# Stock Market

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
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/