Variance Swaps and How They Are Replicated

A variance swap is priced from a static strip of options weighted by one over strike squared, because that strip reproduces the payoff of a log contract.

A variance swap pays its buyer a notional amount multiplied by the difference between the realised variance of the underlying over the contract’s life and a variance strike fixed at inception, with no reference to the path the underlying took between those two dates. The contract is replicated, and its fair strike therefore set, from a static strip of European options spanning a continuum of strikes, because that strip reproduces the payoff of a log contract, and a log contract’s exposure turns out to equal exactly the realised variance of the underlying once it is continuously delta hedged.

Suppose a one-year variance swap is struck at a variance of 400, corresponding to a 20 percent volatility strike since 20 squared is 400, with a variance notional of $10,000 per point of variance. If realised variance over the year comes in at 484, corresponding to 22 percent realised volatility, the buyer receives $10,000 multiplied by the difference between 484 and 400, which is $840,000. The payoff is symmetric in variance, not in volatility, which is why a variance swap buyer benefits more than proportionally from a large spike and less than proportionally from a small one relative to a position sized in volatility terms.

The connection runs through a result from stochastic calculus: for an underlying following a diffusion process, the sum of squared log returns over a period, which is exactly how realised variance is measured, equals twice the difference between the return on a continuously delta-hedged forward position and the return on a static position in the log of the underlying. A trader who is short a log contract and continuously delta hedges it by holding the underlying against the changing delta earns a hedging profit that accumulates as exactly the realised variance over the period, the same mechanism worked through from the option side in gamma scalping and delta-neutral hedging.

The log contract itself does not trade, but its payoff can be replicated with a static, continuously weighted strip of out-of-the-money puts below the forward and out-of-the-money calls above it, each weighted by one over the square of its own strike. That weighting is not arbitrary: it falls directly out of writing the log payoff as an integral over strikes of the second derivative of the payoff function, a standard decomposition that turns any twice-differentiable payoff into a static portfolio of options plus a forward and a bond. Because the strip needs no rebalancing as the underlying moves, only the initial cost of assembling it, that cost sets the fair variance strike at inception.

This construction is exactly the mechanism behind the VIX calculation, which is built from a discretised version of the same strip applied to a rolling 30-day window of S&P 500 options rather than a bespoke bilateral contract.