Higher Order Greeks

Abstract

We examine whether any second order derivatives other than gamma and any third order derivatives are important in explaining changes in the prices of S&P 500 futures options over one week holding periods. We find that while gamma is normally most important, several other higher order derivatives have considerable incremental explanatory power. Particularly important in accounting for option price changes are the derivatives of delta with respect to volatility and time-to-expiration, and the derivatives of gamma with respect to the asset price, and volatility. The first three are more important for away-from-the-money options, while the fourth is most important for at-the-money options. For shorter-term options, consideration of higher order derivatives reduces the mean absolute unexplained price change by 60% for at-the-money options and by at least 75% for away-from-the-money options. We find that in spite of its theoretical problems and inability to explain the cross-sectional option price pattern (the smile), the Black-Scholes model's Greeks accurately describe the time series option price changes once higher order Greeks are incorporated. We further find that making delta-gamma-vega neutral portfolios of S&P 500 options neutral in terms of these four higher order Greeks leads to a substantial reduction in the risk of an unhedged price change.