The volatility risk premium compensates investors for taking volatility risk. Conceptually it is based on the difference between options-implied and expected realized volatility. In equity markets this premium should be positive in the long run and fluctuate overtime depending on the market’s willingness to pay for protection against future changes in price volatility. In practice, measuring the premium overtime is challenging, particularly because expected realized volatility is not known. Using recent realized volatility as a proxy can be highly misleading. However, a realistic estimate can be constructed by considering the trade-off between timeliness and noise ratio of recent price changes and the long-term mean reversion of volatility. This “realistic” volatility risk premium has been positively correlated with subsequent daily volatility index future returns.

The post ties in with SRSV’s summary lecture on implicit subsidies, particularly its final section on volatility markets.

The post is based on proprietary research notes of Macrosynergy Partners and SRSV Ltd.

### What is a volatility risk premium?

The volatility risk premium is __compensation for bearing the price volatility risk of a financial contract__. Technically-speaking it is defined as defined as the difference between risk-neutral (options-implied) and physical expectations of returns variation (view post here) .If the aggregate of risk-averse investors is long the underlying contracts, as is the case in equity markets, this premium should on average be positive, manifesting as a long-term gap between options-implied volatility and realized returns variance.

Generally, the volatility risk premium is estimated based on some form of difference between implied and expected realized volatility. __The premium should fluctuate overtime, being high in times of elevated volatility risk aversion and low when investors are less interested in protection__. The better these fluctuations can be measured, the more suitable the volatility risk premium would be as guidance for managing risk exposure to volatility: a high positive premium would support short volatility positions; a negative premium would support long volatility positions.

N.B.: A short volatility position is a bet on falling volatility, for example by selling call and put options or shorting a volatility index future. A long volatility position is a bet on rising volatility.

### What is a realistic variance risk premium?

In research papers __the variance risk premium is typically calculated as the difference between current implied volatility, and (recent) historic realized price__ volatility. The underlying assumption is that recent realized volatility is a good forecast for realized volatility in the near future.

The choice of the lookback horizon is critical. A __very short lookback, such as a day, produces an excessive noise to signal ratio__ for realized volatility predictions. For example, intra-day price changes are heavily affected by specific circumstances of that specific day, such important data releases, policy announcements or month-end flows. On the other hand, a __long lookback, such as a month or a quarter causes severe time inconsistencies__ that compromise related volatility risk premia as a trading signal in many important situations:

- Consider first the example of a risk shock, i.e. a negative event that greatly increases actual uncertainty and fear in the market. In this case the implied volatility adjusts immediately to the new state of perceived risk, while the realized volatility will take a number of days or even weeks to reflect it. During this adjustment period the volatility risk premium will be inflated, wrongly suggesting that investors should be short volatility.
- A similar problem arises when volatility is trending. Say market uncertainty is gradually subsiding. Then the implied volatility will fall ahead of the measured realized volatility. Measured volatility risk premia will be negative, wrongly inducing investors to go long volatility.
- Finally, when volatility is extremely high or low, recent realized volatility ignores the mean-reverting tendency of volatility and forecasts that are based only on the recent past will overstate expectations in the high-vol state (and understate the volatility risk premium) and understate expectations in the low-vol state (and overstate the volatility risk premium).

All these examples illustrate that __it is essential that realized volatility forecasts for the purpose of volatility risk premia [1] maximize time consistency with the implied volatility measure and [2] consider the time series properties of volatility__. As to [1] complete time consistency is not achievable but short exponential lookback windows come close to being realistic proxies of the current state of market uncertainty. As to [2] some form of modelling of both short- and long-term factors of volatility is of the essence.

The below chart shows implied volatility for S&P500 and Eurostoxx versus a realistic predicted realized volatility (short exponential lookback and adjustment for reversion to a medium-term trend) at daily frequency since 2000. This measure can be updated in real time. It displays two important characteristics:

__In normal times implied volatility stays above realized volatility__, consistent with a positive volatility risk premium- In times of risk shocks implied volatility does
*not*systematically overshoot predicted realized volatility. Similarly, in times of volatility trends the predicted realized volatility does not meaningfully lag the implied volatility.__On balance the predicted realized volatility does seem to catch up with market changes sufficiently quickly__.

### How have realistic variance risk premia evolved since 2000?

We calculate the volatility risk premium as the __difference between implied volatility and our “realistic” predicted realized volatility, as percent of concurrent implied volatility__. Its pattern shows a stationary positive premium in the medium term with regular spikes upward and very deep rare spikes downward.

__The mean of the realistic volatility risk premium since 2000 has been 11% of implied volatility, with a standard deviation of roughly 15%-points__. The empirical distribution has displayed fat tails (relative to normal distribution) and a downside skew. That means on rare occasions the premium has gone deeply negative. There has been a lot of short-term variation, plausibly reflecting both actual changes in risk aversion and measurement issues.

__When it comes to financial crisis events the realistic volatility risk premium shows incidences of both denial and fear__. Denial here means that implied volatility refuses to catch up with the increase in price variance and the volatility risk premium becomes negative. Fear here means that implied volatility surges ahead of realized volatility, indicating investors elevated preference for protection. However, __premia paid for protection, measured as the relative surcharge on current implied volatility has not been generally higher in crisis periods than in normal times__., suggesting that volatility markets have not become “headless” or “unhinged”, but rather scaled their premia to the new level of uncertainty.

### How well have realistic variance risk premia predicted returns?

In a most simple analytical setting, __the statistical probability of positive daily correlation of realistic volatility risk premia with subsequent returns of short-volatility positions through VIX futures since 2000 (and VSTOXX since 2009) has been near 100% on a daily basis__. The balanced accuracy (average of positive and negative hit ratio in respect to next-day returns) of the premia has been 50.7% for the U.S. market. More importantly, the premium served well in protecting a short-volatility strategy from outsized drawdowns. Most large volatility surges occurred at times when the volatility risk premium was either negative or small. By nature, a volatility risk premium indicator cannot predict volatility shocks, but it can detect situations of potential denial and incomplete adjustments to a changed volatility regime.

In contrast to the volatility index futures, the statistical probability of positive daily correlation of realistic volatility risk premia with subsequent equity index future returns has only been 96% on a daily basis for S&P500 and EuroStoxx. The balanced accuracy of directional predictions has only been marginally above 50%, not enough to produce convincing value in a rules-based strategy.