Tailoring the Logistic Distribution to Fit the Empirical Distribution of Financial Asset Returns

Abstract

Risk measures, including Value-at-Risk (VaR) and Conditional VaR (Expected Shortfall), turn out to be quite sensitive to the degree to which distributions are thick tailed and asymmetric. Lack of encoding information about asymmetry and leptokurtosis is a well-known drawback of the Gaussian law. This has led to a search for alternative distributions (Szego (2004)). In this paper, we will tackle the issue of accounting for asymmetry, (possibly severe) excess kurtosis and dependence by following the alternative approach of adjusting bell-shaped distributions using orthogonal polynomials as shape adapters. Our focus will be on polynomial transformations of parent symmetric probability density functions to match the empirical moments of target distributions characterized by possibly substantial heavy-tails and asymmetry. We will demonstrate a simple but powerful novel result, i.e. that the function that achieves the transformation from a given parent to a target distribution, depends on both the orthogonal polynomials associated to the former and the moment-differentials between these two distributions. We will then apply this result to the modelling of heavy-tailed and skewed distributions of financial asset returns.