A Closer Look At Volatility
In previous commentaries I’ve discussed how volatility varies depending on the time interval over which it is measured–namely, it increases with the square root of the time interval. If you measure the volatility (for intuition, read “average percentage change”) of a stock over a one-week period, the volatility over a four-week period should be twice the one-week volatility. If you find the volatility over a one-day period is 2%, the volatility over a 16-day period will be four times as large, or 8%, and so on.
To get volatility values for longer time periods (such as a month, three months, or a year), you measure the volatility over the more numerous short time periods (such as one day) and use that shorter time period volatility as a base from which to compute volatilities over longer time periods using the square root law.
This is a consequence of the model used to compute both option prices and the probabilities of various prices occurring after a given time. If you use this model, the answers you get will not be far off. Of course, instead of using the one-day volatility as a base and deriving the sixteen-day volatility using the square root law, you could simply collect a great deal of data and compute the volatility using sixteen-day time periods. If the underlying adheres to the model, the answers should be about the same.
What if the answers are not the same? The accompanying table shows the volatility of the S&P 500 index over a number of periods. According to the model the four-day volatility should be twice the one-day volatility, the nine-day volatility should be three times the one-day volatility, and so on.
| # of Days | Volatility | Model Multiplier | Actual Multiplier |
|---|---|---|---|
| 1 | .89% | 1.00 | 1.00 |
| 4 | 1.62% | 2.00 | 1.82 |
| 9 | 2.3% | 3.00 | 2.58 |
| 16 | 2.9% | 4.00 | 3.29 |
| 25 | 3.58% | 5.00 | 4.00 |
| 36 | 4.2% | 6.00 | 4.75 |
| 49 | 4.9% | 7.00 | 5.54 |
| 64 | 5.57% | 8.00 | 6.26 |
| 81 | 6.26 | 9.00 | 7.03 |
| 100 | 6.84% | 10.00 | 7.68 |
The volatility measured over one day is .89%, which tells you that the S&P typically moves a little less than 1% a day. The Model Multiplier tells you how much the volatility increases according to the model. So, in 25 days, the volatility should be 5.00 times the one-day volatility. Five times 0.89% is 4.45%, but if you measure the actual percentage changes over 25-day periods, you will find that the 25-day volatility is not 4.45%, but rather 3.56%, only 4.00 times 0.89%, somewhat less than the model thinks.
Moreover, if you examine the table you will see that all of the Model Multipliers are too high. Going out to 100 days, when the volatility should be 10.00 times 0.89%, or 8.9%, you find that the actual volatility is only 6.84%, only 7.68 times the one-day volatility, and only about 77% of what it “should” be. What does this mean to you as an option trader?
The fact that the volatilities do not grow as fast as they should under the model tells you that the S&P has a tendency to reverse, and not to trend as much as the model assumes. Because it trends less, it will, for instance, have less of a chance of being 20% from its current price than the model thinks. This affects option values.
For this market, the model option values, which of course depend on the underlying behaving as it “should” under the model, are too high. To get a firmer intuitive grip on this, consider out-of-the-money (OOTM) options. Because the S&P tends to trend less and reverse more than the model thinks, the OOTMs have less of a chance of expiring in-the-money than the model thinks. So even at “fair” prices, the OOTMs are overpriced.
The table above is one you should create for any underlying whose options you trade. It will tell you if the underlying has a tendency to trend more than the model thinks (in which case the multipliers will be greater than the Model Multipliers), less the model thinks (as above), or just as much as the model thinks (if the Actual Multipliers are about equal to the Model Multipliers). This information will tell you if the model overestimates, underestimates or properly estimates its options’ true values.