Download - MAR KET ANALYSIS · MAR KET ANALYSIS Charles B. Epstein, Editor Applying Chaos Theory to Investment ANALYZING AND FORECASTING FUTURES PRICES Anthony F. Herbst and ... Part Three:

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SM

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FRA

CT

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Cha

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B. E

pste

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Cha

os T

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Inve

stm

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AN

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UR

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PRIC

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Ant

hony

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IN

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Edg

arE

.Pe

ters

____

____

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____

____

____

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____

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INSI

DE

TH

E F

INA

NC

IAL

FU

TU

RE

S M

AR

KE

TS,

3R

D E

DIT

ION

Mar

k J.

Pow

ers

and

Mar

k G

. Cas

telin

oR

EL

AT

IVE

DIV

IDE

ND

YIE

LD

Edg

arE

. Pet

ers

Ant

hony

E. S

pare

SEL

LIN

G S

HO

RT

Jose

ph A

. Wal

ker

TR

EA

SUR

Y O

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AT

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S A

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TH

E F

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EIG

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GE

Dim

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N. C

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fas

TH

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OR

EIG

N E

XC

HA

NG

E A

ND

MO

NE

Y M

AR

KE

TS

GU

IDE

Julia

n W

alm

sley

CO

RPO

RA

TE

FIN

AN

CIA

L R

ISK

MA

NA

GE

ME

NT

Dia

ne B

. Wun

nick

e, D

avid

R. W

ilson

, Bro

oke

Wun

nick

eM

ON

EY

MA

NA

GE

ME

NT

ST

RA

TE

GIE

S FO

R F

UT

UR

ES

TR

AD

ER

SN

auze

r J.

Bal

sara

TH

E M

AT

HE

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TIC

S O

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ON

EY

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ME

NT

Ral

ph V

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TH

E N

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CH

NO

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OF

FIN

AN

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EN

TD

imitr

is N

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rafa

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HE

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DE

R'S

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NU

AL

Will

iam

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ngO

PTIO

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AR

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T M

AK

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Alle

n J.

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on a

cid-

free

pap

er.

Cop

yrig

ht ©

199

4 by

Joh

n W

iley

& S

ons,

Inc.

All

right

s re

serv

ed. P

ublis

hed

sim

ulta

neou

sly

in C

anad

a.

Rep

rodu

ctio

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tran

slat

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of a

ny p

art

of th

is w

ork

beyo

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per

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ed b

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for

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r

info

rmat

ion

shou

ld b

e ad

dres

sed

to th

eP

erm

issi

ons

Dep

artm

ent,

John

Wile

y &

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s, In

c., 6

05 T

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Ave

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New

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k, N

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1015

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s pu

blic

atio

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vide

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ean

d

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orita

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info

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in r

egar

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the

subj

ect

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ter

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sol

d w

ithth

e un

ders

tand

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that

the

publ

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r is

not

eng

aged

inre

nder

ing

lega

l, ac

coun

ting,

or o

ther

pro

fess

iona

l ser

vice

s.If

lega

l adv

ice

or o

ther

expe

rt a

ssis

tanc

e is

req

uire

d, th

ese

rvic

es o

f a c

ompe

tent

prof

essi

onal

per

son

shou

ld b

e so

ught

. Fro

m a

Dec

lara

tion

ofPr

inci

ples

join

tly a

dopt

ed b

y a

Com

mitt

ee o

fthe

Am

eric

an B

ar A

ssoc

iatio

n an

d a

Com

mitt

eeof

Pub

lishe

rs.

Lib

rary

of

Con

gres

s C

atal

ogin

g-in

-Pub

licat

ion

Dat

a:

Pete

rs,E

dgar

E.,

1952

—F

ract

al m

arke

t ana

lysi

sap

plyi

ng c

haos

theo

ry to

inve

stm

ent a

nd

econ

omic

s / E

dgar

E. P

eter

s.

p.cm

.

Incl

udes

inde

x.IS

8N 0

-471

-585

24-6

I. In

vest

men

ts—

Mat

hem

atic

s.2.

Fra

ctal

s.3.

Cha

otic

beh

avio

r in

syst

ems.

I. T

itle.

II. T

itle:

Cha

os th

eory

.

HG

45I5

.3.P

4719

94

332.

6015

l474

—dc

2O93

-285

98

Prin

ted

in th

e U

nite

d S

tate

s of

Am

eric

a

10 9

8 7

6 5

4 3

2

To

Sher

yl



Pre

face

In19

91, I

fin

ishe

d w

ritin

g a

book

ent

itled

, Cha

osan

d O

rder

in th

e C

apita

l

Mar

kets

. It

was

pub

lishe

d in

the

Fall

of th

at y

ear

(Pet

ers,

199

Ia).

My

goal

was

to w

rite

a c

once

ptua

l int

rodu

ctio

n, f

or th

ein

vest

men

t com

mun

ity, t

o ch

aos

the-

ory

and

frac

tal s

tatis

tics.

I a

lso

wan

ted

to p

rese

nt s

ome

prel

imin

ary

evid

ence

that

, con

trar

y to

acc

epte

d th

eory

, mar

kets

are

not

wel

l-de

scri

bed

by th

e ra

n-

dom

wal

k m

odel

, and

the

wid

ely

taug

ht E

ffic

ient

Mar

ketH

ypO

thes

is (

EM

H)

is

not w

ell-

supp

orte

d by

em

piri

cal e

vide

nce.

I ha

ve r

ecei

ved,

in g

ener

al, a

ver

y po

sitiv

e re

spon

se to

that

boo

k. M

any

read

ers

have

com

mun

icat

ed th

eir

appr

oval

—an

d so

me,

thei

rdis

appr

oval

—an

d

have

ask

ed d

etai

led

ques

tions

. The

que

stio

ns f

ell i

nto

two

cate

gori

es: (

I) te

ch-

nica

l, an

d (2

) co

ncep

tual

. In

the

tech

nica

l cat

egor

y w

ere

the

requ

ests

for

mor

e

deta

il ab

out t

he a

naly

sis.

My

book

had

not

bee

n in

tend

ed to

be

ate

xtbo

ok, a

nd

I ha

d gl

osse

d ov

er m

any

tech

nica

l det

ails

invo

lved

in th

ean

alys

is. T

his

ap-

proa

ch im

prov

ed th

e re

adab

ility

of

the

book

, but

it le

ft m

any

read

ers

won

der-

ing

how

to p

roce

ed.

In th

e se

cond

cat

egor

y w

ere

ques

tions

con

cern

ed w

ith c

once

ptua

lis

sues

. If

the

EM

H is

fla

wed

, how

can

we

fix

it? O

r be

tter

still

, wha

t is

avi

able

rep

lace

-

men

t? H

ow d

o ch

aos

theo

ry a

nd f

ract

als

fit i

n w

ith tr

adin

g st

rate

gies

and

with

the

dich

otom

y be

twee

n te

chni

cal a

nd f

unda

men

tal a

naly

sis?

Can

thes

e se

em-

ingl

y di

spar

ate

theo

ries

be

unite

d? C

an tr

aditi

onal

theo

ry b

ecom

eno

nlin

ear?

In th

is b

ook,

Lam

add

ress

ing

both

cat

egor

ies

of q

uest

ions

. Thi

s bo

okis

dif

fer-

ent f

rom

the

prev

ious

one

, but

it r

efle

cts

man

ysi

mila

r fe

atur

es. F

ract

al M

arke

t

Ana

lysi

s is

an

atte

mpt

to g

ener

aliz

e C

apita

l Mar

ket T

heor

y (C

MT

)an

d to

ac-

coun

t for

the

dive

rsity

of

the

inve

stm

ent c

omm

unity

.O

ne o

f th

e fa

iling

s of

trad

i-

tiona

l the

ory

is it

s at

tem

pt to

sim

plif

y "t

he m

arke

t" in

to a

n av

erag

epr

otot

ypic

al VI'



viii

Pre

face

Pre

face

—ix

ratio

nal i

nves

tor.

