Estimating Volatility II
In the last commentary, we saw that when you compute the volatility of IBM over various look-back periods, you get the various one-day volatility values. Here are the look-back periods we used, their one-day volatilities, and the corresponding annualized volatilities:
| Look-back | One-day volatility | Annualized volatility |
|---|---|---|
| 100 | 2.53% | 40% |
| 80 | 2.16% | 34% |
| 60 | 2.05% | 33% |
| 50 | 2.00% | 32% |
| 40 | 2.07% | 33% |
| 20 | 2.42% | 38% |
| 10 | 2.63% | 42% |
Note: there are 252 trading days in the typical year, so to get the annualized volatility from the one-day volatility, you use the square root law and multiply the one-day volatility by the square root of 252, which is about 15.875. Thus 2.00% * 15.875 = 31.75%, or 32% after rounding; and similarly with the others.
So looking back 100 days from last Thursday, August 18, 1999, we find that IBM has fluctuated typically about 2.53% per day, which annualizes to 40%; looking back only 50 days, we find a typical fluctuation of only 2.00% per day, which annualizes to only 32%; and looking back only 10 days we find a typical fluctuation of 2.63% per day, which annualizes to 42%. And the other look-back periods yield likewise figures different from these and from one another.
Keep in mind that the object is to estimate IBM’s volatility between now and expiration of the relevant option. How different are the option values you get when you use these different volatility estimates? Let’s take the IBM call with strike 120 and expiration in October, and see how different the value is if we use the two most extreme one-day values, the 2.00% figure obtained from looking back 50 days from August 18 and the 2.63% figure obtained from looking back only 10 days.
On August 18, IBM closed at 123.875. Using a volatility of 2.00%, the Oct 120 call is worth $8.24; using a volatility of 2.63%, it is worth $9.94. How about the Dec 120 call? Below is a table showing the value of each of these IBM options using both of these volatilities:
| Option | Value @ 2.00% annual 32% | Value @ 2.63% annual 42% |
|---|---|---|
| Oct 120 | $ 8.24 | $ 9.94 |
| Dec 120 | $11.69 | $14.26 |
The plot thickens. As you see, the option values vary quite a lot depending upon which volatility figure you use. The historical data gives you a value for the Dec 120 of $9.94 if you look back only 10 days, and a value of $8.24 if you look back 50 days. But obviously the option can have only one value. Which is correct?
I will continue this discussion on Thursday. In the meantime, you might ask yourself what a good strategy would be in this situation.