Peng, Bin; Peng, Fei
A icle
P icing maximum-minimum bidi ec ional op ions in
inomial CEV model
Jou nal o Economics, Finance and Adminis a i e Science
P o ided in Coope a ion wi h:
Uni e sidad ESAN, Lima
Sugges ed Ci a ion: Peng, Bin; Peng, Fei (2016) : P icing maximum-minimum bidi ec ional op ions
in inomial CEV model, Jou nal o Economics, Finance and Adminis a i e Science, ISSN 2218-0648,
Else ie España, Ba celona, Vol. 21, Iss. 41, pp. 50-55,
h ps://doi.o g/10.1016/j.je as.2016.06.001
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/179776
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Jou nal
o
Economics,
Finance
and
Adminis a i e
Science
21
(2016)
50–55
Jou nal
o
Economics,
Finance
and
Adminis a i e
Science
www.else ie .es/je as
A icle
P icing
maximum-minimum
bidi ec ional
op ions
in
inomial
CEV
model夽
Bin
Penga,∗,
Fei
Pengb
aSEME,
Beijing
Uni e si y
o
Ci il
Enginee ing
and
A chi ec u e,
Beijing,
P.
R.
China
bElec ical
&
Compu e
Enginee ing,
UBC,
Vancou e ,
Canada
a
i
c
l
e
i
n
o
A icle
his o y:
Recei ed
10
June
2016
Accep ed
15
June
2016
JEL
classifica ion:
C14
F17
Keywo ds:
T inomial
CEV
model
Recu si e
algo i hm
Maximum-minimum
bidi ec ional
op ions
a
b
s
a
c
Maximum-minimum
bidi ec ional
op ions
a e
a
kind
o
exo ic
pa h
dependen
op ions.
In
he
cons an
elas ici y
o
a iance
(CEV)
model,
a
combining
inomial
ee
was
s uc u ed
o
app oxima e
he
non-
cons an
ola ili y
ha
is
a
unc ion
o
he
unde lying
asse .
On
his
basis,
a
simple
and
e ficien
ecu si e
algo i hm
was
de eloped
o
compu e
he
isk-neu al
p obabili y
o
each
di e en
node
o
he
unde lying
asse
eaching
a
maximum
o
minimum
p ice
and
he
o al
numbe
o
maxima
(minima)
in
he
inomial
ee.
Wi h
help
o
i ,
he
compu a ional
p oblems
can
be
e ec i ely
sol ed
a ising
om
he
inhe en
complexi ies
o
di e en
ypes
o
maximum-minimum
bidi ec ional
op ions
when
he
unde lying
asse
e ol es
as
he
inomial
CEV
model.
Nume ical
esul s
demons a e
he
alidi y
and
he
con e gence
o
he
app oach
men ioned
abo e
o
he
di e en
pa ame e
alues
se
in
he
inomial
CEV
model.
©
2016
Published
by
Else ie
Espa˜
na,
S.L.U.
on
behal
o
Uni e sidad
ESAN.
This
is
an
open
access
a icle
unde
he
CC
BY-NC-ND
license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
P ecios
de
opciones
bidi eccionales
máximas-mínimas
en
el
modelo
ECV
inomial
Códigos
JEL:
C14
F17
Palab as
cla e:
Modelo
ECV
inomial
Algo i mo
ecu si o
Opciones
bidi eccionales
máximas-mínimas
e
s
u
m
e
n
Las
opciones
bidi eccionales
máximas-mínimas
son
un
ipo
de
opciones
exó icas
dependien es
de
la
ayec o ia.
En
el
modelo
de
elas icidad
cons an e
de
la
a ianza
(ECV),
se
es uc u ó
un
á bol
inomial
combinado
pa a
ap oxima
la
ola ilidad
no
cons an e,
que
es
una
unción
del
ac i o
subyacen e.
En
base
a
es o
se
desa olló
un
algo i mo
sencillo
y
eficaz
pa a
calcula
la
p obabilidad
de
neu alidad
al
iesgo
de
cada
nodo
del
ac i o
subyacen e
llegando
a
un
p ecio
máximo
o
mínimo
y
el
núme o
o al
de
máximos
(mínimos)
del
á bol
inomial.
