Implied and realized volatility

In plain language — Realized volatility summarizes how returns were dispersed in a sample that has already been observed. Implied volatility is the number that, when entered into a valuation model, makes an option's theoretical value consistent with a selected market price.

The abbreviations RV (realized volatility) and IV (implied volatility) often appear on the same chart, but they are not two raw readings of the same quantity. Both require conventions. A valid comparison needs a horizon, timestamp, data set, formula, annualization method, model and option coordinate.

RV looks at a path of returns and depends on how that path is sampled. IV is recovered from the price of a specific option and depends on the model used to invert that price. Neither one guarantees future volatility, and the difference between them is not, by itself, an available return.

RV and IV have different origins, horizons and uncertainties. The diagram follows the data, transformations and checks needed before comparing them.

Realized volatility: a sample statistic

Let P_t denote prices recorded at consistent intervals. A common approach uses log returns:

r_t = ln(P_t / P_{t-1})

s = sqrt[ Σ(r_t − r̄)² / (n − 1) ]

annualized RV = s × sqrt(A)

n is the number of returns and A is the assumed number of intervals in a year. Daily data often use A = 252, but this is a convention, not a natural constant. Weekly, hourly or five-minute data require a consistent factor; markets that trade continuously and markets with defined sessions do not automatically share the same calendar.

This formula is only one possible estimator. A reproducible measure should state at least:

whether it uses closes, mid-prices, trades or another observation;
simple or logarithmic returns;
sampling frequency and time zone;
rolling window or fixed period;
treatment of overnight moves, holidays, missing data and corporate actions;
estimator, degrees of freedom and annualization factor.

Estimators based on high, low, open and close use different information from close-to-close volatility. Very frequent intraday data can incorporate market microstructure effects, bid-ask bounce and synchronization errors. The word “realized” therefore does not remove model risk: it identifies the ex-post side of the measure.

Neutral example

A series of twenty daily returns has a sample standard deviation of 1.25%. Using 252 trading days:

annualized RV ≈ 1.25% × sqrt(252) ≈ 19.8%

The result describes that window, series and formula. A sixty-day window, intraday data or a weighted estimator can produce another number without either result necessarily being “wrong”.

Implied volatility: inverting a price

An option price is observable as a quote or transaction. Its IV is not directly observable. A model M is selected, the other inputs are fixed, and the value σ_imp that satisfies the following relation is found:

selected price = M(S, K, T, r, q, σ_imp; conventions)

Inputs generally include the underlying or forward, strike, time to expiry, rates, dividends or other cash flows, exercise style and settlement method. Black–Scholes (1973) is the classic reference for European options under specific assumptions; trees, models for options on futures and other methods change the inversion problem. Saying “IV is 24%” without identifying the contract, price and model omits part of the definition.

The price to invert is also a choice. Bid, ask, mid and last trade can produce different IVs. A stale last price, a wide market or an option without a reliable bid does not become informative merely because software returns many decimal places. Keeping an IV bid–ask interval and timestamp is often more accurate than reporting one point.

IV is local to the series: underlying, strike, expiry, option type and style. Options in the same expiry may have different IVs across strikes; different expiries form a volatility term structure. All coordinates together form the volatility surface.

Comparing IV and RV without changing the question

The horizons must be comparable. A trailing twenty-session RV summarizes the recent past; the IV of an option with ninety days remaining belongs to another interval. A readable comparison might specify “30-day close-to-close RV, annualized with 252 observations per year” and “mid IV of the roughly 30-day expiry, at a defined moneyness, at 4:00 p.m.”

The difference IV − RV is not proof of mispricing. An option price can incorporate demand for protection, jump risk, model uncertainty, liquidity, carry and compensation required to bear a nonlinear payoff. Future RV is still unknown; costs, hedging and the underlying's path affect the result of a position.

For example, twenty-day RV is 19.8% and the mid IV of a specific 30-day option is 24.0%. The 4.2 volatility-point gap is not a collectible profit. Turning it into a result would require a position, quantities, Greeks, a price path, a hedging rule, executable prices and costs. Vega only approximates the local response to a change in IV while holding other inputs fixed.

IV, expectation and forecast are not synonyms

IV is commonly interpreted as forward-looking information because it comes from prices of options with remaining life. It remains a parameter implied by a pricing relationship, not a direct observation of a future standard deviation. Its interpretation depends on the risk measure, model assumptions and premia embedded in market prices. It is therefore not a certain or guaranteed forecast of the volatility that will be realized.

Even a “model-free” measure, such as an index built from a basket of options under a published methodology, is not definition-free: strike selection, interpolation, target maturity and quote filters are part of the result. “Model-free” means that the calculation does not rely on one parametric option pricing model; it does not mean that rules are absent.

Common errors

Calling IV raw data. The raw input is a price or quote; IV comes from an inversion.
Comparing different windows without saying so. A one-year RV and a seven-day IV do not describe the same horizon.
Treating the last trade as the current market. A stale trade may lie outside the current bid-ask.
Omitting the coordinate. “The underlying's IV” hides strike, expiry and the ATM or delta convention.
Reading IV above RV as an automatically profitable sale. This ignores tails, jumps, hedging, spread, margin and loss risk.
Changing annualization halfway through the analysis. The number can change solely because the calendar changed.

Checklist

Which prices and returns feed RV?
Which window, frequency, estimator and annualization are used?
Which option, timestamp and side of the quote generate IV?
Which model, rate, dividend, forward and day-count convention are used?
Are the IV and RV horizons truly comparable?
Is IV a point, a mid or a bid-ask interval?
Do smile, skew and events make one coordinate insufficient?
Which costs and risks separate the observed gap from realizable P&L?

Sources

Options Industry Council, Technical Information — operational definitions of historical and implied volatility.
Options Industry Council, Volatility & the Greeks — the relationship between volatility, valuation and Greeks.
Options Industry Council, Options Quotes & Calculators — comparison of historical and implied measures and the model-dependent nature of outputs.
CME Group, Understanding Options: a real-world example on the impact that skew can have on an options position — IV, prices and differences across strikes.
CME Group, Understanding Options Analytics: Greeks and Implied Volatility — analytical coordinates and sensitivities.
Fischer Black and Myron Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 1973 — primary source for the classic model.
The Options Clearing Corporation, Characteristics and Risks of Standardized Options — contractual features and risks of standardized options.