LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A variance swap is a forward on annualized realized variance. The fair strike is the delivery price that makes the contract worth zero under a no-arbitrage argument. After a log transform and Itô's formula, that strike is the riskless rate adjusted by a strip of out-of-the-money puts and calls. The lecture follows a Goldman Sachs desk argument and the 1999 variance note it compares on screen, using Carr-Madan replication to turn the log payoff into options.
Why it matters
A forward fixes a maturity and a delivery price. The long side buys at that price at maturity. The short side sells. Here the underlier is variance, not a commodity. The strike, K, is the market's quoted level for variance over the life of the deal. Realized variance is whatever the path produces. The payoff is the difference, scaled by the notional per volatility point. If realized variance finishes above the strike, the long side is paid. If it finishes below, the long side pays. There is nothing to deliver. It is cash. Interest-rate swaps can pay along the way. A variance swap, in this lecture, pays only at maturity, because of convexity: an early spike can look enormous relative to variance still to be realized over the whole window, so settling early would mis-state the contract.
Variance adds. Volatility, the square root, does not. The lecture flags that this is not a pure volatility position, and it points at the VIX discretization used by the exchange without rederiving that grid.
The idea in plain words
Start from a diffusion: drift times the spot, plus volatility times the spot times a Brownian increment. Annualized realized variance is one over T times the integral of squared volatility out to T. That integral is not a traded quote. Set f to the log of the spot, which turns the multiplicative diffusion into an additive one. Itô's formula needs no explicit time derivative here. The first derivative of the log is one over the spot. The second is minus one over the spot squared. Square the diffusion and the usual multiplication table applies: dt squared and dt times dW drop out, and dW squared contributes dt, leaving sigma squared times the spot squared times dt.
Substitute, and the spot terms cancel until the differential of the log equals the spot's relative move minus half of sigma squared dt. Isolate the variance, integrate from zero to T, and the log integral collapses to the log of the ending spot over the starting spot. Multiply by two over T. Annualized realized variance equals two over T times the integral of dS over S, minus two over T times that log ratio. The first piece is a continuously rebalanced stock holding. The second is a short log contract.
How to use it
The hinge is there so the strip uses out-of-the-money options. In-the-money options are described as more expensive and less convenient. If the hinge equals the forward, the adjustment terms drop and you are left with the strip alone. In the VIX-style discretization the lecture cites, the hinge is the first listed strike below the forward, because a real chain is finite. You can then compare the quoted variance strike with what the path later realizes. The lecture does not report that comparison as a number.
What usually goes wrong
Settling the swap on an early spike, as if it were a periodic rate swap, ignores the convexity reason for a single maturity payoff. Treating the contract as a volatility forward ignores the square root. Variance is what adds along the path.
Dropping the hinge and integrating through illiquid in-the-money strikes is theoretically fine and practically the thing the hinge was invented to avoid. Using a continuous strip without a discretization rule pretends every strike trades. The exchange rule named here, the first strike at or below the forward, is a concrete choice, not the only one, and a different grid is a different approximation to the same integral.
Questions people ask
What does the long side want?
Realized variance above the strike. The payoff is that gap times the notional, in cash at maturity. The short side wants the opposite.
Why is there a strip of options?
Because the log contract that isolates variance is not listed. Carr-Madan writes that payoff as weighted out-of-the-money puts and calls around a hinge.
How is the hinge chosen in the VIX-style grid?
As the first listed strike below the forward, then a finite sum replaces the integral. A hinge sitting on the forward would remove the adjustment terms.