MAR KET ANALYSIS - books.mec.biz · FRACTAL MAR KET ANALYSIS Charles B ... Wong TRADING APPLICATIONS OF JAPANESE CANDLESTICK CHARTING Gary S. Wagner and Brad L. Matheny FRACTAL MARKET

WIL

EY

FIN

AN

CE

ED

ITIO

NS

POR

TFO

LIO

MA

NA

GE

ME

NT

FO

RM

UL

AS

Ral

ph V

ince

TR

AD

ING

AN

D I

N V

EST

ING

IN

BO

ND

OPT

ION

SM

. Ant

hony

Won

g

FRA

CT

AL

MA

R K

ET

AN

ALY

SIS

Cha

rles

B. E

pste

in, E

dito

rA

pply

ing

Cha

os T

heor

y to

Inve

stm

ent

AN

ALY

ZIN

GA

ND

FO

RE

CA

STIN

G F

UT

UR

ES

PRIC

ES

Ant

hony

F. H

erbs

tan

dE

cono

mic

sC

HA

OS

AN

D O

RD

ER

IN

TH

E C

API

TA

L M

AR

KE

TS

Edg

arE

.Pe

ters

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

___

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

PER

AT

ION

S A

ND

TH

E F

OR

EIG

N E

XC

HA

NG

E C

HA

LL

EN

GE

Dim

itris

N. C

hora

fas

TH

E F

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

MA

TIC

S O

F M

ON

EY

MA

NA

GE

ME

NT

Ral

ph V

ince

TH

E N

EW

TE

CH

NO

LO

GY

OF

FIN

AN

CIA

L M

AN

AG

EM

EN

TD

imitr

is N

. Cho

rafa

sT

HE

DA

Y T

RA

DE

R'S

MA

NU

AL

Will

iam

F. E

ngO

PTIO

N M

AR

KE

T M

AK

ING

Alle

n J.

Bai

rdT

RA

DIN

G F

OR

A L

IVIN

GD

r. A

lexa

nder

Eld

erC

OR

POR

AT

E F

INA

NC

IAL

DIS

TR

ESS

AN

D B

AN

KR

UPT

CY

, SE

CO

ND

ED

ITIO

NE

dwar

d I.

Altm

anFI

XE

D.I

NC

OM

E A

RB

ITR

AG

EM

. Ant

hony

Won

gT

RA

DIN

G A

PPL

ICA

TIO

NS

OF

JAPA

NE

SE C

AN

DL

EST

ICK

CH

AR

TIN

GG

ary

S. W

agne

r an

d B

rad

L. M

athe

nyFR

AC

TA

L M

AR

KE

T A

NA

LY

SIS:

APP

LY

ING

CH

AO

S T

HE

OR

Y T

O I

NV

EST

ME

NT

AN

D E

CO

NO

MIC

SE

dgar

E. P

eter

sU

ND

ER

STA

ND

ING

SW

APS

John

F. M

arsh

all a

nd K

enne

th R

. Kap

ner

JOH

NW

ILE

Y &

SON

S,IN

C.

GE

NE

NT

ICA

LG

OR

ITH

MS

AN

D I

NV

EST

ME

NT

ST

RA

TE

GIE

SR

icha

rd J

Bau

er, J

rN

ew Y

ork

C

hich

este

r

Bris

bane

T

oron

to

Sin

gapo

re

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s te

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prin

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on a

cid-

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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

n or

tran

slat

ion

of a

ny p

art

of th

is w

ork

beyo

nd

that

per

mitt

ed b

y S

ectio

n 10

701

108

of th

e 19

76 U

nite

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Sta

tes

Cop

yrig

ht A

ct w

ithou

t the

perm

issi

on o

f the

cop

yrig

bt

owne

r is

unl

awfu

l. R

eque

sts

for

perm

issi

on o

r fu

rthe

r

info

rmat

ion

shou

ld b

e ad

dres

sed

to th

eP

erm

issi

ons

Dep

artm

ent,

John

Wile

y &

Son

s, In

c., 6

05 T

hird

Ave

nue,

New

Yor

k, N

Y

1015

8-00

12.

Thi

s pu

blic

atio

n is

des

igne

d to

pro

vide

acc

urat

ean

d

auth

orita

tive

info

rmat

ion

in r

egar

d to

the

subj

ect

mat

ter

cove

red.

It is

sol

d w

ithth

e un

ders

tand

ing

that

the

publ

ishe

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

oin

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

h 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

3

00. A

s n

beco

mes

larg

er,

equa

tion

(5.5

)

appr

oach

es e

quat

ion

(5.2

).E

quat

ions

(5.

