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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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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 tÃªte-Ã -tÃªte 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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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 GÃ¶del

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 wasnâ€™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 didnâ€™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 â€˜On

certain Appearances observed in the

Dissection of the Eyes of Fishes.â€™

"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 wasnâ€™t the first to give a mathematical

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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 ThÃ©orie de la

spÃ©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

lâ€™Ã©tatâ€” 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 â€“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 â€“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 â€“.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 Bachelierâ€™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, âˆ†P, as a proportion of the current

price P, can be written as:

âˆ†P/P .

But this is approximately the same as the change in the

log of the price:

âˆ†P/P â‰ˆ âˆ† (log P) .

What this means is that Bachelier should have written his

equation:

(log P â€“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 Bachelierâ€™s PhD thesis, except the celebrated

mathematician Henri PoincarÃ©, 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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Chaos and Fractals in Financial Markets, Part 2, by J. Orlin Grabbe

for the log of the price

(log P â€“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 â€“.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 (âˆ†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 (âˆ†

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 donâ€™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 donâ€™t

want that.

So, at this point, we have seen Bachelierâ€™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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Letâ€™s take a square and divide it into smaller pieces.

Letâ€™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.

Letâ€™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 doesnâ€™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 = 1.8927â€¦

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 doesnâ€™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 isnâ€™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 canâ€™t beat that.

Historyâ€™s First Fractal

Letâ€™s see if we have this fractal stuff straight. Letâ€™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. Itâ€™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). Itâ€™s called Cantor dust. When the Cantor wind blows,

the dust gets in your lungs and you canâ€™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. (Thatâ€™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.

Butâ€”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 "historyâ€™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 weâ€™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,

weâ€™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 Bachelierâ€™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, whatâ€™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"? Itâ€™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

allâ€”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 highâ€”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, letâ€™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 Mandelbrotâ€™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

deviationâ€”or smaller dispersion around the mean.

Butâ€”it could well be still normal.

So whatâ€™s with Mandelbrot, anyway? What does he

mean? Weâ€™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 fluidâ€™s stickiness: honey is more

viscous than water, for example. "Honey donâ€™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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Chaos and Fractals in Financial Markets, Part 2, by J. Orlin Grabbe

-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, letâ€™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 Â½. The next place represents Â¼ . The third

place represents 1/8 . And so on. The n-th place

represents 1/2n .

So

.1101 = 1 x Â½ + 1 x Â¼ + 0 x 1/8 + 1 x 1/16

= Â½ + Â¼ + 1/16 = 13/16.

The binary

.0011 = 0 x Â½ + 0 x Â¼ + 1 x 1/8 + 1 x 1/16 = 3/16.

Now, this may look like weâ€™ve made things more

complicated, but actually weâ€™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). Letâ€™s expand 1/10 into

binary:

.00011001100110011001â€¦

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 doesnâ€™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:

.10011001100110011001â€¦

.00110011001100110011â€¦

.01100110011001100110â€¦

.11001100110011001100â€¦

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

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

probabilityâ€”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 donâ€™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 doesnâ€™t

play dice." But if God wanted to play dice, he didnâ€™t

need Albert Einsteinâ€™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 doesnâ€™t play

dice use probability theory just as much as everyone

else. So, without further ado, letâ€™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 = Â½

x(n+1) = x(n) â€“ 1, with probability q = Â½ .

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

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 = Â½ , but

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