Estimating Volatility VI
There are 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 1999 to August 18:
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| Figure 1. Five-day percentage changes, IBM, 1/1/99 – 8/18/99.
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The histogram does not quite resemble the well-known bell curve: a peak in the middle and tailing off more or less symmetrically to the left and the right. What does this tell you? For one, IBM behaved on those 157 five-day intervals a little differently from the expected behavior under standard model for percentage changes in the price.
This does not necessarily imply that the five-day behavior of IBM changed, but it should raise your eyebrows. The standard Black Scholes option pricing model assumes the underlying percentage changes over a fixed time period, such as five days, will display approximately as a bell curve.
Of course, you cannot expect a perfect picture, as there will be some random variation in these histograms, depending upon the particular price moves taken by IBM over this time period. Can random variation explain the aberrations that you see in this histogram? That question can be answered, but it requires some mathematics and I want to keep this discussion intuitive.
One thing you might do is look at a longer time interval, say 15 days. There are 147 15-day intervals between January 1 and August 18. Here is the histogram of 15-day percentage changes in IBM 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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This histogram looks a bit more like the bell curve, although it is skewed a bit to the right, reflecting IBM’s tendency to rise during this period. Again, it would require some mathematics to tell whether the deviation from the normal bell curve can be explained by chance variation or not, but roughly speaking, the graph passes muster.
The important idea to grasp in this discussion, and last Thursday’s, is that you can look at the past behavior of IBM in a very intuitive way and get an idea of the kinds of percentage changes it has experienced prior to today without appealing to any mathematics beyond that which you learned in high school. On Thursday I will talk about some intuitive calculations you can do that would, in fact, be a little difficult mathematically.