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If you generate random stock price paths according to a GBM with daily increments, what will be the distribution of the realized volatility? Assume that the realized volatility is measured over daily increments for the whole year.

I assume that the realized volatility is an unbiased estimator of the parameter sigma in the GBM, but I've never seen it proved. As for the higher moments, what can be said ?

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  • $\begingroup$ Is GBM stands for Generalized Brownian Motion and you assume that $\sigma$ is not constant? $\endgroup$
    – zer0hedge
    May 2, 2017 at 12:27
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    $\begingroup$ The realized variance will have a Chi-Squared distribution, centered around the (constant) true variance used in the MonteCarlo simulation. A realized variance calculated from N non-overlapping observations will be Chi-Squar with N d.o.f. $\endgroup$
    – nbbo2
    May 2, 2017 at 14:28
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    $\begingroup$ @noob2 - I think it is N-1 degrees of freedom, but otherwise spot on because of the non-overlapping (independent) increments of equal time length. $\endgroup$ May 2, 2017 at 20:05

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I assume the OP means what is the variance, across N paths of M steps each, of the realised volatility of the GBM.

You would imagine that it is related to the dt-step size implied by M. i.e. for lower M, the variance of the pathwise-estimated volatility increases.

Feels like is related to strike adjustments for variance swaps with discrete observations, not sure if there is literature out for that but worth googling

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