Estimating historical volatility from inconsistent time intervals

Question

Given historical asset prices at consistent time intervals, one can estimate annual volatility as:

SampleStDev(log(S_i/S_i-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 1

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 2

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 $T_n-T_0=n\cdot\Delta T$.