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why people say vol swap is short vol of vol?

Say we consider a simple vol swap with 3% strike with two days maturity here: scenario1: realized vol is 2% on day1 and 4% on day2 scenario2: realized vol is 1% on day1 and 5% on day2 in both scenarios, same mean for vol but scenario2 has higher vol of vol.

In scenario2, vol of vol is higher and vol swap payoff is higher.

That means vol swap is actually long vol of vol, right? Did I miss something here?

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I do not agree with the statement that volswap is short convexity or that varswap is long convexity etc. It depends what your 'base' instrument is: if it is the varswap then the volswap has convexity (actually concavity), if it is the volswap then the varswap has convexity. I.e. convexity is a relative measure. However given the existence of both the varswap and the volswap, then their difference is a measure of convexity or vol of vol, and since the volswap is then the linear instrument the varswap is long convexity, which means the volswap is then "short" convexity / vol-of-vol. But again, it is a relative measure.

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