De
es a
mane a,
los
p oblemas
compu acionales
pueden
esol-
e se
eficazmen e
a
aíz
de
las
complejidades
inhe en es
a
los
dis in os
ipos
de
opciones
bidi eccionales
máximas-mínimas
cuando
el
ac i o
subyacen e
e oluciona
como
el
modelo
ECV
inomial.
Los
esul a-
dos
numé icos
demues an
la
alidez
y
con e gencia
del
en oque
an e io men e
mencionado
pa a
los
pa áme os
de
alo es
es ablecidos
en
el
modelo
ECV
inomial.
©
2016
Publicado
po
Else ie
Espa˜
na,
S.L.U.
en
nomb e
de
Uni e sidad
ESAN.
Es e
es
un
a ´
ıculo
Open
Access
bajo
la
licencia
CC
BY-NC-ND
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
夽This
pape
is
suppo ed
by
Na ional
Na ual
Fouda ion
o
China
(NO:
71002098,
NO:
61102009),
Excellen
You h
ounda ion
o
Beijing,
China
(YETP1652)
and
he
Na u al
Sciences
and
Enginee ing
Resea ch
Council
o
Canada
(NSERC).
∗Co esponding
au ho .
E-mail
add ess:
[email p o ec ed]
(B.
Peng).
1.
In oduc ion
Black
and
Scholes
(1973)
de i ed
he
well-known
op ion
p icing
o mula
by
assuming
he
unde lying
asse
p ice
ollows
a
geome ic
B ownian
mo ion.
Unde
he
cons uc ion,
he
p ice
dis ibu ion
is
logno mal
and
ola ili y
is
cons an .
Howe e ,
in
he
ma ke
eal-
i y,
whe e
o en
he
asse
p ice
beha io
is
a ec ed
by
ola ili y
smile
e ec ,
his
implies
ha
he
asse
p ice
is
gene ally
unlikely
o
be
logno mally
dis ibu ed
he
unde lying
asse
ola ili y
ends
h p://dx.doi.o g/10.1016/j.je as.2016.06.001
2077-1886/©
2016
Published
by
Else ie
Espa˜
na,
S.L.U.
on
behal
o
Uni e sidad
ESAN.
This
is
an
open
access
a icle
unde
he
CC
BY-NC-ND
license
(h p://c ea i ecommons.
o g/licenses/by-nc-nd/4.0/).
B.
Peng,
F.
Peng
/
Jou nal
o
Economics,
Finance
and
Adminis a i e
Science
21
(2016)
50–55
51
o
change
as
asse
p ice
mo es
up
and
down.
Cox
(1975)
ini ially
obse ed
ha
he
cha ac e
o
he
ola ili y
o
he
unde lying
asse
was
linked
o
i s
p ice
le el.
He
no ed
ha
he
o igin
o
he
ola il-
i y
smile
was
he
nega i e
co ela ion
be ween
asse
p ice
changes
and
ola ili y
changes,
on
his
basis
Cox
de eloped
cons an
elas-
ici y
o
a iance
(CEV)
model.
As
documen ed
by
Becke s
(1980)
and
Da il
(1982),
he e
a e
heo e ical
a gumen s
o
and
empi i-
cal
e idences
o
he
alidi y
o
CEV
model.
Many
empi ical
s udies
in
Macbe h
and
Me ille,
1980
and
Emanuel
Mache h
(1982)
ha e
shown
ha
CEV
model
be e
desc ibe
he
e olu ion
o
asse
p ice.
Hence
i
is
ins uc i e
o
apply
CEV
model
o
op ion
p icing.
The
p oblem
o
p icing
a
s anda d
Eu opean
op ion,
when
he
unde lying
asse
alue
is
d i en
by
a
CEV
model
was
sol ed
by
Cox
and
Ross
(1976)
who
de i ed
an
analy ical
solu ion
o
he
op ion
alue.
Things
a e
mo e
complica ed
in
he
case
o
pa h-
dependen
op ions.
