Volatilities of Arbitrary Time Periods
Volatility is usually presented on a annualized basis. For instance, a volatility of 30% tells you that in a year you expect the underlying price to fluctuate by about 30%. So if the underlying price is $100, you expect to see prices on the order of $70 or $130 after a year; and if the underlying price is $1.00 you expect to see prices on the order of $0.70 or $1.30.
This interpretation can be made more precise in the following way: There is about a two-thirds chance (actually 68%) that the price after one year will be between 30% lower and 30% higher. And there is a 95% chance that the price will be between 60% lower and 60% higher (60% is two standard deviations).
What about parts of a year? Suppose the annualized volatility is 40% but you are looking at an option with three months to expiration, and you want to know what kind of fluctuation you should expect over the next three months.
Let’s turn the problem around and assume that you know the volatility over a three-month period is, say, 10%. With the kind of three-month fluctuation that this 10% implies, what kind of variation can you expect to see over a year’s time? There are four three-month periods in a year. Is it reasonable expect the annualized volatility to be four times as large, or 40%?
To move 40% higher, price would have to rise by an average of 10% in each three-month period; likewise, for price to fall by 40% it would have to fall by an average of 10% in each time period. But the 10% volatility does not say that price will move in one direction in different time periods — it merely says that price will fluctuate by about 10% in each time period.
Clearly, to expect a typical fluctuation as large as 40% is not reasonable. How much fluctuation should you expect in a year? The answer is given by the Square Root Law: if you multiply the time period by four, you double the volatility; if you multiply it by nine, you triple the volatility; and so on. If you multiply the time period by any number, you multiply the volatility by the square root of that number.
Since one year consists of four three-month periods, the one-year volatility is twice the three-month volatility, which means that for an annualized volatity of 40%, the three-month volatility must be 20%. What is the two-month volatility? There are six two-month periods in one year, so you multiply the two-month volatility by the square root of six (roughly 2.5) to get the annual volatility. So the answer is given by: What times 2.5 equals 40%? The twp-month volatility must be (roughly) 16%.