How does a volatility swap differ from a variance swap and why is it harder to hedge?

Question

AcadiFi · Accepted Answer

A volatility swap pays (σ_realized − σ_strike) × vega notional, where σ_realized = √variance_realized. The payoff is linear in volatility, unlike a variance swap which is linear in variance (quadratic in vol).

Mathematically, vol = √var is concave in variance. For a dealer who's short a vol swap, the hedge would require holding a portfolio whose payoff tracks √variance — but the natural replicator (log-contract) pays linearly in variance. Therefore a short vol-swap dealer must hold a short variance swap plus a convexity adjustment, which requires dynamic rebalancing and exposes the dealer to model risk (how fast vol diffuses).

```mermaid
flowchart TD
  A[Vol Swap Payoff] --> B[Linear in sigma]
  C[Variance Swap Payoff] --> D[Linear in sigma squared]
  D --> E[Hedgeable via Log Contract]
  B --> F[Requires Convexity Adj]
  F --> G[Model Risk + Rebalance]
```

Example: Arnholdt Capital buys a 3-month vol swap at 18 vol, $50,000 vega notional. Realized vol comes in at 22. Payoff = $50,000 × (22 − 18) = $200,000. If this had been a variance swap with the same vega notional, the variance payoff would have been (22² − 18²) × $50,000/(2 × 18) = (484 − 324) × $1,389 = $222,222.

The difference reflects convexity: variance swaps overpay relative to vol swaps when realized vol is far from strike. Dealers charge a convexity premium on vol swaps — typically 1-2 vol points above the variance-strike implied vol. For FRM, remember: variance swaps replicate cleanly; vol swaps do not.

How does a volatility swap differ from a variance swap and why is it harder to hedge?

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