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?

  • $\begingroup$ Well, implied vol yes. The implied vol on a vol swap is different from the sqrt of the implied variance of the variance swap. But the realized vol of a vol swap is the same as the sqrt of the realized variance. $\endgroup$ – RAY Aug 25 '16 at 7:20
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    $\begingroup$ No, the par rate of a volatility swap is not the square root of the par rate of a variance swap (seen Jensens' inequality). A convexity adjustment is required. In pure diffusion frameworks only variance swaps' par rates have a closed form (under idealised assumptions). Knowing that, I hope the definition makes more sense to you. It is perfectly fine IMHO $\endgroup$ – Quantuple Aug 25 '16 at 7:21
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    $\begingroup$ You are looking at the square root of the expectation of squared log-returns, versus the expectation of the square root of squared log-returns. $\endgroup$ – Quantuple Aug 25 '16 at 7:22
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    $\begingroup$ Ah, so I guess I answered my own question. This is basically saying, the VIX is the volatility price (strike) on a variance swap, rather than the strike of the vol swap. What needs to be clear though is that if you are to buy/sell the VIX futures, your payoff will be much more similar to that of volatility swap than that of a variance swap since the convexity does not come into play here, as the futures settles in the volatility space rather than the variance space. $\endgroup$ – RAY Aug 25 '16 at 7:22
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    $\begingroup$ Great. Thanks a lot. Great helpful discussion here. $\endgroup$ – RAY Aug 25 '16 at 7:43

\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.

  • $\begingroup$ Those integrals should be taken with a grain of salt. The VIX formula is a discrete sum based on the variance swap replication formula, and may truncate not so far away. $\endgroup$ – jherek Jun 12 '20 at 12:43
  • $\begingroup$ @jherek Absolutely, those are theoretical prices so to speak. Real life is a bit more difficult. $\endgroup$ – Frido Rolloos Jun 12 '20 at 12:51

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 $25-20=5$.
  • $\begingroup$ I've just edited the answer to make all the details appear in it instead of referring to the comments. Is that OK? $\endgroup$ – Quantuple Aug 25 '16 at 9:33
  • $\begingroup$ I am not sure I understand the distinction between the futures and the VIX. The futures are settled at 8:30 am on expiration Wednesday at a value calculated from opening quotations of S&P options using a method THAT IS IDENTICAL IN STRUCTURE to the VIX calculation (i.e. the varswap thing and then take square root that). $\endgroup$ – noob2 Aug 25 '16 at 15:55
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    $\begingroup$ We are talking about the futures here only because you can't directly trade the index itself. The idea is the same: the index is in the volatility space (as opposed to the vol squared/variance space), but it's fair price is the square root of weighted variance swap strikes, not weighted vol swaps. $\endgroup$ – RAY Aug 25 '16 at 21:17
  • $\begingroup$ You need to adjust the notional for the var swap if you are going to use unit notional for the vol swap, e.g. 1/(2*20) var notional for the var swap. This gives 5.625 profit, not 225. $\endgroup$ – Chan-Ho Suh Nov 7 '17 at 18:01

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