Scaling Implied Volatility Moves with the Square Root of Time

Summary

The discussion explains how to estimate a one-standard-deviation price move from annualized at-the-money implied volatility. Under the usual square-root-of-time approximation, volatility is multiplied by the square root of the fraction of a year in the forecast horizon, then applied to the underlying price. The example’s calculation uses a trading-day year, but the response points out that a calendar month is closer to twenty trading days than thirty.

The approximation is useful for scaling volatility across horizons when direct market quotes are unavailable or stale. It should not be treated as a rule that option implied volatilities at different expiries must follow square-root scaling: each expiry’s implied volatility reflects market pricing and can differ independently. The answers affirm the square-root calculation as a basic estimate, while emphasizing that the assumed horizon and available expiry-specific prices matter. They do not discuss distribution shape, skew, or the probability of moves beyond one standard deviation.

Key ideas

• Annualized volatility is commonly scaled to a horizon using the square root of elapsed time.
• Apply the horizon-scaled volatility to the underlying price to estimate a one-standard-deviation move.
• A month of trading sessions is shorter than thirty trading days under a standard trading calendar.
• Implied volatility at separate option expiries need not match square-root-of-time scaling.
• The approximation is most useful for interpolation or when relevant market quotes are unavailable or stale.

Tags

# How to assess stock price movement from implied volatility?


# How to assess stock price movement from implied volatility?












Assume that: - The underlying is at 100 - The implied volatility of ATM call/put is 30%.

Then, is it correct that expected 1-standard-deviation move over the next month is calculated as:

$$100 * 30\% \cdot \sqrt\frac{30}{252} = 10.35 ~ \text{points}$$

I am confused as to whether I should be taking the square root or not.

## Answer by Nathan S. (score 1, accepted)

https://quant.stackexchange.com/a/16856

I think you're doing some self study and it looks like you're on a good path. You have this right. It's the square root of time. I would just note as far as your example goes that there are not 30 days in the month of a 252 (trading day approximation) day year. It's closer to 20.

And vanguard2k points out in a comment that you need to consider implied vol from multiple expiries as much as scaling with the square root of time. In other words, the IV (that's implied vol) for the one month expiry and the 3 month expiry need not by related by the square root of 3. Those expiries' prices are independent market phenomena. The square root of time is more like how you can interpolate for points that don't have market prices or get some more information to substitute for reliance on prices that may be stale.

## Answer by onlyvix.blogspot.com (score 1)

https://quant.stackexchange.com/a/16853

Yes, the answer is correct. Volatility scales with the square root of time, so always take square root. A simple trick to remember this, is to calculate the scaling factor as if volatility were linear (30/252) and then take the square root.

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