A volatility swap pays the difference between realised volatility (the square root of realised variance) and a pre-agreed vol strike. Unlike a variance swap — which pays the linear difference in variance — the vol swap's payoff is a concave function of variance. By Jensen's inequality, the fair vol strike is below the square root of the fair variance strike: . The gap is the convexity adjustment. This note derives the adjustment, quantifies it under the Heston model, and explains the practical implications for desks that trade both products.
Learning objectives. By the end of this note you should be able to:
- state the vol swap payoff and contrast it with the variance swap;
- apply Jensen's inequality to derive the direction of the convexity adjustment;
- compute the convexity adjustment under the Heston model;
- describe why vol swaps are harder to replicate than variance swaps.
In this lesson
- 4.1 The volatility swap payoff
- 4.2 Jensen's inequality and the convexity adjustment
- 4.3 The Heston convexity adjustment
- 4.4 Why vol swaps have no static replication
Before you start
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