What Is Implied Volatility? IV, the Term Structure, and Realized Vol

Implied volatility is the volatility number you have to put into an option pricing model to make the model return the price the option is actually trading at. Realized volatility runs the other way: it is measured after the fact from the actual closes. Both are quoted as annualized percentages, which is the only reason the two can be compared at all.

A quoted volatility figure is incomplete without two labels: which expiry it came from, and over which window it was measured. Most of the confusion around implied volatility comes from dropping one of those labels and then subtracting two numbers that were never describing the same horizon. This page covers what implied volatility is, how realized volatility is computed, why one name carries many implied volatilities at once, what the gap between implied and realized is called, and how vol of vol differs depending on which volatility it is built from. It is descriptive throughout.

Implied volatility is a price restated in volatility units

An option pricing model takes a short list of inputs: the price of the underlying, the strike, the time left to expiry, a risk free rate, and a volatility. In the closed form Black-Scholes model with the dividend yield set to zero, those inputs enter through

``` d1 = (ln(S/K) + (r + s^2/2)T) / (s sqrt(T)) d2 = d1 - s sqrt(T) ```

where `s` is the volatility. Every input in that expression can be looked up except `s`.

Implied volatility inverts the problem. The option already has a market price, so you hold the observable inputs fixed and solve for the `s` that makes the model reproduce that price. Nothing is being predicted at that step: it is a change of units, from dollars of premium into an annualized standard deviation quoted in percent, which is what lets options on different underlyings and different strikes sit on a common scale.

It also means implied volatility carries every assumption of the model used to extract it. Black-Scholes assumes a single constant volatility and lognormally distributed returns. If that description were exact, every strike and every expiry on one name would invert to the same number. They do not, and the shape of the disagreement is its own subject: across strikes at one expiry it is options skew, and across expiries it is the term structure.

Realized volatility is measured from closes

Realized volatility, also called historical volatility, needs no model and no option at all. Take the daily log returns of the underlying over a window of N sessions, compute their sample standard deviation (dividing by N minus 1, not by N), and multiply by the square root of 252, the conventional number of trading sessions in a year. The result is a level in annualized percent, in the same units as an implied volatility. The annualization is arithmetic rather than a claim about anything: a daily standard deviation of one percent scales to about 15.9 percent annualized, because the square root of 252 is roughly 15.87.

The window length is where the judgment sits. A 10 session window responds fast and is noisy, while a 60 session window is steadier and keeps one violent day in the sample for three months. The same underlying can print very different realized volatilities on the same date depending only on the window chosen, which is why the window belongs in the label.

A percentile is a third statistic, not a restatement of the second. Saying that a 20 session realized volatility sits in the 80th percentile answers where today ranks inside its own trailing history, and says nothing about whether the level is 12 percent or 60 percent. The rank also depends on how long that history is, so the honest form publishes the number of sessions it was computed over instead of calling it a year when it is not one.

There is no single "the IV" for a name

Implied volatility varies by expiry, so "the IV of a stock" is always a claim about one particular expiry even when the sentence does not say so. A name can carry a low implied volatility on a contract expiring tomorrow and a much higher one on a contract expiring in a month, or the reverse into an earnings date. Plotting the ATM implied volatility of each listed expiry against its days to expiry gives the term structure, whose slope describes how the market is pricing near horizons against far ones.

Two practical consequences follow. The first is that the nearest listed expiry is often extremely short: on a major index it is routinely one day out, and a one day implied volatility is a legitimate number that describes a single session and nothing beyond it. The second concerns the 30 day figure that so much commentary quotes. A number labeled 30 day is built one of two ways, and they are different operations. Some sources interpolate between the two expiries that bracket 30 days to make a constant maturity series; others, Kresmion among them, select the single listed expiry nearest 30 days inside a sensible band and leave the figure blank when no expiry lands there. A selected expiry may have 24 days left, or 38, so honest labeling names the construction and publishes the days to expiry actually used next to the number.

The VIX takes the other route: it blends the two S&P 500 expiries that bracket 30 days, across a wide strip of strikes rather than a single at the money option, to arrive at a constant 30 day figure. That is a different construction from an ATM implied volatility on one listed expiry, so the two stay related without ever matching exactly.

Matching the tenor before comparing

Implied and realized volatility are quoted in the same units, which makes it easy to subtract one from the other and just as easy to subtract the wrong pair. The comparison only carries meaning when the horizons match.

The classic mistake is to take the nearest listed expiry, which can be one day out, and subtract a 20 session realized volatility from it. That subtraction is arithmetically correct and financially meaningless: one leg describes a single session, the other describes a month of sessions, and most of the difference between them is the difference in horizon.

