# Realized Variance: Formula, Annualization, and Measurement Choices

Learn how realized variance sums squared observed returns, why conventions matter, and how to calculate and interpret annualized variance without confusing it with volatility.

Canonical: https://wiki.fcontext.com/options/realized-variance/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

**Realized variance** measures price variation that actually occurred over a stated observation window. A common estimator adds squared log returns and then annualizes the result. Realized volatility is the square root of realized variance.

There is no context-free realized-variance number. The answer depends on the asset, price source, start and end observations, return definition, sampling interval, treatment of the mean, annualization factor, missing data, and corporate actions. A contract's formal specification controls its settlement calculation; a research estimate may use a different convention.

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## Formula and conventions

For prices `P₀, P₁, …, Pₙ`, define each log return as:

`rᵢ = ln(Pᵢ / Pᵢ₋₁)`

One zero-mean daily-return convention is:

`RV = (A / n) × Σᵢ₌₁ⁿ rᵢ²`

where `A` is the assumed number of observations per year, often `252` for daily U.S. trading observations. Annualized realized volatility is `√RV`. Under this convention, a variance of `0.04` corresponds to volatility of `0.20`, or `20%`.

Other estimators subtract the sample mean, divide by `n − 1`, use simple returns, or sample intraday prices. These are not interchangeable. Higher-frequency sampling can capture more intraday movement, but bid-ask bounce, discreteness, asynchronous quotes, and bad ticks create market-microstructure noise. Realized-kernel and related estimators are designed to address parts of that problem.

Cboe's S&P 500 Variance Futures specification provides a concrete contract convention: daily S&P 500 Index log returns, zero assumed daily mean, `252`-day annualization, and specified closing values plus a final special opening quotation. That convention should not be silently substituted for another product or research series.

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## Five-return calculation

Assume five consecutive daily log returns are `+1.00%`, `−0.50%`, `+0.75%`, `−1.25%`, and `+0.25%`. Written as decimals, their squared sum is:

`0.0100² + (−0.0050)² + 0.0075² + (−0.0125)² + 0.0025² = 0.00034375`

Using the zero-mean convention and `252` trading days:

`RV = (252 / 5) × 0.00034375 = 0.017325`

`realized volatility = √0.017325 = 0.1316 = 13.16%`

Variance is `0.017325`, not `13.16%`. If quoted in squared volatility points, the same result is approximately `173.25` because `(13.16)² ≈ 173.19`; small differences arise from rounding. The five-day sample is intentionally short and unstable, so it illustrates arithmetic rather than a dependable risk estimate.

One large observation matters disproportionately because returns are squared. Replacing the `−1.25%` day with `−5.00%` increases annualized variance to `0.13545` and realized volatility to about `36.80%`, even though the other four returns do not change.

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## Measurement and use checklist

- Fix the exact start, end, time zone, calendar, and observation timestamp before calculating.
- Use adjusted or unadjusted prices consistently and handle splits, distributions, stale values, and missing observations explicitly.
- State whether returns are logarithmic or simple and whether the mean is assumed zero or estimated.
- Match the divisor and annualization factor to the selected convention.
- Do not compare a 5-minute estimator directly with a close-to-close estimator without reconciling overnight returns and noise.
- Preserve full precision through the calculation and round only the reported output.
- Separate realized variance to date from expected future variance; before expiration, a variance-linked instrument can contain both.
- Read the governing contract for observation disruptions, corrections, final settlement, multiplier, and quotation units.
- Treat historical realized variance as a measurement, not a forecast or a guaranteed future range.
- Stress jumps and data errors; both can dominate a sum of squared returns.

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## Common misconceptions

- “Realized variance and realized volatility are the same number.” Volatility is the square root of variance.
- “A variance of 400 means 400% volatility.” In volatility-point-squared notation, `400` corresponds to `20` volatility points.
- “More observations always improve the estimate.” Excessive sampling can amplify microstructure noise.
- “Daily return variance includes intraday and overnight risk equally.” Close-to-close returns combine them but do not identify their separate contributions.
- “Subtracting the sample mean never matters.” It may be small for short-horizon financial returns, but it is a defined methodological choice.
- “Realized variance is implied volatility after expiration.” One is computed from observed returns; the other is inferred from option prices under a model.
- “Annualization creates more information.” It rescales an estimate and can make a short, noisy sample look deceptively comparable.
- “Every variance contract uses the textbook formula.” Observation schedules, price sources, disruption rules, units, and settlement terms can differ.

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## Related topics

- [Historical volatility](/options/historical-volatility/)
- [Implied volatility](/options/implied-volatility/)
- [Implied versus realized volatility](/options/implied-vs-realized-volatility/)
- [Corridor variance swap](/options/corridor-variance-swap/)

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## Sources

- [Cboe S&P 500 Variance Futures Contract Specifications](https://cdn.cboe.com/resources/participant_resources/SP_500_Variance_Futures.pdf)
- [Cboe Variance Calculator User Guide](https://www.cboe.com/document/tech-spec/document/technical-specifications/sp-500-variance-futures-variance-calculator-user-guide/)
- [Andersen, Bollerslev, Diebold, and Labys: Modeling and Forecasting Realized Volatility](https://doi.org/10.1111/1468-0262.00418)
- [Barndorff-Nielsen et al.: Designing Realized Kernels to Measure the Ex-Post Variation of Equity Prices](https://doi.org/10.1111/j.1368-423X.2008.00275.x)