Estimate of GBM Return Variance

Question

I typically see people define realized variance as the squared difference in log returns, i.e. $$RVar = \frac{1}{T} \sum_{n=1}^N \log \left( \frac{S_{n}}{S_{n-1}} \right)^2$$ where $t_n - t_{n-1} = \delta t$ and $T=N\delta t$. This is obviously an unbiased estimator of log return variance given that the expected log return is 0. However, under geometric Brownian motion, we have $$\mathbb{E} \log \delta S = \delta t \left(\mu - \frac{1}{2} \sigma^2\right)$$ so $$\mathbb{E} \log \delta S^2 = \delta t \sigma^2 \delta t \left(\mu - \frac{1}{2} \sigma^2\right)$$ where $\mu$ is the growth rate of the underlying. This suggests that $RVol$ is not unbiased for $\sigma^2$, except in a special case. For now, assume $\mu$ is known to avoid statistical degrees of freedom issues. My questions are two:

(1) Why would one not attempt to correct for this bias in a volatility estimate?

(2) Estimating $\sigma$ is of special interest in options. Under risk-neutral measure, $\mu = r$ the risk-free rate. Which would be more correct to use in our unbiased volatility estimate? It seems suspect that volatility stays constant under change of measure, yet the choice of measure effects our volatility estimate.

Answer 1

It is correct that the definition of realized variance that you gave depends on the drift of $S_n$ which seems unwanted at first glance. This is however the official definition that is used in derivative contracts. The reason for that is I think that if we pass to the limit of $N\to\infty$ in the fixed time interval $[0,T]$ (that is "continuous sampling") we obtain the quadratic variation of $\log S\,:$ $$ \frac 1T\sum_{i=1}^N\left(\ln\left(\frac {S_n}{S_{n-1}}\right)\right)^2\to\frac 1T\int_0^T\sigma^2(s)\,ds $$ from which the drift has disappeared. Statisticians call this a consistent estimator (here of quadratic variation). I think that attempts to correct for the "bias" in the discrete case might spoil this consistency and lead to contractual complications in options on realized variance.

If you just want to estimate historical volatility you are of course free to use another estimator. Popular is the sample variance of the returns which comes in different flavours.

Your second question I find odd. A historical time series is given to us and we cannot change the probability measure under which it was produced. All we can do is to estimate volatility from it and use that in a model in which we can choose the measure. The good thing however is that the Girsanov theorem does not change quadratic variation and/or volatility. It changes only the drift.