# Understanding volatility of volatility in realized roughness

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.

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

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 $$H$$.

• @Hans-Peter, thanks for the acknowledge and edit. Mar 9, 2023 at 19:44
• 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. Mar 9, 2023 at 19:47