Estimating Volatility VII
There were 157 five-day intervals between January 1, 1999, and August 18, 1999. Here is the histogram of the 157 five-day percentage changes in IBM from January 1, 1999 to August 18, 1999:
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| Figure 1. Five-day percentage changes, IBM, 1/1/99 – 8/18/99.
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IBM closed today (September 9, 1999) at 134 3/4. Suppose you are interested an IBM call with strike price 140 expiring in five days and in the chance that this option ever closes in the money. This is just the chance that IBM will experience a percentage change of at least (140 – 134.75)/134.75 = 3.89% in the next five days. You can get this answer by a simple counting technique that requires no mathematical calculation.
If you have manual data, you can just do a simple count, but if you have your data in a spreadsheet, you can use it–either way you will see that out of the 157 five-day intervals, 51 showed a closing percentage increase of at least 3.89% sometime during the five days. And the chance you are looking for is just 51/157=32.5%.
Similarly, there are 147 15-day intervals between January 1 and August 18. Here is the 15-day histogram of percentage changes in IBM price since the beginning of this year:
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| Figure 1. 15-day percentage changes, IBM, 1/1/99 – 8/18/99.
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Suppose the 140 call expires in 15 days and you are interested in the same question: what is the chance this call option ever will be in the money? Out of the 147 15-day intervals, you will find that 66 showed a closing percentage increase of at least sometime during the 15 days. So the chance is 66/147 = 44.9%.
This is in fact a calculation that would stump option experts in mathematical option theory, even with all their mathematical artillery. But you can get the answer very simply, and your answer is arguably better than the answer that would be derived mathematically.
Check your understanding of this technique by asking yourself how the answers we got might be biased, and how you might correct for that bias. I’ll talk about that next Tuesday.