Apr

8

We've all had the experience of taking an eye off the screen for a second, perhaps to answer a question or phone call, and finding the price wildly different from where it was a moment ago, almost invariably at our expense. Certainly, one would speculate the moves in a minute are much more variable than you would anticipate if prices were just random and you related the minute moves to a second. Similarly, for an hour relative to a minute, a day related to an hour, and week relative to a day. This is especially the case on Friday's hitting an extreme.

Such thinking is the basis for many statistical tests starting with one developed by working, some 80 years ago, and now encapsulated in closed form with various variance ratio tests, and related tests based on bootstrapping and simulation with actual prices. And yet, random numbers can do funny things. For every minute where there's a full 4-point swing in the S&P for example, there are hours where the price backs and fills with the two hour chances being much less variable than the one hour or one minute swings might suggest.

The subject calls out for some statistical thinking. And I thought I would approach it the way a kid might. My hope was that I might draw some insights that the normal statistical methods of considering this subject would not immediately elicit.

Lets start with ticks. If one tick can be up 0.25 or down 0.25, then after one tick the maximum change is 0.25. After 8 such changes the maximum change is 200 either way. If we looked at the range of 8 tick changes, we would find it varied from 2 to -2 with most bunched at 0, in a likeness to a binomial distribution with the distribution of number up and down given by the binomial coefficients.

If there are 2000 changes in a day, we might expect the variability using squares of 2000 changes to be 2000 times as great as for a one tick move. And if we compare the variability of two days using squares, we would expect it to be twice as great as the variability of one day. However, variability is computed in linear rather than square terms so we expect the two-day change when brought down to linearity to be 1.4 times as great as a one-day change.

A good way to estimate the variability of a distribution is with the formula the standard deviation = the range divided by the square root of the number of observations. This works because the range of a standard normal deviate can be approximated by the square root of the number of observations in a sample. Thus, the largest expected value for two draws from a standard normal is 0.7, the expected value for the range is 1.4 and the largest expected value for 4 draws of a standardized normal deviate is 1, the expected range is 2, and for 9 draws it's 1.5 and 3. The last 9 daily changes in the S&P, starting with the close of 3.23 were -2, - 05, -11, 02, 0, 2, 14, 1, and 4. The range is 19. We divide by 3 to compute an estimated daily standard deviation of daily changes of 6.33.

The last four non-overlapping 2-day changes in S&P starting with the close on 3/23 were respectively -7, -09, +2, and +15. The range is 24. And we would estimate the 2-day standard deviation as 24/2 = 12. The two-day standard deviation is a little less than 2 times the one-day standard deviation.

A good way to see if there is an inordinate tendency to momentum in prices is to compute running totals of the 1-day, 2-day, 3-day, 4-day, and 5-day ranges. If the 4-day range isn't twice as great as the 1-day range, or the 5-day range isn't as big as 2.2 times the 1-day, or the 5-day is not as big as 1.5 times as great as the 2-day range, et al, (square root of 5/square root of 2) then there's a tendency to reversal.

What other uses do you see for the running ranges?

P.S. A good estimate of the shortcut formula for standard deviations vis-à-vis ranges is on p.27 of the highly recommended book Statistical Rules of Thumb by Gerald van Belle. Also, study of the distribution of the expected value of the range led me to the very interesting paper Statistical Selection of the Best System, by David Goldsman, et al. It contains some very nice sequential methods for determining whether one system is better than another or one estimate of a mean, or one estimate of the largest observations to expect and is highly worth working through with a pencil and paper.


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