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Kathleen Janoski, by Wesley Phelan.
MISCELLANEOUS
q

Damage control on the Vince Foster story, and why the
Saint Colby and the Fifth Column death of former CIA director William Colby is not an
occasion for tears.
The dereliction of the journalism profession, the FBI
can't count files, the Montana drug operation, NSA
Some Observations on the Non-News
attacks on my Internet posts, and the heroic efforts of
Kenneth Starr.




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The Home Page of J. Orlin Grabbe

A satire directed at a group of super-critics who have
spent time attacking or distorting the Vince Foster
story as told by Jim Norman and me. (I simply
collected some of the arguments and put them in a
single post, so everyone could see how silly they
An Apology and Goodbye sounded.) However, some brain-dead individuals
apparently believed my tongue-in-cheek statement that
Mike McCurry of the White House Press Office calls
me once a week to direct my efforts, and wrote a lot of
angry letters condemning everything but their own
inattention and gullibility.
Sumitomo services the heroin trade. Intended to
How to Launder Money in the
Copper Market (1) become a series.
Why Bob Woodward's book The Choice is particularly
Woodward's Wayward Book
ill-timed.
The Clinton administration has imprudently taken
credit for much that is taking place in the economy.
How would they deal with a stock market crash? Also
The Clinton Crash
mentions the banking problems created by alterations
in the money flows associated with the drug and arms
trade.
Is the CIA Trying for a Piece of Fifth
Contact fails to contact the Fifth Column.
Column Action?
Operation BOPTROT and Casey's plan to turn
The Governor of Kentucky to be
Indicted Soon? Russian soldiers into drug addicts.
Dole Dumps an Old Friend and Lies
Why Dole resembles Clinton.
About His Finances
John Deutch and Alan Greenspan worry about stock
The Cracks in Clinton's Economy
declines.
The Fifth Column Gets Press And the Prez worries Dick Morris might squeal.
The Uses of Terrorism The trade-off between guns and buttering-up-voters.
Susan McDougal, following bad legal advice, becomes a
The Sniffles of Susan
martyr without a cause.
Why is the Federal Reserve illegally intervening in the
The Symbiosis of Alan Greenspan
and Bill Clinton stock market?
Pinnacle Bank--The Usual Suspects Banking, the Arkansas way.
Wars and Rumors of War Iraq, I3, and why Bill won't pardon Susan.
Musings Presidential pardons and all that.
Bill's Blow, Stock Blowoffs, and
The future sucks.
Millennial Madness
Five Indictments of the Mass Media The alcoholic press consumes a fifth.
The Dickheads Are Getting
Federal agents show up in Nancy, Kentucky.
Desperate
Laundering Numbers Sumitomo & FBI files: the media can't count.
As the worm squirms. Another chorus of "no more
The Starr Detractors
indictments".
The Other Starr Detractors Chris Ruddy and Ambrose Evans-Pritchard.
The media releases us from the burden of political
The Joke Is on You
truth.
That country composed in the main of cellars,
October Country
sub-cellars, coal-bins, closets, attics.
General Convicted for Political
The politics of promotion.
Fund-Raising




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The Home Page of J. Orlin Grabbe

Cocaine distribution from the Ilo Pango Air Force
The Coke Was Stored in Hangars 4
Base, and the roles of the CIA, NSA,
and 5
and the Mossad.
The Starr Indictments: The Media
And that's not all the media is wrong about.
Wrong Again
Bye Bye, Miss American Pie Lemmings and stock market calculations.
Al Gore as President And Tipper as FLOTUS.
Sticky Fingers at the Justice
The government steals and doesn't pay its bills.
Department and U.S. Customs
The chief marketeer of the PROMIS software is
Earl Brian Convicted in California
convicted of financial fraud.
The Starr Indictments II Trick or treat.
Murder and Drugs in Arkansas The FBI sits on evidence in a drug-related murder.
Ten Predictions & Postmortem My March 1996 predictions and postmortem comment.
Sixscore thousand persons that cannot discern between
Twenty Predictions for 1997 their right hand and their left hand; and also much
cattle.



Charles "the Angel of Death" Hayes
The "Angel of Death" is an Internet term applied to Charles S. Hayes because of his role
in bringing about the resignation of many a corrupt politician. Hayes is also the head
and public representative of the Fifth Column group of hackers who track political
payoffs through money-laundering channels.
Angel of Death Gives Deposition to DOJ An argument for the resignation of the entire Justice
in Inslaw Case Department.
The Dickheads Are Getting Desperate Federal agents show up in Nancy, Kentucky.
Sticky Fingers at the DOJ and U.S.
The government steals and doesn't pay its bills.
Customs
The chief marketeer of the PROMIS software is
Earl Brian Convicted in California
convicted of financial fraud.
Don't talk about PROMIS. Don't help the Special
Angel of Death Arrested
Prosecutor.
Chuck Hayes Versus Bill Clinton The progress of the war of Clinton's resignation.
Chuck Hayes Versus Bill Clinton 2 FBI director Louis Freeh doesn't like Chuck Hayes.
The House Committee on Banking asks the NSA
Chuck Hayes and Me, Part 1
about PROMIS software and money-laundering.
Hayes investigates drug-money laundering, while the
Chuck Hayes and Me, Part 2
House Committee on Banking kisses NSA butt.
Why the Fifth Column is necessary: "I don't want
Chuck Hayes and Me, Part 3
another Danny Casolaro on my hands.
The Metaphysics of Political Illusion U.S. Attorney Joseph Famularo is a kook.
My letter to U.S. Attorney Joseph Famularo,
Orlin Grabbe Versus Joseph Famularo
demanding reasons for his falsified claims.
Orlin Grabbe Versus Joseph Famularo
Famularo's chief witness is also a liar.
2
Orlin Grabbe Versus Joseph Famularo
Famularo says it's all Martin Hatfield's fault.
3
FBI agent David Keller overrules the Court and the
The FBI Conspiracy Against Chuck
Hayes Assistant US Attorney.




