Reshaping the Linear-Hyperbolic density by its orthogonal polynomials to embody possibly severe kurtosis and skewness

The empirical distributions of many financial asset returns and stock indexes present a leptokurtic and skewed shape which goes along with accentuated peakedness, fat tails and slimness of shoulders and a peak turned toward the longer tail. As in this case the celebrated Gaussian law fails to provide a valuable paradigm, research has first moved outside, looking at Gaussian-like leptokurtic distributions, and only recently has run back over the idea of reshaping the parent (Gaussian) density from “inside”, via (Hermite) orthogonal polynomials. In this paper the two approaches are combined to investigate the linear-hyperbolic (LH) leptokurtic distribution when it is reshaped by means of “its” orthogonal polynomials. The features of the parent and modified LH distributions are investigated from both theoretic and empirical-evidence standpoints