# 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?

#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?

• What is the underlying? On average, the payoff (realized minus implied variance) is negative. In the time series, you should see the occasional spikes where this position should pay off. You should add how you've come up with your $RV$ and $IV$, though: Are you talking about realized vs. implied variation (think: variance swap) or are you talking about the realized vs. implied (logarithmic) change in the underlying value, i.e. $E^P_t(\left(\ln(S_{t+\Delta t}\right)^2)-\ln(S_t))$ vs $E^Q_t(\left(\ln(S_{t+\Delta t})-\ln(S_t)\right)^2)$? Aug 6, 2021 at 8:19
• Also: I would commonly define the Variance Risk Premium as the difference in expectation, i.e. $E^P(\cdot) - E^Q(\dot)$. Would you agree? Aug 6, 2021 at 11:55
• Agree with both comments above, noting in passing that the sign on the ex-ante RP (ie yields) will generally be opposite to that of the ex-post RP (ie returns)... above implies a tempral mismatch between IV and RV (given the obvious year-end seasonality). Aug 6, 2021 at 21:38
• @Kermittfrog this is the VIX and SPX realized vol (yang zhang). Do you think on average SPX has the highest variance risk premia since it is the asset everyone uses to hedge? Aug 8, 2021 at 6:37