The

rea

sons

for

setti

ng o

ut o

n th

is r

oute

wer

e no

ble.

In th

e tr

adi-

tion

of W

este

rn s

cien

ce, t

he fo

undi

ng fa

ther

s of

CM

T a

ttem

pted

to le

arn

som

e-th

ing

abou

t the

who

le b

y br

eaki

ng d

own

the

prob

lem

into

its

basi

c co

mpo

nent

s.T

hat a

ttem

pt w

as s

ucce

ssfu

l. B

ecau

se o

f the

fars

ight

ed w

ork

of M

arko

witz

,S

harp

e, F

ama,

and

oth

ers,

we

have

mad

e en

orm

ous

prog

ress

ove

r th

e pa

st 4

0 ye

ars.

How

ever

, the

red

uctio

nist

app

roac

h ha

s its

lim

its, a

nd w

e ha

ve r

each

ed th

em.

It is

tim

e to

take

a m

ore

holis

tic v

iew

of h

ow m

arke

ts o

pera

te. I

n pa

rtic

ular

, it i

stim

e to

rec

ogni

ze th

e gr

eat d

iver

sity

that

und

erlie

s m

arke

ts. A

ll in

vest

ors

do n

otpa

rtic

ipat

e fo

r th

e sa

me

reas

on, n

or d

o th

ey w

ork

thei

r st

rate

gies

ove

r th

e sa

me

inve

stm

ent h

oriz

ons.

The

sta

bilit

y of

mar

kets

is in

evita

bly

tied

to th

e di

vers

ityof

the

inve

stor

s. A

mat

ure"

mar

ket i

s di

vers

e as

wel

l as

old.

If a

ll th

e pa

rtic

i-pa

nts

had

the

sam

e in

vest

men

t hor

izon

, rea

cted

equ

ally

to th

e sa

me

info

rmat

ion,

and

inve

sted

for

the

sam

e pu

rpos

e, in

stab

ility

wou

ld r

eign

. Ins

tead

, ove

r th

e lo

ngte

rm, m

atur

e m

arke

is h

ave

rem

arka

ble

stab

ility

. A d

ay tr

ader

can

trad

e an

ony-

mou

sly

with

a p

ensi

on fu

nd: t

he fo

rmer

trad

es fr

eque

ntly

for

shor

t-te

rm g

ains

;th

e la

tter

trad

es in

freq

uent

ly fo

r lo

ng-t

erm

fina

ncia

l sec

urity

. The

day

trad

er r

e-ac

ts to

tech

nica

l tre

nds;

the

pens

ion

fund

inve

sts

base

d on

long

-ter

m e

cono

mic

grow

th p

oten

tial.

Yet

, eac

h pa

rtic

ipat

es s

imul

tane

ousl

y an

d ea

ch d

iver

sifie

s th

eot

her.

The

red

uctio

nist

app

roac

h, w

ith it

s ra

tiona

l inv

esto

r, c

anno

t han

dle

this

dive

rsity

with

out c

ompl

icat

ed m

ultip

art m

odel

s th

at r

esem

ble

a R

ube

Gol

dber

gco

ntra

ptio

n. T

hese

mod

els,

with

thei

r m

ultip

le li

miti

ng a

ssum

ptio

ns a

nd r

estr

ic-

tive

requ

irem

ents

, ine

vita

bly

fail.

The

y ar

e so

com

plex

that

they

lack

flex

ibili

ty,

and

flexi

bilit

y is

cru

cial

to a

ny d

ynam

ic s

yste

m.

The

firs

t pur

pose

of t

his

book

is to

intr

oduc

e th

e F

ract

al M

arke

t Hyp

othe

sis—

a ba

sic

refo

rmul

atio

n of

how

, and

why

, mar

kets

func

tion.

The

sec

ond

purp

ose

ofth

e bo

ok is

to p

rese

nt to

ols

for

anal

yzin

g m

arke

ts w

ithin

the

frac

tal f

ram

ewor

k.M

any

exis

ting

tool

s ca

n be

use

d fo

r th

is p

urpo

se. I

will

pre

sent

new

tool

s to

add

to

the

anal

yst's

tool

box,

and

will

rev

iew

exi

stin

g on

es.

Thi

s bo

ok is

not

a n

arra

tive,

alth

ough

its

prim

ary

emph

asis

is s

till c

once

p-tu

al. W

ithin

the

conc

eptu

al fr

amew

ork,

ther

e is

a r

igor

ous

cove

rage

of a

naly

ti-ca

l tec

hniq

ues.

As

in m

y pr

evio

us b

ook,

I be

lieve

that

any

one

with

a fi

rmgr

ound

ing

in b

usin

ess

stat

istic

s w

ill fi

nd m

uch

that

is u

sefu

l her

e. T

he p

rimar

yem

phas

is is

not

on

dyna

mic

s, b

ut o

n em

piric

al s

tatis

tics,

that

is, o

n an

alyz

ing

time

serie

s to

iden

tify

wha

t we

are

deal

ing

with

.

TH

E S

TR

UC

TU

RE

OF

TH

E B

OO

K

The

boo

k is

div

ided

into

five

par

ts, p

lus

appe

ndic

es. T

he fi

nal a

ppen

dix

con-

tain

s fr

acta

l dis

trib

utio

n ta

bles

. Oth

er r

elev

ant t

able

s, a

nd fi

gure

s co

ordi

nate

d

to th

e di

scus

sion

, are

inte

rspe

rsed

inth

e te

xt. E

ach

part

bui

lds

on th

e pr

evio

us

part

s, b

ut th

e bo

ok c

an b

e re

adno

nseq

uent

ially

by

thos

e fa

mili

ar w

ith th

e co

n-

cept

s of

the

first

boo

k.

Par

t One

: Fra

ctal

Tim

e S

erie

s

Cha

pter

1 in

trod

uces

frac

tal t

ime

serie

s an

d de

fines

both

spa

tial a

nd te

mpo

ral

frac

tals

. The

re is

a p

artic

ular

em

phas

is o

n w

hat

frac

tals

are

, con

cept

ually

and

phys

ical

ly. W

hy d

o th

ey s

eem

cou

nter

intu

itive

, eve

n th

ough

frac

tal g

eom

etry

is

muc

h cl

oser

to th

e re

al w

orld

than

the

Euc

lidea

n ge

omet

ry w

eal

l lea

rned

in

high

sch

ool?

Cha

pter

2 is

a b

rief r

evie

w o

f Cap

ital

Mar

ket T

heor

y (C

MT

) an

d

of th

e ev

iden

ce o

f pro

blem

s w

ith th

e th

eory

. Cha

pter

3 is

, in

man

y w

ays,

the

hear

t of t

he b

ook:

I de

tail

the

Fra

ctal

Mar

ket

Hyp

othe

sis

as a

n al

tern

ativ

e to

the

trad

ition

al th

eory

dis

cuss

ed in

Cha

pter

2.

As

a F

ract

alM

arke

tH

ypot

hesi

s,

it co

mbi

nes

elem

ents

of f

ract

als

from

Cha

pter

1w

ith p

arts

of t

radi

tiona

l CM

T

in C

hapt

er 2

. The

Fra

ctal

Mar

ket H

ypot

hesi

s se

tsth

e co

ncep

tual

fram

ewor

k

for

frac

tal m

arke

t ana

lysi

s.

