Estimating Gram-Charlier Expansions Under Positivity Constraints

Gram-Charlier expansions have became popular in Finance as an improvement over the normality assumption. The reason is that in Gram-Charlier expansions, parameters appear which directly control the skewness and kurtosis. Those expansions, of polynomial nature, have the unfortunate drawback of sometimes yielding negative densities. The goal of this paper is to show how positivity constraints can be numerically implemented. We successfully apply our method to the estimation of risk neutral densities that arise within a financial option pricing context. The method can be easily adapted to other setups involving polynomial approximations