The
analy ical
solu ion
o
he
p icing
p oblem
is
no
a ailable
and
nume ical
app oxima ions
mus
be
used.
Boyle
and
Tian
(1999)
used
Mon e
Ca lo
simula ions.
Subsequen ly,
he
ex ended
Babbs
me hod
on
he
basis
o
Black-Scholes
model
o
cons uc
a
inomial
ee
me hod
o
e alua ing
he
lookback
op ions
unde
he
CEV
model,
as
was
ansla ed
in o
Chinese
by
He
(2001).
Du ydo
and
Line sky
(2001)
de i ed
he
close- o m
o mulae
o
ba ie
op ions
unde
he
CEV
model
wi h
help
o
he
nume ical
in e sion
o
he
Laplace
ans o m
o
he
op ion
p ice
ol-
lowing
o dina y
di e en ial
equa ion.
Bin
and
Fei
(2006)
applied
he
in ui ion
binomials
ee
me hod
o
p ice
Asian
op ion
unde
he
cons an
elas ici y
o
a iance.
Among
hese
nume ical
ech-
niques,
inomial
ee
e alua ion
app oach
s ill
plays
a
ema kable
ole
bo h
o
i s
highly
flexible
and
implemen a ion
in
sol ing
he
p icing
p oblem
o
pa h-dependen
op ions
unde
he
CEV
model.
Howe e ,
in
he
amewo k
o
he
inomial
ee
o
app oxima e
he
cons an
elas ici y
o
a iance
model,
he
ansi ion
p obabili-
ies
a e
no
longe
cons an
o e
he
ee.
This
p ope y
in oduces
a
u he
complica ion
in
he
e alua ion
p ocess.
In
addi ion,
he e
is
li le
wo k
on
he
exo ic
pa h-dependen
such
as
maximum-
minimum
bidi ec ional
op ions.
The
objec i e
o
his
pape
is
o
s udy
he
applica ion
o
he
inomial
ee
app oach
o
maximum-minimum
bidi ec ional
op ions
when
he
asse
p ice
e ol es
as
a
CEV
model.
Especially,
ecu si e
algo i hm
based
on
a
o wa d
induc ion
p ocedu e
is
de eloped
o
compu e
he
isk-neu al
p obabili y
o
each
di e en
node
o
he
unde lying
asse
eaching
a
maximum
o
minimum
p ice
in
he
binomial
ee.
The
emainde
o
he
pape
is
o ganized
as
ollows:
Sec ion
2
illus a es
he
inomial
me hod
used
o
app oxima e
he
CEV
model
unde
he
equi alen
ma ingale
measu e.
In
Sec ion
3
he
inomial
app oxima ion
is
applied
o
p ice
maximum
and
minimum
bidi ec ional
op ions
and
nume ical
esul s
a e
gi en.
Conclusions
a e
p esen ed
in
he
final
sec ion.
2.
A
inomial
CEV
model
I
is
known
ha
Black-Scholes
model
wi h
cons an
ola ili y
does
no
hold
empi ically
o
asse
p ices.
Al e na i e
s ochas ic
models
ha e
been
s udied
and
applied
o
op ion
p icing.
Fo
exam-
ple,
Cox
(1975)
and
Da il
(1982)
s udy
a
gene al
class
o
s ochas ic
model
known
as
he
Cons an
Elas ici y
o
Va iance
(CEV)
di usion
model
in
a
isk-neu al
wo ld.
dS
=
Sd
+
SdB

=
ıS0.5a−1(1)
whe e
B
is
a
s anda d
Wiene
p ocess
unde
he
Q-measu e,
is
isk-
less
a e
o
in e es ,
he
s anda d
de ia ion
o
e u n

is
a
unc ion
o
he
unde lying
p ice
ins ead
o
a
cons an
ı
and
a
a e
cons an s,
0
≤
a
<
2.
We
conside
a
disc e e
app oxima ion
o
he
asse
p ice
e o-
lu ion
desc ibed
in
(1)
using
he
inomial
app oach.
