Gram-Charlier Densities (Revised version)

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.

Number of Pages in PDF File: 41

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

JEL Classification: C40, C63, G13, F31

working papers series