From Colin Bennet's trading volatility (pg 117), he says:
"A forward on a volatility future is short vol of vol. This means it is possible to back out the implied vol of vol from the price of this volatility future. This implied vol of vol can be used to price options on variance or even options on volatility futures themselves"
So how does one do this exactly? any ideas?
Vol of vol is a much used and abused term, but I think Bennet is specifically talking about the difference between the square root of the forward start variance swap strike $$ \sqrt{\kappa^2_t} = \sqrt{E_t \left( \frac{1}{T'-T}\int_T^{T'} \sigma^2_u du \right)} $$ and the (theoretical) VIX future $$ VIX_t := E_t \sqrt{ E_T \left(\frac{1}{T'-T} \int_T^{T'} \sigma^2_u du \right)}. $$ (See also my answer in this thread for definitions of different types of volatility contracts.)
The forward start variance swap, which is the appropriately weighted difference between two vanilla variance swaps (one with maturity $T'$ and the other with maturity $T$), can be synthesised from vanilla options, and hence theoretically observable.
The VIX future is of course also an observable. Hence the `vol of vol' can be inferred without using a model.
Suppose now that you want to price an option on the VIX future, i.e. you want to price $$ \left(VIX_T - K\right)_+ = \left( \sqrt{ E_T \left( \frac{1}{T'-T}\int_T^{T'} \sigma^2_u du \right)} - K \right)_+. $$ (Note that an option on the VIX future with maturity date equal to the futures maturity date is equal to an option on the VIX index.)
If you assume a lognormal distribution for the random variable $VIX_T$, then if you know $E_t [ VIX_T ]$, which you do because that is $VIX_t$, and if you know $E_t [ VIX^2_T ]$, which you do because that is the forward start variance swap strike $\kappa^2_t$, then you have the vol of vol $\alpha$ in the lognormal model, given by $$ \alpha^2 (T-t) = \log(\kappa^2_t/VIX^2_t). $$
Once you have $\alpha$ and the risk neutral drift of the VIX future (I'll leave that to you to figure out) you can price options on VIX using Black-Scholes formula. See also this thread for the lognormal approximation applied to pricing options on realised volatility.
