Estimating Gram-Charlier Expansions with Positivity Constraints.

Gram-Charlier expansion have become popular in Finance as a generalization over the normality assumption. Even though Gram-Charlier expansions allow for a certain flexibility over skewness and kurtosis they have the unfortunate drawback of sometimes yielding negative densities. The goal of this paper is to show how positivity constraints can be numerically inplemented. We successfully apply our method to the estimation of risk neutral densities that arise within a financial option pricing context.