Variance Swaps — Construction, Replication, and Realised P&L

A variance swap is an OTC contract that pays the difference between realised variance (measured as the annualised sum of squared log returns over the life of the swap) and a pre-agreed variance strike. It is the canonical instrument for trading volatility directly: unlike a straddle or a vega-neutral option position (which has residual delta and gamma risk), a variance swap is vega-positive with nearly zero delta at inception and a well-defined, constant dollar vega.

The replication of a variance swap via the log contract (from the previous lesson) shows that the fair variance strike is model-free — determined entirely by observable option prices, without a model for the stock price dynamics.

• state the variance swap payoff in annualised variance terms;
• derive the fair variance strike from the spanning formula;
• compute the realised variance from a time series of daily log returns;
• describe the P&L of a long variance swap position.

1. 2.1 The variance swap payoff
2. 2.2 The fair variance strike
3. 2.3 Replication: the log contract plus delta
4. 2.4 The variance risk premium