Estimating Volatility V
Figure 1 is a histogram of one-day percentage changes in IBM from January 4, 1999 to August 18, 1999:
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| Figure 1. One-day percentage changes, IBM, 1/4/99 – 8/18/99.
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The histogram resembles the well-known bell curve: a peak in the middle and tailing off more or less symmetrically to the left and the right. Most histograms derived in this way from securities data will have a shape similar to this.
Using this histogram, which is a picture of the historical data back to the beginning of this year, you can estimate, for instance, the chance that a one-day percentage change will be greater than 2%, or less than -1%, as follows: There are 161 one-day percentage changes since January 4, 1999, and of these precisely 32 were greater than 2%; so your estimate of the chance that a one-day future change will be greater than 2% is 32/161, or about 19.9%. Similarly, of the 161 one-day percentage changes since January 4, 1999, precisely 45 were less than -1%, so your estimate of the chance that a future one-day percentage change will be less than -1% is 45/161, or about 28%.
These estimates do not rely on any mathematical approximations, but are derived directly from the data and how the price IBM has historically behaved. Of course, the history we used for these estimates only went back to January 4, 1999. We might look at more data to get better estimates.
In particular, the last year has been a time of generally rising prices for IBM, and so you might want to include a longer time period with more usual price behavior. In general, if you feel the recent time period is special in a way that may not persist, you will want to include more data.
On the other hand, if you know that IBM has experienced some structural change recently that will affect the way its price behaves in the future, you will want to limit your data to those days after this change took place. These are issues that can be handled using sophisticated statistical techniques, but an intuitive approach will almost always give you an answer very close to what the statisticians would find.
Next week we will look at more histograms, including those showing longer-term percentage price changes that directly impact option values.