Variance Swap (Detailed)
A Variance Swap is an over-the-counter derivatives contract in which counterparties exchange a fixed variance strike against the subsequently realized variance of an underlying asset, providing pure, path-independent exposure to volatility without the need for delta hedging.
A variance swap provides a clean payoff that depends only on the realized variance of the underlying over the swap's life, not on the path the underlying takes or the direction of its movement. This path independence distinguishes variance swaps from options, whose value is affected by where the underlying is at any point in time relative to the strike.
The payoff formula is: (Realized Variance - Strike Variance) x Vega Notional / (2 x Strike Volatility). In practice, this is often simplified to: (Realized Variance - Strike Variance) x Variance Notional. The realized variance is calculated using the daily log returns of the underlying, squared and annualized, over the swap's life. The strike variance is agreed upon at inception and represents the market's expectation of future variance.
Why use variance rather than volatility directly? Variance is the square of volatility and has the mathematical property that a portfolio of options can perfectly replicate a variance swap payoff through a log contract — a theoretical strip of options at all strikes. This replication relationship is what allows market makers to hedge variance swaps using options, giving the instruments practical tradability despite their exotic nature.
The convexity difference between variance and volatility is important. Variance swaps have a convex payoff in volatility: a move from 20 to 30 in volatility generates a larger variance payoff (900 - 400 = 500 units of variance) than the linear equivalent. This convexity makes variance swaps attractive to buyers who want to be long tail risk.
Historically, the strike on variance swaps has consistently been set above the subsequently realized variance on equity indices, reflecting the volatility risk premium. Sellers of variance — typically banks and institutional investors — have captured this premium systematically over long periods, though the strategy suffers severe losses during volatility spikes such as 2008 and 2020.
Practical applications include: hedge funds expressing a volatility view without managing options delta hedging, banks hedging their vega exposure from structured products, and corporations hedging earnings uncertainty through counterpart banks. The VIX index is closely related to the fair strike of a 30-day S&P 500 variance swap, making variance swap theory central to understanding volatility indices.