Historical Volatility from 30 Days

This interview question asks how to turn a short series of daily stock prices into the volatility input needed for a Black–Scholes option pricing model. The candidate must outline a clean, practical workflow: transforming prices into an appropriate return series, extracting a measure of dispersion from that series, and then expressing it at the correct time scale for use in a continuous-time model. It appears frequently in quant analyst and derivatives pricing roles, especially where equity options or volatility products are involved, and serves as a gateway to more advanced volatility modeling.

The solution leans on basic time-series ideas, particularly the distinction between prices and returns, and the preference for log returns in continuous-time finance. It requires familiarity with sample variance and standard deviation, and with the conventions for annualizing short-horizon volatility. Interviewers listen for correct use of log returns, a clear treatment of degrees of freedom, appropriate scaling to annual units, and awareness of practical conventions such as trading days per year. They also look for concise reasoning, not just formulas, to show the candidate understands each step conceptually.

What it tests

Volatility estimation for financial models relies on the statistical properties of returns, not raw prices, because returns are stationary and their variability directly reflects risk. The key is to use logarithmic returns, which are additive over time and compatible with the continuous compounding assumption underlying models like Black-Scholes. Sample variance of these returns captures the dispersion, but since data is sampled discretely (daily), the variance must be annualized to match the time units required by the model. This is done by scaling the daily variance by the number of trading days in a year, reflecting the assumption that returns are independent and identically distributed over time.