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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
PDF compression, OCR, web-optimization with CVISION's PdfCompressor
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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
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.
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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
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.
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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
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
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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)
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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
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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
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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.
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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
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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.
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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
Recommended