We also consider the structural approaches of Malz respectively Heston who assume a jump-diffusion or a stochastic volatility model for the underlying process.
Methods based on polynomial expansions have difficulties to describe the exchange rate data at hand.
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.