Dynamic selection of Gram-Charlier expansions with risk targets: an application to cryptocurrencies

This paper implements a procedure for dynamically selecting the Gram-Charlier approximation that best fits the empirical distribution of cryptocurrency returns at any point in time. The endogenous selection of the Gram-Charlier expansion length exploits its property for approximating frequency distributions through a flexible number of parameters that allows capturing changes at the tails provoked by new extreme events. The procedure is based on the differences between the cumulative distribution function of Gram-Charlier distributions with a particular focus on the fitting of the distribution left tail for risk assessment purposes. The method is tested through backtesting techniques for a group of major cryptocurrencies. The results show that the selection of the Gram-Charlier expansion order on the basis of cumulative distribution function dynamics, provides, in most cases, a significant improvement for conditional coverage compared to the use of fixed-order Gram-Charlier expansions. The method seems to be a useful tool for risk management purposes, especially for highly volatile assets such as cryptocurrencies.

Automated summary

This study implements a procedure for dynamically selecting the Gram-Charlier approximation that best fits the empirical distribution of cryptocurrency returns and tests it using backtesting techniques. The results demonstrate that dynamic selection of the Gram-Charlier expansion order can significantly improve conditional coverage compared to fixed-order Gram-Charlier expansions.