How VIX Relates to Variance Swaps, Volatility Swaps, and VIX Futures

# Is the VIX more similar to a volatility swap or a variance swap?


# Is the VIX more similar to a volatility swap or a variance swap?












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?

## Answer by RAY (score 11, accepted)

https://quant.stackexchange.com/a/29831

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 $25-20=5$.

## Answer by user34971 (score 30)

https://quant.stackexchange.com/a/40768

\begin{align*} \text{Variance strike} &= \mathrm{E}_t \left[ \int_t^T \sigma_u^2 du \right ] \\ \text{Volswap strike} &= \mathrm{E}_t \left[ \sqrt{\int_t^T \sigma_u^2 du} \right ] \\ \text{VIX} &= \sqrt{\mathrm{E}_t \left[ \int_t^T \sigma_u^2 du \right ]} \\ \text{VIX future} &= \mathrm{E}_t \left [\sqrt{\mathrm{E}_T \left[ \int_T^{T'} \sigma_u^2 du \right ]} \right ] \\ \text{Forward variance strike} &= \mathrm{E}_t \left[ \int_T^{T'} \sigma_u^2 du \right ] \\ \text{Forward start volswap strike} &= \mathrm{E}_t \left [\sqrt{ \int_T^{T'} \sigma_u^2 du} \right ] \end{align*}

The VIX index is the square root of the variance swap strike.

The VIX future is actually somewhere in between the forward volswap strike and the square root of the forward variance swap strike as can be seen by Jensen's inequality and the tower law.