From: John Conover <john@email.johncon.com>
Subject: Re: Will the stock market
Date: 5 Feb 1999 19:33:36 -0000
Bill Terrell writes:
> conover@netcom.com wrote:
>
> > One of the advantages of using a random walk fractal as a first order
> > approximation for equity values is that the assumption can be
> > verified, emperically-just subtract yesterday's value from today's,
> > for all days, and assemble the results into a frequency
> > distribution. Its a very nice Gaussian distribution, as expected-for
> > all stocks through the century, by the day. The data for all stocks is
> > available on CD.
>
> There are those who intelligently argue a fractal model which is distinctly
> non-random-walk and non-normal in its distribution, including no less than Benoit B.
> Mandelbrot; see this month's (February) issue of Scientific American.
Oh, sure, Bill. But as a first order approximation, a random walk is
pretty good, (depending on who is telling the story, of course.) There
is a graph of the distributions of the daily marginal increments of
the DJIA, NYSE, and S&P 500, for 27 years at:
http://www.johncon.com/john/correspondence/981229233103.31169.html
and it does seem to indicate, as you suggest, that there is a slight
persistence in equity indices. A random walk has statistically
independent increments-ie., a 50/50 chance that the next increment
will be like the previous increment. The indices, however, seem to
have about a 60% chance-meaning that there is a slight
"predictability" or "forecastabililty" from one day to the next. The
Hurst exponent, Fast Fourier Transform, and entropy studies of the
indices seem to support the contention, also, (the entropy is too low,
implying that the increments are not as random as a random walk would
suggest.)
The Hurst exponent, in addition, seems to indicate that there is a
four year "cyclic" phenomena in the indices-which could be interpreted
as the signature of a chaotic mechanism, possibly a strange attractor,
(but there could be structural reasons, too.) There are graphs of the
Hurst exponent for the NYSE at:
http://www.johncon.com/john/correspondence/980807151309.11811.html
http://www.johncon.com/john/correspondence/980807152940.11914.html
http://www.johncon.com/john/correspondence/980807154817.12009.html
But as a first order approximation, a random walk fractal seems
adequate-at least as a conceptual description from a stochastic point
of view-and the mathematics of the statistics is simple, (ie.,
"bubbles" in the stock market follow a 1 / sqrt (t) frequency
distribution, etc.)
John
--
John Conover, john@email.johncon.com, http://www.johncon.com/