Le
n
be
he
numbe
o
ime
in e als
be ween
he
ime
and
he
ma u i y
T,
[ ,T]
is
di ided
in o
n
equal
pieces,
each
o
wid h
 .
The
unde lying
asse
may
mo e
up
a
le el,
down
a
le el
o
s ay
he
le el
wi h
geome ic
a e age
be ween
up
le el
and
down
le el.
In
he
case
o
he
asse
p ice
dynamics
d i en
by
CEV
model,
he
ola ili y
is
no
a
cons an
bu
a ies
wi h
he
le el
o
he
unde lying
p ice.
This
implies
ha
he
app oxima ion
inomial
ee
is
non- ecombining
and
he
numbe
o
nodes
p oduced
a
each
e ical
laye
is
3i
i
=
0.
.
..
.
..n,
hus
compu a ional
complexi y
becomes
unmanage-
able
e en
wi h
a
small
numbe
o
ime
s eps.
In
o de
o
e ec i ely
sol e
he
compu a ional
p oblems
a ising
om
he
inhe en
complexi ies
o
he
cons an
elas ici y
o
a iance
model,
we
need
o
ans o m
he
a iable
S
go e ned
by
(1)
wi h
non-cons an
ola ili y
so
ha
he
ans o med
p ocess
has
cons an
ola ili y.
Conside ing
he
ans o med
p ocess
X =
S1−0.5a/(1
−
0.5a)ı
and
using
he
I o’s
Lemma,
we
ha e:
dX
=∂X
∂S dS
+∂X
∂ d
+1
2
∂2X
∂S2(ıS0.5a)2d
=S−0.5a
ıdS
−1
2
0.5aS−0.5a−1
ı(ıS0.5a)2d
=S−0.5a
ıdS
−0.5aıS0.5a−1
2d
(2)
Subs i u ing
equa ion
(1)
in o
he
abo e
equa ion
and
using
he
ac
ha
S
=ı(1
−
0.5a)X1/(1−0.5a).
Equa ion
(2)
becomes:
dX
=S−0.5a
ı Sd
+
ıS0.5adB−0.5aıS0.5a−1
2d
= (1
−
0.5a)X
−a2
4(1
−
0.5a)Xd
+
dB
(3)
Now,
i
is
easy
o
build
up
a
compu a ionally
simple
inomial
ee
o
app oxima e
he
X-p ocess.
The
alue
X o
he
p ocess
a
ime
,
a e
one
pe iod
a
ime
+
1,
can
ise
o
X +√ ,
o
dec ease
o
X −√ ,
o
s ay
he
same
wi h
X .
Con inuing
in
his
way,
we
see
ha
he
alue
o
he
inomial
X-p ocess
a e
equal
o
Xj
+i =
X +
(j
−
i) ,
i
=
0,
1,
.
.
.,
n,
j
=
0,
1,
.
.
.,
2i(4)
Whe e
Xj
+i ep esen s
he
alue
o
he
inomial
X-p ocess
a
ime
+
i
a e
j/2
up
s eps
and
i-j/2
down
s eps.
o
i
=
0,1,.
.
..,n
and
j
=
0,1,.
.
..,2i.
Gi en
he
alues
on
he
X
la ice,
we
can
eco e
he
dynamics
o
he
unde lying
p ice
on
i s
la ice.
As
be o e,
conside
he
alue
S jo
he
p ocess
a
ime
,
a e
one
pe iod
a
ime
+
1,
can
ise
o
Sj+
+i1 =
(X +√ ),
o
dec ease
o
Sj−
+i1 =
(X −√ ),
o
s ay
he
same
wi h
S .
Con inuing
in
his
way,
we
no e
ha
he
alue
o
he
binomial
app oxima ing
S-p ocess
a
ime
+
i
a e
j/2
up
s eps
and
i-j/2
down
s eps.
is
exp essed
as
ollows:
Sj
+i =
(Xj
+i )
=
(X +
(j
−
i) )
(5)
Once
we
ha e
de eloped
he
inomial
ee
ha
app oxima es
he
S-p ocess,
i
emains
o
compu e
he
p obabili ies
o
an
upwa d
and
downwa d
mo e
o
he
e olu ion
o
he
asse
p ice
in
he
app oxima ing
inomial
ee.
