Probability Distributions for Stocks

We list here a few links that might be of interest and will continue to add to this page as time permits
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Typical, but not very good definition...

Find The Right Fit With Probability Distributions
Discusses discrete and continuous distributions
An attempt to chart uncertainty
Some common distributions: uniform, normal, lognormal, (many of the most popular models of stocks assume that the prices are distributed lognormally).
Poisson, student's t, beta, ..

Like so many shoes in our statistical shoe closet, we try to choose the best fit for the occasion, but we don't really know what the weather holds for us. We may choose a normal distribution then find out it underestimated left-tail losses; so we switch to a skewed distribution, only to find the data looks more "normal" in the next period. The elegant math underneath may seduce you into thinking these distributions reveal a deeper truth, but it is more likely that they are mere human artifacts. For example, all of the distributions we reviewed are quite smooth, but some asset returns jump discontinuously. 

The normal distribution is omnipresent and elegant and it only requires two parameters (mean and distribution). Many other distributions converge toward the normal (e.g., binomial and Poisson). However, many situations, such as hedge fund returns, credit portfolios and severe loss events, don't deserve the normal distributions.
This Demonstration calculates the probability that the random price of the exchange-traded fund, SPY, will be higher or lower after one month than a particular future possibility.  It uses some 'model' with four parameters and shows what the cumulative distribution looks like with different parameters.

Normal Probability Distribution in Stocks
Stock market analysts try to make predictions about the behavior of stocks. The more data that they have about the stocks, the better their predictions will be. One of the most important pieces of information that analysts have is the distribution of stock returns, because it tells them how likely each possible scenario is. The normal distribution is commonly used in analyses of the stock market, but some analysts have questioned its applicability.


The normal distribution may fit much of the data, but it has limitations in its relevance to investors' choices. Portfolio choices are influenced by investors' perceptions of risk. Because the normal distribution is symmetrical, the possibility of gains is treated in the same way as the risk of losses when calculating the variance of a portfolio. However, investors see only losses as risks, so only downward movements should negatively affect a portfolio's volatility.

Our comment: This is true, but relates to the use of the distribution in efficient portfolio theory where 'risk' is represented by the variance/co-variance matrix of possible investments.  This criticism is more relevant to distributions that are not symmetric, rather than to those that are, such as the normal distribution because the 'risk' of gain is mirrored by the risk of loss in symmetric distributions.

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