Why Volatility Swap Convexity Depends on the Comparison Benchmark

Summary

The document examines why a volatility swap may be described as short volatility of volatility even when a two-day example appears to give it a larger payoff when daily realized volatility varies more. The answer explains that convexity is relative to the instrument used as the benchmark. Compared with a variance swap, a volatility swap has concavity; compared with a volatility swap, the variance swap has convexity.

The difference between the two contracts can therefore express exposure to convexity, or volatility of volatility, with the variance swap treated as the convex instrument when the volatility swap is the linear reference. The initial example alone does not settle the label: the conclusion depends on the payoff comparison and chosen base instrument. The response provides a conceptual distinction rather than a detailed payoff derivation or pricing treatment.

Key ideas

• Whether a volatility swap is called short convexity depends on the comparison instrument.
• Relative to a variance swap, a volatility swap has concavity.
• Treating the volatility swap as the linear reference makes the variance swap the convex instrument.
• The difference between variance and volatility swaps can represent convexity or volatility-of-volatility exposure.

Tags

# vol swap and vol of vol


# vol swap and vol of vol












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?

## Answer by user34971 (score 4)

https://quant.stackexchange.com/a/53894

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.