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FBI Agent David "Killer" Keller Rides
Keller struggles with the English language.
Again
Chuck Hayes and Me, Part 4 Bert Lance Visits Kentucky.
Chuck Hayes and Me, Part 5 The deal with Gingrich.
Jailed Chuck Hayes Claims FBI Setup,
Notes from underground.
by James Norman
Famularo's Case Against Chuck Hayes
Department of Justice under scrutiny.
Begins to Crumble
Hayes Claims He Will Sue FBI Agent By Sherry Price of Pulaski Week.
Three Motions Filed by Chuck Hayes The Kentucky saga continues.
More Gobbledegook from FBI Agent David "Killer"
Murder on the World Wide Web
Keller.
Federal Justice in London, Kentucky Indicted jailer, but not Chuck Hayes, free on bond.
Why Congress should withdraw all funding from the
The Dickheads Are Still Desperate
FBI.
Hayes' testimony confirms Fifth Column-induced
Judge Jennifer B. Coffman's Kangaroo
retirements. Former drug dealer and former
Court
mental patient testify on behalf of government.
By Gail Gibson. Witness' past prompts delay in
Chuck Hayes vs. Lawrence W. Myers
murder-plot trial
Sex with Hillary What one has to do to get a press conference.
A request to look into the activities of the Justice
Letter to Chairman, House Judiciary
Committee Department/FBI in Kentucky.
How You Can Help Charles Hayes Three things to do.
Background of Lawrence "Myers" of Media Bypass
Police Reports on Lawrence W. Meyers
fame. What he did to his friend.
Judge Jennifer Coffman's Kangaroo
I be de judge.
Court Continues
Jury decides Charles Hayes never worked for the
The Make-Believe World of Charles
Hayes CIA.
Lawrence Meyers Pleads Guilty to
Court record from 1986.
Grand Theft
The U.S. Attorney hides Lawrence "Myers"
Charles Hayes: Motion for a Directed
Verdict or Mistrial background.
Letter of Charles Hayes to Montana
Drugs and money laundering in Montana.
Senate
Government Used False CIA Affidavits, Lawrence Myers dismissed from military for mental
Defense Says condition.
Letter of Chip Tatum to Montana
More on FBI drug-smuggling in Montana.
Senate
Spook Wars in Cyberspace, by Dick
Is the FBI Railroading Charles Hayes?
Russell
Judge has tête-à-tête with President at the
Is Bill Clinton Obstructing Justice in
Hayes Case? Watergate Hotel.
Mexican Madness Pendejo in paradise.
Chuck Hayes Testimony Regarding His
CIA Files
Letter re Charles Hayes
Hayes on CIA, PROMIS, Fifth Column,
Foster, and DOJ
Chuck Hayes Prognosticates from
Prison, by Rich Azar
Charles Hayes: A Prison Interview, by
The Future of Politics.
Wesley Phelan


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The Home Page of J. Orlin Grabbe
Letter to Chuck Hayes
Why is this convicted thief and psychiatric case
Court Testimony Regarding Lawrence
Myers running free while Chuck Hayes is in jail?
Judge Jenny places presentence investigation report
Charles S. Hayes Sentencing Transcript
under seal!
James Norman's Letter to Warden of
Manchester Prison
New Address for Chuck Hayes
Judge Jennifer B. Coffman Coaches the How to deal with the testimony of (Convicted Felon)
Prosecution Lawrence Myers and (FBI Agent) Steve Brannon.

On July 25, 1997, despite a presentence investigation report that recommended that Charles Hayes be sentenced to time
served (i.e. immediately released), Judge Jennifer B. Coffman sentenced Charles Hayes to the federal maximum of 10 years in
prison.

How You Can Help Charles Hayes


Allegations Regarding Vince Foster, the NSA, and
Banking Transactions Spying
A series exploring the murder of Vince Foster, who--among other duties-- was an
overseer of an NSA project to spy on banking transactions. The series begins with a
memo by Jim Norman, Senior Editor at Forbes, to Mike McCurry of the White House
Press Office inquiring about the espionage activities of Vince Foster. Then (reflecting
my own on-going investigation) it gradually branches out into relevant background
issues necessary to understand pieces of the story. Finally, it converges back to Foster's
final days.
Most of the people who have an opinion on this issue have never done any investigation
themselves, and the few who have long ago buried their heads in the flora of Ft. Marcy
Park and never looked at the larger picture.
Foster was under counter-intelligence investigation when he died. Nearly every
investigator misses this part of the picture. Did these activities lead to his death? Or did
others take advantage of the circumstances to rid themselves of someone who might
talk? Neither question can be answered until the actual killer(s) is (are) identified.
Jim Norman sends a memo to the White House, which leaks it to a Starr
Part 1
assistant.
Charles O. Morgan threatens to sue anyone who says Systematics has a
Part 2
relationship with the NSA.
Part 3 Vince Foster oversaw covert money laundering at Systematics.
Hillary Clinton and Web Hubbell represented Systematics during the BCCI
Part 4
takeover of First American.
Part 5 Philosophical musings on the meaning of this investigation.
Part 6 The curious tenacles (and strange denials) of Systematics.
Part 7 Virtual realities in the media and banking. Israel as a virtual nuclear power.
Part 8 Caspar Weinberger's Swiss account. Israel's nuclear spying.
Part 9 Cover-up by House Banking Committee investigators.
BCCI, money laundering, and the nuclear weapons programs of Pakistan and
Part 10
Israel.
Part 11 Money laundering and the intelligence community.




