Reshaping the Linear hyperbolic density by its orthogonal polynomials to embody possibly severe kurtosis

The empirical distributions of many financial asset returns and stock indexes show a leptokurtic and skewed shape with shifts in the probability clusters associated with accentuated peakedness, fat tails, slimness of shoulders and a peak turned toward the longer tail. Since the familiar Gaussian law failed to provide a valid paradigm in these cases, research initially looked elsewhere, examining Gaussian-like leptokurtic distributions. However, recent research has turned to the idea of reshaping the parent (Gaussian) density from “inside”, by using specific (Hermite) orthogonal polynomials. This paper combines the two approaches to investigate the linear-hyperbolic (LH) leptokurtic distribution which is conveniently reshaped by means of its own orthogonal polynomials. The features of the parent and modified LH distributions are investigated from both analytic and empirical-evidence standpoints.