Estimating Volatility from Unequal Observation Intervals
Summary
The document explains how to estimate annualized return variance when price observations are separated by unequal, known time intervals. Under a geometric Brownian motion assumption and zero mean, one answer forms a time-scaled estimate from squared log returns divided by each interval, then normalizes by the total elapsed time. For equal intervals, this reduces to the familiar variance of log returns divided by the interval length.
A second answer models each observed return as an aggregation of independent, identically distributed daily returns. With normally distributed increments and known observation-window lengths, it derives maximum-likelihood estimates for drift and variance, weighting each squared residual by its interval. The approaches make assumptions about the return process and interval lengths; the likelihood treatment also assumes known aggregation windows and normal increments. The material gives formulas rather than empirical comparisons, and it does not discuss random or uncertain observation timing.
Key ideas
- Unequal observation intervals must be accounted for when converting observed returns into annualized variance.
- Under a zero-mean geometric Brownian motion assumption, each squared log return is scaled by its interval and the total elapsed time.
- With aggregated normal increments, maximum-likelihood estimation uses each observation window in estimating drift and variance.
- The presented formulas rely on known time intervals and stated distributional assumptions.
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Full text
# Estimating historical volatility from inconsistent time intervals
# Estimating historical volatility from inconsistent time intervals
Given historical asset prices at consistent time intervals, one can estimate annual volatility as:
SampleStDev(log(Si/Si-1)) / sqrt(interval)
What's the correct way to do this when the time intervals are inconsistent (e.g. S2 was observed 1 day after S1, but S3 was observed 5 days later)? Should longer time intervals get greater weight in the estimate?
## Answer by Kermittfrog (score 2, accepted)
https://quant.stackexchange.com/a/70805
To make @nbbo2's answer more precise, let's assume that we observe various sums $z_{k,h}$ of independent and identically distributed random variables $x_i$ (i.e. returns),
$$ z_{k}\equiv \sum_{i=k-h_k+1}^k x_i $$ where $h_k$ is the horizon of aggregation. For simplicity, let's aggregate over one, two, three, ... individual daily returns, $x_i$. On a "normal" day, we'd have $h=1$, and on a Monday, we usually have $h=3$.
Assuming normally distributed individual return contributions, and "smallest time fraction" $\Delta$, e.g. $\Delta = 1/255$, we have
$$ x_i\sim N(\mu\Delta,\sigma^2\Delta)\Rightarrow z_k\sim N(\mu\Delta_k,\sigma^2\Delta_k) $$ where $\Delta_k=\sum_{i=k-h_k+1}^k\Delta=\Delta \times h_k$.
We can now find the maximum-likelihood-estimators. Given $n$ observations $z_1,\ldots,z_n$ and knowledge of all the individual observation windows $\Delta_1,\ldots,\Delta_n$ the log likelihood is
$$ l(z)=-\frac{n}{2}\ln(2\pi)-\frac{n}{2}\ln(\sigma^2)-\frac{n}{2}\sum_k^n\ln(\Delta_k)-\frac{1}{2}\sum_k^n\frac{(z_k-\mu\Delta_k)^2}{\sigma^2\Delta_k} $$
The maximum likelihood estimators are found as:
$$ \hat{\mu}=\frac{\sum_k z_k}{\sum_k \Delta_k} $$ and $$ \hat{\sigma^2}=\sum_k\frac{\left(z_k-\hat{\mu}\Delta_k\right)^2}{n\Delta_k} $$
## Answer by nbbo2 (score 0)
https://quant.stackexchange.com/a/70763
Assume GBM. Suppose there are $n+1$ observations, indexed by $j=0,\cdots,n$. The stock price is $S_j$ at time $T_j$ where time is measured in yearly units.
The annualized variance (assuming zero mean) is
$V=\frac{1}{T_n-T_0}\sum_{j=1}^n \ln(S_j/S_{j-1})^2/(T_j-T_{j-1})$
this is true whether the intervals are equal (i.e. $T_j-T_{j-1}=\Delta T, \forall j$) or unequal but fixed in length (as long as they are nonrandom).
When the intervals are equal it simplifies to the more familiar form
$V=\frac{1}{\Delta T} Var(\ln(S_j/S_{j-1})) $ since $T_n-T_0=n\cdot\Delta T$.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.