Index Option Trader

A Quick Lesson In Mathematics & Statistics: Part 1

I felt compelled to write to entry after receiving numerous emails about the implications of standard deviation and linear regression while looking at charts. First off, linear regression can be defined as: the best fit of sample data points to a linear model by minimizing the sum of the squares of deviations between the points and the line. Basically what were talking about here is the mean price, or the price on a line of best fit. Standard deviation on the other hand is a measure of the dispersion of a set of data from its mean. In other words, how far something is (in this case, price) is away from the mean — or line of best fit.

In a standard normal distribution at 3 standard deviations, 99.7% of all values of in a sample of data will fall within a given range away from the mean. The 99.7% figure is known as a confidence interval.

Now for the investing application:

Essentially what this means is that 99.7% of the time, price (of any underlying - stock, index, etc.) will stay within 3 standard deviations away from the mean, or average price. And, well… There are other times they will not, as displayed in the image below:

The above chart is a weekly SPY chart, showcasing the standard deviation indicator at the bottom. As you can see, the standard deviation measure stayed sub 3 most right into the choppy markets of Summer 2007. Standard deviation is also used as a measure of volatility, and rightly so. Moving through the rest of 2007, the SPY price moved roughly 4 standard deviations away from its mean price - Pretty rare… But, just when you might have thought we were moving into lower deviations, the SPY price (at the height of quasi-triple digit oil, the beginnings of the sub-prime and credit market write-offs, and poor economic numbers) fell of a cliff and shot up to 6 standard deviations away from the mean. A 6 sigma (a statistical term for standard deviation) event is extremely rare. The only other time in recent history that the SPY breached the 6 sigma mark was back in the turbulence of 2001-02 where it was breached a couple of times.

What I am trying to say is that this decline, or shall I say the extent of this decline, is very, very rare. When markets correct, traders look for a correction up to 3 sigma as a normal event. The fact that this market sold off as much as it did could be largely related to capitulation, a exhaustive selling phenomenon, which occurred after the poor fundamental data of late was digested. When capitulation occurs, the market becomes an emotional mess. Technicals and fundamentals are thrown out of the window as investors bail ship due to an overwhelming amount of fear. This activity usually continues until the sellers are completely exhausted.

Stay tuned for Part 2, which will cover the post-capitulation market dynamics, the slope of hope, and the wall of worry.