Realized Variance, Drift Bias, and Quadratic Variation
Summary
The document examines whether the sum of squared log returns is an unbiased estimate of variance under geometric Brownian motion. At finite sampling intervals, the estimate can reflect drift as well as volatility. The response explains why this convention remains standard for realized variance, including in derivative contracts: as sampling becomes continuous over a fixed period, the sum converges to quadratic variation, whose expression depends on volatility and not drift.
That limiting property motivates keeping the conventional estimator rather than applying a discrete-sample bias correction that could disrupt consistency or complicate contracts. For historical volatility estimation, the response notes that sample variance of returns is another option. It also clarifies that historical data cannot be reassigned to a different probability measure; under Girsanov’s theorem, changing measure alters drift but leaves quadratic variation and volatility unchanged. The discussion is conceptual and does not compare estimators empirically.
Key ideas
- At finite sampling intervals, squared log returns can reflect drift as well as volatility.
- With increasingly frequent sampling, realized variance converges to the quadratic variation of log prices.
- The conventional realized variance definition is used in derivative contracts, and discrete bias corrections may affect its limiting behavior.
- Sample variance of returns is an alternative for estimating historical volatility.
- Changing probability measure changes drift but not quadratic variation or volatility.
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# Estimate of GBM Return Variance
# Estimate of GBM Return Variance
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 by Kurt G. (score 1)
https://quant.stackexchange.com/a/80940
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.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.