Working Paper Series no. 56: Gram-Charlier Densities

Abstract

The Gram-Charlier expansion, where skewness and kurtosis directly appear as parameters, has become popular in Finance as a generalization of the normal density. We show how positivity constraints can be numerically implemented, thereby guaranteeing that the expansion defines a density. The constrained expansion can be referred to as a Gram-Charlier density. First, we apply our method to the estimation of risk neutral densities. Then, we assess the statistical properties of maximum-likelihood estimates of Gram-Charlier densities. Lastly, we apply the framework to the estimation of a GARCH model where the conditional density is a Gram-Charlier density.

Eric Jondeau and Michael Rockinger
July 1999

Classification JEL : C40, C63, G13, F31.

Keywords : Hermite expansions; Semi-nonparametric estimation; Risk-neutral density; GARCH model