Defining the Variance Risk Premium from Implied and Realized Variance

Summary

This note defines a variance risk premium over a future interval as realized variance minus implied variance for that same interval, with implied variance measured at the interval’s start. It emphasizes aligning the realized measure with the period that was implied, using a lag where needed. The supplied example starts from implied and realized volatility observations and computes the difference between their squared values; the displayed premium values are negative.

The author asks whether this calculation is correct and how to convert it into a volatility risk premium, including whether an absolute value followed by a square root is appropriate. The document provides no answer to those questions, so it does not establish a valid conversion from variance differences to a volatility premium. In particular, it gives no method for transforming the sign or interpreting the square root of a negative variance difference.

Key ideas

• The stated variance risk premium is realized variance minus implied variance over the same future interval.
• Implied variance is measured at the beginning of the interval being evaluated.
• Volatility observations must be squared to form variances before taking their difference.
• The document raises but does not resolve how to convert the variance premium into a volatility premium.

Tags

# Computing the Variance Risk Premium


# Computing the Variance Risk Premium












The Variance Risk Premium (VRP) is defined as:

$$VRP(t,t+\Delta t) \equiv RV(t,t+\Delta t)^2 - IV_t(t,t+\Delta t)^2$$

where $RV^2$ is the realized variance between $t$ and $t + \Delta t$ and $IV_t^2$ is the implied variance for the same period, implied as of $t$ (i.e. at the beginning of that period). We also lag RV back to line up with what was being implied.

I am unsure if I am getting the correct output. Below is the data, transformations and graph. Could someone confirm i am getting the desired output.

Secondly, if I would like to get the volatility risk premium, what is the conversion i would need to do?

Thank you in advance.

#The Data below is in volatility

```
tail(df)
               IV     RVt_1          VRP
2021-06-28 0.1576 0.1197841 -0.010489536
2021-06-29 0.1602 0.1202760 -0.011197727
2021-06-30 0.1583 0.1225323 -0.010044727
2021-07-01 0.1548 0.1243042 -0.008511502
2021-07-02 0.1507 0.1257307 -0.006902284
2021-07-06 0.1644 0.1242254 -0.011595423
```

Where VRP = RVt_1^2 - IV^2. RV and IV in the database are in vols - Realized vols and implied vols

#Plot of the graph over time.

Lastly here are some of the stats.

```
mean(df$VRP) -0.005147035

median(df$VRP)-0.01194805
```

If I wanted to convert these to vols, can i take the absolute value before taking the square root?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.