Inferring Volatility of Volatility from VIX Futures and Variance Swaps
Summary
The document describes how to infer a measure of volatility of volatility by comparing a VIX future with a forward-start variance swap. The forward variance strike represents the expected future variance, while the VIX future reflects the expectation of the square root of that variance. Both quantities are observable or can be synthesized from vanilla options, so their difference provides an estimate without first specifying a volatility model.
For pricing an option on a VIX future, the response then assumes the future VIX level is lognormally distributed. Its mean is linked to the VIX future, and its second moment to the forward variance strike; these moments determine the model’s volatility parameter. That parameter can then be used in a Black–Scholes-style option valuation. The inference depends on the lognormal assumption, and the document leaves the appropriate risk-neutral drift to the reader. The resulting estimate is therefore model-dependent when used to price options, even though its inputs are market-observable.
Key ideas
- A forward-start variance swap can be constructed from variance swaps with different maturities.
- The forward variance strike and VIX future encode different moments of future variance.
- Their market values can be used to infer a volatility-of-volatility measure.
- A lognormal assumption turns those moments into a parameter for pricing VIX options.
- The option valuation also requires an appropriate risk-neutral drift.
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# Deriving vol of vol from volatility futures price
# Deriving vol of vol from volatility futures price
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?
## Answer by user34971 (score 1)
https://quant.stackexchange.com/a/70604
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.