Par

t Tw

o: F

ract

al (

R/S

) A

naly

sis

Hav

ing

defin

ed th

e pr

oble

m in

Par

t One

, I o

ffer

tool

s fo

r an

alys

is in

Par

t

Tw

o—in

par

ticul

ar, r

esca

led

rang

e (R

IS)

anal

ysis

.M

any

of th

e te

chni

cal

ques

tions

I re

ceiv

ed a

bout

the

first

boo

k de

alt w

ithR

/S a

naly

sis

and

re-

ques

ted

deta

ils a

bout

cal

cula

tions

and

sig

nific

ance

test

s.P

arts

Tw

o an

d

Thr

ee a

ddre

ss th

ose

issu

es. R

/S a

naly

sis

is a

rob

ust

anal

ysis

tech

niqu

e fo

r un

-

cove

ring

long

mem

ory

effe

cts,

frac

tal s

tatis

tical

str

uctu

re,

and

the

pres

ence

of c

ycle

s. C

hapt

er 4

sur

veys

the

conc

eptu

al b

ackg

roun

dof

R/S

ana

lysi

s an

d

deta

ils h

ow to

app

ly it

. Cha

pter

5 g

ives

bot

h st

atis

tical

test

sfo

r ju

dgin

g th

e

sign

ifica

nce

of th

e re

sults

and

exa

mpl

es o

f how

R/S

ana

lysi

s re

acts

tokn

own

stoc

hast

ic m

odel

s. C

hapt

er 6

sho

ws

how

R/S

ana

lysi

s ca

n be

used

to u

ncov

er

both

per

iodi

c an

d no

nper

iodi

c cy

cles

.

Par

t Thr

ee: A

pply

ing

Fra

ctal

Ana

lysi

s

Thr

ough

a n

umbe

r of

cas

e st

udie

s, P

art T

hree

det

ails

how

R/S

anal

ysis

tech

-

niqu

es c

an b

e us

ed. T

he s

tudi

es, i

nter

estin

g in

thei

r ow

nrig

ht, h

ave

been

se-

lect

ed to

illu

stra

te th

e ad

vant

ages

and

dis

adva

ntag

es o

fusi

ng R

IS a

naly

sis

on

diffe

rent

type

s of

tim

e se

ries

and

diffe

rent

mar

kets

. Alo

ng th

e w

ay,

inte

rest

ing

thin

gs w

ill b

e re

veal

ed a

bout

tick

dat

a, m

arke

t vol

atili

ty, a

ndho

w c

urre

ncie

s ar

e

diffe

rent

from

oth

er m

arke

ts.



Pre

face

Pre

face

Par

t Fou

r: F

ract

al N

oise

Hav

ing

used

R/S

ana

lysi

s to

fin

d ev

iden

ce to

sup

port

the

Frac

tal M

arke

t Hy-

poth

esis

, 1 s

uppl

y m

odel

s to

exp

lain

thos

e fi

ndin

gs. P

art F

our

appr

oach

es m

arke

tac

tivity

fro

m th

e vi

ewpo

int o

f st

ocha

stic

pro

cess

es; a

s su

ch, i

t con

cent

rate

s on

frac

tal n

oise

. In

Cha

pter

13,

usi

ng R

/S a

naly

sis,

dif

fere

nt "

colo

red"

noi

ses

are

anal

yzed

and

com

pare

d to

the

mar

ket a

naly

sis.

The

fin

ding

s ar

e re

mar

kabl

ysi

mila

r. I

n ad

ditio

n, th

e be

havi

or o

f vo

latil

ity is

giv

en a

sig

nifi

cant

exp

lana

tion.

Cha

pter

14

disc

usse

s th

e st

atis

tics

of f

ract

al n

oise

pro

cess

es, a

nd o

ffer

s th

em a

san

alte

rnat

ive

to th

e tr

aditi

onal

Gau

ssia

n no

rmal

dist

ribu

tion.

The

impa

ct o

ffr

acta

l dis

trib

utio

ns o

n m

arke

t mod

els

is d

iscu

ssed

. Cha

pter

15

show

s th

e im

-pa

ct o

f fr

acta

l sta

tistic

s on

the

port

folio

sel

ectio

n pr

oble

m a

ndop

tion

pric

ing.

Met

hods

for

ada

ptin

g th

ose

mod

els

for

frac

tal d

istr

ibut

ions

are

rev

iew

ed.

Part

Fou

r is

a v

ery

deta

iled

sect

ion

and

will

not

be

appr

opri

ate

for

all r

eade

rs.

How

ever

, bec

ause

the

appl

icat

ion

of tr

aditi

onal

CM

T h

as b

ecom

e in

grai

ned

into

mos

t of

the

inve

stm

ent c

omm

unity

, I b

elie

ve th

at m

ost r

eade

rssh

ould

rea

d th

esu

mm

ary

sect

ions

of

each

cha

pter

, if

noth

ing

else

, in

Part

Fou

r.C

hapt

er 1

3, w

ith

its s

tudy

of

the

natu

re o

f vo

latil

ity, s

houl

d be

of

part

icul

ar in

tere

st.

Whi

le r

eadi

ng th

e bo

ok, m

any

of y

ou w

ill w

onde

r,w

here

is th

is le

adin

g?

Will

this

hel

p m

e m

ake

mon

ey?

Thi

s bo

ok d

oes

not

offe

r ne

w tr

adin

g te

ch-

niqu

es o

r fi

nd p

ocke

ts o

f in

effi

cien

cy th

at th

e sa

vvy

inve

stor

can

pro

fit f

rom

.

It is

not

a b

ook

of s

trat

egy

for

mak

ing

bette

rpr

edic

tions

. Ins

tead

, it o

ffer

s a

new

vie

w o

f ho

w m

arke

ts w

ork

and

how

to te

st ti

me

seri

es f

or p

redi

ctab

ility

.

Mor

e im

port

antly

, it g

ives

add

ition

al in

form

atio

nab

out t

he r

isks

inve

stor

s

take

, and

how

thos

e ri

sks

chan

ge o

ver

time.

If

know

ledg

e is

pow

er, a

s th

e ol

d

clic

hé g

oes,

then

the

info

rmat

ion

here

sho

uld

beco

nduc

ive,

if n

ot to

pow

er, a

t

leas

t to

bette

r pr

ofits

.

Con

cord

, Mas

sach

uset

ts

ED

GA

R E

. PE

TE

RS

Par

t Fiv

e: N

oisy

Cha

os

Par

tFi

ve o

ffer

s a

dyna

mic

al s

yste

ms

alte

rnat

ive

to th

e st

ocha

stic

pro

cess

es o

fPa

rt F

our.

In

part

icul

ar, i

t off

ers

nois

y ch

aos

as a

pos

sibl

e ex

plan

atio

n of

the

frac

-ta

l str

uctu

re o

f m

arke

ts. C

hapt

er 1

6, w

hich

giv

es R

/S a

naly

sis

of c

haot

ic s

ys-

tem

s, r

evea

ls r

emar

kabl

e si

mila

ritie

s w

ith m

arke

t and

oth

er ti

me

seri

es. A

part

icul

ar e

mph

asis

is p

lace

d on

dis

tingu

ishi

ng b

etw

een

frac

tal n

oise

and

noi

sych

aos.

A r

evie

w is

giv

en o

f th

e B

DS

(Bro

ck—

Dec

hert

—Sc

hein

kman

) te

st, w

hich

,w

hen

used

in c

onju

nctio

n w

ith R

/S a

naly

sis,

can

giv

e co

nclu

sive

evi

denc

ew

ay o

r th

e ot

her.

Cha

pter

17

appl

ies

frac

tal s

tatis

tics

to n

oisy

chao

s, r

econ

cilin

gth

e tw

o ap

proa

ches

. An

expl

anat

ion

is o

ffer

ed f

or w

hy e

vide

nce

of b

oth

frac

tal

nois

e an

d no

isy

chao

s ca

n ap

pear

sim

ulta

neou

sly.

