Variance Swap

Most instruments pay off based on where a price goes. A variance swap pays off based on how much it moved along the way — up or down, doesn't matter.

The two sides agree on a "strike" level of volatility, say 20%. At the end, the actual (realised) volatility of the stock's daily moves is measured. Came out at 30% — a turbulent period? The buyer of variance collects. A calm 12%? The buyer pays.

This turns turbulence itself into an asset. Investors buy variance as crash insurance (markets get wild when they fall), and sellers harvest the premium that insurance buyers persistently overpay — most of the time.

The contract

Payoff at expiry, with vega notional expressed as variance notional \(N_{var}\):

Note it settles on variance (vol squared): a move from 20 → 30 vol pays more than 20 → 10 costs. That convexity is why the market quotes variance, which dealers can replicate exactly, rather than volatility, which they can't.

Quoting in vega

Traders think in "vega notional" — P&L per volatility point near the strike: \(N_{vega} = 2 K_{var} N_{var}\). A trade of "100k vega at 20 strike" makes ≈ $100k per vol point of realised above 20 (more, due to convexity).

Uses and abuses

• Hedging: long variance offsets equity drawdowns (vol spikes when markets crash — strong negative correlation).
• Carry harvesting: implied variance usually exceeds subsequent realised — selling variance collects this "variance risk premium", with occasional violent losses (short variance in 2008 or the 2018 "Volmageddon" was ruinous).
• Dispersion: selling index variance vs. buying single-name variance trades correlation.

Replication: the log contract

The theoretical heart: realised variance can be replicated model-free by delta-hedging a portfolio of options whose payoff is \(-\tfrac{2}{T}\ln(S_T/S_0)\). By the Carr–Madan expansion, that log payoff decomposes into a strip of out-of-the-money options weighted by \(1/K^2\):

The fair strike is thus readable off the entire volatility smile — no model of dynamics required, only continuous paths. This same formula (discretised) is how the VIX is computed: the VIX is essentially a 30-day variance-swap strike.

Where replication breaks

• Jumps: a single crash makes discrete realised variance exceed what the hedge captures; dealers cap payoffs (e.g. at 2.5× strike) to bound this.
• Wings: the \(1/K^2\) weighting demands deep OTM puts that may not exist or trade at extreme spreads; the missing wing is a model reserve.

P&L accrual

A seasoned variance swap decomposes into realised-so-far plus implied-remaining, weighted by elapsed time \(t\):

The formulas above are standard textbook formulations, simplified for teaching. They explain the mechanism — they are not a valuation tool, and they will not reproduce a dealer’s price.