From: John Conover <john@email.johncon.com>

Subject: Stock Market

Date: Sun, 19 Sep 1993 23:54:30 -0700 (PDT)

FYI, it has been shown that stock market fluctuations are "brown noise," in nature. (After Brownian motion of molecules of a gas, which is a related problem-both have a power spectrum that is proportional to the inverse square of the frequency.) This can be handled with information theoretic analysis. In the mid-50's Claude Shannon (who invented information theory) reportedly became wealthy using information theory to analyze (using cross-entropy and mutual information concepts) the market. Now information theory is well established science of the market (and economics, in general.) Here is how it works (and this is algorithm that programed traders use.) Suppose that, based on past history, a stock has shown a probability, p, of winning a dollar (or million, etc.) in a week. Likewise, the probability of losing the dollar, (or the million) is 1 -p. The trick is to find the optimum amount to bet. If you look at it, it is really very simple. You don't bet, you make no money-and if you bet all you own, you could loose everything. So the optimum is in between-the optimal bet to maximize your capital growth is to bet the fraction 2p - 1 of your present capital. This will maximize the logarithmic growth of your capital, given by Shannon's information capacity C(p) = 1 - H(p) of the binary symmetric channel with an error probability of p. Here H(p) is the entropy function, H(p) = -[p log p + (1 - p) log (1 - p)]. Thus, if p = .6 (ie., a very good stock,) you should bet 20% of your capital, and using base 2 logarithms, H(p) equals 0.97 bits per bet. Thus, 2^c = 1.02, or you will make 2% per bet. Typical programmed traders would hold this stock for about 4.5 days. Not bad, 2% a week, or a hundred percent per year. It is one of the few theories that can model the "innovation process," and was first isolated in 1956 by John L. Kelly, Jr. BTW, these economic applications of entropic principles can be used in dice games and roulette as well, etc., as well. -- John Conover, john@email.johncon.com, http://www.johncon.com/

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