The

res

ult i

s cl

osel

y tie

d to

the

Frac

tal M

arke

t Hyp

othe

sis

and

the

theo

ry o

f m

ultip

le in

vest

men

t hor

izon

s.C

hapt

er 1

8 is

a r

evie

w o

f th

e fi

ndin

gs o

n a

conc

eptu

al le

vel.

Thi

s fi

nal

chap

ter

unite

s th

e Fr

acta

l Mar

ket H

ypot

hesi

s w

ith th

e em

piri

cal w

ork

and

theo

retic

al m

odel

s pr

esen

ted

thro

ugho

ut th

e bo

ok. F

or r

eade

rs w

ho u

nder

-st

and

a pr

oble

m b

ette

r w

hen

they

kno

w th

e so

lutio

n, it

may

be

appr

opri

ate

tore

ad C

hapt

er 1

8 fi

rst.

The

app

endi

ces

offe

r so

ftw

are

that

can

be

used

for

ana

lysi

s an

d re

prod

uce

tabl

es o

f th

e fr

acta

l dis

trib

utio

ns.



L

Ack

now

ledg

men

ts

Iw

ould

like

to th

ank

the

follo

win

g pe

ople

for

thei

rinv

alua

ble

advi

ce a

nd a

ssis

-

tanc

e: F

rom

Pan

Ago

ra A

sset

Man

agem

ent:

Ric

hard

Cro

wel

l, Pe

ter

Rat

hjen

s,

John

Lew

is, B

ruce

Cla

rke,

Ter

ry N

orm

an, A

lan

Bro

wn,

and

Cliv

e L

ang.

Als

o

War

ren

Spro

ul, E

arl K

eefe

r, R

ober

t Mel

len,

Gui

doD

eBoe

ck, a

nd R

on B

rand

es

for

help

, ide

as, a

nd r

efer

ence

s. T

hank

s al

so to

Kri

stin

e ("

with

a K

") L

ino,

Dav

id A

rrig

hini

, Jim

Rul

lo, a

nd C

huck

LeV

ine

ofth

e Pa

nAgo

ra A

sset

Allo

ca-

tion

team

, for

thei

r in

dulg

ence

and

hel

p.I

wou

ld a

lso

like

to th

ank,

bel

ated

ly, m

y or

igin

alW

iley

edito

r, W

endy

Gra

u,

who

per

suad

ed m

e to

wri

te b

oth

book

s an

d sa

w m

eth

roug

h th

e fi

rst p

ublic

a-

tion.

I w

ould

als

o lik

e to

than

k M

yles

Tho

mps

on, m

y cu

rren

tedi

tor,

for

see

ing

me

thro

ugh

this

one

.Fi

nally

, I w

ould

onc

e ag

ain

like

to th

ank

my

wif

e,Sh

eryl

, and

my

child

ren,

Ian

and

Luc

ia, f

or th

eir

cont

inue

d su

ppor

t dur

ing

the

man

y ho

urs

I ne

eded

to

mys

elf

to c

ompl

ete

this

pro

ject

.

E.E

.P.

XII

I



L

Con

tent

s

PA

RT

ON

E F

RA

CT

AL

TIM

E S

ER

IES

Intr

oduc

tion

to F

ract

al T

ime

Ser

ies

3

Fra

ctal

Spac

e, 4

Frac

tal T

ime,

5Fr

acta

l Mat

hem

atic

s, 9

The

Cha

os G

ame,

10

Wha

t Is

a Fr

acta

l?, 1

2T

he F

ract

al D

imen

sion

, 15

Frac

tal M

arke

t Ana

lysi

s, 1

7

2Fa

ilure

of th

e G

auss

ian

Hyp

othe

sis

18

Cap

ital

Mar

ket T

heor

y, 1

9St

atis

tical

Cha

ract

eris

tics

of M

arke

ts, 2

1T

he T

erm

Str

uctu

re o

f V

olat

ility

, 27

The

Bou

nded

Set

, 37

Sum

mar

y, 3

8

3A

Fra

ctal

Mar

ket H

ypot

hesi

s39

Effi

cien

tM

arke

ts R

evis

ited,

39

Stab

le M

arke

ts v

ersu

s E

ffic

ient

Mar

kets

, 41

The

Sou

rce

of L

iqui

dity

, 42

Info

rmat

ion

Sets

and

Inv

estm

ent H

oriz

ons,

43

Stat

istic

al C

hara

cter

istic

s of

Mar

kets

, Rev

isite

d, 4

3T

he F

ract

al M

arke

t Hyp

othe

sis,

44

Sum

mar

y, 4

9

xv



xvi

Con

tent

sC

onte

nts

Xvi

i

PA

RT

TW

O F

RA

CT

AL

(R/S

) A

NA

LYS

IS10

Vol

atili

ty: A

Stu

dy in

Ant

iper

sist

ence

143

Rea

lized

Vol

atili

ty, 1

46

4M

easu

ring

Mem

ory—

The

Hur

st P

roce

ss a

nd R

/S A

naly

sis

53Im

plie

dV

olat

ility

, 148

Bac

kgro

und:

Dev

elop

men

t of

R/S

Ana

lysi

s, 5

4Su

mm

ary,

150

The

Jok

er E

ffec

t, 60

11P

robl

ems

with

Und

ersa

mpl

ing:

Gol

d an

d U

.K. I

nfla

tion

151

Ran

dom

ness

and

Pers

iste

nce:

Int

erpr

etin

g th

e H

urst

Exp

onen

t, 61

R/S

Ana

lysi

s: A

Ste

p-by

-Ste

p G

uide

, 61

Typ

e I

Und

ersa

mpl

ing:

Too

Litt

le T

ime.

152

An

Exa

mpl

e: T

he Y

en/D

olla

r E

xcha

nge

Rat

e, 6

3T

ype

II U

nder

sam

plin

g: T

oo L

ow a

Fre

quen

cy, 1

54T

wo

Inco

nclu

sive

Stu

dies

, 154

5T

estin

gR

/S A

naly

sis

65S

umm

ary,

158

The

Ran

dom

Nul

l Hyp

othe

sis,

66

12C

urre

ncie

s:A

Tru

e H

urst

Pro

cess

159

Sto

chas

ticM

odel

s, 7

5Su

mm

ary,

85

The

Dat

a, 1

60Y

en/D

olla

r, 1

61M

ark/

Dol

lar,

163

6F

indi

ng C

ycle

s: P

erio

dic

and

Non

perio

dic

86P

ound

/Dol

lar,

163

Peri

odic

Cyc

les,

88

Yen

/Pou

nd, 1

65

Non

peri

odic

Cyc

les,

93

Sum

mar

y. 1

66

Sum

mar

y, 1

02PA

RT

FO

UR

FR

AC

TA

L N

OIS

E

PA

RT

TH

RE

E A

PP

LYIN

G F

RA

CT

AL

AN

ALY

SIS

13F

ract

iona

l Noi

se a

nd R

/S A

naly

sis

169

7C

ase

Stu

dy M

etho

dolo

gy10

7T

he C

olor

of N

oise

, 170

Met

hodo

logy

, 108

Pink

Noi

se: 0

<H

<0.

