I am reading the following paragraph on the VIX wikipedia article and I find it confusing:
The VIX is calculated as the square root of the par variance swap rate for a 30-day term[clarify] initiated today. Note that the VIX is the volatility of a variance swap and not that of a volatility swap (volatility being the square root of variance, or standard deviation).
This makes zero sense to me, since a volatility swap is precisely the square root of a variance swap which is what VIX is aiming to represent/estimate.
Would someone have a better/cleaner explanation than this, and perhaps update the wikipedia paragraph?
The price/value of the VIX index is more akin to the strike/price of a variance swap expressed in vol units than to the strike/price of a vol swap.
However, if you are to trade a VIX future (i.e. a delta one contract on the VIX index), the exposure you gain is more comparable to the one of a vol swap in the following sense:
Consider a notional of 1 and a fixed investment horizon $[0,T]$. Ignore second order effects (e.g. daily margining).
If you buy a variance swap at $t=0$ at a price of 20% (variance strike in volatility units) and that the realised volatility over the contract's life ends up being 25%, you will lock a profit: $25^2-20^2=225$.
If you buy a volatility swap at 20% at $t=0$ (volatility strike) and that the realised volatility over the contract's duration ends up being 25%, your profitt will be: $25-20=5$
If you enter a VIX future at 20 (variance swap par rate expressed in vol units) at $t=0$ and unwind your position at 25 at $t=T$, you will have made $25-20=5$.