9.4 The variance risk premium

$E^{\mathbb Q}_t[RV]$ is the variance-swap rate, which 9.6 shows is observable from option prices. Carr and Wu (2009) use the opposite sign, $RV-SW$, so in their convention the premium is negative. The ex-post realised premium is $SW_t-RV_{t,T}$.

9.4.1The sign of the premium#

Proof

$E^{\mathbb P}[M\,RV]=E^{\mathbb P}[M]E^{\mathbb P}[RV]+\operatorname{Cov}(M,RV)$. Divide by $E^{\mathbb P}[M]$ and subtract $E^{\mathbb P}[RV]$.∎

$M$ is high in bad states (marginal utility is high when wealth has fallen). Realised variance is also high in bad states: equity volatility rises when prices fall (the leverage effect, M02). So $\operatorname{Cov}(M,RV)>0$ and the premium is positive. Selling variance means selling a payoff that pays out exactly when investors most value money. A positive average return to selling it is compensation for that, not an inefficiency.

9.4.2Evidence from the US and India#

Evidence

• Carr and Wu (2009), "Variance Risk Premiums", Review of Financial Studies 22(3): 1311–1341. They synthesise variance-swap rates from option prices (the replication of 9.6) for 5 stock indexes and 35 individual stocks, and compare them with subsequent realised variance. Average realised-minus-swap premiums are strongly negative for the S&P 500 and Nasdaq-100 indexes, and the premiums are not explained by the classical equity factors (market beta, size, value, momentum).
• Bakshi and Kapadia (2003), "Delta-Hedged Gains and the Negative Market Volatility Risk Premium", RFS 16(2): 527–566. Delta-hedged long S&P 500 option positions lose money on average, which is (9.11) with $\sigma_i>\sigma_r$ on average.
• Coval and Shumway (2001), "Expected Option Returns", Journal of Finance 56(3): 983–1009. Zero-beta (crash-neutral) straddles on the S&P 500 earn large negative returns.
• Bollerslev, Tauchen and Zhou (2009), "Expected Stock Returns and Variance Risk Premia", RFS 22(11): 4463–4492. The premium predicts subsequent equity index returns at quarterly horizons.
• India. Sankar, Ramachandran and Lukose (2020), "Dynamics of variance risk premium: Evidence from India", International Review of Economics & Finance 70: 321–334, test whether variance risk is priced in a market with heavy retail participation and reject the hypothesis that it is not, using model-free and model-dependent measures across sampling frequencies and periods. They also find that past continuous variance, not jumps, predicts short-term synthetic variance-swap returns. A direct comparison of India VIX with subsequent realised Nifty volatility is a computation you can make yourself (India VIX and Nifty closes from NSE's daily index files, cached and read with toolkit.data.load(..., source="cache"); Problem 17). I did not find a peer-reviewed number for the average India VIX minus realised gap that I could verify, so none is quoted here.