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A variance swap is a derivative contract whose payoff depends on the difference between realised variance and a predetermined strike variance. Unlike delta-hedged options, a variance swap provides pure, path-independent exposure to volatility. It is the cleanest way to trade the spread between implied and realised vol.
At expiry, the payoff is: (Realised Variance − Strike Variance) × Variance Notional. Realised variance is computed from daily log returns: σ² = (252/N) × Σ(ln(S_i/S_{i-1}))². If the strike was set at 20%² = 400 (variance points) and realised variance is 25%² = 625, the buyer profits by (625 − 400) × notional.
Strike variance: (20%)² = 0.0400. Realised variance over 3 months: (24%)² = 0.0576. Variance notional: $250,000 per variance point. Payoff to the buyer: (0.0576 − 0.0400) × $250,000 = $4,400. Equivalently, in vol terms: the buyer was "long vol" at 20% and realised vol came in at 24%, generating 4 vol points of profit × vega notional.
A delta-hedged option has gamma concentrated near the strike. If the stock moves far from the strike, gamma drops and the position becomes insensitive to further moves. A variance swap has uniform vol exposure across all stock levels. It does not care where the moves happen — only that they happen. Realised variance is the sum of squared daily returns, regardless of path. This makes variance swaps the theoretically ideal instrument for expressing a pure volatility view.
A variance swap can be replicated by a portfolio of options across all strikes, weighted by 1/K². This is the basis of the VIX calculation — the VIX is essentially the square root of the fair strike of a 30-day variance swap on the S&P 500. The replication requires a continuum of strikes (impossible in practice), so pricing involves interpolation and extrapolation of the volatility smile.
Variance swaps are convex in volatility: a move from 20% to 30% realised vol generates more profit than a move from 20% to 10% generates loss. This is because variance is vol squared. Buyers of variance swaps benefit from this convexity; sellers bear it. The strike of a variance swap is therefore set above the ATM implied vol to compensate the seller for this convexity disadvantage.
A variance swap is a derivative that pays the difference between realised variance (computed from actual daily stock returns) and a predetermined strike variance. The buyer profits when the stock is more volatile than expected. It provides pure exposure to volatility without the path-dependent complications of delta-hedged options.
A delta-hedged option has gamma concentrated near its strike. If the stock moves far away, gamma drops and the position becomes insensitive to further volatility. A variance swap has uniform volatility exposure regardless of where the stock trades. It captures all daily moves equally, making it a purer measure of total realised volatility.
The VIX is calculated as the square root of the fair strike of a 30-day variance swap on the S&P 500. Both use the same replication logic: a portfolio of out-of-the-money options across all strikes, weighted by 1 over K squared. The VIX is therefore a model-free measure of implied volatility, derived from the variance swap framework.
FinLingo covers variance products in Level 4 — 6 units on variance and volatility swaps. Level 1 is free.
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