4) a

nd (

5.5)

adj

ust

for

the

dist

ribu

tion

68T

estin

g R

/S A

naly

sis

The

Ran

dom

Nul

l Hyp

othe

sis

69

1.5 1 0.5 0

0.5

11.

52

2.5

Log

(Nui

nber

of

Obs

erva

tions

)

Tab

le 5

.1L

og(R

IS)

Val

ueE

stim

ates



70T

estin

g R

/S A

naly

sis

The

Ran

dom

Nul

l Hyp

othe

sis

71

of th

e va

rian

ce o

f th

e no

rmal

dis

trib

utio

n to

fol

low

the

gam

ma

dist

ribu

tion;

that

is, t

he s

tand

ard

devi

atio

n w

ill s

cale

at a

slo

wer

rat

e th

an th

e ra

nge

for

smal

l val

ues

of n

. Hen

ce, t

he r

esca

led

rang

e w

ill s

cale

at a

fast

er r

ate

(H w

illbe

gre

ater

than

0.5

0) w

hen

n is

sm

all.

Man

delb

rot a

nd W

allis

(l96

9a,b

,c)

re-

ferr

ed to

the

regi

on o

f sm

all n

as

"tra

nsie

nt"

beca

use

n w

as n

otla

rge

enou

gh

for

the

prop

er b

ehav

ior

to b

e se

en. H

owev

er, i

n ec

onom

ics,

we

rare

ly h

ave

enou

gh d

ata

poin

ts to

thro

w o

ut th

e sm

alle

r n:

that

may

be

all t

hat w

eha

ve.

Man

delb

rot a

nd W

allis

wou

ld n

ot s

tart

inve

stig

atin

g sc

alin

gbe

havi

or u

ntil

H =

20.

The

oret

ical

ly, A

nis

and

Llo

yd's

for

mul

a w

as e

xpec

ted

to e

xpla

in th

ebe

havi

or s

een

from

the

Mon

te C

arlo

exp

erim

ents

.T

able

5.1

and

Fig

ure

5.2

show

the

resu

lts. T

here

is s

ome

prog

ress

,but

equ

a-

tions

(5.

4) a

nd (

5.5)

stil

l gen

erat

e R

IS v

alue

s fo

r sm

all n

that

are

high

er th

an

the

sam

pled

val

ues.

The

re is

a p

ossi

bilit

y th

at th

e re

sults

are

cau

sed

by a

bia

s, o

rigi

natin

gin

the

pseu

do-r

ando

m n

umbe

r ge

nera

tor,

that

dou

ble

scra

mbl

ing

does

not

red

uce.

Perh

aps

a sa

mpl

e si

ze o

f 30

0 is

stil

l not

eno

ugh.

To

test

for

sam

ple

bias

, an

inde

pend

ent s

erie

s of

num

bers

was

use

d. T

his

seri

es w

as 5

00 m

onth

lyS&

P 50

0

chan

ges,

nor

mal

ized

to m

ean

zero

and

uni

t var

ianc

e. T

hese

num

bers

wer

esc

ram

bled

10

times

bef

ore

star

ting.

The

n th

ey w

ere

rand

omly

scr

ambl

ed30

0

0.5 0

FIG

UR

E 5

.2R

IS v

alue

s,M

onte

Car

lo s

imul

atio

n ve

rsus

Ani

s an

dLl

oyd'

s eq

uatio

n.

Tab

le 5

.2Lo

g (R

/S)

Val

ue E

stim

ates

Num

ber

ofO

bser

vatio

ns

Scr

ambl

edS

&P

500

Mon

te C

arlo

0.45

510.

4577

10 200.

6474

0.71

2325

0.88

9150

0.88

121.

0577

100

1.1097

125

250

1.1012

1.2591

1.2710

times

, and

R/S

val

ues

wer

eca

lcul

ated

as

befo

re. T

able

5.2

show

s th

e re

sults

.

The

y ar

e vi

rtua

llyin

dist

ingu

isha

ble

from

the

Gau

ssia

nge

nera

ted

seri

es. T

he

resu

lts a

re e

ven

mor

e re

mar

kabl

ew

hen

we

cons

ider

that

mar

ket r

etur

nsar

e no

t

norm

ally

dis

trib

uted

; the

y ar

efa

t-ta

iled

with

a h

igh

peak

at

the

mea

n, e

ven

afte

r sc

ram

blin

g. F

rom

thes

ere

sults

, we

can

say

that

the

Ani

s an

d L

loyd

for

-

mul

a is

mis

sing

som

ethi

ngfo

r va

lues

of

n le

ss th

an 2

0.W

hat t

hey

are

mis

sing

is u

nkno

wn.

