In The Volatility Smile book by Derman & Miller at pag. 113, I don't understand the statistical uncertainty in the measurements of volatility and how to interpret the notation.
The authors state that the volatility of the hedging error is, approximately,
$$ \sigma_{HE} \approx \frac{\sigma}{\sqrt n}\frac{\partial C}{\partial \sigma} $$
Then to interpret it, they write:
Suppose we measure the volatility of one lognormal stock process by taking n discrete measurements of the price. The statistical uncertainty in the measurement of the volality estimate is $d\sigma=\sigma/\sqrt n$. [...]
My questions:
- How can I link the "differential" $d\sigma$ to the statistical uncertainty?
- How can I calculate it? Is it simply related to the fact that $E[r_i^2]=\sigma^2$ and to estimated to variance one takes the averages of the n squared log-returns $r_i$ and then you calculate the variance of the variance?
Maybe my doubts are trivial or due to a lack of understanding the notation. I don't know.
Please, let me know if further details are needed. Thanks for your help.
Edit: possible answers to be checked
Assuming $r_i$ to be zero mean and with $\sigma^2$ variance log-returns.
$\sigma$ is a constast in the usual BSM setting. So, if we are estimating it, how much could this costant move by? It could move by its estimation error.
An estimate of the variance (square of volatility) could be $\bar \sigma^2 = n^{-1}\sum_{i=1}^{n}r_i^2$. Its expected value is $\sigma^2$ and its variance (measurement error) is $\sigma^2/n$ Thus, without taking in consideration the Jensen effect, the measurement error of the volatility is the square root of $\sigma^2/n$, that is $\sigma/\sqrt n$.
