EDEGEWORTH BLACK-SCHOLES OPTION PRICING FORMULA ALI YOUSEF ABSTRACT. In this paper we consider the Black-Scholes option pricing problem. We use Edgeworth second order approximation to approximate the underlying asset’s return distribution. We state and prove Lemma 3.2 which finds a closed form for the solution of the Black-Scholes partial differential equation under Edgeworth approximation. The theoretical form of Edgeworth option pricing formula found in Lemma 3.2 depends mainly on the skewness and kurtois of the asset’s return distribution which indeed corrects the classical Black-Scholes formula that was presented by Black and Scholes in 1973. Moreover, we find a simple form of delta hedging. In the end we find standardized Edgeworth expansions for the following distributions; lognormal, student t–distribution and chi–squared distributions. 1. INTRODUCTION The Edgeworth expansion was first introduced by Edgeworth in 1905, see [11], as an expansion representing one distribution function in terms of another distribution, in such a way that the cumulants of the other distribution function should be known. Since its foundation in 1905, it became the center of many statistical studies and has been applied in many fields like, economics, finance and also in engineering, see [13]. The asymptotic behaviour were developed by [25], and its validity region was discuused by [4]. 1 In the following section, we give some accounts about Edgeworth second order expansion and its validity region. 2. EDGEWORTH SECOND ORDER EXPANSION AND ITS ASYMPTOTIC PROPERTIES Let   1 , , n X X be a sequence of IID random varaibles of size n from a continuos distribution function F , such that the mean   , the variance 2   , the skewness  and the kurtosis  are all finite, where,   3 3 EX        and   4 4 EX        . Let 1 1 n n i i X n X     ,   n n Z nX     and    n n F x PZ x   . Then the Central Limit Theorem states that the limiting distribution of n Z , that is, 1 Key words and phrases. Black–Scholes formula, Chi–Squared, delta hedging, Edgeworth expansion, Lognormal, Student t–distribution. 23 J. CONCRETE AND APPLICABLE MATHEMATICS, VOL. 11, NO. 1, 23-46, 2013, COPYRIGHT 2013 EUDOXUS PRESS LLC