The transformed Gram Charlier distribution: Parametric properties and financial risk applications

In this paper we study an extension of the Gram-Charlier (GC) density in Jondeau and Rockinger (2001) which consists of a Gallant and Nychka (1987) transformation to ensure positivity without parameter restrictions. We derive its parametric properties such as unimodality, cumulative distribution, higher-order moments, truncated moments, and the closed-form expressions for the expected shortfall (ES) and lower partial moments. We obtain the analytic kth order stationarity conditions for the unconditional moments of the TGARCH model under the transformed GC (TGC) density. In an empirical application to asset return series, we estimate the tail index; backrest the density, VaR and ES; and implement a comparative analysis based on Hansen's skewed-t distribution. Finally, we present extensions to time-varying conditional skewness and kurtosis, and a new class of mixture densities based on this TGC distribution.

Automated summary

This paper extends the GC density and ensures positivity through a transformation. It investigates the parametric properties of the density and conditional properties under the TGARCH model. In an empirical application, it estimates tail index, reestimates density, VaR and ES, and conducts a comparative analysis.