-neutral density in order to infer the coefficients of the rational function of polynomials. A suitable alternative to this method will be to use various points of local expansion, resulting from the capability of the Padé to respect such a confluence. By manipulating the base option pricing formula with risk-neutral density (see Cox and Ross, 1976), both of these methods is implicitly satisfying the martingale constraint (see Longstaff, 1995 and Jurczenko, et al., 2006). We then investigate from simulated option prices the shape of the risk-neutral density Padé approximations to compare their radius of convergence