Choosing Time Windows When Estimating the Variance Risk Premium
Summary
The document contrasts two ways to calculate a variance risk premium: subtracting realized variance over the same future interval covered by implied variance, or subtracting variance realized over the preceding interval. The answer explains that there is no universally correct choice without specifying the research question. Implied variance is an ex-ante risk-neutral expectation, while realized variance is one ex-post outcome, so a single observation cannot isolate the premium from forecast error and path-specific noise.
Long-run averages can reduce the influence of any one realized path. Comparing current implied variance with past realized variance instead treats recent past volatility as a proxy for future volatility. That assumption can be plausible when volatility is persistent across regimes, but it may fail around future events, mean reversion, or broader changes in conditions. The two window choices therefore serve different empirical purposes and rely on different assumptions; the document does not prescribe one estimator for every application.
Key ideas
- Implied variance is an ex-ante risk-neutral expectation, whereas realized variance records an ex-post path.
- Comparing implied and realized variance over matched future intervals combines expectation with forecast error and path-specific outcomes.
- Long-term averaging can reduce dependence on any one realized path.
- Using past realized variance as a proxy for future variance assumes some persistence in volatility.
- Events, mean reversion, and changing conditions can weaken that proxy assumption.
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# Confusion about time horizons for components of the variance risk premium
# Confusion about time horizons for components of the variance risk premium
My original understanding of how to compute the variance risk premium is that it is given by the following
$$VRP_{t,t+\Delta t} = IV_{t,t+\Delta t} - RV_{t,t+\Delta t}$$
such that the realized volatility must be "pulled back" to make the quantities contemporaneous.
This agreed upon with another post previously posted in Quant SE: Computing the Variance Risk Premium
And the above makes sense logically...
However, this is contradicted by Bollerslev et al. (2009, RFS) "Expected Stock Returns and Variance Risk Premia", whom states that the variance risk premium is given by:
$$VRP_{t,t+\Delta t} = IV_{t,t+\Delta t} - RV_{t-\Delta t,t}$$
Main Question: Why are the periods for the risk-neutral expectation and true realized variances not contemporaneous when being used to calculate the variance risk premium? Which method is correct? Or are they just different methods used?
## Answer by Soumirai (score 1, accepted)
https://quant.stackexchange.com/a/80674
Simple answer is, there is no right answer. IV is market expectation of future RV, plus some risk premium. When you compare past IV to past RV, you compare an ex-ante expectation to one ex-post realized path, among many other possible. So you can't pinpoint the ex-ante risk premium component on that single path. But you can do long term averages to average out the singleness of each path. When you compare current IV to past RV, you make the strong assumption that past RV is a good proxy for future RV, which partly makes sense (vol works in regimes and recent behavior is likely to persist in the near future), and partly doesn't (ignores future events, mean reversion, and the fact that the world changes)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.