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The Cabazon Indian reservation, PROMIS, BCCI, and the death of Danny
Part 12
Casolaro.
NSA's PROMIS virus, Mellon money laundering, and Vince Foster's blue NSA
Part 13
notebook.
Part 14 Letter of advice to the Pittsburgh-based nuclear network.
Part 15 What role is really being played by Richard Mellon Scaife?
Part 16 The NSA worm. Federal Reserve money laundering.
Sheila Foster Anthony effects a $286,000 wire transfer to her sister-in-law Lisa
Part 17
Foster four days before Vince Foster's death.
Part 18 Gaping holes in the investigation of the death of Vince Foster.
The modern triangular trade, the parallels between Arkansas and Pittsburgh,
Part 19
and why Jim Norman is in trouble.
Part 20 Genealogy--mine and the Mellons.
Part 21 The Meadors hearing and the problems of Mellon bank.
James Hamilton doesn't get a chance to deny the $286,000 wire transfer by Sheila
Part 22
Anthony.
Part 23 Bush spies on Perot. The blonde hairs found on Foster's body.
The Foster hit appears unprofessional, and to have taken the White House by
Part 24
surprise.
Part 25 The global money laundering operation.
Part 26 Abner Mikva resigns. Did Foster have knowledge of too many felonies?
Earl Brian, who sold the PROMIS software around the world, is under
Part 27
indictment in California.
Part 28 SIOP: the Single Integrated Operational Plan for Nuclear War.
Part 29 How Jackson Stephens brought the Global Laundry to America.
The National Programs Office and the drugs- for-arms operation at Mena and
Part 30
other secured facilities.
Part 31 Death and the empire of Jackson Stephens.
Part 32 Questions concerning the death of Vince Foster.
Part 33 How to launder money. Clinton's CIA connection. The Mossad panics.
Mossad agents forced to leave the country after putting out a contract on a Foster
Part 34
investigator.
Part 35 The Mossad team in Vince Foster's apartment, and Foster's final movements.
Mike Wallace accepts a $150,000 bribe from the DNC to debunk the notion Vince
Part 36
Foster was murdered.
The House Committee on Banking asks the NSA about PROMIS software and
Part 37
money-laundering in Arkansas.
Chuck Hayes investigates drug-money laundering, while the House Committee on
Part 38
Banking kisses NSA butt.
Why the Fifth Column is necessary: "I don't want another Danny Casolaro on
Part 39
my hands."
Part 40 Bert Lance visits Kentucky.
Part 41 The deal with Gingrich.


Other Foster Resources
America's Dreyfus Affair: The Case of
Parts 1-6. Essay by David Martin.
the Death of Vince Foster

Charles R. Smith on Vince Foster and the
Vince Foster and the NSA
Clipper Chip.


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Hugh Turley shows there is consistency in the
Parallels Between the Deaths of Tommy
Burkett and Vince Foster technology of assisted "suicide".
Ft. Marcy Park Witness Patrick
Second Amended Complaint, October 1998
Knowlton Lawsuit
Clinton Meets Columbo Oval office showdown.
The Starr Report with the Knowlton
Microsoft Word document. Includes exhibits.
Appendix
Vince Foster FBI Files Online




Miscellaneous Articles and Essays
I do consulting. Unlike the free information on this web page, it
My Resume
comes at a high price.
Speech given to the Eris Society in August 1993. Since reprinted
In Praise of Chaos in Liberty (March 1994) and the SIRS Renaissance electronic
data base (1995).
Eloge du Chaos French translation of "In Praise of Chaos"
Cuba de mi Amor Music to your ears.
Memories of Pasadena A personal memoir. Appeared in Laissez Faire City Times, Vol 5,
No 2, Jan. 8, 2001.



Short Stories
Published in Laissez Faire City Times, Vol 2, No 34, Oct 19,
Keys
1998.
Published in Laissez Faire City Times, Vol 2, No 35, Oct 26,
Agrarian Life
1998.
A Hundred Eighty Dollars Published in Laissez Faire City Times, Vol 2, No 36, Nov. 2,
1998.
Problems in spiritual reckoning. Published in Liberty (August
Karma Accountant
1993).
Cover Girl Published in Laissez Faire City Times (November 1997).
How to Survive in the East Village. Appeared in Art Times
The Hat
(November 1992).
The relationship between sex and quantum mechanics.
Connections Generated some fan mail from physicists. First appeared in
Beet #6 (Spring 1992).
Saving the world and saving oneself. Published in Liberty
Feed the Children
(May 1995).
Published in Laissez Faire City Times, Vol 3, no 40, Oct. 11,
Dolphin Man
1999.
Published in Laissez Faire City Times, Vol 3, no 42, Oct. 25,
The Age of the Feuilleton
1999.
Published in Laissez Faire City Times, Vol 3, no 46, Nov. 29,
Waiting for Gödel
1999.




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That's All Folks!
But don't forget to check out the links page . . .




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Chaos and Fractals in Financial Markets, Part 2, by J. Orlin Grabbe


Chaos and Fractals in Financial Markets
Part 2
by J. Orlin Grabbe
The French Gambler and the Pollen
Grains
In 1827 an English botanist, Robert Brown got his hands
on some new technology: a microscope "made for me by
Mr. Dolland, . . . of which the three lenses that I have
generally used, are of a 40th, 60th, and 70th of an inch
focus."
Right away, Brown noticed how pollen grains suspended
in water jiggled around in a furious, but random, fashion.
To see what Brown saw under his microscope, make
sure that Java is enabled on your web browser, and then
click here.

