MAR KET ANALYSIS · MAR KET ANALYSIS Charles B. Epstein, Editor Applying Chaos Theory to Investment...
167
WILEY FINANCE EDITIONS PORTFOLIO MANAGEMENT FORMULAS Ralph Vince TRADING AND IN VESTING IN BOND OPTIONS M. Anthony Wong FRACTAL MAR KET ANALYSIS Charles B. Epstein, Editor Applying Chaos Theory to Investment ANALYZING AND FORECASTING FUTURES PRICES Anthony F. Herbst and Economics CHAOS AND ORDER IN THE CAPITAL MARKETS Edgar E. Peters ___________________________________________________________________ INSIDE THE FINANCIAL FUTURES MARKETS, 3RD EDITION Mark J. Powers and Mark G. Castelino RELATIVE DIVIDEND YIELD Edgar E. Peters Anthony E. Spare SELLING SHORT Joseph A. Walker TREASURY OPERATIONS AND THE FOREIGN EXCHANGE CHALLENGE Dimitris N. Chorafas THE FOREIGN EXCHANGE AND MONEY MARKETS GUIDE Julian Walmsley CORPORATE FINANCIAL RISK MANAGEMENT Diane B. Wunnicke, David R. Wilson, Brooke Wunnicke MONEY MANAGEMENT STRATEGIES FOR FUTURES TRADERS Nauzer J. Balsara THE MATHEMATICS OF MONEY MANAGEMENT Ralph Vince THE NEW TECHNOLOGY OF FINANCIAL MANAGEMENT Dimitris N. Chorafas THE DAY TRADER'S MANUAL William F. Eng OPTION MARKET MAKING Allen J. Baird TRADING FOR A LIVING Dr. Alexander Elder CORPORATE FINANCIAL DISTRESS AND BANKRUPTCY, SECOND EDITION Edward I. Altman FIXED.INCOME ARBITRAGE M. Anthony Wong TRADING APPLICATIONS OF JAPANESE CANDLESTICK CHARTING Gary S. Wagner and Brad L. Matheny FRACTAL MARKET ANALYSIS: APPLYING CHAOS THEORY TO INVESTMENT AND ECONOMICS Edgar E. Peters UNDERSTANDING SWAPS John F. Marshall and Kenneth R. Kapner JOHN WILEY & SONS, INC. GENENTIC ALGORITHMS AND INVESTMENT STRATEGIES Richard J Bauer, Jr New York • Chichester • Brisbane • Toronto • Singapore PDF compression, OCR, web-optimization with CVISION's PdfCompressor
Transcript of MAR KET ANALYSIS · MAR KET ANALYSIS Charles B. Epstein, Editor Applying Chaos Theory to Investment...
-
WIL
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FIN
AN
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ED
ITIO
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POR
TFO
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MA
NA
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ME
NT
FO
RM
UL
AS
Ral
ph V
ince
TR
AD
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AN
D I
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EST
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IN
BO
ND
OPT
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SM
. Ant
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Won
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FRA
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AN
ALY
SIS
Cha
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B. E
pste
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Cha
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Inve
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AN
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Ant
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HA
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IN
TH
E C
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TA
L M
AR
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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
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pare
SEL
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Jose
ph A
. Wal
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SUR
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TH
E F
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fas
TH
E F
OR
EIG
N E
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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
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AK
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Alle
n J.
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OR
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lexa
nder
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OR
POR
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E F
INA
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AN
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EM
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G A
PPL
ICA
TIO
NS
OF
JAPA
NE
SE C
AN
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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
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rd J
Bau
er, J
rN
ew Y
ork
•C
hich
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r•
Bris
bane
•T
oron
to•
Sin
gapo
re
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Thi
s te
xt is
prin
ted
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
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
d
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
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-
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'
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-
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.
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-
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.
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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
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-
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
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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
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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
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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
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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.
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-
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
PDF compression, OCR, web-optimization with CVISION's PdfCompressor
http://www.cvisiontech.com/pdf_compressor_31.html
-
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
