Compuing Five Day Volatility
Suppose you are looking at an IBM option with five trading days to expiration. How much variation can you expect to see in the next five days? You can, of course, compute the 1-day volatility in the conventional way using, say, the last 100 days of data, and multiply that 1-day volatility by the square root of five to get the estimated 5-day volatility, getting 2.232 x 2.53%=5.66%.
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Thus you would estimate the standard deviation of the percentage change in IBM price over the next five days to be 5.66%. This is not a poor estimate. You can use any knowledge you might have of the normal distribution to compute the chance that IBM will close more than 3% higher than today, and make other such calculations, these calculations will not be wrong — in fact, they are the calculations that most people do.
However another more intuitive way to get a reasonable answer directly from the data is this: look at all the 5-day periods in the last year and compute the percentage change in each period, and then compute the standard deviation of these percentages. This method goes directly to the data, to the actual behavior of IBM over the last year, and does not require that you have faith in any mathematics other than the standard deviation. And, using the data in this way, you can make any computation you might need using an Excel spreadsheet.
The closing data for IBM starting January 1999 are:
| Date | Â | Close |
| 990104 | Â | 91.5 |
| 990105 | Â | 94.8125 |
| 990106 | Â | 94.375 |
| 990107 | Â | 95.09375 |
| 990108 | Â | 93.78125 |
| 990111 | Â | 94.625 |
| 990112 | Â | 92.53125 |
| 990113 | Â | 92.75 |
| 990114 | Â | 90.3125 |
| 990115 | Â | 96.46875 |
| 990119 | Â | 96.125 |
| 990120 | Â | 97.25 |
| 990121 | Â Â Â Â | 98.5 |
Thus, you would look at the first five days in the data, Jan. 4 through Jan 8, and compute the percentage change for these five days, (93.78125-91.5)/91.5 = 2.5%. This one sample of the percentage change in five days. Then you would look at the next five days, Jan. 5 through Jan. 11, getting -0.2%. And so on. With one year’s worth of data, you will have 248 samples of the percentage change in IBM in five days.
What can you do with these 248 samples? For one, you could collect them into a histogram to see how large the average is and how large the extremes are. For another, you could compute the standard deviation of these 5-day percentage changes, which would be a good estimate of the volatility of IBM over a five day period, as it has actually occurred during this year of data.
More on Thursday.