Gram-Charlier Densities

M. ROCKINGER, E. Jondeau

Journal of Economic Dynamics and Control

October 2001, vol. 25, n°10, pp.1457-1483

Departments: Finance

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

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