Cornish-Fisher versus Gramm-Charlier: An Appraisal

Both Cornish-Fisher and Gramm-Charlier expansions are means to transforming a Gaussian distribution into a non-Gaussian distribution, the skewness and the kurtosis of which can be controlled if the transformations are properly implemented. This may be useful for modelling distributions for a wide range of issues, especially in risk assessment and asset pricing. The expansions differ in their nature: Cornish-Fisher is a transformation of a random variable, or of quantiles, meanwhile Gramm-Charlier is a transformation of a probability density. Both transformations must be implemented with care, as their domain of validity does not cover the whole range of possible skewnesses and kurtosis. It appears that the domain of validity of Cornish-Fisher is much wider that the domain of validity of Gramm-Charlier. This, and the fact that Cornish-Fisher provides easily the quantiles of the distribution, gives it an advantage over Gramm-Charlier in several configurations.