To
ensu e
non-nega i e
and
less
han
one
p obabili ies,
i
is
op imal
o
ha e
he
unde lying
asse
p ice
o
cen al
mo e
equi alen
o
he
expec ed
asse
p ice
o
he
unde lying
s ochas ic
p ocess
we
define
S¯
j
+i ,
j
=
0,
1,
.
.
.2i
o
be
g ea es
wi h
up
jumps
such
ha
e  Sj
+(i−1) =
S¯
j
+i <
0
wi h
up
p obabili y
pj
+(i−1) and
Sˆ
j
+i o
be
he
smalles
wi h
down
jumps
such
ha
e  Sj
+(i−1) −
Sˆ
j
+i >
0.
wi h
down
p obabili y
52
B.
Peng,
F.
Peng
/
Jou nal
o
Economics,
Finance
and
Adminis a i e
Science
21
(2016)
50–55
qj
+(i−1) and
S˜
j
+i o
be
middle
mo e
be ween
he
g ea es
and
he
smalles ,
such
ha
e  Sj
+(i−1) −
S˜
j
+i =
0.
The
p obabili ies
ha
he
unde lying
asse
wi h
p ice
Sj
+(i−1)
makes
an
up
jump
and
down
jump
a e
specified
by
he
ollowing
se
o
equa ion,
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
pj
+(i−1) S
៝
j
+i −
S˜
j
+i −
qj
+(i−1) S˜
j
+i −
Sˆ
j
+i =
0
pj
+(i−1) S¯
j
+i 2
−S˜
j
+i 2
+
qj
+(i−1) Sˆ
j
+i 2
−S˜
j
+i =
2 Sj
+(i−1) ˛
(6)
Sol ing
o
he
p obabili ies
om
he
abo e
equa ion,
one
ob ain:
pj
+(i−1) =
2 Sj
+(i−1) ˛
S¯
j
+i −
S¯
j
¯
+i S¯
j
+i −
Sj
¯
+i ,
qj
+(i−1) =
2 Sj
+(i−1) ˛
S¯
j
¯
+i −
Sj
¯
+i Sj
+i −
Sj
¯
+i (7)
Ob iously,
he
p obabili y
ha
he
unde lying
asse
wi h
p ice
Sj
+(i−1) makes
pa allel
mo e
is
1
−
pj
+(i−1) −
qj
+(i−1) .
The
abo e
calcula ion
o
he
ansi ion
p obabili ies
pj
+(i−1) and
qj
+(i−1) ep esen s
legi ima e
p obabili ies
which
allows
o
mul-
iple
jumps
in
he
app oxima ing
inomial
X-p ocess.
This
is
impo an
because,
in
he
egion
nea
o
S
=
0,
he
magni ude
o
each
jump
could
be
e y
small
and,
as
a
consequence,
he
ansi ion
p obabili ies
could
exceed
one
o
smalle
han
ze o.
3.
P icing
maximum-minimum
bidi ec ional
op ions
We
now
apply
he
inomial
CEV
model
depic ed
in
abo e
sec ion
o
p icing
he
maximum-minimum
bidi ec ional
op ions
In oking
he
no-a bi age
p inciple,
he
alue
a
ime
o
op ions
is
gi en
by
discoun ing
a
he
isk- ee
in e es
a e
he
sum
o
all
he
op ion
payo s
mul iplied
by
he
co esponding
p obabili y
o
occu ing.
Thus
we
need
o
see
he
.payo
co esponding
o
each
node
o
he
ee
a
ma u i y.
Fo
floa ing
s iking
maximum-
minimum
bidi ec ional
op ions,
i s
payo
a
ma u i y
is
equal
o
he
di e ence
be ween
he
cu en
unde lying
asse
and
he
mini-
mum
p ice
egis e ed
by
he
unde lying
asse
du ing
he
op ion
li e
ime
plus
he
di e ence
be ween
he
maximum
unde lying
ass
p ice
egis e ed
du ing
he
op ion
li e
ime
and
cu en
asse
p ice.
