# Convexity

Variance swap payoffs are linear with variance, but convex with volatility.

The vega notional represents only the average P&L  for a 1% change in volatility.

A long variance swap position will always profit more from an increase in volatility than it will lose for a corresponding decrease in volatility.

This difference between the magnitude of the gain and the loss increases with the change in volatility  --> convexity!

This convexity is the reason that variance swaps strikes trade above ATM volatility.

Because of the convexity, a variance swap will always outperform a contract linear in volatility of the same strike.

The convexity premium should depend on the expected variability of the realised volatility.

The higher the variability of volatility --> the more beneficial the convexity. For the long:

- maximum loss when RV = 0  --> convexity works in his favour --> maximum vega loss = K/2.

For the short:

- losses potentially unlimited unless the variance swap is capped: standard cap = 2.5*K --> maximum vega loss = 2.625*K.