Estimating Gram-Charlier Expansions with Positivity Constraints.

Abstract

The Gram-Charlier expansion, where skewness and kurtosi 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.Hermite expansions ; Semi-nonparametric estimation ; Risk-neutral density ; GARCH model.