Variance swap

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A variance swap is a financial derivative whose payoff is equal to the difference between the square of annualized realized volatility (that is, the annualized realized variance), $\sigma_{realized}^{2}$ , of returns on the underlying price over that period and a fixed quantity, $\sigma_{strike}^{2}$ , sometimes known as the variance strike.

I.e., in the above notation, the payoff is $\sigma_{realized}^{2}-\sigma_{strike}^{2}$ . Effectively, it is a forward contract on the realized variance.

The annualized realized variance is calculated based on a prespecified set of sampling points over the period. It does not coincide with the classic statistical definition of variance, but follows the usual market convention of not subtracting the mean.

The variance swap may be hedged and hence priced using a portfolio of European call and put options with weights inversely proportional to the square of strike.

The advantage of variance swaps is that they provide pure exposure to the variability of the underlying price, as opposed to call and put options which carry directional risk (delta).

The payout of a variance swap is often capped. It is market practice to determine the number of contract units as $\frac{Vega Notional}{2\sigma_{strike}}$ to approximate the payoff of a volatility swap.

Closely related contracts include straddle, volatility swap, correlation swap, gamma swap, conditional variance swap, corridor variance swap, forward-start variance swap,option on realised variance and dispersion trade.

Using the Heston model, a closed-form solution can be derived for the fair variance swap rate.