Implied volatility vs expected move: the square root of time rule

Implied volatility is quoted as an annual percentage. Almost nobody holds an option for a year. Converting the annual number into a range for the expiry you actually trade is a one-line calculation, and getting it wrong is the most common numerical mistake in options trading. This guide covers the conversion, the intuition behind it, and the places the simple rule needs care.

The rule

A stock at 30% implied volatility is not expected to move 30% by Friday. Over a year, one standard deviation is 30%. Over 30 days it is 30% × √(30/365) = 8.6%. Over 7 days it is 4.2%. Over one day it is 30% × √(1/365) = 1.57%.

The handy shortcut for one trading day is to divide annual IV by 16, since √252 ≈ 15.9: a 32% IV stock has a daily one-standard-deviation move of about 2%.

Why the square root

Daily moves are, to a first approximation, independent of each other. When independent random moves add up, their variances add, not their sizes. Variance over n days is n times the daily variance, so the standard deviation over n days is √n times the daily standard deviation. Doubling the horizon adds only 41% to the range; quadrupling it doubles the range.

That has a practical consequence for option buyers: the range grows more slowly than the time you pay for. A 30-day option costs roughly 2.4 times a 7-day option at the same IV, not 4.3 times, because it covers a range only 2.07 times wider. This is why time decay accelerates near expiry: the last week contains a disproportionate share of the option's remaining range.

Calendar days or trading days?

Both conventions exist. Dividing by 365 with calendar days and dividing by 252 with trading days give nearly identical answers for the same expiry, because the ratio of the two counts is close to constant. What you must not do is mix them: 5 trading days divided by 365 understates the range by about 17%. OptionsMovement counts calendar days as a fraction to the 4:00 pm ET close and divides by 365.

Which IV to use

Every strike has its own implied volatility (that is what "skew" and "smile" mean), so "the" IV of a stock is a choice. For the expected move, the at-the-money IV is the right one: it is the volatility of the straddle that prices the range. Using a headline IV number that averages across strikes, or an "IV30" index, gives a nearby but different figure. Using the IV of an out-of-the-money put, which is usually higher, overstates the range.

Where the simple rule needs care

Very short expiries

Under about a week, quoted IV is noisy because the time value it is computed from is tiny. The straddle price is a more reliable input; the straddle guide shows how to use it, and the calculator switches methods at 8 days.

Scheduled events

The √T rule assumes volatility is spread evenly over the period. An earnings report is a lump of variance on one day, so the IV of an expiry that contains it is higher than an expiry that does not, and the range does not scale smoothly across the event. The earnings guide shows how to separate the event from the background.

Lognormal vs normal

Strictly, the model treats log returns as normal, so the up range is slightly larger than the down range in dollar terms. For expected moves under about 10% the difference is negligible and the symmetric range is fine. For a 40% expected move (biotech catalyst, meme stock) it is not: the upside range in dollars is materially larger than the downside range, and a calculator that uses symmetric bounds will misplace both edges.

IV rank and IV percentile

Because IV alone does not say whether options are cheap or dear for this stock, traders compare it with its own history. IV rank puts today's IV on a 0 to 100 scale between the 52-week low and high; IV percentile is the share of days in the past year with IV below today's. Both are useful context, and neither changes the expected move: a stock at 60% IV has the same expected move whether that is its rank-10 low or rank-90 high. Rank tells you about the price of the range; the range is still the range.

A quick reference