Con e sely,
he
fixed
s ike
maximum-minimum
bidi ec ional
op ions
pay
o
he
maximum
be ween
ze o
and
he
di e ence
be ween
he
maximum
unde lying
asse
p ice
and
a
fixed
s ike
p ice
plus
he
maximum
be ween
ze o
and
he
di e ence
be ween
a
fixed
s ike
p ice
and
he
minimum
asse
p ice.
Howe e ,
hings
a e
complica ed
by
he
ac
ha ,
o
each
e minal
node
o
he
ee
co espond,
in
gene al,
di e en
alues
o
he
op ion
payo .
This
is
because,
e en
when
some
ajec o ies
o
he
unde lying
asse
p ice
end
wi h
he
same
e minal
alue,
hey
could
ha e
egis e ed
a
di e en
minimum
and
maximum
p ice
du ing
he
op ion
li e ime.
In
he
amewo k
o
he
inomial
CEV
model
de eloped
by
Sec-
ion
2
we
see
ha
he
ansi ion
p obabili ies
a e
no
cons an
o e
he
ee,
and
as
a
consequence,
all
he
pa hs
o
he
unde lying
asse
wi h
he
same
e minal
alue
may
ha e
he
di e en
p obabili ies
o
occu ing.
In
his
amewo k,
he
e alua ion
p oblem
is
no
sol ed
e en
i
we
can
compu e
he
numbe
o
pa hs
wi h
he
same
e minal
alue
bu
wi h
a
di e en
minimum
o
maximum
p ice
egis e ed
du ing
he
op ion
li e.
This
p ope y
in oduces
a
u he
complica ion
in
he
e alua ion
p ocess.
To
copy
wi h
his
p oblem,
we
de i e
a
ecu si e
algo i hm
based
on
a
o wa d
induc ion
p o-
cedu e
ha
allows
o
simply
compu e
he
isk-neu al
p obabili y
o
each
di e en
payo
o
he
op ions
a
ma u i y.
The
algo i hm
wo ks
as
ollows:
a
fi s ,
we
w i e
(i,
i-j)
o
he
inomial
la -
ice
whe e
he
unde lying
asse
has
a
p ice
Sj
+i (i
=
0,1,.
.
..,n;
j
=
0,1,.
.
..,2i).
No e
ha
i-j
is
he
di e ence
be ween
he
down
s eps
and
he
up
s eps
aken
by
he
unde lying
asse
p ice
and,
as
we
see
below,
i
is
use ul
ep esen a ion
o
posi ion
o
he
asse
p ice
a
ime
i.
Fu he ,
we
use
index
k
and
h
(k,
h
=
0,1,.
.
..i)
espec i ely
o
speci y
he
lowes
and
highes
laye
o
ho izon al
nodes
eached
by
he
unde lying
asse
p ice
a e
i
ime
s eps.
In
o he
wo ds,
k
=
0,1,.
.
..i
means
ha
he
asse
wi h
cu en
p ice
Sj
+i has
egis e ed
minimum
p ice
equal
o
S0
+k ,
h
=
0,1,.
.
..i
means
ha
he
asse
wi h
cu en
p ice
Sj
+i has
egis e ed
maximum
p ice
equal
o
S2h
+h we
use
he
iple
(i,j,
k)
and
(i,j,
h)
o
espec i ely
speci y
he
s a e
o
he
wo ld
in
which
he
unde lying
asse
eached
a
minimum
p ice
equal
o
S0
+k and
a
maximum
p ice
equal
o
S2h
+h ,.
o e
a
pa h
wi h
j/2
up
s eps
and
i-j/2
down
s eps.
Clea ly,
gi en
i
and
j,
he
alue
o
he
index
k
anges
o e
he
in e nal
max (i
−
j,
0),i
−
j/2;
and
he
alue
o
he
index
h
anges
o e
he
in e nal max (j
−
i,
0),j/2,
whe e ◦ e u ns
he
lowe
in ege
is
closes
o
a
eal
numbe .