What was going on was a puzzle. Many people
wondered: Were these tiny bits of organic matter
somehow alive? Luckily, Hollywood wasn™t around at
the time, or John Carpenter might have made his
wonderful horror film They Live! about pollen grains
rather than about the infiltration of society by liberal
control-freaks.
Robert Brown himself said he didn™t think the movement
had anything to do with tiny currents in the water, nor
was it produced by evaporation. He explained his
observations in the following terms:
"That extremely minute particles of solid
matter, whether obtained from organic or
inorganic substances, when suspended in
pure water, or in some other aqueous fluids,
exhibit motions for which I am unable to
account, and from which their irregularity
and seeming independence resemble in a
remarkable degree the less rapid motions of
some of the simplest animalcules of
infusions. That the smallest moving
particles observed, and which I have termed
Active Molecules, appear to be spherical, or
nearly so, and to be between 1-20,0000dth
and 1-30,000dth of an inch in diameter; and

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Chaos and Fractals in Financial Markets, Part 2, by J. Orlin Grabbe

that other particles of considerably greater
and various size, and either of similar or of
very different figure, also present analogous
motions in like circumstances.
"I have formerly stated my belief that these
motions of the particles neither arose from
currents in the fluid containing them, nor
depended on that intestine motion which
may be supposed to accompany its
evaporation."[1]
Brown noted that others before him had made similar
observations in special cases. For example, a Dr. James
Drummond had observed this fishy, erratic motion in
fish eyes:
"In 1814 Dr. James Drummond, of Belfast,
published in the 7th Volume of the
Transactions of the Royal Society of
Edinburgh, a valuable Paper, entitled ˜On
certain Appearances observed in the
Dissection of the Eyes of Fishes.™
"In this Essay, which I regret I was entirely
unacquainted with when I printed the
account of my Observations, the author
gives an account of the very remarkable
motions of the spicula which form the
silvery part of the choroid coat of the eyes
of fishes."
Today, we know that this motion, called Brownian
motion in honor of Robert Brown, was due to random
fluctuations in the number of water molecules
bombarding the pollen grains from different directions.
Experiments showed that particles moved further in a
given time interval if you raised the temperature, or
reduced the size of a particle, or reduced the "viscosity"
[2] of the fluid. In 1905, in a celebrated treatise entitled
The Theory of the Brownian Movement [3], Albert
Einstein developed a mathematical description which
explained Brownian motion in terms of particle size,
fluid viscosity, and temperature. Later, in 1923, Norbert
Wiener gave a mathematically rigorous description of
what is now referred to as a "stochastic process." Since
that time, Brownian motion has been called a Wiener
process, as well as a "diffusion process", a "random
walk", and so on.
But Einstein wasn™t the first to give a mathematical


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Chaos and Fractals in Financial Markets, Part 2, by J. Orlin Grabbe

description of Brownian motion. That honor belonged to
a French graduate student who loved to gamble. His
name was Louis Bachelier. Like many people, he sought
to combine duty with pleasure, and in 1900 in Paris
presented his doctoral thesis, entitled Th©orie de la
sp©culation.
What interested Bachelier were not pollen grains and
fish eyes. Instead, he wanted to know why the prices of
stocks and bonds jiggled around on the Paris bourse. He
was particularly intrigued by bonds known as rentes sur
l™©tat” perpetual bonds issued by the French
government. What were the laws of this jiggle?
Bachelier wondered. He thought the answer lay in the
prices being bombarded by small bits of news. ("The
British are coming, hammer the prices down!")
The Square Root of Time
Among other things, Bachelier observed that the
probability intervals into which prices fall seemed to
increased or decreased with the square-root of time
(T0.5). This was a key insight.
By "probability interval" we mean a given probability
for a range of prices. For example, prices might fall
within a certain price range with 65 percent probability
over a time period of one year. But over two years, the
same price range that will occur with 65 percent
probability will be larger than for one year. How much
larger? Bachelier said the change in the price range was
proportional to the square root of time.
Let P be the current price. After a time T, the prices will
(with a given probability) fall in the range
(P “a T0.5, P + a T0.5), for some constant a.
For example, if T represents one year (T=1), then the last
equation simplifies to
(P “a , P + a), for some constant a.
The price variation over two years (T=2) would be
a T0.5 = a(2)0.5 = 1.4142 a
or 1.4142 times the variation over one year. By contrast,
the variation over a half-year (T=0.5) would be
a T0.5 = a(0.5) 0.5 = .7071 a
or about 71 percent of the variation over a full year. That
is, after 0.5 years, the price (with a given probability)


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would be in the range
(P “.7071a , P + .7071a ).
Here the constant a has to be determined, but one
supposes it will be different for different types of prices:
a may be bigger for silver prices than for gold prices, for
example. It may be bigger for a share of Yahoo stock
than for a share of IBM.
The range of prices for a given probability, then,
depends on the constant a, and on the square root of
time (T0.5). This was Bachelier™s insight.
Normal Versus Lognormal
Now, to be sure, Bachelier made a financial mistake.
Remember (from Part 1 of this series) that in finance we
always take logarithms of prices. This is for many
reasons. Most changes in most economic variables are
proportional to their current level. For example, it is
plausible to think that the variation in gold prices is
proportional to the level of gold prices: $800 dollar gold
varies in greater increments than does gold at $260.
The change in price, ∆P, as a proportion of the current
price P, can be written as:
∆P/P .
But this is approximately the same as the change in the
log of the price:
∆P/P ≈ ∆ (log P) .
What this means is that Bachelier should have written his
equation:
(log P “a T0.5, log P + a T0.5), for some constant a.
However, keep in mind that Bachelier was making
innovations in both finance and in the mathematical
theory of Brownian motion, so he had a hard enough
time getting across the basic idea, without worrying
about fleshing out all the correct details for a
non-existent reading audience. And, to be sure, almost
no one read Bachelier™s PhD thesis, except the celebrated
mathematician Henri Poincar©, one of his instructors.
The range of prices for a given probability, then, depends
on the constant a, and on the square root of time (T0.5),
as well as the current price level P.
To see why this is true, note that the probability range