How

ever

,em

piri

cally

, I w

as a

ble

to d

eriv

e a

corr

ectio

n to

the

Ani

s

and

Llo

yd f

orm

ula.

Thi

sco

rrec

tion

mul

tiplie

s (5

.4)

and

(5.5

)w

itha

corr

ectio

n

fact

or a

nd y

ield

s:

((n

0.

5)/

r)

/ r(5

.6)

The

res

ults

of

this

em

piri

cally

deri

ved

corr

ectio

n ar

e sh

own

in T

able

5.1

and

Figu

re 5

.3. T

he c

orre

ctio

n co

mes

very

clo

se to

the

sim

ulat

ed R

/S v

alue

s.

From

this

poi

nt f

orw

ard,

all

expe

cted

RIS

val

ues

unde

r th

era

ndom

nul

l hy-

poth

esis

will

be

gene

rate

dus

ing

equa

tion

(5.6

).

The

Exp

ecte

d V

alue

of t

he H

urst

Exp

onen

t

Usi

ngth

e re

sults

of

equa

tion

(5.6

), w

e ca

nno

w g

ener

ate

expe

cted

val

ues

of th

e

Hur

st e

xpon

ent.

Judg

ing

from

Tab

le 5

.1 a

nd F

igur

e 5.

3, w

e ca

nex

pect

that

the

Hur

st e

xpon

ent w

ill b

esi

gnif

ican

tly h

ighe

r th

an 0

.50

for

valu

es le

ss th

an

500

show

ing,

aga

in, t

hat H

0.50

for

an

inde

pend

ent p

roce

ssis

an

asym

p-

totic

lim

it. T

he e

xpec

ted

Hur

st e

xpon

ent w

ill, o

f co

urse

, var

y,de

pend

ing

on

the

valu

es o

f n

we

use

to r

unth

e re

gres

sion

. In

theo

ry, a

ny r

ange

will

be

appr

o-

pria

te a

s lo

ng a

s th

e sy

stem

unde

r st

udy

and

the

E(R

/S)

seri

es c

over

the

sam

e

valu

es o

f n.

In

keep

ing

with

the

prim

ary

focu

s of

this

book

, whi

ch is

fin

anci

al

2

1.5

0.5

11.

52

2.5

33,

54

Log(

Num

ber

of O

bser

vatio

ns)



72T

estin

g R

/S A

naly

sis

The

Ran

dom

Nul

l Hyp

othe

sis

73

FIG

UR

E 5

.3R

/S v

alue

s, M

onte

Car

lo s

imul

atio

n ve

rsus

cor

rect

ed A

nis

and

Lloy

deq

uatio

n.

econ

omic

s, w

e w

ill b

egin

with

n =

10.

The

fina

lva

lue

of n

will

dep

end

on th

esy

stem

und

er s

tudy

. in

Pet

ers

(199

Ia),

the

mon

thly

ret

urns

of t

he S

&P

500

wer

e fo

und

to h

ave

pers

iste

nt s

calin

g fo

r n

0.5

0), t

hen

the

grap

h w

ould

be

upw

ardl

y sl

opin

g. C

on-

vers

ely,

if th

e pr

oces

s w

as a

ntip

ersi

sten

t (H

< 0

.50)

, the

gra

ph w

ould

be

FIG

UR

E6.

5W

eirs

tras

s fu

nctio

n, V

sta

tistK

.

2.5 2

1.5

0.5

25

2

1.5

C)

C,,

LI



94F

indi

ng C

ycle

s: P

erio

dic

and

Non

perio

dic

one

revo

lutio

n of

the

Ear

th a

roun

d th

e su

n, a

nd th

e tim

e it

take

s fo

r ou

rpl

anet

to

rota

te o

nce

on it

s ax

is. W

e ha

ve d

evel

oped

cloc

ks a

nd c

alen

dars

that

pre

cise

lydi

vide

thes

e fr

eque

ncie

s in

to in

crem

ents

cal

led

year

s, d

ays,

or

min

utes

. The

sea

-

sona

l pat

tern

see

ms

abso

lute

ly p

erio

dic.

