Variance swaps, the log contract and what the VIX is

A variance swap pays the difference between realised and strike variance, and it is the only pure volatility instrument that can be replicated statically. That one fact — variance replicates, volatility does not — explains the entire structure of the volatility derivatives market, including why the VIX is computed the way it is.

Equation 3.1

The variance swap

Linear in *variance*, not in volatility — which is what makes it replicable.

\text{payoff} = N_{\text{var}}\left(\sigma_{\text{realised}}^2 - K_{\text{var}}^2\right)

$N_{\text{var}}$: Variance notional. Vega notional is $2K_{\text{var}}N_{\text{var}}$ near the strike.
$\sigma^2_{\text{realised}}$: Annualised sum of squared log returns over the life.

Why variance and not volatility. A delta-hedged option earns realised variance weighted by its dollar gamma, which depends on where the spot happens to be. To get a pure variance exposure you need a portfolio whose dollar gamma is the same at every price — and that portfolio turns out to be a strip of options weighted by $1/K^2$ , which is exactly the log contract. There is no corresponding trick for volatility: $\sqrt{\text{variance}}$ is a non-linear function of the thing you can replicate, so a volatility swap must be priced with a convexity adjustment and hedged dynamically. Variance is the natural unit because it is the one the hedging mathematics produces.

Figure 3.2 · Why the low strikes matter in the variance strip. The replication coefficient scales as 1/K², normalised here to one at strike 100. Lower strikes receive larger weights before option prices and strike spacing enter the sum. This is one reason downside put prices matter so much to the variance rate.

Derivation 3.3

The replicating strip

Start from the log contract and rewrite it in terms of traded options.

$\mathbb{E}\left[\int_0^T \sigma_t^2\,dt\right] = 2\,\mathbb{E}\left[\int \frac{dS}{S} - \ln\frac{S_T}{S_0}\right]$
Itô on $\ln S$ : the drift term cancels under the risk-neutral measure.
$-\ln\frac{S_T}{S_0} \ \text{replicated by} \ \int_0^{F}\frac{P(K)}{K^2}dK + \int_F^{\infty}\frac{C(K)}{K^2}dK$
Puts below the forward, calls above, each weighted by $1/K^2$ .
$K_{\text{var}}^2 = \frac{2}{T}\left[\int_0^{F}\frac{P(K)}{K^2}dK + \int_F^{\infty}\frac{C(K)}{K^2}dK\right]$

\text{a static portfolio of options, plus a dynamic stock hedge}

Proposition 3.4

What the VIX is

The VIX is the square root of a 30-day variance swap rate on the S&P 500, computed from exactly the strip above — a weighted sum over all listed strikes with non-zero bids, not the implied volatility of any single option. That is why it is described as model-free, and why skew feeds into it: the out-of-the-money puts carry real weight in the sum.

Holds when

A steeper skew raises the VIX even with at-the-money implied volatility unchanged.
It is a 30-day constant-maturity figure, interpolated between the two nearest expiries.
It is an expectation under the risk-neutral measure, so it sits above realised on average.

Variance swaps, the log contract and what the VIX is

After this lesson you should be able to

The variance swap

The replicating strip

What the VIX is

The rest of this lesson is in Premium