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In the paper Buy Rough Sell Smooth by Glasserman and He (2018), on page 5 equation (8) they define an estimate of the volatility of volatility ν, by setting $\log(ν)= β_1/2$. I would like to understand why this should be considered volatility of volatility, as $β_1$ is simply one of the regression terms in the model $\log z_2 (ℓ) = β_1 + β_2 \log ℓ + ε$ to estimate H aka the roughness or the estimated Hurst exponent, where $H = β_2 /2$. Neither the paper or the original reference Volatility is Rough by Gatheral et al. (2014) go into details on ν.

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kwinto's answer is correct. This paper [1] might be helpful in understanding the chain of reasoning (specifically equations (3) and (4) in the cited paper).

Gatheral et al. observe the discrete volatility process $m(q,\ell)$, $$m(q,\ell) = \frac{1}{N}\sum_{k=1}^N|\log(\sigma_{k\ell})-\log(\sigma_{(k-1)\ell})|^q.$$ They fit a linear regression of $\log(m(q,\ell))$ against $\log\ell$, $$\log m(q,\ell) \approx \beta_1+\beta_2\log\ell$$ and observed that the fit is good. Through another regression they then observe that $\beta_2$ has the form $Hq$ with $H\approx0.1$. For convenience, one reparameterizes $\beta_1=\log \nu/q$ as kwinto points out. The results of both regressions together suggest that $\log\sigma_t$ has the form: $$d\log\sigma_t^2=\nu dW_t^H$$ where $\nu$ is thus the volatility of volatility as equation (4) of the original paper says.

[1] Is Volatility Rough?, Fukasawa et al., https://arxiv.org/pdf/1905.04852.pdf

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Look at eq.(7), its RHS is an explicit expansion of what is defined in LHS of eq.(2) with $q=2$. Now, look at RHS of eq.(2), it defines $E[|Z|^2] \ l^{2H}$. So, $\nu = \sqrt{E[|Z|^2]}$ is the volatility by definition. Volatility of what? Right, of $ \log \sigma $.

Alternatively, just look at eq. (3)-(4). In (3) $\sigma_t$ is the volatility of $\log S_t$. In (4) $\nu$ is the volatility of $\log \sigma_t$, hence $\nu$ is the vol of vol.

$\beta_1$ and $\beta_2$ are just reparametrizations of $\nu$ and $H$.

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  • $\begingroup$ @Hans-Peter, thanks for the acknowledge and edit. $\endgroup$
    – kwinto
    Mar 9, 2023 at 19:44
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    $\begingroup$ I am happy to hear any feedback from those who downvoted this answer. Otherwise, it does not make a good impression on the community, which downvotes the correct answer from a new user for no particular reason. $\endgroup$
    – kwinto
    Mar 9, 2023 at 19:47

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