Gram–Charlier-Like Expansions of the Convoluted Hyperbolic-Secant Density

Abstract

Since financial series are usually heavy tailed and skewed, research has formerly considered well-known leptokurtic distributions to model these series and, recently, has focused on the technique of adjusting the moments of a probability law by using its orthogonal polynomials. This paper combines these approaches by modifying the moments of the convoluted hyperbolic secant. The resulting density is a Gram–Charlier-like (GC-like) expansion capable to account for skewness and excess kurtosis. Multivariate extensions of these expansions are obtained on an argument using spherical distributions. Both the univariate and multivariate (GC-like) expansions prove to be effective in modeling heavy-tailed series and computing risk measures.