From: John Conover <john@email.johncon.com>
Subject: forwarded message from Kimberly Bodelson
Date: Wed, 22 Jan 1997 21:34:30 -0800
In case you are curious, Per Bak is a notable person in the field of
computer mediated financial instrument trading, (ie., programmed
trading.) Brownian motion, or random walk fractals (what most
programmed trading methodologies are based on,) are but a simple
subset of a mathematically sophisticated science. Random walk modeling
provides modest gains over analytical methods of stock fundamentals.
One of the issues is that random walk fractals assume that a Gaussian
distribution of the normalized increments will adequately model equity
prices-like, over a 40 year period, or so. As it turns out, this is an
adequate model to somewhat beyond 3 sigma, or about 1 part in 1,000,
(which is very good by engineering standards.) However, beyond 3
sigma, we have more drops, (October of 1927 and 1989 for example,)
than can be accounted for, according to the statistics of a Gaussian
distribution. These, traditionally, have followed "bubble" run ups in
stock prices, (similar to where we are at the present-but the run down
does not always happen; sometimes the market just stalls for exceeding
long periods, for example, 1930-1955, and 1966-1984.)
There have been a lot of theoretical proposals for the minute
discrepancy; it is created by a chaotic strange attractor, (fractal
systems are a subset of chaotic systems-which exhibit non-periodic
cyclic phenomena, like attractors that jump back and forth, strangely
and unpredictably, like the weather, for example); correlation
phenomena, (from catastrophe theory, like social upheaval, for
example); etc.
Bak's proposal is interesting because it is simple and makes sense,
both mathematically and economically.
John
BTW, the reason for the interest in the discrepancy is that by
anyone's analysis, we are in a bubble, and folks want to be ready to
measure it when, and if, it happens to understand the phenomena.
BTW, there are an enormous number of noise traders.
BTW, FYI, noise trading depends on the fact that a fractal with
Gaussian distributed increments is mean reverting. People's heights
are mean reverting, for example. Although the offspring of a tall
person may be taller than average, his height will be less than the
parents, on average, ie., offspring's height will revert to the
mean. (A graph of such things has a bell curve distribution, of which
a Gaussian distribution is one such-but not the only one. If you think
about it, that is the way bell curves are made. Something can not be
mean reverting, or ergotic in the technical vernacular, without having
some kind of bell curve distribution.) Vice versa for small
parents. So, what you do is measure the daily increments of the entire
stock exchange, and sort by the stocks dropped the most in value, and
invest in the 25-40 stocks that dropped the most. (If you think about
it, and you were betting on the greater heights of offspring, you
would bet on parents that are shorter than average, because their
offspring, on the average will be taller, and you make money on folks
who are taller than their parents-as an analogy.) Why 25-40 stocks?
because you want your capital to grow with an average normalized
increment that is the root mean square of the normalized increments,
squared. If you do not do this, (its the old f = 2P - 1 Shannon
probability stuff,) you will have great wealth, instantly, and then
loose it all. (In this scheme, the volatility of your wealth is very
high-you moderate it by the fact that volatilities, which you are
playing, add root mean square, and that is how you lower the
volatility of your wealth. A fact that is used in electronics
extensively. Lower the band width of a system the signal is not
effected, but the noise goes down with the square root of the
bandwidth.)
So, how well do noise traders do? Very well in a bubble run up, (they
are probably creating it,) with a typical gain of 2-3X per year, or
better. But suppose the bubble pops, and one day all 25-40 stock's
prices get cut in half? And if you are playing the optimal, (f = 2P -
1, measured with an accurate statistical estimate)? You would have to
settle for a more conservative 40% growth per year, (recent numbers,)
and the big drop would effect your portfolio value by only about 10%.
Which is better? Noise trading in the short run, but you are doomed to
loose. The best that can be done in the long run is f = 2P - 1. So, I
would imagine that there will be some unhappy noise traders,
eventually. (Eventually is 15 years, plus, according to statistical
estimation theory.)
BTW, there is another way of doing it, and that is the theory of bold
play, which can even be better than noise trading in up bubbles. But
that is another story ...
(I'll give you a hint. Set how much money you want, and if you reach
it you quit playing, or you go broke, and quit playing-which ever
comes first. The value you want is called K. If your capital is less
than K / 2, you bet half your capital. Else, you bet K - your
capital. You will probably loose, but you will maximize your chances
of making K, before going broke. Your capital will assume a "devil's
staircase" as a function of time. It is an interesting function-for
example, if you want to increase your capital by a factor of 10, the
expected number of plays is less than 25, but your chances of making
your goal is about 1 in 10, in 50/50 odds games. Interestingly, it
works with unfair games, too. It is the method of choice for
professionals gaming in Las Vegas, or Reno, who are trying to break
the bank. The devil's stair case is an interesting mathematical
anomaly. Not well behaved at all. It is a fractal with a weird
dimension, and is self-similar and self-affine, meaning it has no
scale; its derivative vanishes almost everywhere except at values
which form a Cantor set.)
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From: ksb@santafe.edu (Kimberly Bodelson)
To: activities-announce@santafe.edu
Subject: SFI Colloquium Series--Per Bak (Fri, 1/24, 4:00 PM)
Date: Wed, 22 Jan 97 10:18:50 MST
The Santa Fe Institute Colloquium Series
is pleased to present
A Talk by Per Bak
presented by
Per Bak
Niels Bohr Institute.
Friday, January 24, 4:00 p.m.
SFI Main Conference Room
ABSTRACT
Large variations in stock prices happen with sufficient frequency to raise
doubts about existing models, which fail to account for non-Gaussian
statistics. A simple model is constructed, and it is argued that the the
variations may be due to a crowd effect, where agents imitate each others
behavior. The interplay between "rational" traders whose behavior is derived
from an analysis of fundamental analysis of the stock, and "noise traders",
whose behavior is governed solely by studying the market dynamics, is
investigated. When the relative number of noise traders is large, "bubbles"
often occur. When the number of rational traders is large, the market is
generally locked within the price range they define.
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Kimberly S. Bodelson 505-984-8800 (phone)
Santa Fe Institute 505-982-0565 (fax)
1399 Hyde Park Road ksb@santafe.edu
Santa Fe, NM 87501 http://www.santafe.edu
USA ||
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John Conover, john@email.johncon.com, http://www.johncon.com/