This paper introduces the effect of the crossed products of Hermite polynomials on Gram-Charlier densities. This allows capturing the impact of the interaction between skewness and kurtosis and evaluating this new parameter as an additional source of information for risk management. We show that our modified Gram-Charlier density presents an improved accuracy, especially at distribution tails. Risk quantification is assessed for S&P500 losses with backtesting procedures for Value-at-Risk and Median Shortfall.
