Scaling issues for risky asset modelling

Abstract

In this paper we investigate scaling properties of risky asset returns and make a strong case (1) against the need for multifractal models and (2) in favor of the requirement of heavy tailed distributions. Amongst the standard empirical properties of risky asset returns are an autocorrelation function for the returns which dies away rapidly and is statistically insignificant beyond a few lags, and also autocorrelation functions of squares and absolute values of returns which die away very slowly, persisting over years, or even decades. Together these indicate that, assuming returns come from a stationary process, they are not independent, but at most short-range dependent, while various functions of the returns are long-range dependent. These scaling properties are well known, although commonly ignored for modeling convenience. However, much more can be inferred from the scaling properties of the returns. It turns out that the empirical scaling functions are initially linear and ultimately concave, which is strongly suggestive of returns distributions with infinite low order moments or alternatively that multifractal behavior is a modeling requirement. Modifications of the commonly used models cannot readily meet these requirements. The evidence will be presented and its significance discussed, along with a class of models which can incorporate the empirically observed features.