Estimating and annualising volatility

In short

• Volatility = the standard deviation of continuously compounded returns, quoted on an annual basis by convention.
• Recipe: prices → $\ln(P_t/P_{t-1})$ → sample standard deviation (divide by n − 1) → multiply by $\sqrt{250}$.
• 250 is the approximate number of trading days in a year. Weekly data use $\sqrt{52}$; monthly data use $\sqrt{12}$.
• The daily mean times 250 estimates the expected annual continuous return, but a short sample makes that estimate close to useless.
• n + 1 prices give only n returns.