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for the log of the price
(log P “a T0.5, log P + a T0.5)
translates into a probability range for the price itself as
( P exp(- a T0.5), P exp( a T0.5 ) ) .
(Here "exp" means exponential, remember? For
example, exp(-.7) = e -.7 = 2.718281-.7 = .4966. )
Rather than adding a plus or minus something to the
current price P, we multiply something by the current
price P. So the answer depends on the level of P. For a
half-year (T=0.5), instead of
(P “.7071a , P + .7071a )
we get
( P exp(- .7071 a ), P exp( .7071 a ) ) .
The first interval has a constant width of 1.4142 a, no
matter what the level of P (because P + .7071 a - (P
-.7071 a) = 1.4142 a). But the width of the second
interval varies as P varies. If we double the price P, the
width of the interval doubles also.
Bachelier allowed the price range to depend on the
constant a and on the square root of time (T0.5), but
omitted the requirement that the range should also
depend on the current price level P.
The difference in the two approaches is that if price
increments (∆P) are independent, and have a finite
variance, then the price P has a normal (Gaussian
distribution). But if increments in the log of the price (∆
log P) are independent, and have a finite variance, then
the price P has a lognormal distribution.
Here is a picture of a normal or Gaussian distribution:




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Chaos and Fractals in Financial Markets, Part 2, by J. Orlin Grabbe




The left-hand tail never becomes zero. No matter where
we center the distribution (place the mean), there is
always positive probability of negative numbers.
Here is a picture of a lognormal distribution:




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The left-hand tail of a lognormal distribution becomes
zero at zero. No matter where we center the distribution
(place the mean), there is zero probability of negative
numbers.
A lognormal distribution assigns zero probability to
negative prices. This makes us happy because most
businesses don™t charge negative prices. (However, US
Treasury bills paid negative interest rates on certain
occasions in the 1930s.) But a normal distribution
assigns positive probability to negative prices. We don™t
want that.
So, at this point, we have seen Bachelier™s key insight
that probability intervals for prices change proportional
to the square root of time (that is, the probability interval
around the current price P changes by a T0.5), and have
modified it slightly to say that probability intervals for
the log of prices change proportional to the square root
of time (that is, the probability interval around log P
changes by a T0.5).
How Big Is It?
Now we are going to take a break from price
distributions, and pursue the question of how we
measure things. How we measure length, area, volume,
or time. (This will lead us from Bachelier to
Mandelbrot.)
Usually, when we measure things, we use everyday
dimensions (or at least the ones we are familiar with
from elementary plain geometry). A point has zero
dimension. A line has one dimension. A plane or a
square has two dimensions. A cube has three
dimensions. These basic, common-sense type
dimensions are sometimes referred to as topological
dimensions.
We say a room is so-many "square feet" in size. In this
case, we are using the two-dimensional concept of area.
We say land is so-many "acres" in size. Here, again, we
are using a two-dimensional concept of area, but with
different units (an "acre" being 43,560 "square feet").
We say a tank holds so-many "gallons". Here we are
using a measure of volume (a "gallon" being 231 "cubic
inches" in the U.S., or .1337 "cubic feet").
Suppose you have a room that is 10 feet by 10 feet, or
100 square feet. How much carpet does it take to cover
the room? Well, you say, a 100 square feet of carpet, of
course. And that is true, for ordinary carpet.


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Let™s take a square and divide it into smaller pieces.
Let™s divide each side by 10:




We get 100 pieces. That is, if we divide by a scale factor of 10, we get 100 smaller
squares, all of which look like the big square. If we multiply any one of the smaller
squares by 10, we get the original big square.
Let™s calculate a dimension for this square. We use the same formula as we used for
the Sierpinski carpet:
N = rD .
Taking logs, we have log N = D log r, or D = log N/ log r.
We have N = 100 pieces, and r = 10, so we get the dimension D as
D = log(100)/log(10) = 2.
(We are using "log" to mean the natural log, but notice for this calculation, which
involves the ratio of two logs, that it doesn™t matter what base we use. You can use
logs to the base 10, if you wish, and do the calculation in your head.)
We called the dimension D calculated in this way (namely, by comparing the
number of similar objects N we got at different scales to the scale factor r) a
Hausdorff dimension. In this case, the Hausdorff dimension 2 is the same as the
ordinary or topological dimension 2.



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So, in any case, the dimension is 2, just as you suspected all along. But suppose you
covered the floor with Sierpinski carpet. How much carpet do you need then?
We saw (in Part 1) that the Sierpinski carpet had a Hausdorff dimension D = 1.8927¦
A Sierpinski carpet which is 10 feet on each side would only have N = 101.8927 =
78.12 square feet of material in it.
Why doesn™t a Sierpinski carpet with 10 feet on each side take 100 square feet of
material? Because the Sierpinski carpet has holes in it, of course.
Remember than when we divided the side of a Sierpinski carpet by 3, we got only 8
copies of the original because we threw out the center square. So it had a Hausdorff
dimension of D = log 8/ log 3 = 1.8927. Then we divided each of the 8 copies by 3
again , threw out the center squares once more, leaving 64 copies of the original.
Dividing by 3 twice is the same as dividing by 9, so, recalculating our dimension, we
get D = log 64/ log 9 = 1.8927.
An ordinary carpet has a Hausdorff dimension of 2 and a topological (ordinary)
dimension of 2. A Sierpinski carpet has a Hausdorff dimension of 1.8927 and a
topological dimension of 2. [4]
Benoit Mandelbrot defined a fractal as an object whose Hausdorff dimension is
different from its topological dimension. So a Sierpinski carpet is a fractal. An
ordinary carpet isn™t.
Fractals are cheap and sexy. A Sierpinski carpet needs only 78.12 square feet of
material to cover 100 square feet of floor space. Needing less material, a Sierpinski
carpet costs less. Sure it has holes in it. But the holes form a really neat pattern. So a
Sierpinski carpet is sexy. Cheap and sexy. You can™t beat that.
History™s First Fractal
Let™s see if we have this fractal stuff straight. Let™s look at the first known fractal,
created in 1870 by the mathematical troublemaker George Cantor.
Remember that we create a fractal by forming similar patterns at different scales, as
we did with the Sierpinski carpet. It™s a holey endeavor. In order to get a carpet whose
Hausdorff dimension was less than 2, we created a pattern of holes in the carpet. So
we ended up with an object whose Hausdorff dimension D (which compares the
number N of different, but similar, objects at different scales r, N = rD ) was more than
1 but less than 2. That made the Sierpinski carpet a fractal, because its Hausdorff
dimension was different from its topological dimension.
What George Cantor created was an object whose dimension was more than 0 but less
than 1. That is, a holey object that was more than a point (with 0 dimensions) but less
than a line (with 1 dimension). It™s called Cantor dust. When the Cantor wind blows,
the dust gets in your lungs and you can™t breathe.
To create Cantor dust, draw a line and cut out the middle third:


0________________________________________________________1



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0__________________1/3 2/3_________________1


Now cut out the middle thirds of each of the two remaining pieces:


0____1/9 2/9____ 1/3 2/3____7/9 8/9 ____1


Now cut out the middle thirds of each of the remaining four pieces, and proceed in this
manner for an infinite number of steps, as indicated in the following graphic.




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What's left over after all the cutting is Cantor dust.
At each step we changed scale by r = 3, because we
divided each remaining part into 3 pieces. (Each of these
pieces had 1/3 the length of the original part.) Then we
threw away the middle piece. (That™s how we created the
holes.) That left 2 pieces. At the next step there were 4
pieces, then 8, and so on. At each step the number of
pieces increased by a factor of N = 2. Thus the Hausdorff
dimension for Cantor dust is:
D = log 2 / log 3 = .6309.
Is Cantor dust a fractal? Yes, as long as the topological
dimension is different from .6309, which it surely is.
But”what is the topological dimension of Cantor dust?
We can answer this by seeing how much of the original
line (with length 1) we cut out in the process of making
holes.
At the first step we cut out the middle third, or a length
of 1/3. The next step we cut out the middle thirds of the
two remaining pieces, or a length of 2(1/3)(1/3). And so
on. The total length cut out is then:
1/3 + 2(1/32) + 4(1/33) + 8(1/34) + . . . = 1.
We cut out all of the length of the line (even though we
left an infinite number of points), so the Cantor dust
that's left over has length zero. Its topological dimension
is zero. Cantor dust is a fractal with a Hausdorff
dimension of .6309 and a topological dimension of 0.
Now, the subhead refers to Cantor dust as "history™s first
fractal". That a little anthropocentric. Because nature has
been creating fractals for millions of years. In fact, most
things in nature are not circles, squares, and lines.
Instead they are fractals, and the creation of these
fractals are usually determined by chaos equations.
Chaos and fractal beauty are built into the nature of
reality. Get used to it.
Today, there are roughly of order 103
recognized fractal systems in nature, though
a decade ago when Mandelbrot's classic
Fractal Geometry of Nature was written,
many of these systems were not known to
be fractal. [5]
Fractal Time
So far we™ve seen that measuring things is a complicated


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business. Not every length can be measured with a tape
measure, nor the square footage of material in every
carpet measured by squaring the side of the carpet.
Many things in life are fractal, and follow power laws
just like the D of the Hausdorff dimension. For example,
the "loudness" L of noise as heard by most humans is
proportional to the sound intensity I raised to the
fractional power 0.3:
L = a I0.3 .
Doubling the loudness at a rock concert requires
increasing the power output by a factor of ten, because
a (10 I)0.3 = 2 a I0.3 = 2 L .
In financial markets, another subjective domain, "time"
is fractal. Time does not always move with the rhythms
of a pendulum. Sometimes time is less than that. In fact,
we™ve already encounted fractal time with the Bachelier
process, where the log of probability moved according to
a T0.5 .
Bachelier observed that if the time interval was
multiplied by 4, the probability interval only increased
by 2. In other words, at a scale of r = 4, the number N of
similar probability units was N = 2. So the Hausdorff
dimension for time was:
D = log N/ log r = log 2/ log 4 = 0.5 .
In going from Bachelier to Mandelbrot, then, the
innovation is not in the observation that time is fractal:
that was Bachelier™s contribution. Instead the question is:
What is the correct fractal dimension for time in
speculative markets? Is the Hausdorff dimension really
D = 0.5, or does it take other values? And if the
Hausdorff dimension of time takes other values, what™s
the big deal, anyway?
The way in which Mandlebrot formulated the problem
provides a starting point:
Despite the fundamental importance of
Bachelier's process, which has come to be
called "Brownian motion," it is now
obvious that it does not account for the
abundant data accumulated since 1900 by
empirical economists, simply because the
empirical distributions of price changes are
usually too "peaked" to be relative to


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samples from Gaussian populations. [6]
What does Mandelbrot mean by "peaked"? It™s now time
for a discussion of probability.
Probability is a One-Pound Jar of Jelly
Probability is a one-pound jar of jelly. You take the jelly
and smear it all over the real line. The places where you
smear more jelly have more probability, while the places
where you smear less jelly have less probability. Some
spots may get no jelly. They have no probability at
all”their probability is zero.
The key is that you only have one pound of jelly. So if
you smear more jelly (probability) at one location, you
have to smear less jelly at another location.
Here is a picture of jelly smeared in the form of a
bell-shaped curve:




The jelly is smeared between the horizontal (real) line all
the way up to the curve, with a uniform thickness. The
result is called a "standard normal distribution".
("Standard" because its mean is 0, and the standard
deviation is 1.) In this picture, the point where the
vertical line is and surrounding points have the jelly
piled high”hence they are more probable.
As we observed previously, for the normal distribution
jelly gets smeared on the real (horizontal) line all the
way to plus or minus infinity. There may not be much
jelly on the distant tails, but there is always some.
Now, let™s think about this bell-shaped picture. What
does Mandelbrot mean by the distribution of price
changes being "too peaked" to come from a normal


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distribution?
Does Mandelbrot™s statement make any sense? If we
smear more jelly at the center of the bell curve, to make
it taller, we can only do so by taking jelly from some
other place. Suppose we take jelly out of the tails and
intermediate parts of the distribution and pile it on the
center. The distribution is now "more peaked". It is more
centered in one place. It has a smaller standard
deviation”or smaller dispersion around the mean.
But”it could well be still normal.
So what™s with Mandelbrot, anyway? What does he
mean? We™ll discover this in Part 3 of this series.

Click here to see the Answer to Problem 1 from Part 1.
The material therein should be helpful in solving
Problem 2.

Meanwhile, here are two new problems for eager
students:
Problem 3: Suppose you create a Cantor dust using a
different procedure. Draw a line. Then divide the line
into 5 pieces, and throw out the second and fourth
pieces. Repeat this procedure for each of the remaining
pieces, and so on, for an infinite number of times. What
is the fractal dimension of the Cantor dust created this
way? What is its topological dimension? Did you create
a new fractal?
Problem 4: Suppose we write all the numbers between 0
and 1 in ternary. (Ternary uses powers of 3, and the
numbers 0, 1, 2. The ternary number .1202, for example,
stands for 1 x 1/3 + 2 x 1/9 + 0 x 1/27 + 2 x 1/81.) Show
the Cantor dust we created here in Part 2 (with a
Hausdorff dimension of .6309) can be created by taking
all numbers between 0 and 1, and eliminating those
numbers whose ternary expansion contains a 1. (In other
words, what is left over are all those numbers whose
ternary expansions only have 0s and 2s.)
And enjoy the fractal:




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Notes
[1] Robert Brown, "Additional Remarks on Active
Molecules," 1829.
[2] Viscosity is a fluid™s stickiness: honey is more
viscous than water, for example. "Honey don™t jiggle so
much."
[3] I am using the English title of the well-known Dover
reprint: Investigations on the Theory of the Brownian
Movement, Edited by R. Furth, translated by A.D.
Cowpter, London, 1926. The original article was in
German and titled somewhat differently.
[4] I am admittingly laying a subtle trap here, because of
the undefined nature of "topological dimension". This is
partially clarified in the discussion of Cantor dust, and
further discussed in Part 3.
[5] H. Eugene Stanley, Fractals and Multifractals, 1991
[6] Benoit Mandelbrot, "The Variation of Certain
Speculative Prices," Journal of Business, 36(4), 394-419,
1963.

J. Orlin Grabbe is the author of International Financial
Markets, and is an internationally recognized derivatives
expert. He has recently branched out into cryptology,
banking security, and digital cash. His home page is
located at http://www.aci.net/kalliste/homepage.html .


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-30-
from The Laissez Faire City Times, Vol 3, No 24, June
14, 1999




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What Robert Brown Saw Under His Microscope


What Robert Brown Saw Under His
Microscope
"true" if statistics of random walk are to be shown size in pixels of the simulation-area

The applet above (written by Anne M. Denton) shows a small circle inside of a large circle. Think of the
large circle as Robert Brown's microscope lens. The smaller circle inside is one of the pollen grains being
observed by Brown. The red line is the center of the circle. The pollen grain is being bombarded by water
molecules, which causes it to move about erratically.
Actually, however, Brown couldn't see the water molecules, and it would not be until 1905 that it was
accepted that the erratic movement was caused by molecular bombardment. So to really see what Brown
saw, click on "flip view" above , and you will just see the red center moving erratically. That's what
Brown and others saw. What kind of strange animal was this?
Using the buttons above, you can stop and start the applet, or reset it. You can also change the
parameters, such as mu (which is the viscosity of the fluid). If you slide mu to the left, so that the fluid
becomes more like water than like honey, the particle moves about more freely.
In 1905 Einstein showed that the distance r traveled by the particle was proportional to the square root of
time T:
r = a T0.5.
So on a log r versus log T scale, we get a line with a slope of 0.5.
Return to Part 2




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Chaos and Fractals in Financial Markets, Part 1, by J. Orlin Grabbe


Chaos and Fractals in Financial Markets
by J. Orlin Grabbe

Answer to Exercise 1

Exercise 1: Iterate the following system: x(n+1) = 2x(n)
mod 1. [By "mod 1" is meant that only the fractional part
of the result is kept. For example, 3.1416 mod 1 =
.1416.] Is this system chaotic?

To get a feel for how the system behaves, let™s first
iterate a few values. Start with x(0) = .1. We get x(1) =
2(.1) mod 1 = .2 mod 1 = .2 . This is the first iteration.
This and the following iterations are listed in the table:

Iteration Value of x

1 .2

2 .4

3 .8

4 .6

5 .2

6 .4

7 .8

8 .6

9 .2

10 .4

11 .8

And so on. The values of x cycle through .6, .2, .4, .8,


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over and over.
Now let x(0) = 1.25. What values to we get? The first
iteration is x(1) = 2(1.25) mod 1 = 2.50 mod 1 = .50. The
second iteration is x(2) = 2(.5) mod 1 = 1 mod 1 = 0. The
third iteration is x(3) = 2(0) mod 1 = 0 mod 1 = 0. Once
at zero, the system stays there.