Spr

ing

is f

ollo

wed

by S

umm

er, A

u-

tum

n, a

nd W

inte

r, in

that

ord

er. W

e ha

ve b

ecom

eac

cust

omed

to im

plyi

ng th

e

wor

d pe

riod

ic e

very

tim

e w

e us

e th

e w

ord

cycl

e. Y

et, w

e kn

owth

at s

ome

thin

gs

have

cyc

les,

but

we

cann

ot b

e su

re e

xact

ly h

ow lo

ng e

ach

cycl

e la

sts.

The

sea

-

sona

l pat

tern

of

the

Ear

th's

wea

ther

is p

erfe

ctly

pre

dict

able

,but

we

know

that

exce

ptio

nally

hig

h te

mpe

ratu

res

can

be f

ollo

wed

by

mor

e of

the

sam

e,ca

usin

g a

"hea

t wav

e."

We

also

kno

w th

at th

e lo

nger

the

heat

wav

e la

sts,

the

mor

e lik

ely

that

it w

ill c

ome

to a

n en

d. B

ut w

e do

n't k

now

exa

ctly

whe

n.W

e no

w k

now

that

thes

e no

nper

iodi

c cy

cles

can

hav

e tw

o so

urce

s:

1.T

hey

can

be s

tatis

tical

cyc

les,

exe

mpl

ifie

d by

the

Hur

st p

heno

men

aof

pers

iste

nce

(lon

g-ru

n co

rrel

atio

ns)

and

abru

pt c

hang

es in

dir

ectio

n;

2.T

hey

can

be th

e re

sult

of a

non

linea

r dy

nam

ic s

yste

m, o

rde

term

inis

tic

chao

s.F

IGU

RE

6.6

aF

ract

al ti

me

serie

s: H

= 0

.72.

We

will

now

bri

efly

dis

cuss

the

diff

eren

ces

betw

een

thes

e tw

o sy

stem

s.

Stat

istic

alC

ycle

s

2

The

Hur

st p

roce

ss, e

xam

ined

clo

sely

in C

hapt

er 4

, is

a pr

oces

s th

at c

anbe

de-

scri

bed

as a

bia

sed

rand

om w

alk,

but

the

bias

can

cha

nge

abru

ptly

, in

dire

ctio

n

or m

agni

tude

. The

se a

brup

t cha

nges

in b

ias,

mod

eled

by

Hur

st a

s th

e jo

ker

in h

is

prob

abili

ty p

ack

of c

ards

, giv

e th

e ap

pear

ance

of

cycl

es. U

nfor

tuna

tely

,de

spite

the

robu

stne

ss o

f th

e st

atis

tical

str

uctu

re, t

he a

ppea

ranc

e of

the

joke

r is

a

dom

eve

nt. B

ecau

se th

e cu

tting

of

the

prob

abili

ty d

eck

occu

rs w

ithre

plac

emen

t,

ther

e is

no

way

to p

redi

ct w

hen

the

joke

r w

ill a

rriv

e. W

hen

Man

delb

rot

(198

2)

said

that

"th

e cy

cles

mea

n no

thin

g" if

eco

nom

ic c

ycle

s ar

e a

Hur

st p

roce

ss,h

e

mea

nt th

at th

e du

ratio

n of

the

cycl

e ha

d no

mea

ning

and

was

not

apr

oduc

t of

the

time

seri

es a

lone

. Ins

tead

, the

arr

ival

of

the

joke

r w

as d

ue to

som

e ex

ogen

ous

even

t tha

t may

or

may

not

be

pred

icta

ble.

In

light

of

this

, Hur

st"c

ycle

s" h

ave

no

aver

age

leng

th, a

nd th

e lo

g/lo

g pl

ot c

ontin

ues

to s

cale

inde

fini

tely

. Fig

ure

6.6(

a)

show

s a

sim

ulat

ed ti

me

seri

es w

ith H

0.72

. The

tim

e se

ries

"lo

oks

like"

ast

ock

mar

ket c

hart

, with

pos

itive

and

neg

ativ

e ru

ns a

nd th

e us

ual a

mou

ntof

"noi

se."

Fig

ure

6.6(

b) is

an

RIS

plo

t for

the

sam

e se

ries

. Alth

ough

the

seri

es is

over

8,0

00 o

bser

vatio

ns in

leng

th, t

here

is n

ote

nden

cy to

dev

iate

fro

m th

e tr

end

line.

The

re is

no

aver

age

cycl

e le

ngth

.

0.5

11.

52

Log(

Num

ber

of O

bser

vatio

ns)2

.53

FIG

UR

E 6

.bb

R/S

ana

lysi

s, fr

acta

l tim

e se

ries:

H =

0.7

2.

