Options-implied volatility of U.S. equity prices is measured by the volatility index, VIX. Options-implied volatility of volatility is measured by the volatility-of-volatility index, VVIX. Importantly, these two are conceptually and empirically different sources of risk. Hence, there should also be two types of risk premia: one for the uncertainty of volatility and for the uncertainty of variation in volatility. The latter is often neglected and may reflect deep uncertainty about the structural robustness of markets to economic change. A new paper shows the importance of both risk factors for investment strategies, both theoretically and empirically. For example, implied volatility and “vol of vol” typically exceed the respective realized variations, indicating that a risk premium is being paid. Also, high measured risk premia for volatility and “vol-of-vol” lead to high returns in investment strategies that are “long” these factors.

*The post ties in with SRSV’s lecture on risk premia and implicit subsides.
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*The below are excerpts from the paper. Emphasis and cursive text have been added.*

### Implied volatility and implied “vol of vol”

“[*Equity*] market volatility is often captured by the volatility index (VIX). Calculated in real time from the cross-section of S&P500 option prices, __the VIX index provides a risk-neutral forecast of the index volatility over the next 30 days__. The VIX index exhibits substantial fluctuations, which in the data and in many economic models drive the movements in asset prices and the risk premia.”

“Computed from VIX options in an analogous way to the VIX, __the volatility-of-volatility index (VVIX) directly measures the risk-neutral expectations of the volatility of volatility in the financial markets__. In the data, we find that the VVIX has separate dynamics from the VIX, so that fluctuations in volatility of volatility are not directly tied to movements in market volatility.”

“[*The figure below*] shows the time-series of the VIX and VVIX from February 2006 to December 2016. There are some common prominent moves in both series, such as the peak during…the financial crisis. Notably, however, the VVIX also peaks during other times of economic uncertainty, such as the summer of 2007 (quant meltdown, beginning of the subprime crisis), May 2010 (Eurozone debt crisis, flash crash), August 2011 (U.S. debt ceiling crisis), and August 2015 (sell-off driven by the Chinese stock market crash). The movements in VIX during these events are far smaller than the spikes in the VVIX. The suggests that the __VVIX captures important uncertainty-related risks in the aggregate market, distinct from the VIX itself__.”

### Model view: why vol and “vol of vol” matter”

“Time-varying…volatility is a significant risk factor…Volatility-of-volatility risks are [*also*] a significant risk factor which affects the time-series and the cross-section of index and VIX option returns, above and beyond volatility risks.”

“Our no-arbitrage model…for stock index returns, as well as for equity and volatility option prices…extends the one-factor stochastic volatility specification of equity returns… Specifically, __we introduce a separate time-varying volatility-of-volatility risk factor which drives the conditional variance of the variance of market returns__. We use the model to characterize the payoffs to delta-hedged equity and VIX options.”

“We show that __the expected payoff on the delta-hedged index option positions capture risk compensations for both volatility and volatility-of-volatility risks, while for VIX options, the expected gains primarily reflect the compensation for volatility-of-volatility risk__….We can further decompose the risk premia components of the expected gains on the index options into the product of the market price of risk, the risk exposure, and the time-varying quantity of each source of risk….The two risk compensations are given by the product of the market price of risk, the exposure of the asset to the corresponding risk, and the quantity of risk. In particular, options are positive-beta assets with respect to both volatility and volatility-of-volatility risks.”

“The model delivers clear, testable predictions for expected option returns and their relation to volatility and volatility-of-volatility risks.

- In the model,
__if investors dislike volatility and volatility of volatility, so that the market prices of these risks are negative__, delta-hedged equity and VIX option gains are negative on average. - In the cross-section
__, the average returns are more negative for option strategies which have higher exposure to the volatility and volatility-of-volatility risks__. - Finally, in the time series,
__higher volatility and volatility of volatility predict more negative delta-hedged option gains__in the future.”

### Empirical evidence

“The evidence is consistent with a no-arbitrage model which features time-varying market volatility and volatility-of-volatility factors which are priced by the 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, and demand a risk compensation for the exposure to these risks.”

“All of our variables are at the monthly frequency. The implied variance measures are given by the index values at the end of the month, and the realized variance measures are calculated over the past month and annualized. The __sample for the benchmark measures runs from February 2006 to December 2016__…We…compute the realized volatilities for the stock market and the VIX….__Realized variance is defined as the sum of squared high-frequency log returns__ over the trading day…We follow the standard approach of considering 5 minute return intervals…The 5 minute realized variance is very accurate, difficult to beat in practice, and is typically the ideal sampling choice in most applications combining accuracy and parsimony.”

“We find that the model implications for volatility and volatility-of-volatility risks are strongly supported in the cross-section of index and VIX option returns.

- The empirical evidence suggests that
__fluctuations in the volatility of volatility are not directly related to the level of the volatility itself__. This is consistent with our two-volatility model specification. - Using predictive regressions, we show that the VIX is a significant predictor of the future realized variance of market returns, while the
__VVIX significantly forecasts future realized variation in the VIX index__ - On average, the risk-neutral volatilities of the market return and market volatility captured by the VIX and VVIX exceed the realized volatilities of returns and the VIX. The
__difference between the risk-neutral and physical volatilities of market returns is known as the variance premium__(variance-of-variance premium for the VIX), and the findings of positive variance and variance-of-variance premium suggest that investors dislike variance and variance-of-variance risks, and demand a premium for being exposed to these risks.

- The average delta-hedged returns on out-of-the-money equity index calls and puts are significantly negative in our sample…The
__negative average returns on index and VIX options directly suggest that the market prices of volatility and volatility-of-volatility risks are negative__. - Empirically, we document that
__average option returns are significantly and negatively related to our proxies for volatility and volatility-of-volatility risks__. Hence, using the cross-section of equity index options and VIX options, we find strong evidence for a negative market price of volatility and volatility-of-volatility risks. - Finally, we consider a predictive role of our volatility measures for future option returns. In the model, expected delta-hedged gains are time-varying and are driven by the volatility and volatility of volatility (by volatility of volatility for VIX options). In particular, as option betas are all positive, when the market prices of volatility-related risks are negative,
__both volatility measures should forecast future returns with a negative sign. This model prediction is consistent with the data__.”