The Key Idea
Historical volatility is the annualised standard deviation of log returns. A fundamental property is that the result is the same regardless of the sampling interval — daily, weekly, or monthly — as long as the data covers the same period and you use the correct annualisation factor.
The annualisation factor scales one period's standard deviation up to a full year:
Daily: σ_annual = σ_daily × √252
Weekly: σ_annual = σ_weekly × √52
Monthly: σ_annual = σ_monthly × √12
This works because variance (σ²) scales linearly with time, so n independent periods have variance n × σ², and standard deviation scales with √n.
Gold Futures (GC=F) — Oct 2008 to Jul 2012 Full-period annualised volatility
The three estimates differ only due to compounding and the finite sample size — they converge to the same value as the sample grows.
Rolling Annualised Volatility Daily 40-day vs Weekly 5-week vs Monthly 6-month windows
Weekly and monthly series are noisier because each window has fewer observations than the daily series, but all three track the same underlying volatility level.