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.
Risk-neutral probabilities, observable from option prices, combine objective probabilities and risk adjustments across economic states. We consider a recursive-utility framework to separately identify objective probabilities and risk adjustments using only observed market prices. We find that a preference for early resolution of uncertainty plays a key role in generating sizeable risk premia to explain the cross-section of risk-neutral and objective probabilities in the data. Failure to incorporate a preference for the timing of the resolution of uncertainty (e.g., expected utility models) can significantly overstate the implied probability of, and understate risk compensations for, adverse economic states.
In the data, asset prices exhibit large negative moves at frequencies of about 18 months. These large moves are puzzling as they do not coincide, nor are they followed by any significant moves in the real side of the economy. On the other hand, we find that measures of investor’s uncertainty about their estimate of future growth have significant information about large moves in returns. We set-up a recursive-utility based model in which investors learn about the latent expected growth using the cross-section of signals. To model the learning of the agents about the unobserved expected growth, we specify a belief-updating model (Kalman Filter is a special case) which incorporates the recency bias of investors in forecast formation. The uncertainty (confidence measure) about investor’s growth expectations, as in the data, is time-varying and subject to large moves. In the confidence risks model, recency bias in conjunction with confidence risk fluctuations lead to large moves in asset prices. In calibrations we show that the model can account for the large return move evidence in the data, distribution of asset prices, predictability of excess returns and other key asset market facts.