Estimating Volatility Uncertainty and Its Effect on Hedging Error
Summary
The document explains an approximation from options literature that links the volatility of hedging error to the uncertainty in an estimated volatility. For a lognormal stock process sampled at discrete intervals, the stated uncertainty is proportional to volatility divided by the square root of the number of observations. Multiplying that uncertainty by the option value’s sensitivity to volatility connects estimation error to hedging error.
The answer supports this interpretation using squared, zero-mean log returns: their sample average estimates variance, and the document gives the variance of that estimator as volatility squared divided by the sample count. Taking the square root yields the stated volatility uncertainty, while ignoring the Jensen effect. This is a simplified explanation; it assumes the stated return setup and does not develop corrections for nonzero means, dependence, or other departures from the model.
Key ideas
- The stated hedging-error approximation scales with an option’s sensitivity to volatility.
- With the document’s assumptions, volatility estimation uncertainty is volatility divided by the square root of the sample count.
- The estimate begins with the average of squared, zero-mean log returns.
- The derivation ignores the Jensen effect and does not address broader model departures.
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Full text
# Volatility of Hedging Error and Statistical Uncertainty of Estimates
# Volatility of Hedging Error and Statistical Uncertainty of Estimates
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 $\sigma/\sqrt n$.
## Answer by Enrico (score 4, accepted)
https://quant.stackexchange.com/a/83795
Answers
- $\sigma$ 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 $\sigma/\sqrt n$.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.