50, 1

72

Dat

a, 1

09B

lack

Noi

se: 0

.50<

H <

1.0,

183

Stab

ility

Ana

lysi

s, 1

10T

he M

irro

r E

ffec

t, 18

7Fr

actio

nal D

iffe

renc

ing:

AR

FIM

A M

odel

s, 1

88

8D

owJo

nes

Indu

stria

ls, 1

888—

1990

: An

Idea

l Dat

a S

et11

2S

umm

ary,

196

Num

ber

of O

bser

vatio

ns v

ersu

s L

engt

h of

Tim

e, 1

1214

Frac

tal

Sta

tistic

s19

7

Tw

enty

-Day

Ret

urns

, 113

Frac

tal (

Stab

le)

Dis

trib

utio

ns, 1

99Fi

ve-D

ay R

etur

ns, 1

17St

abili

ty u

nder

Add

ition

, 206

Dai

ly R

etur

ns, 1

19C

hara

cter

istic

s of

Fra

ctal

Dis

trib

utio

ns, 2

08St

abili

ty A

naly

sis,

125

Mea

suri

ng a

, 209

Raw

Dat

a an

d Se

rial

Cor

rela

tion,

127

Mea

suri

ng P

roba

bilit

ies,

214

Sum

mar

y, 1

30In

fini

te D

ivis

ibili

ty a

nd I

GA

RC

H, 2

14Su

mm

ary,

216

9S

&P

500

Tic

k D

ata,

198

9—19

92: P

robl

ems

with

Ove

rsam

plin

g13

215

App

lyin

g F

ract

al S

tatis

tics

217

The

Una

djus

ted

Dat

a, 1

33Po

rtfo

lio S

elec

tion,

218

The

AR

(l)

Res

idua

ls, 1

37O

ptio

n V

alua

tion,

224

Impl

icat

ions

. 141

Sum

mar

y, 2

32

L



xviii

Con

tent

s

PA

RT

FIV

EN

oisy

Cha

os

16N

oisy

Cha

os a

nd R

/S A

naly

sis

235

Info

rmat

ion

and

Inve

stor

s, 2

37C

haos

, 239

App

lyin

g R

/S A

naly

sis,

241

Dis

tingu

ishi

ng N

oisy

Cha

os f

rom

Fra

ctio

nal N

oise

, 246

FRA

CT

AL

TIM

E S

ER

IES

17Fr

acta

lS

tatis

tics,

Noi

sy C

haos

, and

the

FM

H25

2

Fre

quen

cyD

istr

ibut

ions

, 253

Vol

atili

ty T

erm

Str

uctu

re, 2

57Se

quen

tial S

tand

ard

Dev

iatio

n an

d M

ean,

259

Mea

suri

ng a

, 263

The

Lik

eiih

ood

of N

oisy

Cha

os, 2

64O

rbita

l Cyc

les,

265

Self

-Sim

ilari

ty, 2

68A

Pro

posa

l: U

nitin

g G

AR

CH

, FB

M, a

nd C

haos

, 270

18U

nder

stan

ding

Mar

kets

271

Info

rmat

ion

and

Inve

stm

ent H

oriz

ons,

272

Stab

ility

, 272

Ris

k, 2

73L

ong

Mem

ory,

274

Cyc

les,

274

Vol

atili

ty, 2

75T

owar

d a

Mor

e C

ompl

ete

Mar

ket T

heor

y, 2

75

App

endi

x1:

The

Cha

os G

ame

277

App

endi

x2:

GA

US

S P

rogr

ams

279

App

endi

x 3:

Fra

ctal

Dis

trib

utio

n T

able

s28

7

Bib

liogr

aphy

296

Glo

ssar

y30

6

Inde

x31

3



L

1 Intr

oduc

tion

to F

ract

alT

ime

Ser

ies

Wes

tern

cultu

re h

as lo

ng b

een

obse

ssed

by

the

smoo

th a

ndsy

mm

etri

c. N

ot a

ll

cultu

res

are

sim

ilarl

y ob

sess

ed, b

ut th

e W

est (

mea

ning

Eur

opea

n de

rive

d) h

as

long

reg

arde

d pe

rfec

t for

ms

as s

ymm

etri

c, s

moo

th,

and

who

le. W

e lo

ok f

or

patte

rns

and

sym

met

ry e

very

whe

re. O

ften

, we

impo

se p

atte

rns

whe

re n

one

ex-

ists

, and

we

deny

pat

tern

s th

at d

o no

t con

form

to o

urov

eral

l con

cept

ual f

ram

e-

wor

k. T

hat i

s, w

hen

patte

rns

are

not s

ymm

etri

cal

and

smoo

th, w

e cl

assi

fy

them

as

illus

ions

.T

his

conf

lict c

an b

e tr

aced

bac

k to

the

anci

ent G

reek

s.T

o de

scri

be o

ur

phys

ical

wor

ld, t

hey

crea

ted

a ge

omet

ry b

ased

on

pure

,sym

met

ric,

and

sm

ooth

form

s. P

lato

sai

d th

at th

e "r

eal"

wor

ld c

onsi

sted

of

thes

esh

apes

. The

se f

orm

s

wer

e cr

eate

d by

a f

orce

, or

entit

y,ca

lled

the

"Goo

d."

The

wor

ld o

f th

e G

ood

coul

d be

glim

psed

onl

y oc

casi

onal

ly, t

hrou

gh th

em

ind.

The

wor

ld w

e in

habi

t

is a

n im

perf

ect c

opy

of th

e re

al w

orld

, and

was

cre

ated

by a

dif

fere

nt e

ntity

,

calle

d th

e "D

emiu

rge.

" T

he D

emiu

rge,

a le

sser

bein

g th

an th

e G

ood,

was

doom

ed to

cre

ate

infe

rior

cop

ies

of th

e re

al w

orld

.T

hese

cop

ies

wer

e ro

ugh,

asym

met

ric,

and

sub

ject

to d

ecay

. In

this

way

, Pla

tore

conc

iled

the

inab

ility

of

the

Gre

ek g

eom

etry

, lat

er f

orm

aliz

ed b

y E

uclid

, to

desc

ribe

our

wor

ld. T

he

prob

lem

was

not

with

the

geom

etry

, but

with

our

wor

ldits

elf.

3



4In

trod

uctio

n to

Fra

ctal

Tim

e S

erie

sF

ract

al T

ime

5

FR

AC

TA

L S

PA

CE

Fra

ctal

geom

etry

is th

e ge

omet

ry o

f th

eD

emiu

rge.

Unl

ike

Euc

lidea

n ge

ome-

try,

it th

rive

s on

rou

ghne

ssan

d as

ymm

etry

. Obj

ects

are

not

vari

atio

ns o

n a

few

perf

ect a

nd s

ymm

etri

cal f

orm

s,bu

t are

infi

nite

ly c

ompl

ex. T

he m

ore

clos

ely

they

are

exa

min

ed, t

he m

ore

deta

il is

rev

eale

d. F

or e

xam

ple,

a tr

ee is

afr

acta

l

form

. Im

agin

e a

pine

tree

like

the

Dou

glas

fir

, com

mon

ly u

sed

for

Chr

istm

as

tree

s. C

hild

ren

ofte

n dr

awD

ougl

as f

irs

as tr

iang

les

(the

bra

nche

s)w

ith

rect

angu

lar

base

s (t

he tr

ee tr

unks

),gi

ving

the

tree

s as

muc

h sy

mm

etry

as

pos-

sibl

e. L

ogos

of

Chr

istm

as tr

ees

have

the

sam

e ap

pear

ance

or

may

subs

titut

e

cone

s fo

r th

e tr

iang

les.

Yet

, Dou

glas

fir

s ar

e no

t tri

angl

es o

r co

nes.

The

y ar

e a

netw

ork

of b

ranc

hes

qual

itativ

ely

sim

ilar

to th

e sh

ape

of th

e ov

eral

l tre

e,bu

t

each

indi

vidu

al b

ranc

h is

dif

fere

nt.

The

bra

nche

s on

bra

nche

s (s

ucce

ssiv

e ge

n-

erat

ions

of

bran

ches

) be

com

epr

ogre

ssiv

ely

smal

ler.

Yet

, with

in e

ach

gene

ra-

tion

ther

e is

act

ually

a r

ange

of

size

s. A

nd, e

ach

tree

is d

iffe

rent

.

