Overview

The variance swap is the purest volatility instrument: it pays the difference between realised variance and a pre-agreed strike, with no optionality. Unlike a vanilla option (which mixes vol and spot exposure), a variance swap is a direct bet on future realised vol. Its replication via a strip of vanilla options makes it the foundation of model-free volatility measurement — the methodology behind the CBOE VIX index.

This topic develops the variance swap from first principles: the model-free log-contract replication, the realized variance P&L, the Heston model's closed-form solution, and the convexity adjustment that distinguishes vol swaps from var swaps.

By the end of this topic you should be able to:

• replicate any twice-differentiable payoff as a weighted strip of calls and puts;
• derive the variance swap's fair strike from the log contract;
• compute the Heston variance swap value in closed form;
• explain the convexity adjustment between variance and volatility swaps;
• describe the VIX formula and its connection to variance swap theory.

1. Contents
2. The big picture