volatility - Construction of VIX and VVIX

I just read the CBOE's Whitepapers for VIX and VVIX and notice that they are constructed in the same way, i.e. a range of calls and puts on the respective underlyings (S&P500 in case of VIX, and VIX itself in case of VVIX) weighted inversely to their strikes squared. I understand that the motive is to create a constant gamma portfolio.

The question is, as the underlyings follow different forms of processes (assume GBM for S&P500, CIR for VIX), how come the construction be the same for both indices? I thought that different processes would lead to different greeks, which in turn would affect the way we build the constant gamma portfolio? Or does it not matter at all?

Answer

Strictly speaking, indices such as the VIX are built to approximate the expected variance (of log-returns) that would effectively realise under a pure diffusion setting (i.e. no jumps) $$ \frac{dX_t}{X_t} = \mu(t) dt + \sigma(t,.) dW_t^{\mathbb{Q}} $$

Writing out the equations (*) yields the famous static replication formula in terms of strike-weighted OTMF options that you refer to, along with the constant Gamma portfolio interpretation you mention.

Although many people claim that this constitutes a model-free estimate of future variance, this is not completely true since pure diffusion is assumed all the way (but this does not preclude the fact that the diffusion coefficient $\sigma(t,.)$ could exhibit its own source of stochasticity, i.e. that the true diffusion process could be Heston or local volatility or GBM... hence the model-free adjective).

IMHO, you should really see volatility indices such as the VIX as expected realised variances assuming pure diffusion, in a similar way you look at the implied volatility of an option as the figure you should use in a (wrong) GBM setting to retrieve the (right) observed market price.

I hope this clears your confusion.

(*) This requires approximating the sample variance of the log-returns observed over $[0,t]$ as the quadratic variation $\langle \ln X \rangle_t$

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[Edit] More details on the derivation + constant Vega feature in this excellent note by Fabrice Rouah.