How is the variance risk premium different from the volatility risk premium?

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AcadiFi · Accepted Answer

The variance risk premium (VRPvar) is the difference between the variance swap rate K_var and expected realized variance under the physical measure. The volatility risk premium (VRPvol) is the analogous quantity in vol units. Because volatility is the square root of variance and square root is concave, Jensen's inequality gives E[sqrt(var)] < sqrt(E[var]). That means a variance swap trades strictly above the at-the-money implied volatility squared, creating a convexity adjustment that matters for hedging.

Consider Riverstone Systematic running a short-variance program. They receive K_var = 400 (implied vol 20) and pay realized variance. If realized variance comes in at 324 (vol 18), the vol-space P&L looks like 2 points, but the variance-space P&L is 76 variance points. A single 40-vol spike day can swamp months of quiet carry because variance is quadratic in returns.

A worked contrast: in March 2020, 1-month VIX reached 82 while realized vol ran near 75 for the tail period — so the VRP in vol space briefly turned negative. But variance-space P&L was catastrophically negative for sellers because realized variance (5625) dwarfed recent swap strikes (400-900). For the FRM exam: remember that the variance swap strike equals a specific integral of out-of-the-money option prices, which is exactly how the VIX index is constructed.

How is the variance risk premium different from the volatility risk premium?

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