1.5

0.5

0

95



96F

indi

ng C

ycle

s: P

erio

dic

and

Non

perio

dic

Cha

otic

Cyc

les

Non

linea

rdy

nam

ical

sys

tem

s ar

e de

term

inis

tic s

yste

ms

that

can

exh

ibit

er-

ratic

beh

avio

r. W

hen

disc

ussi

ng c

haos

, it i

s co

mm

on to

ref

er to

cha

otic

map

s.M

aps

are

usua

lly s

yste

ms

of it

erat

ed d

iffe

renc

e eq

uatio

ns, s

uch

as th

e fa

mou

sL

ogis

tic E

quat

ion:

X

0, t

here

als

o ex

ists

b >

0 s

uch

that

:

(14.

1)

Thi

s re

latio

nshi

p ex

ists

for

all d

istr

ibut

ion

func

tions

. F(x

) is

a g

ener

al c

har-

acte

ristic

of t

he c

lass

of s

tabl

e di

strib

utio

ns,

rath

er th

an a

pro

pert

y of

any

one

dist

ribut

ion.

The

cha

ract

eris

tic fu

nctio

ns o

f F c

anbe

exp

ress

ed in

a s

imila

r m

anne

r: (14.

2)

The

refo

re, f

(b1*

t), f

(b2*

t), a

nd f(

b*t)

all

have

the

sam

e sh

aped

dis

trib

utio

n,

desp

ite th

eir

bein

g pr

oduc

ts o

f one

ano

ther

.T

his

acco

unts

for

thei

r "s

tabi

lity.

"

The

act

ual r

epre

sent

atio

n of

the

stab

le d

istr

ibut

ions

is ty

pica

lly d

one

inth

e

man

ner

of M

ande

ibro

t (19

64),

usi

ngth

e lo

g of

thei

r ch

arac

teris

tic fu

nctio

ns:

(14.

3)

The

sta

ble

dist

ribut

ions

hav

e fo

ur p

aram

eter

s: a

,c,

and

& E

ach

has

its

own

func

tion,

alth

ough

only

two

are

cruc

ial.

Firs

t, co

nsid

er th

e re

lativ

ely

unim

port

ant

c an

d &

is th

e lo

ca-

tion

para

met

er. E

ssen

tially

,th

e di

strib

utio

n ca

n ha

ve d

iffer

ent m

eans

than

0 (

the

stan

dard

nor

mal

mea

n), d

epen

ding

on

& In

mos

t cas

es,

the

dist

ribut

ion

unde

r

stud

y is

nor

mal

ized

, and

= 0

; tha

t is,

the

mea

nof

the

dist

ribut

ion

is s

et to

0.



Para

met

er c

is th

e sc

ale

para

met

er. I

t is

mos

t im

port

ant w

hen

com

pari

ng r

eal

dist

ribu

tions

. Aga

in, w

ithin

the

norm

aliz

ing

conc

ept,

c is

like

the

sam

ple

devi

a-tio

n; it

is a

mea

sure

of

disp

ersi

on.

Whe

n no

rmal

izin

g, it

isco

mm

on to

sub

trac

tth

e sa

mpl

e m

ean

(to

give

a m

ean

of 0

) an

d di

vide

by

the

stan

dard

dev

iatio

n,so

that

uni

ts a

re in

term

s of

the

sam

ple

stan

dard

dev

iatio

n. T

heno

rmal

izin

g op

era-

tion

is d

one

to c

ompa

rean

em

piri

cal d

istr

ibut

ion

to th

est

anda

rd n

orm

al d

istr

i-bu

tion

with

mea

n=

0an

d st

anda

rd d

evia

tion

of1.

c is

use

d to

set

the

units

byw

hich

the

dist

ribu

tion

isex

pand

ed a

nd c

ompr

esse

dab

out &

The

def

ault

valu

e of

c is

1. T

hese

two

para

met

ers'

onl

ypu

rpos

e is

set

ting

the

scal

e of

the

dist

ribu

-tio

n, r

egar

ding

mea

n an

ddi

sper

sion

. The

y ar

e no

t rea

llych

arac

teri

stic

to a

nyon

e di

stri

butio

n, a

nd s

o ar

e le

ss im

port

ant.

Whe

n c

=I

and

6 =

0,th

e di

stri

bu-

tion

is s

aid

to ta

kea

redu

ced

form

.Pa

ram

eter

s a

and

13 d

eter

min

eth

e sh

ape

of th

e di

stri

butio

nan

d ar

e ex

-tr

emel

y im

port

ant.