Euc

lidea

n ge

omet

ry c

anno

t rep

licat

e a

tree

.U

sing

Euc

lidea

n ge

omet

ry, w

e

can

crea

te a

nap

prox

imat

ion

of a

tree

, but

it a

lway

slo

oks

artif

icia

l, lik

e a

child

's d

raw

ing

or a

logo

. Euc

lidea

n ge

omet

ryre

crea

tes

the

perc

eive

d sy

mm

e-

try

of th

e tr

ee, b

ut n

ot th

eva

riet

y th

at a

ctua

lly b

uild

s its

str

uctu

re.

Und

erly

ing

this

per

ceiv

ed s

ymm

etry

is a

cont

rolle

d ra

ndom

ness

, and

incr

easi

ngco

mpl

ex-

ity a

t fin

er le

vels

of

reso

lutio

n. T

his

"sel

f-si

mila

r" q

ualit

y is

the

defi

ning

cha

r-

acte

rist

ic o

f fr

acta

ls. M

ost n

atur

al s

truc

ture

s,pa

rtic

ular

ly li

ving

thin

gs, h

ave

this

cha

ract

eris

tic.

A s

econ

d pr

oble

m, w

hen

we

appl

yE

uclid

ean

geom

etry

to o

ur w

orld

, is

one

of d

imen

sion

ality

. We

live

in a

thre

e-di

men

sion

al s

pace

, but

onl

y so

lidfo

rms

are

trul

y th

ree-

dim

ensi

onal

,ac

cord

ing

to th

e de

fini

tions

that

are

the

basi

s of

Euc

lidea

n ge

omet

ry. I

n m

athe

mat

ical

term

s, a

nob

ject

mus

t be

diff

eren

tiabl

e

acro

ss it

s en

tire

surf

ace.

Aw

iffl

e ba

ll, f

or in

stan

ce, i

s no

t ath

ree-

dim

ensi

tmal

obje

ct, a

lthou

gh it

res

ides

in a

thre

e-di

men

sion

al s

pace

.

In a

dditi

on, o

ur p

erce

ptio

n of

dim

ensi

on c

an c

hang

e, d

epen

ding

on

our

dis-

tanc

e fr

om a

n ob

ject

. Fro

m a

dist

ance

, a D

ougl

as f

ir lo

oks

like

atw

o-di

men

sion

al

tria

ngle

. As

we

com

e cl

oser

, it a

ppea

rs a

s a

thre

e-di

men

sion

al c

one.

Clo

ser

still

,

we

can

see

its b

ranc

hes,

and

it lo

oks

like

a ne

twor

k of

one-

dim

ensi

onal

line

s.

Clo

ser

exam

inat

ion

reve

als

the

bran

ches

as

thre

e-di

men

sion

al tu

bes.

Euc

lidea

n

geom

etry

als

o ha

s di

ffic

ulty

with

the

dim

ensi

onal

ity o

f cr

eatio

nsof

the

Dem

iurg

e an

d w

ith in

crea

sing

com

plex

ity.

By

cont

rast

, Euc

lidea

n st

ruct

ures

be-

com

e si

mpl

er a

t sm

alle

r an

dsm

alle

r sc

ales

. The

thre

e-di

men

sion

also

lid r

e-

duce

s to

a tw

o-di

men

sion

al p

lane

. The

two-

dim

ensi

onal

pla

ne is

mad

e up

of

one-

dim

ensi

onal

line

s an

d, f

inal

ly, n

ondi

men

sion

alpo

ints

. Our

per

cept

ion

of th

e

tree

, on

the

othe

r ha

nd, w

ent

from

two-

dim

ensi

onal

toth

ree-

dim

ensi

onal

to

one-

dim

ensi

onal

, and

bac

k to

thre

e-di

men

sion

al.

Thi

s is

dif

fere

nt f

rom

the

Euc

lidea

n pe

rcep

tion.

Euc

lidea

n ge

omet

ry is

onl

y us

eful

as

a gr

oss

sim

plif

icat

ion

of th

e w

orld

of

the

Dem

iurg

e. F

ract

al g

eom

etry

, by

cont

rast

,is

char

acte

rize

d by

sel

f-si

mila

rity

and

incr

ease

d co

mpl

exity

und

erm

agni

fica

tion.

Its

maj

or a

pplic

atio

n as

a g

eom

e-

try

of s

pace

has

bee

n in

gen

erat

ing

real

istic

look

ing

land

scap

es v

ia c

ompu

ters

.

The

Der

niur

ge c

reat

ed n

ot o

nly

frac

tal s

pace

but f

ract

al ti

me

as w

ell.

Al-

thou

gh o

ur p

rim

ary

focu

s w

ill b

e on

frac

tal t

ime

seri

es, f

ract

al s

pace

will

help

us u

nder

stan

d fr

acta

l tim

e.W

e w

ill s

ee th

e di

ffer

ence

bet

wee

nth

e sm

ooth

ness

of th

e E

uclid

ean

wor

ld a

nd th

e ro

ughn

ess

of o

ur w

orld

, whi

ch li

mits

the

usef

ul-

ness

of

Euc

lid's

geo

met

ry a

s a

met

hod

of d

escr

iptio

n.

FRA

CT

AL

TIM

E

Thi

sco

nflic

t bet

wee

n th

e sy

mm

etry

of

Euc

lidea

nge

omet

ry a

nd th

e as

ymm

e-

try

of th

e re

al w

orld

can

be

furt

her

exte

nded

to o

ur c

once

pt o

f tim

e.T

radi

tion-

ally

, eve

nts

are

view

ed a

s ei

ther

ran

dom

or

dete

rmin

istic

. In

the

dete

rmin

istic

view

, all

even

ts th

roug

h tim

e ha

ve b

een

fixe

d fr

om th

e m

omen

t of

crea

tion.

Thi

s vi

ew h

as b

een

give

n a

theo

logi

cal

basi

s by

den

omin

atio

ns s

uch

as th

e

Cal

vini

sts,

and

sci

entif

ic e

ndor

sem

ent b

yce

rtai

n "b

ig b

ang"

theo

rist

s. I

n co

n-

tras

t, ni

hilis

t gro

ups

cons

ider

all

even

tsto

be

rand

om, d

eriv

ing

from

no

stru

c-

ture

or

orde

r th

roug

h tim

e.In

fra

ctal

tim

e, r

ando

mne

ss a

ndde

term

inis

m, c

haos

and

ord

er c

oexi

st. I

n

frac

tal s

hape

s, w

e se

e a

phys

ical

repr

esen

tatio

n of

how

thes

e op

posi

tes

wor

k

toge

ther

. The

pin

e tr

ee h

as g

loba

l str

uctu

rean

d lo

cal r

ando

mne

ss. I

n ge

nera

l,

we

know

wha

t a p

ine

tree

look

s lik

e, a

nd w

e ca

n pr

edic

t the

gen

eral

org

loba

l

shap

e of

any

pin

e tr

ee w

ith a

hig

h de

gree

of a

ccur

acy.

How

ever

, at t

he in

divi

d-

ual b

ranc

h le

vel,

each

bra

nch

isdi

ffer

ent.

We

do n

ot k

now

how

long

it is

, or

its

diam

eter

. Eac

h tr

ee is

dif

fere

nt, b

ut s

hare

sce

rtai

n gl

obal

pro

pert

ies.

Eac

h ha

s

loca

l ran

dom

ness

and

glo

bal d

eter

min

ism

.In

this

sec

tion,

we

will

exa

min

e

how

the

conc

ept o

f fr

acta

l tim

e ev

olve

d,an

d w

hat i

t mea

ns.

Mos

t cul

ture

s fa

vor

the

dete

rmin

istic

view

of

time.

We

like

to th

ink

that

we

have

a p

lace

in th

eun

iver

se, t

hat w

e ha

ve a

des

tiny.

