What Is Realized Volatility and How to Compute It

Close-to-close and intraday estimators — how to compute realized volatility, why the simple formula is noisy, and when to use something better.

Realized volatility is the annualized standard deviation of an asset's observed returns over a past window. It is the market's actual price variation measured after the fact, as opposed to the forward-looking number implied by options prices. Understanding the realized volatility calculation matters for every quantitative task that touches risk: sizing positions, evaluating strategies, and — most practically — deciding whether an option is cheap or expensive by comparing it to what the underlying has actually done. A useful starting point is the options pricing calculator, which takes volatility as an explicit input and lets you see directly how the number drives price.

The close-to-close realized volatility calculation

The textbook estimator uses daily closing prices. For a window of n trading days, compute the log return on each day i:

r_i = ln(P_i / P_i−1)

Then estimate the daily standard deviation:

σ_daily = √[ (1 / (n−1)) · Σ(r_i − r̄)² ]

and annualize by multiplying by √252, the conventional number of trading days in a year:

σ_ann = σ_daily × √252

Worked example: suppose a stock closes at 100, 101, 99, 102, 100, 103 over five consecutive days. The four daily log returns are approximately 0.00995, −0.02020, 0.02985, −0.01980. The mean is about 0.00. The variance is roughly (0.00990 + 0.04080 + 0.08910 + 0.03920) / 3 = 0.05967, giving a daily standard deviation near 0.01987, or 1.987%. Multiplied by √252 ≈ 15.87, that is a realized vol of about 31.5% annualized — from just four observations. Small samples are noisy; that number can swing dramatically with one more day of data.

Why √252 annualizes daily volatility

Volatility scales with the square root of time, not time itself, because returns are assumed to be independent from day to day. If the variance of a single day's return is σ_d², then the variance over 252 independent days is 252 · σ_d². Taking the square root gives σ_ann = σ_d · √252. The same logic applies to any frequency: if you measure vol from weekly closes you multiply by √52; from monthly closes, by √12. This is also why understanding the distinction between variance versus volatility matters — variance adds linearly across time, volatility does not.

One practical note: some practitioners divide by n rather than n−1 (population rather than sample variance). For windows of 20 or more days the difference is negligible. For windows of 5 or 10 days it is material. Use n−1 when the window is short.

The limitation of close-to-close: ignoring the intraday range

The close-to-close estimator uses one data point per day — the closing price — and discards everything that happened in between. On a day when a stock opens, rallies 4%, sells off 4%, and closes flat, the estimator records zero volatility for that session. That is a significant information loss. Two problems follow:

• Inefficiency. The estimator needs a longer window to achieve the same statistical precision as estimators that use high, low, and open data. In formal terms, it has higher variance of the variance estimate.
• Gap blindness in reverse. Overnight gaps — when the close and the next open differ materially — are captured, but the intraday path is not. Heavy intraday movers on low overnight gaps look artificially calm.

For highly liquid instruments where intraday data is available, range-based estimators are almost always preferable.

Range-based estimators: Parkinson, Garman-Klass, and beyond

Range-based estimators incorporate the daily high (H) and low (L) — and sometimes the open (O) — alongside the close. They are more efficient because the high-low range contains information about the full intraday path that a single end-of-day return discards.

• Parkinson (1980). Uses only the high and low. σ² ≈ (1 / 4 ln 2) · (1/n) · Σ[ln(H_i/L_i)]². The efficiency gain over close-to-close is roughly 5×, meaning you need one-fifth as many observations for equivalent precision.
• Garman-Klass (1980). Adds the close (relative to open) to the Parkinson range term: σ² ≈ (1/n) · Σ[ 0.5·(ln H_i/L_i)² − (2 ln 2 − 1)·(ln C_i/O_i)² ]. Efficiency roughly 7–8× close-to-close.
• Rogers-Satchell (1991). Allows for a non-zero drift and is unbiased even when returns have a trend: σ² ≈ (1/n) · Σ[ ln(H_i/C_i)·ln(H_i/O_i) + ln(L_i/C_i)·ln(L_i/O_i) ]. Useful for indices or trending assets over longer windows.
• Yang-Zhang (2000). Combines an overnight return variance term, a Rogers-Satchell intraday term, and a close-to-open term with a weighting constant k (typically 0.34). It is unbiased, independent of drift, and handles opening gaps correctly. It is generally the most robust single estimator for daily OHLC data.

The tradeoff: range-based estimators assume continuous trading and no jumps. When a stock halts and reopens with a large gap inside the session, the high-low range can overstate vol. In practice the bias is small for liquid instruments on normal days.

Choosing a lookback window: bias versus responsiveness

The window length is a parameter, not a constant, and the right choice depends on the application.

• Short windows (5–10 days) respond quickly to regime changes but have high estimation variance — a single large day can dominate the reading. Useful for short-dated options or momentum strategies where recent vol is what matters.
• Medium windows (20–30 days) are the most common convention. A 21-day window corresponds roughly to one calendar month of trading days and balances noise against staleness.
• Long windows (60–252 days) give a stable, slow-moving estimate of structural vol. Appropriate for risk budgeting, longer-dated options, and mean-reversion benchmarks.

An exponentially weighted moving average (EWMA) is an alternative: rather than weighting all observations equally, it downweights older returns geometrically. The RiskMetrics standard uses a decay factor λ = 0.94, meaning yesterday's squared return enters with a weight of 0.94 relative to today's. EWMA is responsive without the cliff-edge behavior of a rolling window that drops a large observation on day n+1.

Using realized volatility to value options

The most direct use of realized vol is comparing it to implied vol. If an option is quoted at an implied volatility of 28% but the underlying has realized 18% over the past 20 days, the option is — in a backward-looking sense — expensive. If the realized reading were 35%, the option would look cheap. This spread, implied minus realized, is what systematic volatility sellers attempt to harvest as a risk premium.

The comparison is not straightforward. Implied vol is forward-looking by definition: the market prices in anticipated events — earnings, macro data, central bank decisions — that may not appear in recent realized readings. The relationship between the two is explored in detail in the piece on VIX versus realized volatility, which covers how the implied-realized spread behaves across different market regimes. The practical takeaway: realized vol gives you the empirical anchor; implied vol is the market's forecast. The gap between them is the signal, not either number in isolation.