The

setw

o pa

ram

eter

s ar

e de

pend

ent o

n th

e ge

nera

ting

pro-

cess

; c a

nd 6

are

not

. 13

isth

e sk

ewne

ss p

aram

eter

.It

take

s va

lues

suc

h th

at

1 s

13+

1. W

hen

13=

0,

the

dist

ribu

tion

issy

mm

etri

cal a

roun

d 6.

Whe

nth

e sk

ewne

ss p

aram

eter

isle

ss th

an 0

, the

dis

trib

utio

nis

neg

ativ

ely

skew

ed;

whe

n it

is g

reat

er th

an 0

,th

e di

stri

butio

n is

pos

itive

lysk

ewed

.Pa

ram

eter

a, t

he c

hara

cter

istic

expo

nent

, det

erm

ines

the

peak

edne

ssat

6an

d th

e fa

tnes

s of

the

tails

.T

he c

hara

cter

istic

expo

nent

can

take

the

valu

eso

u)

)_a

*(lo

g(U

)lo

g(U

i))(l

4.7a

)

log(

P(U

2

I, t

he r

esid

ual r

isk,

decr

ease

s as

the

num

ber

of a

sset

s, N

,in

crea

ses.

Int

eres

tingl

y, if

alp

ha e

qual

s 1,

ther

e is

no

dive

rsif

icat

ion

effe

ct; i

fal

pha

is le

ss th

an 1

, inc

reas

ing

the

port

folio

siz

e in

crea

ses

the

nonm

arke

t ris

k.Fa

ma

and

Mill

er (

1972

) us

ed th

e fo

llow

ing

exam

ple.

Sup

pose

that

cr=

1an

d X

=1/

Nfo

r al

l sto

cks,

i, in

the

port

folio

. In

othe

r w

ords

, all

stoc

ks a

reeq

ually

wei

ghte

d w

ith r

isk

of 1

.0. E

quat

ion

(15.

5) th

en r

educ

es to

:

(15.

6)

Tab

le 1

5.!

and

Figu

re 1

5.1

show

the

dive

rsif

icat

ion

effe

ct f

or v

ario

us a

and

N, u

sing

equ

atio

n (1

5.6)

. The

rea

der

can

also

gen

erat

e th

ese

num

bers

sim

ply

ina

spre

adsh

eet.

As

pred

icte

d, f

or a

l. In

the

cont

ext o

f fr

acta

l sta

tistic

s, th

is m

akes

per

fect

sen

se. A

ntip

ersi

sten

tse

ries

hav

e m

ore

jagg

ed ti

me

seri

es th

an d

o pe

rsis

tent

or

rand

om o

nes.

Add

ing

toge

ther

ant

iper

sist

ent s

yste

ms

wou

ld o

nly

resu

lt in

a n

oisi

er s

yste

m.

On

the

othe

r ha

nd, m

arke

t exp

osur

e is

not

a m

atte

r of

div

ersi

fica

tion;

it is

the

wei

ghte

d av

erag

e of

the

b's

of th

e in

divi

dual

sec

uriti

es in

the

port

folio

.T

here

fore

, as

in th

e tr

aditi

onal

mar

ket m

odel

, div

ersi

fica

tion

redu

ces

nonm

ar-

ket r

isk,

not

mar

ket r

isk.

The

ada

ptat

ion

of tr

aditi

onal

CM

T to

sta

ble

dist

ribu

tions

was

inge

niou

s, b

utfe

ll m

ostly

on

deaf

ear

s. I

t was

sim

ply

too

com

plic

ated

com

pare

d to

the

stan

-da

rd G

auss

ian

case

. At t

he ti

me,

ther

e w

as n

ot e

noug

h co

nclu

sive

evi

denc

e to

show

that

the

mar

kets

wer

e no

t Gau

ssia

n.N

ow, w

e ha

ve m

ore

conv

inci

ng e

vide

nce.

How

ever

, the

ada

ptat

ion

has

itsow

n pr

oble

ms.

For

emos

t am

ong

them

is th

e re

tent

ion

of th

e se

nsiti

vity

fac

tor,

b, f

rom

the

trad

ition

al m

arke

t mod

el. T

his

was

usu

ally

est

ablis

hed

as a

line

arre

latio

nshi

p be

twee

n in

divi

dual

sec

uriti

es a

nd th

e m

arke

t por

tfol

io, I

. Thi

s re

-la

tions

hip

was

ret

aine

d be

caus

e, a

t the

tim

e, F

ama,

Rol

l, an

d Sa

mue

lson

wer

e



222

App

lyin

g F

ract

al S

tatis

tics

Por

tfolio

Sel

ectio

n22

3

not

awar

e of

Hur

st's

wor

k an

d th

e im

port

ance

of

pers

iste

nce

and

antip

ersi

s-te

nce.

