The Variance Risk Premium as a Strategy

There is a number the options market quotes every day that is, on average, a lie — and you can get paid for knowing it.

When you read an at-the-money implied volatility, you are reading the market's price for variance over the life of that option. Call it the risk-neutral expectation of variance: what variance would have to be for the option to be fairly priced. Now hold that number next to what variance actually turns out to be — the variance the underlying genuinely realizes over the same window. Across decades, across nearly every liquid equity index, the implied number sits above the realized number. Implied volatility runs richer than the volatility that shows up. That persistent gap is the variance risk premium, and this chapter is about turning it from an observation into a position.

Let us build the intuition before the algebra. Why would the market systematically overpay for variance? Because most participants are not buying options to gamble; they are buying insurance. A pension fund holding equities buys puts the way you buy fire insurance on a house — not because it expects the house to burn, but because it cannot survive the case where it does. Insurance is, by design, priced above its expected payout; the seller demands compensation for warehousing a risk that arrives precisely when everything else is going wrong. So the option seller, the variance seller, is the insurer. The premium they collect over the long run is their fee for absorbing crash risk. That is the variance risk premium: not a free lunch, but a paid one, with a bill that comes due in the tail.

Now the formal object. Define realized variance over a window as the sum of squared log-returns — the same log-returns from your very first foundations chapter, now squared and accumulated. The variance risk premium is the difference between the risk-neutral expectation of that realized variance and the physical, real-world expectation of it. The cleanest way to read the risk-neutral side off the market is not a single at-the-money implied vol but the model-free implied variance: the price of a portfolio of out-of-the-money options across all strikes, weighted by one over strike-squared. That construction is exactly what the VIX-style methodology computes, and squaring a VIX-style index gives you the risk-neutral expected variance directly. The premium, then, is roughly that risk-neutral variance minus the variance you expect to actually realize — and empirically that difference is positive, and it is large, especially the part that loads on downside strikes.

So how do you trade it? The canonical expression is a short, delta-hedged variance position. You sell variance — through a variance swap if you have access to one, or synthetically through a strip of options or a short straddle — and you continuously delta-hedge to strip out the directional bet. What is left after hedging is almost pure exposure to the difference between the implied variance you sold at and the variance the underlying realizes. If you sell at an implied of, say, twenty and the asset only realizes sixteen, you keep the gap. The delta-hedging is what isolates the variance; without it you are just short the underlying's direction and the variance premium is buried in noise. A variance swap formalizes this: its payoff is realized variance minus the strike variance, so selling it is the premium in its purest form.

Here is the discipline this course exists to instill. A wide variance risk premium is not automatically an opportunity. Remember what the premium is — compensation for crash risk. The gap is widest precisely when the market is most frightened, which is also when realized variance is most likely to come in above implied and hand the seller a catastrophic loss. So the same number, a fat VRP, is simultaneously a signal to sell and a warning that the next realized move could be enormous. Distinguishing those two readings — genuine harvestable edge versus a flashing tail-risk alarm — is the entire game. It is the throughline of this institute restated in derivatives language: a number is only evidence an asset will move if you have ruled out the boring explanations, and here the boring explanation is that you are simply being paid the going rate to stand in front of a crash.