The Variance Risk Premium

Here is a fact that has held for decades across stock indices: the volatility priced into options is, on average, a bit higher than the volatility the market actually delivers. Buyers of options systematically overpay, and sellers systematically get paid. The size of that overpayment is the variance risk premium (VRP), and harvesting it is one of the oldest trades in the book.

Why does the gap exist? Because an option is insurance, and insurance always costs more than its expected payout. People will pay up to protect a portfolio against a crash, just as they pay for fire insurance they hope never to use. The seller takes on the rare, ugly risk and, in return, pockets a premium that on average exceeds the losses. Implied volatility is the price of that insurance; realized volatility is the claim that eventually comes due.

The formula

The variance risk premium is simply the difference between the variance the market prices in and the variance that actually occurs:

$$\text{VRP} = \sigma_{\text{implied}}^2 - E\!\left[\sigma_{\text{realized}}^2\right].$$

Here $\sigma_{\text{implied}}^2$ is the implied variance baked into option prices today (what you're paid for as a seller), and $E[\sigma_{\text{realized}}^2]$ is the expected realized variance, how much the underlying is actually expected to jump around over the life of the option (what you pay out). When the first term is bigger, the premium is positive and belongs to the seller. Quoted in volatility points instead of variance, equity indices have historically shown implied running a few points above realized most of the time.

The picture below is the whole trade in one bar. What an option costs you (the implied variance) splits into two parts: the variance that actually gets realized and paid back, and the premium the seller keeps on top.

Worked example

Suppose the one-month implied volatility on a stock index is $20\%$. Squaring it, the implied variance you'd sell is $0.20^2 = 0.04$. You sell a one-month variance swap (or an at-the-money straddle) at that level.

Over the next month the market turns out calm, and realized volatility comes in at $15\%$, so realized variance is $0.15^2 = 0.0225$. Your profit per unit of variance notional is the gap:

$$0.04 - 0.0225 = 0.0175 .$$

In volatility terms you sold at $20$ and only $15$ came due, a five-point win. That is the premium doing its job. Historically the average has been smaller, a few points, but the sign is reliably in the seller's favour.

Now the other month. Bad news hits and realized volatility spikes to $30\%$, giving realized variance $0.09$. Your P&L is $0.04 - 0.09 = -0.05$, a big loss that dwarfs a whole string of the small wins above. This is the deal you signed up for.

Where it misleads

The premium is real, but it is not free money. It is payment for a genuinely nasty risk profile.

• Small, steady gains punctuated by rare disasters. Selling volatility is often described as picking up pennies in front of a steamroller. The distribution of returns is the opposite of a lottery ticket: you win a little most of the time and lose a lot occasionally. The average is positive; the path can be brutal.
• The premium is largest exactly when it's most dangerous. Implied volatility, and the premium, spikes during panics (VIX blowouts). Selling then looks lucrative but is when losses cluster, because realized volatility can overshoot even elevated implied.
• It varies with the regime. The gap widens in complacent markets and can vanish or invert in stressed ones. Sizing a short-vol book off the long-run average will over-bet in quiet times right before the storm.

This premium is the economic engine behind variance swaps, short-straddle and short-strangle programs, and the level of the VIX relative to what markets actually deliver. It is also why implied volatility is best read as a price, set by supply and demand for protection, rather than as an unbiased forecast of future movement.