Estimating Volatility III
In Tuesday’s (8/24/99) commentary we saw that option values vary quite a lot depending upon which volatility figure you use. For two IBM call options we had the following:
| Option | Value @ 2.00% annual 32% | Value @ 2.63% annual 42% |
|---|---|---|
| Oct 120 | $ 8.24 | $ 9.94 |
| Dec 120 | $11.69 | $14.26 |
The historical data gives you different volatilities for different look-back periods. The above table illustrates two look-back periods, with a value for the IBM Dec 120 of $9.94 if you look back only 10 days, and a value of $8.24 if you look back 50 days. Other look-back periods give different values, but obviously the option can have only one value. What is correct?
The answer depends on the future behavior of IBM. If the volatility between August 18 and the October expiration turns out to be 2.00%, the Oct 120 call will have been worth $8.24 on August 18; if the volatility turns out to be 2.63%, the option will have been worth $9.94 on August 18. More than likely, the volatility over that period will not turn out to be exactly equal to either 2.00% or 2.63%, but some other value more or less close to these. It is impossible to say for sure exactly what it will be.
The problem is, you are trying to guess the future, and, although the future is not totally inscrutable, you can’t expect to predict it precisely, either. The question to answer is, what is the best guess about the volatility between now and expiration, given the data leading to the table above?
One consideration is the length of the look-back period. If you look back only 10 days and estimate the volatility of the future based upon only these 10 days, your estimate could be severely distorted due to an aberrant 10-day period. Volatility is not constant, but rather increases and decreases in a more or less random fashion. It is entirely possible to have a 10-day period during which the market is very quiet, and thus the volatility is exceptionally low.
This seems to be the case with IBM, looking back only 10 days from August 18, and illustrates the risk incurred when using a small sample size. Unless you feel that IBM has in fact entered a period of low volatility, well represented by the 10 days prior to August 18, using that 10-day look-back period is probably not a good policy.
There are statistical tools for determining an adequate sample size, but I would prefer that you not use them, as you would then be relying on a tool developed by someone else that you might not understand. Rather, we will develop intuitive ways to look at the data that will give results that are just as reliable and that will make good sense even to the mathematically untrained. I will do this in the next few commentaries.