Skip to content
Financial Context Wiki

Historical Volatility: Calculation, Windows, and Interpretation

For educational purposes only; not investment advice.

Direct answer

Section titled “Direct answer”

Historical volatility (HV), also called statistical or realized volatility in some contexts, measures how dispersed an asset’s past periodic returns were under a stated method. A common close-to-close estimate takes the sample standard deviation of daily returns and multiplies it by √252 to express the result on an annualized scale.

HV measures variability, not performance or direction. A stock can fall steadily with low HV or finish unchanged after large moves with high HV. Every useful HV figure must identify its return definition, sampling frequency, lookback window, price adjustment, and annualization factor.

Calculation and choices

Section titled “Calculation and choices”

Start with a consistent price series. For U.S. stocks, adjusted closing prices help prevent splits and distributions from appearing as economic price shocks. Intraday data answer a different question and contain bid-ask bounce, thin trading, and other market-microstructure effects.

Two common return definitions are:

  • Simple return: Rₜ = Pₜ / Pₜ₋₁ - 1
  • Log return: rₜ = ln(Pₜ / Pₜ₋₁)

They are close for small moves but not identical. Log returns add across time; simple returns map directly to holding-period percentage gains and losses. A comparison must use the same choice throughout.

For n returns, the sample standard deviation is:

s = √[Σ(rₜ - r̄)² / (n - 1)]

With daily observations, a conventional annualized estimate is HV = s × √252. The square-root rule relies on assumptions such as approximately additive variance and a stable return process. It standardizes the unit; it does not turn a 20-day sample into a one-year forecast. Weekly and monthly estimates may use √52 and √12, respectively, but trading calendars and vendor rules differ.

A rolling 20-day HV adds the newest return and removes the oldest each day. It can drop sharply when a large old move leaves the window even if nothing unusual happened today. Short windows react quickly but are noisy; long windows are smoother but slow to recognize regime changes.

Close-to-close HV ignores the path within each session. Range-based estimators use high and low prices; Parkinson’s estimator is one published example. Such methods have their own assumptions and should not be mixed silently with close-to-close figures.

Worked calculation

Section titled “Worked calculation”

Suppose six adjusted closes are $50.00, $50.50, $49.75, $51.25, $50.80, and $52.00. They produce five simple daily returns:

1.000%, -1.485%, 3.015%, -0.878%, 2.362%

Their mean is 0.803%. Using n - 1 = 4, the sample standard deviation is 1.964%. Annualizing with 252 trading days gives:

1.964% × √252 = 31.17%

Using log returns from the same prices produces an annualized estimate of 30.95%. Neither is a mistake; they answer the calculation under different documented return conventions. With only five observations, both are extremely unstable and are examples rather than reliable estimates.

Now suppose a dashboard shows 20-day HV of 38%, 60-day HV of 27%, and one-month at-the-money IV of 46%. This says recent realized variability exceeded the longer-window estimate and option prices embed a still-higher forward-looking input. It does not prove IV will fall or that selling options will profit: the upcoming path may include a jump larger than either historical window captured.

Risks and validation

Section titled “Risks and validation”
  • Window risk: one outlier can dominate a short sample; a long sample can dilute a new regime.
  • Data risk: unadjusted splits, missing observations, stale prices, and duplicate rows create false returns.
  • Method risk: simple versus log returns, n versus n - 1, demeaning, and annualization choices change the result.
  • Sampling risk: daily closes omit intraday ranges and overnight/intraday decomposition.
  • Model risk: square-root scaling becomes less dependable with serial dependence, volatility clustering, or jumps.
  • Trading risk: HV is not an option value, a loss limit, or evidence that an IV-HV gap is mispricing.

For reproducible analysis, retain the price source, adjustment flag, timezone, observation dates, missing-data rule, formula, window, and annualization constant. Compare series only after matching those fields.

Common misconceptions

Section titled “Common misconceptions”
  • “31% HV means the stock moved 31% last year.” It is an annualized standard deviation estimate, not cumulative return.
  • “HV predicts next year’s range.” It describes a selected past sample; future volatility can differ immediately.
  • “Higher HV means a negative return.” Volatility has no direction.
  • “Annualizing adds information.” Multiplication by √252 only changes scale.
  • “One platform’s HV must match another’s.” Inputs and conventions may differ.
  • “IV above HV is a complete sell signal.” IV looks forward through option prices while HV looks backward; the gap omits payoff, event, liquidity, and tail risks.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”