The contract

A variance swap pays, at expiry, the realised variance of the underlying's returns over the life minus a strike \(K_{var}\) agreed at the start, times a notional. No premium changes hands, so the strike is set to make the contract worth zero. It is the cleanest way to be long or short volatility: an option's exposure to volatility depends on where the spot is, but a variance swap's does not.

$$ \text{payoff} = N_{var}\left(\sigma_{real}^2 - K_{var}\right), \qquad \sigma_{real}^2 = \frac{252}{n}\sum_{i=1}^{n} \ln^2\!\frac{S_i}{S_{i-1}} $$

The replication

Realised variance can be manufactured from a log contract: by Itô's lemma, \(-2\ln(S_T/S_0)\) delta-hedged accumulates exactly \(\int_0^T \sigma_t^2\, dt\). A log contract is not traded, but any smooth payoff can be built from options, and the log payoff needs every strike weighted by \(1/K^2\):

$$ K_{var} = \frac{2 e^{rT}}{T}\left[ \int_0^{F} \frac{P(K)}{K^2}\, dK + \int_{F}^{\infty} \frac{C(K)}{K^2}\, dK \right] $$

Puts below the forward, calls above, all out of the money. No model enters: the fair strike is a weighted sum of market prices, which is why variance swaps are quoted directly off the surface and why the VIX index is computed with this very formula.

What the weights say

The \(1/K^2\) weight puts most of the value in low strikes. A put skew therefore lifts the fair variance above the at-the-money variance, often by several volatility points on an equity index. Even with a flat smile the fair volatility strike sits slightly above at-the-money volatility, because the payoff is convex in volatility: a variance swap is long volatility of volatility.

Checking with a model

Under Heston the expected average variance is known in closed form, \(\theta + (v_0 - \theta)\frac{1 - e^{-\kappa T}}{\kappa T}\), and the strip built from Heston option prices reproduces it. That agreement is the point: the strip is model-free, so any arbitrage-free model must give the same answer. Jumps are the exception, since the log contract no longer captures variance exactly when the path is discontinuous, a correction of order the cube of the jump size.