Iteration Value of x

1 .50

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

10 0

11 0

Because all values are taken mod 1, we drop (chop off)
anything to the left of the decimal point. So we are
really only concerned with numbers between 0 and 1.
Suppose we write each of the numbers between 0 and 1,
not in the decimal system, but in the binary system. The
binary system uses only 0s and 1s, which represent
powers of 2.
Consider, for example, the binary number
.1101 .
The first place to the right of the "decimal point"


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represents ½. The next place represents ¼ . The third
place represents 1/8 . And so on. The n-th place
represents 1/2n .
So
.1101 = 1 x ½ + 1 x ¼ + 0 x 1/8 + 1 x 1/16
= ½ + ¼ + 1/16 = 13/16.
The binary
.0011 = 0 x ½ + 0 x ¼ + 1 x 1/8 + 1 x 1/16 = 3/16.
Now, this may look like we™ve made things more
complicated, but actually we™ve made them extremely
simple. What happens when we multiply by 2? Well,
since the n-th decimal place represents 1/2n , if we
multiply it by 2, we have
2 x 1/2n = 1/2n-1 .
This is the (n-1)-th place to the right of the decimal
point, or one place closer.
So take .1101. To multiply it by 2, we move the decimal
point one place to the right and get
1.101 .
We then apply mod 1, which chops off everything to the
left of the decimal point and get
.101 .
Multiply by 2 again, we get
.01.
Then multiply by 2 again, we get
.1
Finally, one more multiplication, and we get
0.
So the first observation we can make about the system
x(n+1) = 2 x(n) mod 1, is that if the binary expansion
of x(0) is finite, the system converges to 0. This is
obvious, because if we chop off one binary digit with
each iteration, we eventually run out of binary digits, if


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the number of binary digits is finite.
Any fractional power of 2 will have a finite decimal
expansion, and hence these values will converge to zero.
(A fractional power of 2 is a fraction p/q where q = 2n
for some n. For example, 3/256, which in binary is
.00000011.)
In our first example, we started with x = .1 (where here
.1 was a decimal value, or 1/10). Let™s expand 1/10 into
binary:
.00011001100110011001¦
After the first 4 iterations, which chops off the leading
0001, we are left with a repeating 1001, 1001, 1001, etc.
This is an infinite binary number, so it doesn™t get any
shorter as we chop off places. So we cycle through the
same four numbers over and over. Namely, the four
numbers corresponding to:
.10011001100110011001¦
.00110011001100110011¦
.01100110011001100110¦
.11001100110011001100¦
These correspond to the decimal numbers .6, .2, .4, .8, as
we saw previously in the table.
All rational numbers (p/q, where q is not zero) that
are not fractional powers of 2 (i.e., q is not a power of
2) will eventually cycle.
For example, the starting decimal x(0) = .71 begins to
cycle after 21 iterations. We have x(22) = x(2) = .84,
x(23) = x(3) = .68, and so on.
The starting decimal x(0) = .7182 begins to cycle after
502 iterations. We have x(503) = x(3) = .7456.
What if we start with an an irrational number, such as the
square root of two, or π? Clearly these have
non-repeating binary expansions, so the system is
chaotic if x(0) is irrational.
Return to Part 2
-30-
from The Laissez Faire City Times, Vol 3, No 24, June


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14, 1999




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Chaos and Fractals in Financial Markets, Part 3, by J. Orlin Grabbe


Chaos and Fractals in Financial Markets
Part 3
by J. Orlin Grabbe

Hazardous World
Many things in life are random. They are governed by
probability, by chance, by hazard, by accident, by the
god Hermes, by fortune. So we measure them by
probability”by our one-pound jar of jam.
Places where there is more jam are more likely to
happen, but the next outcome is uncertain. The next
outcome might be a low probability event. Or it might be
a high probability event, but there may be more than one
of these.
Radioactive decay is measured by probability. The
timing of the spontaneous transformation of a nucleus (in
which it emits radiation, loses electrons, or undergoes
fission) cannot be predicted with any certainty.
Some people don™t like this aspect of the world. They
prefer to believe there are "hidden variables" which
really determine radioactive decay, and if we only
understood what these hidden variables were, it would
all be precisely predictable, and we could return to the
paradise of a Laplacian universe.
Well, if there are hidden variables, I sure wish someone
would identify them. If wishes were horses, David Bohm
would ride.[1] Albert Einstein liked to say, "God doesn™t
play dice." But if God wanted to play dice, he didn™t
need Albert Einstein™s permission. It sounds to me like
"hidden" is just another name for probability. "Was it an
accident?" "No, it was caused by hidden forces." Hidden
variable theorists all believe in conspiracy.
But, guess what? People who believe God doesn™t play
dice use probability theory just as much as everyone
else. So, without further ado, let™s return to our
discussion of probability.
Coin Flips and Brownian Motion


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We can create a kind of Brownian motion (or Bachelier
process) by flipping coins. We start with a variable x =
0. We flip a coin. If the coin comes up heads, we add 1
to x. If the coin comes up tails, we subtract 1 from x. If
we denote the input x as x(n) and the output x as x(n+1),
we get a dynamical system:
x(n+1) = x(n) + 1, with probability p = ½
x(n+1) = x(n) “ 1, with probability q = ½ .
Here n represents the current number of the coin flip,
and is our measure of time. So to create a graph of this
system, we put n (time) on the horizontal axis, and the
variable x(n) on the vertical axis. This gives a graph of a
very simple type of Brownian motion (a random walk),
as seen in the graphic below. At any point in time (at any
value of n), the variable x(n) represents the total number
of heads minus the total number of tails. Here is one
picture of 10,000 coin flips:




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Much of finance is based on a simple probability model like this one. Later
we will change this model by changing the way we measure probability,
A Simple Stochastic Fractal
Using probability, it is easy to create fractals. For example, here is a
dynamical system which creates a Simple Stochastic Fractal. The system has
two variables, x and y, as inputs and outputs:
x(n+1) = - y(n)
y(n+1) = x(n)
with probability p = ½ , but

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