Estimating the Variance Risk Premium from Implied and Historical Densities

Summary

The document explains how to estimate a variance risk premium when the available inputs are probability densities rather than volatility estimates. It defines the premium, loosely, as the difference between objective and risk-neutral variance of future returns, while noting that only the risk-neutral variance is directly observed at a given time. The proposed historical estimate relies on two assumptions: investors have been correct on average about future variance, and the premium is stationary.

For each date in the sample, the method calls for an implied density and a historical density, each expressed over returns by converting from price levels using that date’s spot price. Calculate each density’s variance by integrating across its domain, take the difference, and average those differences through time. This gives a density-based analogue of comparing implied and realized volatility. The answer is a procedure rather than an empirical demonstration; the assumptions about average forecasting accuracy and stationarity are material limits, and no particular density-estimation method is prescribed.

Key ideas

• The variance risk premium compares objective and risk-neutral variance of future returns.
• Only the risk-neutral variance is directly observable at the valuation time in the setup described.
• Estimate the premium by comparing implied and historical density variances across dates.
• Convert price-domain densities to return-domain densities using the spot price for each date.
• Interpreting the historical average difference depends on assumptions of forecast accuracy on average and premium stationarity.

Tags

# How to quantify the Variance Risk Premium (VRP) with probability density functions?


# How to quantify the Variance Risk Premium (VRP) with probability density functions?












The VRP is usually displayed by charts like this one:

It's easy to see that, for most of the time, options are priced by using volatility which will reveal itself larger than the realized one. So VRP is simply the arithmetic difference between implied (or model-free) volatility and realized volatility.

However, I'm wondering what's the best way to measure and quantify VRP when we have density functions instead of volatility measures. In the following case, for example, we have two arrays with probability densities and an array with strike prices:

How would you quantify the VRP?

## Answer by Igor Pozdeev (score 3, accepted)

https://quant.stackexchange.com/a/46881

First, VRP is (loosely speaking) the difference between the implied and objective variance of future returns:

$VRP_t = Var_t^P[R_{t+1}] - Var_t^Q[R_{t+1}]$,

of which only the second, risk-neutral variance is observed at time $t$. Assuming that (1) investors have been correct on average about the future variance, and that (2) the premium is stationary, one can quantify the magnitude of VRP by taking the difference of these historical averages you are talking about.

With that said, the way to do the same with the densities is to:

- have an implied and historical density for each time period in your sample;

- convert the domain of each to returns using the spot price at that date;

- calculate the variances as the integral over the domain;

- average the differences.

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.