Volatility-of-Volatility Risk

We show that time-varying volatility of volatility is a significant risk factor which affects both the cross-section and the time-series of index and VIX option returns, above and beyond volatility risk itself. Volatility and volatility-of-volatility movements are identified in a model-free manner from index and VIX option prices, and correspond to the VIX and VVIX indices in the data. The VIX and VVIX have separate dynamics and are only weakly related in the data. Delta-hedged returns for index and VIX options are negative on average, and are more negative for strategies which are more exposed to volatility and volatility-of-volatility risks. In the time series, volatility and volatility of volatility significantly predict delta-hedged returns with a negative sign. The evidence in the data is consistent with a no-arbitrage model which features time-varying market volatility and volatility-of-volatility factors which are priced by investors. In particular, volatility and volatility of volatility have negative market prices of risk, so that investors dislike increases in volatility and volatility of volatility.