Studying dynamics in risk-neutral skewness using a Gauss-Hermite expansion on S&P 500 index options

Abstract: Building on a new method of pricing options by modelling the underlying risk-neutral distribution with 'physicist' Hermite polynomials, we assess the properties of these distributions over time. We employ a set of S&P 500 index options ranging from 2007 to 2016. First, we provide a detailed overview of the estimation process for Gauss-Hermite risk-neutral densities. Second, we derive a simple method of directly expressing risk-neutral skewness and kurtosis from the estimated parameters of the distributions. Owing to high correlations between these moments, we focus on skewness in our analysis. We also highlight some problems associated with the tail-shape of Gauss-Hermite risk-neutral densities extracted from real-world option quotes. By careful filtering we can however remove those problematic densities. Third, we create a time-series index of skewness, which we find to be closely linked to the CBOE SKEW index. Fourth, we find that risk-neutral skewness is a mean-reverting, volatile and auto- correlated time series. However, the variation in skewness over time cannot be sufficiently modelled by autocorrelation or using returns of the underlying. Also, while we do find evidence that time-varying skewness is positively related to out-of-sample pricing errors, a large portion of the variation of risk-neutral densities remains unexplained. We conclude that while Gauss-Hermite risk-neutral densities are sufficiently well behaved to study for risk-neutral moments given appropriate filtering, our analysis suggests that they might not contain more information than other methods used in this context.