Annual Vol Is Nice. What About Tomorrow?

Options are all about flexibility. You pick a direction, then you pick an option strategy based on risk tolerance, available capital, and volatility (“vol”). But here’s the rub—vol is almost always expressed as an annual number, and it measures the potential percentage returns of the stock’s price. If someone tells you the Nasdaq 100 Index (NDX) has a 19% vol, it means that in one year’s time, the NDX’s value will theoretically be within +19% and -19% of its current price 68% of the time. If NDX is trading for 7,300, that’s between 5,913 and 8,687. But what if you want to estimate where NDX might be in a day, or a week?

The Finer Points

First, you need to convert that annual vol number into a different period of time, say, one day. To do that, multiply it by the square root of time. That’s all. And for most articles on vol, that would be the end. But here we’re going to give you a deeper dive.

Why use the square root of time? First, think about how a stock’s price moves up and down. The percentage returns are positive and negative, too. So if you took the average of those positive and negative stock returns, the result could be close to zero. That would suggest the stock’s vol was low, even though it was moving up and down every day. For example, say a stock goes up +10% one day and down -10% the next. The average return is zero, but that’s huge vol.

To solve this problem, you square the stock price’s returns to make them all positive, and then average those squared returns to get what’s known as variance. But who thinks in terms of squared numbers? So, you take the square root of that variance to get it back into something usable. The square root of variance returns is the standard deviation of those returns, which is what traders refer to as vol.

Stock return variance is linearly related to time. You double time, for example, and variance doubles. Because you take the square root of variance to calculate vol, it’s related to the square root of time. You double time, and vol increases by the square root of 2. And oh, that 68% thing? That’s Chebyshev’s inequality theorem. Theoretically, data will fall between +1 and -1 standard deviations 68% of the time, +2 and -2 standard deviations 95% of the time, and +3 and -3 standard deviations 99% of the time. Vol is a standard deviation of returns, and theoretically follows Chebyshev.

And the Number of Days?

Do you use trading days (approximately 262) or calendar days (365)? For an annual number, it doesn’t matter. The square root of 262/262 is one, as is the square root of 365/365. But to convert vol to a one-day estimate, it makes sense to use trading days. For example, if it’s Tuesday and you want to see how much NDX might move on Wednesday, multiply VXN (the CBOE volatility index for the NDX) by the square root of 1/262, then multiply that by the NDX’s value. That is: 0.19 x (square root of (1/262)) x 7,300 = 85.69. Theoretically, NDX could be between 7,214.31 and 7,385.69 68% of the time in one day.

But if your time frame includes weekends, you may want to use calendar days to account for changes even when the market is closed. If you want to see how much NDX could change in 90 days, take 0.19 x (square root of (90/365)) x 7,300 = 688.73. The NDX might theoretically be within 6,611.27 and 7,988.73 68% of the time in 90 days.

As a rule of thumb, within a trading week—Monday through Friday—use trading days. For more than a week, use calendar days.

You can yank out a calculator to do this and impress your friends. But it’s simpler to look at the Probability Analysis page on the Analyze tab on the thinkorswim® platform from TD Ameritrade. And now you understand some of the theory behind those numbers.