Turning thirty daily prices into an annual vol

Question

You have $30$ consecutive daily closing prices for a stock. How do you estimate the historical volatility to plug into Black-Scholes?

Short answer

Take the sample standard deviation of the $29$ daily log returns and annualise. Compute $u_i = \ln(S_i/S_{i-1})$ , then $s = \sqrt{\frac{1}{28}\sum(u_i - \bar u)^2}$ , and set $\hat\sigma = s\sqrt{252}$ (trading days per year). Thirty days gives a noisy estimate, and adjustments for dividends or weekends may be needed.

Adapted from Crack, Heard on the Street, Q2.47, p.31; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Reading the Implied Vol Surface — Moneyness, Term Structure, and Quoting Conventions
Derivation — From Binomial Trees to Black-Scholes in the Continuous Limit
The Black-Scholes Formula Derived via Change of Measure

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