# Is Variance Swap Convexity fairly priced?

Convexity:

- extra value of the convex variance swap payout

- extra volatility paid for OTM options needed to replicated the VS --> represented by convex skew.

Convex payout --> Variance swap must be worth more than a Volatility swap at the same strike level.

How much more?

- it depends on how much volatility is likely to change (= vol of vol) and hence how much effect VS convexity will have.

How can we quantify it?

1.

Short-dated VS have some value as a predictor of future volatility.

As well as looking at the regression R2, we can calculate the std error of the future volatility as estimated by the VS strike.

This "average error" is giving info about the historical variability of future RV around the estimate provided by (0.9*) VS strike.

Back-testing --> this std error is ~ 6% (6 vegas) --> the std dev of (implied variance - future realised variance) is 6%.

This 6 vegas represents an ‘expected’ range of values for future volatility --> determining magnitude of convexity of the VS payoff.

How much is the 6% volatility of volatility worth?

- It depends on the strike of the variance swap --> for strikes in 15-20% region, convexity is worth an extra 1-1.5%.

2.

We can calculate an "implied" variability of vol by considering spread between short-dated ATM IV and VS strikes.

The difference between SX5E ATM IV and VS strikes suggest an average implied vol of vol of ~ 7%.

This is fairly close to the 6% average error of the variance swap in predicting volatility.

This 6% vol of vol for SX5E is a long term average --> like volatility, it changes over time (min of 2%, max of 10%).

There is a strong correlation between volatility and the spread between VS levels and ATM volatility.

It makes sense if we believe that vol of vol is correlated with volatility.

--> Large absolute changes in volatility are more likely when volatility high than when volatility is low.

Estimates for the expected vol of vol are used for pricing options on variance.

They are also of use for pricing VS caps --> Capped VS = VS - Call on variance

For long position --> Capped VS < VS   --> Capped VS trade with strikes slightly below their uncapped equivalents.

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