I want to estimate the weekly standard deviation of a lognormal process in a usual setup.
$$ \frac{dS}{S} = (\dots) dt + \sigma dW $$ where $\sigma$ is a constant and $W$ a brownian motion.
The usual estimator of the standard deviation is $$ \hat{s} = \sqrt{\frac{\sum_{i=1}^n (r_i - \overline{r})^2}{n-1}} $$ where $r_i = \ln{\frac{S_i*5\ days}{S_{(i-1)*5\ days}}}$ and $\overline{r}$ the average of those returns.
I have a daily timeseries and I am not trying to capture some kind of "day specific effect", so I'd like to use all the rolling increments $r_i = \ln{ \frac{S_i}{S_{i-5\ days}} } $ to have more samples. My issue is that those returns are correlated.
Is there an unbiased estimator for the correlated returns?
Thanks for your help.