Non normal distributions are a fact of life. In the financial world, many distributions display tail risk, i.e. (negative) skewness and excess kurtosis. Being able to model such risk is useful in various and important fields: risk measurement, fund management performance evaluation, asset pricing…
One way to model tail risk is to introduce discontinuities, such as jumps, to describe the distribution of values or returns. It is however possible, and often convenient, to model tail risk in a continuous space.
Both Cornish-Fisher and Gramm-Charlier expansions (which is the simple form of a family of Edgeworth 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.
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