Yet

, we

see

rand

om,

cata

stro

phic

eve

nts

that

can

thw

art u

s fr

omfu

lfill

ing

our

purp

ose.

Nat

ural

dis

-

aste

rs c

an d

estr

oy o

ur e

nvir

onm

ent.

Eco

nom

ic d

isas

ters

can

take

wha

t we

own.

Indi

vidu

als

we

do n

ot k

now

can

rob

us

of o

urliv

es, b

y ac

cide

nt o

r w

ith m

alic

e.

Con

vers

ely,

goo

d fo

rtun

e ca

n ar

rive

by

bein

g at

the

righ

t pla

ce a

t the

rig

ht

time.

A c

hanc

e m

eetin

g ca

n op

en n

ew d

oors

.Pi

ckin

g th

e ri

ght l

otte

ry n

umbe

rs

can

brin

g us

a f

ortu

ne.



6

Gre

atev

ents

als

o se

em to

res

t on

chan

ce. N

ewto

n sa

w a

n ob

ject

fal

ling

(leg

-en

d sa

ys it

was

an

appl

e) a

nd f

orm

ulat

ed th

e ca

lcul

us a

nd th

e la

ws

of g

ravi

ty.

Flem

ing

left

a p

etri

dis

h ex

pose

d an

d di

scov

ered

pen

icill

in. D

arw

in d

ecid

ed to

go o

n a

voya

ge a

nd f

orm

ulat

ed a

theo

ry o

f ev

olut

ion

beca

use

of h

is e

xper

ienc

es

on th

e jo

urne

y.T

hese

and

sim

ilar

even

ts s

eem

ed to

hap

pen

by c

hanc

e, a

nd th

ey c

hang

ed h

is-

tory

. Yet

,th

e ca

lcul

us in

depe

nden

tly o

f N

ewto

n, a

t alm

ost t

he

sam

e tim

e—in

fac

t, w

e us

e L

iebn

ez's

not

atio

n. W

alla

cede

velo

ped

the

theo

ry o

fna

tura

l sel

ectio

n in

depe

nden

tly o

f D

arw

in, t

houg

h la

ter.

Bec

ause

of

a pa

per

byW

alla

ce, D

arw

in f

ound

the

ener

gy to

wri

te O

rigi

n of

Spe

cies

so

he w

ould

rec

eive

cred

it as

the

theo

ry's

ori

gina

l dev

elop

er. I

n ou

r ow

n fi

eld

of f

inan

cial

eco

nom

ics,

wha

t is

know

n as

the

Cap

ital A

sset

Pri

cing

Mod

el (

CA

PM)

was

dev

elop

ed in

de-

pend

ently

by

no f

ewer

than

thre

e pe

ople

—Sh

arpe

(19

64),

Lin

tner

(19

65),

and

Mos

sin

(196

6)—

at a

lmos

t the

sam

e tim

e. T

his

wou

ld im

ply

that

thes

e di

scov

erie

sw

ere

mea

nt to

hap

pen.

His

tory

dem

ande

d it.

It w

as th

eir

dest

iny.

It h

as b

een

diff

icul

t to

reco

ncile

ran

dom

ness

and

ord

er, c

hanc

e an

d ne

ces-

sity

, or

free

will

and

det

erm

inis

m. I

s th

is d

icho

tom

y on

ce a

gain

the

Dem

iurg

eim

perf

ectly

cop

ying

the

Goo

d?E

vent

s ar

e pe

rcei

ved

as e

ither

ran

dom

, and

ther

efor

e un

pred

icta

ble,

or

de-

term

inis

tic, a

nd p

erfe

ctly

pre

dict

able

. Unt

il th

e be

ginn

ing

of th

is c

entu

ry, i

tw

as g

ener

ally

acc

epte

d th

at th

e un

iver

se r

an li

ke a

clo

ck. E

vent

ually

,sc

ient

ists

wer

e to

dis

cove

r th

e eq

uatio

ns u

nder

lyin

g th

e un

iver

se, a

nd b

ecom

eab

le to

pre

-

dict

its

even

tual

cou

rse.

Tim

e w

as o

f no

con

sequ

ence

in N

ewto

nian

mec

hani

cs;

theo

retic

ally

, tim

e co

uld

be r

ever

sed,

bec

ause

New

ton'

s eq

uatio

ns w

orke

d fi

new

heth

er ti

me

ran

forw

ard

or b

ackw

ard.

Irr

ever

sibl

e tim

e, th

e fi

rst b

low

to th

isde

term

inis

tic v

iew

, cam

e in

the

mid

-I9t

h ce

ntur

y fr

om th

e em

ergi

ng f

ield

of

ther

mod

ynam

ics.

The

rmod

ynam

ics

bega

n as

the

stud

y of

hea

t was

te p

rodu

ced

by m

achi

nes.

It

was

som

e tim

e be

fore

ther

mod

ynam

ics,

an

appl

ied

scie

nce,

was

take

n se

riou

sly

by s

cien

tists

. The

initi

al s

tudy

foc

used

on

how

ene

rgy

is c

onve

rted

into

use

ful

wor

k. I

n a

syst

em s

uch

as th

e st

eam

eng

ine,

ste

am tu

rns

whe

els

and

perf

orm

sa

func

tion

such

as

pow

erin

g a

boat

's p

addl

e w

heel

. Not

all

the

ener

gypr

oduc

edis

con

vert

ed in

to w

ork.

Som

e is

lost

or

is d

issi

pate

d as

fri

ctio

n. T

he s

tudy

of

thes

e "d

issi

pativ

e sy

stem

s" e

vent

ually

gre

w to

incl

ude

flui

d dy

nam

ics.

Flu

iddy

nam

ics,

whi

ch in

vest

igat

ed th

e he

atin

g an

d m

ixin

g of

flu

ids,

gav

e us

tim

e-

irre

vers

ible

sys

tem

s.Su

ppos

e tw

o lit

ers

of f

luid

are

sep

arat

ed b

y a

wat

erpr

oof,

rem

ovab

le p

arti-

tion.

On

one

side

is a

lite

r of

red

flu

id; o

n th

e ot

her,

a li

ter

of b

lue.

We

deci

deto

use

the

term

ent

ropy

as

the

mea

sure

of

the

degr

ee o

f m

ixin

g of

the

red

and

blue

flu

ids.

As

long

as

the

part

ition

is in

pla

ce, w

eha

ve lo

w e

ntro

py. I

f w

elif

t

the

part

ition

, the

red

and

blue

flu

ids

will

flo

w in

to o

nean

othe

r, a

nd th

e le

vel o

f

entr

opy

will

ris

e as

the

flui

ds b

ecom

e m

ore

mix

ed. E

vent

ually

, whe

nth

e re

d

and

blue

bec

ome

thor

ough

lym

ixed

, all

of th

e fl

uid

will

beco

me

purp

le.

Whe

n fu

lly m

ixed

, the

flu

idha

s re

ache

d a

stat

e of

equi

libri

um. I

t can

not

beco

me

"mor

e m

ixed

." I

tha

s re

ache

d a

leve

l of

max

imum

ent

ropy

. How

ever

,

we

cann

ot "

unm

ix"

the

flui

d. D

espi

te th

e fa

ctth

at th

e m

ixin

g of

the

flui

ds is

unde

rsla

ndab

le in

dyn

amic

al te

rms,

it is

tim

e-de

pend

ent a

ndir

reve

rsib

le. T

he

flui

d's

stat

e of

hig

h en

trop

y, o

run

cert

aint

y, w

hich

com

esfr

om th

e m

axim

um

mix

ing

of tw

o st

ates

(in

this

cas

e, th

e st

ates

are

labe

led

"red

" an

d "b

lue"

),

cann

ot b

e de

scri

bed

bytim

e-re

vers

ible

, New

toni

aneq

uatio

ns. T

he f

luid

will

neve

r be

com

eun

mix

ed; i

ts e

ntro

py w

ill n

ever

decl

ine,

eve

n if

we

wai

t for

ete

r-

nity

. In

ther

mod

ynam

ics,

time

has

an a

rrow

that

poin

ts o

nly

tow

ard

the

futu

re.

