I don't believe stock movement is as "random" as the EMH and random walk theorists would have you believe. Stocks, and any market for that matter still behave according to the laws of supply and demand and more subtly to the "laws" of standard deviation. If you pull up a chart of any stock you like and add Bollinger Bands, you'll find that stocks stay inside the bands about 95% of the time (set at 2 standard deviations away from the mean). Very rarely will stocks go more than 3 standard deviations from the mean. Once in a blue moon (like the crash of '87), stocks fall much further than 3 standard deviations. But on average, I think its remarkable that stocks (or any traded security) stay so confined to a seemingly random number such as 2 standard deviations. My question is, what are the psychological principles behind the standard deviation numbers? There must be some. There are statistical principles behind it (Central limit theorem). But pyschologically speaking, why would stock prices "stop" at 2-3 standard deviations when a stock is going up. Why not 8 or 10? Perhaps human emotions are tied to the same standard deviation numbers. Greed and hysteria reach a certain point and then stop. Fear on the other hand can be much greater than greed. Thus, stocks can go down much faster than they go up. The market can fall 20% in one day, but it's never gone up even close to 20% in a day.

I am preparing an article for my site in 3 parts : Part I - Understanding variation (volatility) concept in general Part II - Applying variation concept in industrial activities - from the viewpoint of Walter Shewart (the father of Statistical Process Control and Quality Engineering) Part III - Adapting the above framework to Stock Market There should be some answers to your question.

No a random distribution can have any form : rectangular, triangular, bell curve,... and even no mean and no variance with so called Levy's law. It is probably the mean of a sample that can tend to Gauss Law with Central Limit Theorem (under some assumptions that are not automatically filled) that makes falsely believe that a random distribution (always) follows a gauss curve. In fact if a random distribution always follow Gauss Curve the Central Limit Theorem would just be useless. Or one must define the term random because in reality it has not been really defined even in mathematics so that it is fuzzy concept. But let's suppose that you define that random means gauss curve it would be a mathematical definition (arbitrary definition but still useful if it serves as reference law) , it doesn't imply that reality has to conform with your mathematics (which is kind of wishfull thinking even when disguised under the expression "Natural Law") and in truth it doesn't or Mandelbrott wouldn't have talked "about Fractals walk down on Wall street" in scientific american review : http://www.elliottwave.com/education/SciAmerican/Mandelbrot_Article2.htm

"a random distribution can have any form : rectangular, triangular, bell curve,... " i will check that, i can hardly see how a rectangular distribution could be the result of a random distributon, it seems to me we call that a constant distribution as to triangular-shaped distribution i guess if one just adds more data it will growth bell shaped i agree however that i should have better defined my understanding of what a random distribution is

You can check with this page : Probabilistic Solutions to Project Scheduling -- Simulation Using Excel http://www.sytsma.com/cism640/CPMSTUFF/cpmsim.htm <IMG SRC=http://www.sytsma.com/cism640/CPMSTUFF/cpmsim2.gif> Computationally, these random variates are computed in the following manner in Excel: Normal: =SQRT(-2*LN((RAND())*2*COS(2*PI*RAND())*STDEV+AVERAGE where STDEV and AVERAGE are the desired parameters of this set of random variables. An example from the SoftProject sheet is taken from cell F17: =SQRT(-2*LN(RAND()))*COS(2*PI()*RAND())*H17+G17 where H17 contains the desired standard deviation parameter and G17 contains the desired mean of the set of random variates. Uniform: =LOWER+RAND()*(UPPER-LOWER) where LOWER is the smallest random variate desired and UPPER is the largest. An example from the SoftProject sheet is taken from cell F19: =M19+RAND()*(N19-M19) where the LOWER and UPPER values are found in cells M19 and N19 respectively. Triangular: =IF(RAND()<MO-LO)/(HI-LO),LO+SQRT((MO-LO)*(HI-LO)*RAND()),HI-SQRT((HI-MO)*(HI-LO)*(1-RAND()))) where HI is the rightmost point of the triangle, LO is the leftmost point, and MO is the modal or topmost point. Notice that RAND() must be the same random number, so it is computed and then placed into the equation. If the formula is keyed in exactly as above, three random numbers will be used, and the random variate produced will be in error. An example from the SoftProject sheet will illustrate from cell F18: =IF(L16<(J16-I16)/(K16-I16),I16+SQRT((J16-I16)*(K16-I16)*L16),K16-SQRT((K16-J16)*(K16-I16)*(1-L16))) where: the single random number is located in cell L16 MO, the modal or topmost point of the triangle is located in cell J16 LO, the smallest and leftmost point of the triangle is located in cell I16 HI, the highest and rightmost point of the triangle is located in cell K16 Exponential: Although no SoftProject example can be given for the exponential distribution, it is frequently used in projects to model waiting time for materials or people. Waiting times tend to be heavily positively skew. =-AVG*LN(RAND()) where AVG is the desired average (a constant, such as 36, for example) for the exponentially distributed random variate and LN is an Excel function to find the natural logarithm. The SoftProject example uses only the normal, uniform, and triangular distributions.

There is nothing psychological or remarkable about it. If you calculated SD for some initial period, say 100 days and for the next 100 days referred the prices to it, you'd find much more irregularities, that is instances where price is outside 1SD. In a way, the mean is like a moving average - it moves with the price. SD calculated from that mean will also increase, so if the price stays at new level for a couple of days the mean and SD will rise high enough to bring the price back inside 1SD boundary. Note how BB behave during breakout - the bands expand by including new price in calculations - so it's actually very hard for the price to stay outside 1SD and if it does it is only for first few days. The attached chart is for BB 20 periods and 2SD. Note the bands expanded on breakout. THey did not quite catch up with the price because today's value is only one of 20 this high, the remaining prices are low. However, if the price remains at today's level or continues to rise, tomorrow there will be two significantly larger values in 20 used for calcualtions of the mean and SD. next day 3 and in a couple of days the bands will be wide enough to include the price.

the fact that a series of observable events will stay within a set of boundaries (1, 2, or 3 sd's) sort of argues <i>against</i> randmoness, doesn't it? it's more like a constrained range of possibilities; like a fence around a playground. It's not <i>unpredictibility</i>, it's predictability minus precision.

The stock prices do occasionally peek through the bands, even in subdued markets, but, as you indicate, not often, and not by much. This is because Bollinger Bands are dynamic - they are continually adjusting to fluctuations in volatility. As volatility increases the bands widen so that the probability that the price will remain inside the bands is always 95%. For this reason, no matter how excited traders get there is a 95% probability that their trades will occur inside the bands. The crash of '87 was an extreme situation that generated trading probabilities beyond 95%. It isn't a matter of psychology; it's a matter of the Bollinger Bands formula.

Actually, it's just the opposite. Price stays within the bands because the bands are constantly adjusting to accommodate the price, just as someone inside a sphere can go pretty much wherever he likes, though he can't break out of the sphere. Price doesn't follow the Bands; the Bands follow the price.