# 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?

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 $\langle \ln X \rangle_t$

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

• May be I have overlooked something, I still feel it is model free, assuming that the daily changes are small. See also this question: quant.stackexchange.com/questions/18007/…. – Gordon Apr 15 '16 at 14:00
• @Gordon, lovely answer on that post. However this assumes no dividend as you rightfully point out (we could actually say no jumps, even if these would be deterministic in timing and size) and relatively small daily returns (which theoretically justifies using quadratic variation rather than sample variance). While the second assumption is 'model free', I feel like the first one is not (you cannot use this replication under Bates for instance, or simply BS + discrete cash dividends). But it really depends on what one calls a model, I'm getting picky here :) – Quantuple Apr 15 '16 at 14:39
• Also note that in your answer you used: $E(S_{t_i} \vert S_{t_{i-1}}) = S_{t_{i-1}} e ^{\int_{t_{i-1}}^{t_i} r_s ds}$ which is why I have written the dynamics $dX_t/X_t = \mu(t) dt + \sigma(t,.)dW_t^{\mathbb{Q}}$ in my answer (my $\mu(t)$ is equivalent to your $r(t)$ under the risk-neutral measure). But I also insisted on the fact that $\sigma(t,.)$ could be anything as long as price paths remain continuous. At the end of the day, although I seem to specify a particular form of dynamics by expliciting an SDE, it remains a very general dynamics... but with continuous paths. – Quantuple Apr 15 '16 at 14:47
• Thanks. With jump and dividend payments, it is completely a different story. – Gordon Apr 15 '16 at 15:05
• @Veeken, I don't agree! You're talking about some kind of discretisation bias, and indeed there is one (replacing an integral by a discrete sum). But if there were jumps, even with a continuous strip of otmf instruments, we would still need to account for an additional term : the log contract is not enough to infer realised variance. – Quantuple Apr 26 '16 at 6:15

Historically, there has been little correlation between the VVIX and the VIX except at extreme values of the VIX. You are correct it will definitely lead to different greeks, but it does not matter at lot as your Portfolio objective would be fulfilled as methodology for both are same.

The motive is indeed to construct a constant gamma portfolio.

A position in a VVIX portfolio replicates the volatility of VIX forward prices. VVIX portfolio prices have usually been at a premium relative to future realized volatility. The discount is a volatility risk premium. For nearby expirations, these prices have also tended to surge at the same time as VIX.

But anyway, it doesn't really matter much so don't complicate things by fretting about it.