Optimal portfolio positioning within generalized Johnson distributions

Abstract

Many empirical studies have shown that …nancial asset returns do not always exhibit Gaussian distributions. For example, it is well known that most hedge fund returns have signi…cant skewness and kurtosis, which are far from being those of a normal distribution. The introduction of the family of Johnson distributions allows to overcome the problem of …tting empirical …nancial data, since it contains a large class of probability distributions. Additionally, this class can be extended to a quite general family of distributions by considering all possible regular transformations of the standard Gaussian law. In this framework, we consider the portfolio optimal positioning problem, which has been …rst addressed by Brennan and Solanki (1981) and by Leland (1980). We assume that the investor wants to maximize the expected utility of his portfolio value, which corresponds to a given function of some speci…c portfolio of common risky assets. Such problem has been previously examined for Gaussian log returns (see Carr and Madan, 2001;Prigent, 2006). We determine and analyze the optimal portfolio for log return having Johnson distributions. As a by-product, we introduce the notion of Johnson stochastic processes. The solution is characterized for arbitrary utility functions and illustrated in particular for a CRRA utility. Our …ndings show how the pro…les of …nancial structured products must be selected when taking account of non Gaussian log-returns.