The

fir

st b

low

had

bee

nst

ruck

aga

inst

the

cloc

kwor

kvi

ew o

f th

e un

iver

se.

The

sec

ond

blow

cam

e w

ithth

e em

erge

nce

of q

uant

umm

echa

nics

. The

re-

aliz

atio

n th

at th

e m

olec

ular

str

uctu

reof

the

univ

erse

can

be

desc

ribe

d on

ly b

y

stat

es o

f pr

obab

ility

furt

her

unde

rmin

ed th

ede

term

inis

tic v

iew

. But

con

fusi

on

rem

aine

d. W

as th

e un

iver

sede

term

inis

tic o

r ra

ndom

?

Slow

ly, i

t has

bec

ome

appa

rent

that

mos

t nat

ural

sys

tem

s ar

ech

arac

teri

zed

by

loca

l ran

dom

ness

and

glo

bal

dete

rmin

ism

. The

se c

ontr

ary

stat

esm

ust c

oexi

st.

Det

erm

inis

m g

ives

us

natu

ral l

aw.R

ando

mne

ss in

duce

s in

nova

tion

and

vari

ety.

A h

ealth

y, e

volv

ing

syst

emis

one

that

not

onl

y ca

nsu

rviv

e ra

ndom

sho

cks,

but

also

can

abs

orb

thos

e sh

ocks

toim

prov

e th

e ov

eral

l sys

tem

,w

hen

appr

opri

ate.

.

For

inst

ance

, it h

as b

een

post

ulat

ed b

y W

est a

nd G

oldb

erge

r(1

987)

that

phy

s-

ical

fra

ctal

str

uctu

res

are

gene

rate

d by

nat

ure

beca

use

they

are

mor

e er

ror-

tole

rant

than

sym

met

rica

l str

uctu

res

in th

eir

crea

tion.

Tak

eth

e m

amm

alia

n

lung

. Its

mai

n br

anch

, the

trac

hea,

div

ides

into

two

subb

ranc

hes.

The

se tw

o

halv

es c

ontin

ue b

ranc

hing

. At

each

bra

nchi

ng g

ener

atio

n,th

e av

erag

e di

amet

er

decr

ease

s ac

cord

ing

to a

pow

erla

w. T

hus,

the

diam

eter

of

each

gen

erat

ion

is d

e-

pend

ent o

n th

e di

amet

ers

of th

e pr

evio

us g

ener

atio

n.In

add

ition

, eac

h br

anch

gene

ratio

n ac

tual

ly h

as a

ran

geof

dia

met

ers

with

in it

. The

ave

rage

diam

eter

of

each

gen

erat

ion

scal

es d

own

acco

rdin

g to

a p

ower

law

, but

any

indi

vidu

al b

ranc

h

can

be d

escr

ibed

only

pro

babi

listic

ally

. We

have

glo

bal d

eter

min

ism

(the

ave

r-

age

bran

ch s

ize)

and

loca

l ran

dom

ness

(th

edi

amet

er o

f in

divi

dual

bra

nche

s).

Why

doe

s na

ture

fav

or th

isst

ruct

ure,

whi

ch a

ppea

rsin

all

mam

mal

ian

lung

s?

Wes

t and

Gol

dber

ger

have

show

n th

at th

is f

ract

al s

truc

ture

is m

ore

stab

le a

nd

erro

r-to

lera

nt th

an o

ther

str

uctu

res.

Rem

embe

r th

at e

ach

bran

chge

nera

tion

is

depe

nden

t on

the

gene

ratio

nsbe

fore

it. I

f di

amet

ers

scal

edex

pone

ntia

lly, n

ot

only

wou

ld a

n er

ror

in th

efo

rmat

ion

of o

ne g

ener

atio

n af

fect

the

next

bra

nchi

ng

Intr

oduc

tion

to F

ract

al T

ime

Ser

ies

Fra

ctal

Tim

e7



8In

trod

uctio

n to

Fra

ctal

Tim

e S

erie

sfr

acta

l Mat

hem

atic

s-

gene

ratio

n, b

ut th

e er

ror

wou

ld g

row

with

eac

h su

cces

sive

gen

erat

ion.

A s

mal

ler

ror

mig

ht c

ause

the

lung

to b

ecom

e m

alfo

rmed

and

non

func

tiona

l. H

owev

er,

by f

ract

al s

calin

g, th

e er

ror

has

less

impa

ct b

ecau

se o

f th

e po

wer

law

, as

wel

l as

the

loca

l pro

babi

listic

str

uctu

re. B

ecau

se e

ach

gene

ratio

n ha

s a

rang

e of

dia

me-

ters

, one

mal

form

ed b

ranc

h ha

s le

ss im

pact

on

the

form

atio

n of

the

othe

rs.

Thu

s, th

e fr

acta

l str

uctu

re (

glob

al d

eter

min

ism

, loc

al r

ando

mne

ss)

is m

ore

erro

r-to

lera

nt d

urin

g fo

rmat

ion

than

oth

er s

truc

ture

s.T

o lo

ok a

head

, if

we

chan

ge th

is c

once

pt f

rom

a s

tatic

str

uctu

re (

the

lung

) to

a dy

nam

ic s

truc

ture

like

the

stoc

k m

arke

t, w

e ca

n m

ake

som

e in

tere

stin

g co

n-je

ctur

es. C

hang

e br

anch

gen

erat

ion

to "

inve

stm

ent h

oriz

on."

The

sto

ck m

arke

tis

mad

e up

of

inve

stor

s, f

rom

tick

trad

ers

to lo

ng-t

erm

inve

stor

s. E

ach

has

adi

ffer

ent i

nves

tmen

t hor

izon

that

can

be

orde

red

in ti

me.

A s

tabl

e m

arke

t is

one

in w

hich

all

the

part

icip

ants

can

trad

e w

ith o

ne a

noth

er, e

ach

faci

ng a

ris

kle

vel l

ike

the

othe

rs',

adju

sted

for

thei

r tim

e sc

ale

or in

vest

men

t hor

izon

. We

will

see

in C

hapt

er 2

that

the

freq

uenc

y di

stri

butio

n of

ret

urns

is th

e sa

me

for

day

trad

ers

as it

is f

or o

ne-w

eek

or e

ven

90-d

ay r

etur

ns, o

nce

an a

djus

tmen

t is

mad

e fo

r sc

ale.

Tha

t is,

fiv

e-m

inut

e tr

ader

s fa

ce th

e sa

me

risk

of

a la

rge

even

tas

doe

s a

wee

kly

trad

er. I

f da

y tr

ader

s ha

ve a

cra

sh a

t the

ir ti

me

scal

e, li

ke a

four

-sig

ma

even

t, th

e m

arke

t rem

ains

sta

ble

if th

e ot

her

trad

ers,

who

hav

e di

f-fe

rent

trad

ing

hori

zons

, see

that

cra

sh a

s a

buyi

ng o

ppor

tuni

ty, a

nd s

tep

in a

ndbu

y. T

hus,

the

mar

ket r

emai

ns s

tabl

e be

caus

e it

has

no c

hara

cter

istic

tim

esc

ale,

just

as

the

lung

had

no

char

acte

rist

ic d

iam

eter

sca

le. W

hen

the

mar

ket's

entir

e in

vest

men

t hor

izon

sho

rten

s, a

nd e

very

one

beco

mes

a o

ne-m

inut

e tr

ader

(inv

esto

rs h

ave

lost

thei

r fa

ith in

long

-ter

m in

form