How

ever

, giv

en a

larg

e en

ough

por

tfol

io, i

t can

be

expe

cted

that

the

di-

vers

ific

atio

n ef

fect

des

crib

ed a

bove

, rel

ativ

e to

a m

arke

t por

tfol

io, w

ill b

efa

irly

sta

ble.

Thu

s, o

ptim

izin

g a

port

folio

rel

ativ

e to

a m

arke

t ind

ex w

ould

be

mor

e st

able

than

a s

trai

ght m

ean/

vari

ance

optim

izat

ion.

A s

econ

d pr

oble

m li

es in

the

valu

e of

a it

self

. The

ada

ptat

ion

assu

mes

that

all

of th

e se

curi

ties

in th

e po

rtfo

lio h

ave

the

sam

e va

lue

of a

. Thi

s is

nec

essa

ry b

e-ca

use

the

sum

of

stab

le P

aret

ian

vari

able

s w

ith th

e sa

me

char

acte

rist

ic e

xpo-

nent

, a, w

ill r

esul

t in

a ne

w d

istr

ibut

ion

that

stil

l has

the

sam

ech

arac

teri

stic

expo

nent

, a. T

his

is th

e ad

ditiv

e pr

oper

ty d

iscu

ssed

in C

hapt

er 1

4.H

owev

er, I

have

sho

wn

that

dif

fere

nt s

tock

s ca

n ha

ve d

iffe

rent

Hur

st e

xpon

ents

and

, the

re-

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

____

__

fore

,di

ffer

ent v

alue

s of

a. (

See

Pete

rs (

l991

a, 1

992)

.) U

nfor

tuna

tely

, the

re is

no

theo

ry f

or th

e ef

fect

of

addi

ng to

geth

er d

istr

ibut

ions

with

dif

fere

nt v

alue

s of

a.

It s

eem

s re

ason

able

that

this

pro

cess

sho

uld

now

be

revi

site

d an

d th

at f

ur-

ther

wor

k sh

ould

be

done

to g

ener

aliz

e th

e ap

proa

ch a

nd m

inim

ize

the

effe

cts

of th

ese

still

trou

bles

ome

prob

lem

s.

Tab

le 1

5.1

The

Effe

cts

of D

iver

sific

atio

n: N

onm

arke

t Ris

k0.

35 0.3

0.25

0

N

Alp

ha(a

)

2.00

1.75

1.50

1.25

1.00

0.50

100.

1000

0.17

780.

3162

0.56

231.

0000

3.16

2320

0.05

000.

1057

0.22

360.

4729

1.00

004.

4721

300.

0333

0.07

800.

1826

0.42

731.

0000

5.47

7240

0.02

500.

0629

0.15

810.

3976

1.00

006.

3246

500.

0200

0.05

320.

1414

0.37

611.

0000

7.07

1160

0.01

670.

0464

0.12

910.

3593

1.00

007.

7460

700.

0143

0.04

130.

1195

0.34

571.

0000

8.36

6680

0.01

250.

0374

0.11

180.

3344

1.00

008.

9443

900.

0111

0.03

420.

1054

0.32

471.

0000

9.48

6810

00.

0100

0.03

160.

1000

0.31

621.

0000

10.0

000

110

0.00

910.

0294

0.09

530.

3088

1.00

0010

.488

112

00.

0083

0.02

760.

0913

0.30

211.

0000

10.9

545

130

0.00

770.

0260

0.08

770.

2962

1.00

0011

.401

814

00.

0071

0.02

460.

0845

0.29

071.

0000

11.8

322

150

0.00

670.

0233

0.08

160.

2857

1.00

0012

.247

416

00.

0063

0.02

220.

0791

0.28

121.

0000

12.6

491

170

0.00

590.

0212

0.07

670.

2769

1.00

0013

.038

418

00.

0056

0.02

030.

0745

0.27

301.

0000

13.4

164

190

0.00

530.

0195

0.07

250.

2693

1.00

0013

.784

020

00.

0050

0.01

880.

0707

0.26

591.

0000

14.1

421

250

0.00

400.

0159

0.06

320.

2515

1.00

0015

.811

430

00.

0033

0.01

390.

0577

0.24

031.

0000

17.3

205

350

0.00

290.

0124

0.05

350.

2312

1.00

0018

.708

340

00.

0025

0.01

120.

0500

0.22

361.

