The Square-Root-of-Time Rule
Why a year of risk is not twelve months of risk added up.
What Is the Rule?
Volatility measures how widely a stock's return can swing over a stretch of time. The square-root-of-time rule says how that spread grows as the stretch gets longer: with the square root of the time, not in proportion to it.
\(\sigma_T = \sigma_1\sqrt{T}\)
Here \(\sigma_1\) is the volatility over one period (one trading day, say) and \(T\) is the number of periods. Four days carry twice the volatility of one day, not four times. A year of 252 trading days carries about 15.9 times the daily volatility.
Why it works: when each day's return is independent of the others, their variances add. The variance over \(T\) days is \(T\sigma_1^2\), and volatility, the standard deviation, is its square root. Gains on some days partly cancel losses on others, so the spread grows more slowly than time.
A stock with 1.5% daily volatility has an annual volatility of about \(1.5\% \times \sqrt{252} \approx 23.8\%\).
\(\sigma_T = \sigma_1\sqrt{T}\)
Here \(\sigma_1\) is the volatility over one period (one trading day, say) and \(T\) is the number of periods. Four days carry twice the volatility of one day, not four times. A year of 252 trading days carries about 15.9 times the daily volatility.
Why it works: when each day's return is independent of the others, their variances add. The variance over \(T\) days is \(T\sigma_1^2\), and volatility, the standard deviation, is its square root. Gains on some days partly cancel losses on others, so the spread grows more slowly than time.
A stock with 1.5% daily volatility has an annual volatility of about \(1.5\% \times \sqrt{252} \approx 23.8\%\).
Remember
Four things to keep in mind about how volatility scales with time.
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Variances add over time; volatilities do not. Add the variances, then take the square root.
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Four times the time means twice the volatility, and about twice the price of an at-the-money option.
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To annualize daily volatility, multiply by the square root of 252, about 15.9. For a month, use the square root of 21, about 4.6.
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The rule assumes each day’s return is independent of the last. Trends, mean reversion and volatility that comes in clusters all bend it.
Why It Matters for Options
The chart follows a $100 stock with 25% annual volatility, with interest rates set to zero to keep the numbers simple.
The spread of outcomes. Forty simulated price paths fan out, but the one-standard-deviation band widens only with the square root of time: about ±$7 after a month, about ±$25 after a year. If risk grew in step with time, a year would carry ±$87.
Option prices. An at-the-money option is worth roughly \(0.4\,S\,\sigma\sqrt{T}\). A 21-day call costs $2.88 and an 84-day call $5.75: four times the time, twice the price.
Time decay. Because the value follows \(\sqrt{T}\), an at-the-money option loses value slowly at first and faster near expiration. A 42-day call worth $4.07 loses $1.19 over its first 21 days and the remaining $2.88 over its last 21.
Expected moves. Traders turn an implied volatility into a one-standard-deviation move the same way, \(S\,\sigma\sqrt{T}\). At 25% implied volatility, a $100 stock's one-month move is about $7.22.
The spread of outcomes. Forty simulated price paths fan out, but the one-standard-deviation band widens only with the square root of time: about ±$7 after a month, about ±$25 after a year. If risk grew in step with time, a year would carry ±$87.
Option prices. An at-the-money option is worth roughly \(0.4\,S\,\sigma\sqrt{T}\). A 21-day call costs $2.88 and an 84-day call $5.75: four times the time, twice the price.
Time decay. Because the value follows \(\sqrt{T}\), an at-the-money option loses value slowly at first and faster near expiration. A 42-day call worth $4.07 loses $1.19 over its first 21 days and the remaining $2.88 over its last 21.
Expected moves. Traders turn an implied volatility into a one-standard-deviation move the same way, \(S\,\sigma\sqrt{T}\). At 25% implied volatility, a $100 stock's one-month move is about $7.22.
FAQ
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What is the square-root-of-time rule?
It says that volatility over a period grows with the square root of the period’s length: σ over T days equals the one-day σ times √T. It holds when the returns of separate days are independent and have the same variance, so their variances add.
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Why use 252 days and not 365?
252 is the usual count of trading days in a year, and daily volatility is usually measured from trading-day returns. Markets are closed on weekends and holidays, so little new price information arrives then. What matters is consistency: measure and scale with the same kind of day.
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Why do variances add but volatilities do not?
For independent returns, the variance of a sum is the sum of the variances. Volatility is the square root of variance, and square roots do not add: two days at 1% volatility each give √(1² + 1²), about 1.41%, not 2%.
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When does the rule break down?
When returns are not independent. If prices trend, so that up days tend to follow up days, volatility over long horizons is larger than the rule predicts; if they revert to a mean, it is smaller. Volatility that comes in calm and stormy spells, and sudden jumps, also bend it. Over short horizons the rule is a good approximation; over long ones, check it against the data.
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How does the rule show up in option prices?
An at-the-money option is worth roughly 0.4 × share price × volatility × √T, so its value grows with the square root of the time to expiration. That is why a three-month option usually costs well under three times a one-month option, and why time decay speeds up as expiration nears.
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