The diagnostic that catches it is worth keeping. If a supposed premium comes out negative for every name on every date in a sample, the tenor is the first suspect rather than the market, because the most persistent finding in the options literature runs the other way: implied has tended to sit above subsequently realized volatility more often than not. A result that is one sign everywhere, without exception, is more likely a construction error than a discovery.

Even a properly matched pair keeps one asymmetry that cannot be engineered away. A 30 day implied volatility is about the coming 30 days, and a 20 session realized volatility is about the last 20 sessions. Matching the tenor aligns the length of the two windows, not their position in time.

The gap has a name

The difference between implied volatility and realized volatility over matched horizons is what the literature calls the variance risk premium. The name refers to variance, meaning volatility squared, which is how the academic work defines it, while a great deal of practical work computes the same idea as a plain difference in volatility points. The sign convention is implied minus realized, so a positive reading means options were priced above what the underlying went on to deliver, and a negative reading means the reverse.

A negative reading on a single name is not automatically a defect. A trailing realized window holding one enormous session, an earnings gap or a single day collapse, carries that session at full weight until it rolls out, and a cluster of such gaps can hold a whole group of names negative for weeks. What separates that from a broken calculation is scope and persistence: an explainable negative on some names on some dates is ordinary, while every name negative on every date points back at the tenor.

Two rules keep the figure honest. It is undefined unless both legs exist, and defaulting a missing realized leg to zero quietly converts "we do not know" into "the premium is the entire implied volatility". And because it is a difference of two levels, anyone holding the gap and either leg can recover the third by arithmetic.

Vol of vol, and which one you are looking at

Volatility itself moves, and the dispersion of its own daily changes is called vol of vol. There are two of them, and letting one stand in for the other is a common error because both answer the same English sentence. Vol of implied vol is built from the daily changes in a 30 day implied volatility and describes how unstable the market's expectation has been; it needs stored option chain history, so it exists only for names captured long enough to have one. Vol of realized vol is built from the daily changes in a realized volatility series, describes how unstable the outcome has been, and needs nothing but closes.

Their levels are not comparable, for a mechanical reason worth stating plainly. A 20 session realized volatility is itself a rolling statistic, so consecutive readings share 19 of their 20 sessions and their day to day changes are heavily autocorrelated. The realized version is smoothed by construction, so its changes and the implied version's changes are different kinds of object. Each series is read against its own history, never against the other one.

Kresmion computes modeled dealer greeks exposure nightly for a curated universe of US names from listed option open interest, using closed form Black-Scholes with the assumptions stated on the tool. The live SPY panel on this page carries the derived layer: the realized volatility windows, where the 20 session reading ranks inside its own trailing history, the vol of realized vol, and the vol of implied vol, which is a dispersion of daily changes from which no level can be recovered. The at-the-money implied level itself, its term structure by expiry, the 25 delta skew, and implied set against realized sit in Kresmion's greeks tool behind a free account, since those figures sit a good deal closer to the raw option chain than the derived ones do.

Key takeaways

Frequently asked questions

Does implied volatility predict what a stock will do?

No, it describes the backdrop. Implied volatility is the price of options restated in volatility units, so it reports what option buyers and sellers are paying and receiving now, not what the underlying will go on to do. It also says nothing about direction: a high implied volatility is consistent with a large move either way.

Why does one stock have several implied volatilities at the same time?

Because implied volatility is extracted contract by contract, and the model that extracts it assumes something the market does not obey. Options at different strikes on one expiry invert to different numbers, which is skew, and options at different expiries invert to different numbers, which is the term structure. Quoting a single IV means picking one point on that surface, usually the at the money option of a chosen expiry.

What does it mean when implied volatility sits below realized volatility?

It means options were priced for less movement than the underlying delivered over the measured window. Before reading anything into it, check that the two horizons match, since a very short dated implied set against a month of realized produces that result almost mechanically. Where the tenors do match, a trailing window holding one or two enormous sessions is the next thing to check.

Which realized volatility window is the right one?

There is no single right answer, only a tradeoff that should be stated. Short windows such as 10 sessions react quickly and jump around, and long windows such as 60 sessions are steadier but keep a single extreme day in the sample far longer. The practical rule is to name the window every time the number is quoted, and to match it to the implied tenor whenever the two are compared.

This page is information, not investment advice.

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Source: closed form Black-Scholes, with the modeling assumptions stated on Kresmion's greeks exposure tool; realized volatility computed as the annualized sample standard deviation of daily log returns from adjusted closes; the variance risk premium as defined in the options literature.

Kresmion Research.