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- Convexity
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.