0000

20.0

000

450

0.00

220.

0102

0.04

710.

2171

1.00

0021

.213

250

00.

0020

0.00

950.

0447

0.21

151.

0000

22.3

607

550

0.00

180.

0088

0.04

260.

2065

1.00

0023

.452

160

00.

0017

0.00

820.

0408

0.20

211.

0000

650

0.00

150.

0078

0.03

920.

1980

1.00

0025

.495

170

00.

0014

0.00

730.

0378

0.19

441.

0000

26.4

575

750

0.00

130.

0070

0.03

650.

1911

1.00

0027

.386

180

00.

0013

0.00

660.

0354

0.18

801.

0000

28.2

843

850

0.00

120.

0064

0.03

430.

1852

1.00

0029

.154

890

00.

0011

0.00

610.

0333

0.18

261.

0000

30.0

000

950

0.00

110.

0058

0.03

240.

1801

1.00

0030

.822

11,

000

0.00

100.

0056

0.03

160.

1778

1.00

0031

.622

8

010

0 20

0 30

0 40

0 50

0 60

0 70

0 80

0 90

0 10

00 1

100

Num

berof

Ass

ets

FIG

UR

E 1

5.1

Div

ersi

ficat

ion.



224

App

lyin

g F

ract

al S

tatis

tics

OP

TIO

N V

ALU

AT

ION

InC

hapt

er 1

0, w

e di

scus

sed

the

Bla

ck

Scho

les

(197

3) f

orm

ula.

It i

s im

port

ant

to r

emem

ber

that

the

basi

c fo

rmul

a is

for

"Eur

opea

n" o

ptio

ns

optio

ns th

atca

n be

exe

rcis

ed o

nly

at e

xpir

atio

n. W

e di

scus

sed

the

use

of e

quat

ion

(10.

1) to

stud

y vo

latil

ity, b

ut it

s or

igin

alpu

rpos

e w

as to

cal

cula

te th

e fa

ir p

rice

of

anop

tion.

The

for

mul

a se

ems

to w

ork

reas

onab

lyw

ell w

hen

the

optio

n is

at-

the-

mon

ey, o

r cl

ose,

but

mos

t opt

ions

trad

ers

find

the

form

ula

to b

e un

relia

ble

whe

n op

tions

are

dee

p ou

t-of

-the

-mon

ey.

Opt

ions

will

alw

ays

have

a v

alue

,ev

en w

hen

the

Bla

ck

Scho

les

form

ula

says

they

sho

uld

be w

orth

vir

tual

lyze

ro.

The

re a

re m

any

expl

anat

ions

for

this

syst

emat

ic d

epar

ture

fro

m th

e fo

rmul

a.T

he m

ost r

easo

nabl

e on

e is

the

fatn

ess

of th

e ne

gativ

e ta

il in

the

obse

rved

fre-

quen

cy d

istr

ibut

ion

of s

tock

ret

urns

. The

mar

ket

know

s th

at th

e lik

elih

ood

ofa

larg

e ev

ent i

s la

rger

than

the

norm

al d

istr

ibut

ion

tells

us,

and

pri

ces

the

op-

tion

acco

rdin

gly.

An

addi

tiona

l pro

blem

lies

in th

e di

scon

tinui

tyof

pri

cing

itse

lf. T

he n

orm

aldi

stri

butio

n is

a c

ontin

uous

one.

If

stoc

k re

turn

s ar

e go

vern

ed b

y th

e no

rmal

dist

ribu

tion,

then

, whe

na

stoc

k pr

ice

mov

es f

rom

50

to 4

5, it

is s

uppo

sed

topa

ss th

roug

h al

l of

the

pric

es in

bet

wee

nto

get

ther

e. H

owev

er, e

xper

ienc

esh

ows

that

all

secu

rity

pri

ces

are

subj

ect t

o di

scon

tinui

ties.

A s

tock

will

oft

enju

mp

over

the

inte

rven

ing

pric

es d

urin

gex

trem

e m

oves

, as

will

cur

renc

ies

orbo

nds.

Mer

ton

(197

6) p

ropo

sed

the

clas

sof

Poi

sson

-dri

ven

jum

p pr

oces

ses

for

larg

e m

ovem

ents

aga

inst

a ba

ckgr

ound

of

Gau

ssia

n ch

ange

s fo

r sm

all

mov

e-m

ents

. Thi

s pr

oces

s is

infi

nite

ly d

ivis

ible

,as

are

sta

ble

dist

ribu

tions

. How

ever

,M

cCul

loch

(19

85)

has

poin

ted

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