Estimating Rough Volatility and Interpreting Volatility of Volatility
Summary
The discussion explains why a regression intercept used in rough volatility analysis can be interpreted as volatility of volatility. Researchers measure increments in log volatility across sampling intervals, calculate their absolute moments, and regress the logarithm of those moments on the logarithm of the interval length. The slope is related to the roughness or Hurst exponent, while the intercept can be reparameterized to recover the scale parameter ν.
The replies connect this empirical scaling relation to a model in which log volatility follows a fractional Brownian motion, with ν setting the volatility scale of log volatility itself. For the second moment, ν corresponds to the square root of the relevant moment scale, clarifying why it is called volatility of volatility. The explanation relies on the cited model relationships and regression interpretation; it is not a full derivation or a practical estimation guide. The fitted regression terms are reparameterizations, and the discussion does not address sampling choices or estimation uncertainty.
Key ideas
- The method regresses log absolute moments of log volatility increments on log sampling intervals.
- The regression slope is related to the roughness or Hurst exponent.
- The intercept can be reparameterized to estimate the scale parameter ν.
- ν measures the volatility of log volatility, rather than the volatility of the underlying asset price.
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# Understanding volatility of volatility in realized roughness
# 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.
## Answer by Hans-Peter Schrei (score 4, accepted)
https://quant.stackexchange.com/a/74867
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
## Answer by kwinto (score 4)
https://quant.stackexchange.com/a/74827
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 $H$.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.