Taming the Skew: Higher-Order Moments in Modeling Asset Price Processes in Finance

It is widely acknowledged that many financial markets exhibit a considerably greater degree of kurtosis (and sometimes also skewness) than is consistent with the Geometric Brownian Motion model of Black and Scholes (1973). Among the many alternative models that have been proposed in this context, two have become especially popular in recent years: models of jump-diffusions, and models of stochastic volatility. This paper explores the statistical properties of these models with a view to identifying simple criteria for judging the consistency of either model with data from a given market; our specific focus is on the patterns of skewness and kurtosis that arise in each case as the length of the interval of observations changes. We find that, regardless of the precise parameterization employed, these patterns are strikingly similar within each class of models, enabling a simple consistency test along the desired lines. As an added bonus, we find that for most parameterizations, the set of possible patterns differs sharply across the two models, so that data from a given market will typically not be consistent with both models. However, there exist exceptional parameter configurations under which skewness and kurtosis in the two models exhibit remarkably similar behavior from a qualitative standpoint. The results herein will be useful to empiricists, theorists and practitioners looking for parsimonious models of asset prices.