Historical Volatility
Historical volatility (HV), also known as realized volatility, is the annualized standard deviation of an asset's past logarithmic price returns over a specified lookback period. It measures how much an asset's price has actually fluctuated over that period and serves as the primary empirical input for assessing whether options are relatively cheap or expensive compared to implied volatility.
Key takeaways
- Historical volatility is backward-looking, measuring past price variability; implied volatility is forward-looking, reflecting the market's expectation of future volatility embedded in option prices.
- The relationship between historical and implied volatility defines the 'vol risk premium': IV typically exceeds HV by 2-4 volatility points on average, reflecting the premium investors pay for options-based insurance.
- HV is sensitive to the lookback window: short windows (5-10 days) produce volatile estimates; long windows (1 year) produce smoother but potentially stale estimates.
- Volatility is itself time-varying and mean-reverting — periods of low volatility tend to be followed by rising volatility (and vice versa), described by GARCH-family models.
- Realized volatility ('RV') calculated from intraday high-frequency data provides more precise estimates than end-of-day close-to-close HV measures.
Explanation
Historical volatility is the most fundamental empirical measure of an asset's price variability, computed from the time series of the asset's past logarithmic price changes. The use of log returns (rather than arithmetic returns) ensures that the volatility estimate captures proportional price movements symmetrically and remains consistent with the lognormal price assumption underlying the Black-Scholes model. Log return at time t: r_t = ln(P_t / P_{t-1}).
The annualization convention multiplies the daily standard deviation by the square root of the number of trading days in a year (typically 252 for equities). This scaling is derived from the assumption that daily returns are independent and identically distributed — the square root of time rule for variance aggregation. In practice, returns exhibit serial correlation and conditional heteroskedasticity (GARCH effects), meaning the square-root-of-time rule is an approximation rather than an exact transformation. For short horizons and highly serially correlated assets (some commodities, for example), the approximation can materially misstate multi-day volatility.
The comparison between historical volatility and implied volatility is central to options trading strategy. The 'volatility risk premium' (VRP) — the systematic tendency for implied volatility to exceed subsequent realized volatility — creates a persistent structural opportunity for volatility sellers. An options trader who consistently sells straddles or strangles when IV significantly exceeds recent HV is exploiting this premium, collecting the difference between the volatility priced into options and the lower volatility that actually materializes. Academic studies confirm the VRP has been positive and statistically significant across equity, currency, and commodity markets over extended periods, though with significant time variation.
GARCH (Generalized Autoregressive Conditional Heteroskedasticity) models, introduced by Robert Engle (Nobel Prize, 2003), formalize the time-varying nature of volatility. The GARCH(1,1) model — the industry standard for volatility modeling — expresses today's conditional volatility as a weighted average of the long-run average variance, yesterday's squared residual (news about volatility), and yesterday's conditional variance (persistence). This model captures two fundamental stylized facts of financial time series: volatility clustering (high-volatility periods tend to follow each other) and mean reversion (volatility gravitates back toward its long-run average). GARCH-based volatility forecasts are widely used by options traders, risk managers, and algorithmic trading systems.
Formula
HV = σ_daily × √252; σ_daily = √(Σ(r_t - r̄)² / (n-1)); r_t = ln(P_t / P_{t-1})
Example
A derivatives trader calculates the 30-day historical volatility of Apple (AAPL) shares using the past 30 trading days' log returns. After computing daily log returns and their standard deviation (σ_daily = 1.42%), they annualize: HV_30 = 1.42% × √252 = 22.5%. The trader observes that the 30-day at-the-money implied volatility for AAPL options is 27.0% — a vol spread of 4.5 points (IV − HV). Given the historical average vol premium of approximately 3 points for AAPL, the current premium of 4.5 points suggests options are somewhat rich. The trader sells a 1-month strangle (selling both an OTM call and put), collecting approximately $3.80 in premium, positioning to profit if realized vol over the next 30 days remains below 27% (the breakeven IV at which the position generates zero profit).
Related terms
Aggregation Algorithmic Trading At The Money Black Scholes Model Contango Correlation Declaration Date Equity Implied Volatility Mean Reversion Paycollect Premium