Wi h
help
o
hese
impo an
ela ions,
we
can
iden i y
all
he
di e en
minimum
p ices,
S0
+k
and
all
he
di e en
maximum
p ices,
S2h
+h egis e ed
by
he
unde lying
asse
p ice
ha
has
a
cu en
alue
Sj
+i .
We
s a
by
compu ing
he
p obabili y
o
each
s a e
o
na u e
ep esen ed
by
he
iple
(i,j,
k)
in
he
inomial
ee.
To
do
his,
we
need
o
dis inguish
wo
cases,
he
fi s
conside s
he
s a es
o
na u e
cha ac e ized
by
he
iple
(i,
j,
k)
such
ha
k
=
i-j.
In
his
case
he
unde lying
asse
p ice
was
a
he
node
(i,
i
−
j),
Sj
+i is
loca ed
on
he
same
ho izon al
se
o
nodes
o
he
minimum
p ice
eached
by
he
unde lying
asse
du ing
he
fi s
ime
s eps.
This
means
ha
a
ime
+
(i
−
1)
he
unde lying
asse
p ice
was
a
he
node
(i
−
1,
i
−
1
−
j)
and
he
las
s ep
om
(i
−
1,
i
−
1
−
j)
o
(i,
i
−
j)
is
a
down
s ep.
I
a
ime
+
(i
−
1)
he
unde lying
asse
p ice
was
a
node
(i
−
1,
i
−
1
−
(j
−
1))
he
lowes
laye
o
nodes
eached
by
he
unde lying
asse
p ice
would
be
loca ed
a
he
le el
k
=
i
−
j
and
his
is
equi alen
o
he
hypo hesis
k
=
i
−
j.
his
means
he
unde lying
asse
could
each
he
minimum
p ice
S0
+k a
ime
+
i
wi h
he
las
pa allel
mo e.
Hence
he
s a e
o
na u e
(,
i,
j,
k)
is:
(i,
j,
k)=(i
−
1,
j,
k
−
1)∩
Ld∪(i
−
1,
j,
k)∩
Ld
∪(i
−
1,
j
−
1,
k)∩
Lm(8)
Whe e
Ldand
Lm espec i ely
ep esen s
he
e en
ha
he
las
ime
s ep
aken
by
he
unde lying
asse
p ice
is
a
down
s ep
and
s ay
he
same.
Hence,
he
p obabili y
o
he
s a e
o
na u e
(,
i,
j,
k)
is
p ob (i,
j,
k)=
p ob[(i
−
1,
j,
k
−
1)∩
Ld∪(i
−
1,
j,
k)∩
Ld
∪(i
−
1,
j
−
1,
k)∩
Lm]
=
p ob (i
−
1,
j,
k
−
1)×
qj
+(−1)i +
p ob (i
−
1,
j,
k)
×qj
+(i−1)
+p ob (i
−
1,
j
−
1,
k)×1
−
pj−1
+(i−1) −
qj−1
+(i−1) 
(9)
The
second
case
conside s
he
s a es
o
na u e
such
ha
k
>
i
−
j.
in
his
si ua ion
he
unde lying
asse
p ice
a
he
node
B.
Peng,
F.
Peng
/
Jou nal
o
Economics,
Finance
and
Adminis a i e
Science
21
(2016)
50–55
53
(i,
i
−
j),
Sj
+i is
g ea e
han
he
minimum
p ice
eached
du ing
he
fi s
i
−
1
ime
s eps.
Indeed,
he
lowes
laye
o
nodes
ouched
by
he
unde lying
asse
is
loca ed
a
le el
k,
ha
is,
below
he
cu en
node
(i,
i
−
j)
loca ed
a
le el
i
−
j.
in
his
si ua ion,
he
le el
k
was
al eady
eached
by
he
unde lying
asse
p ice
a
ime
+
(i
−
1) .
I
he
asse
p ice
is
a
he
node
(i
−
1,
i
−
1
−
j)
and
he
i h
s ep
is
a
down
s ep
and
i
he
asse
p ice
is
a
he
node
(i
−
1,
i
−
1
−
(j
−
1))
and
he
i h
s ep
is
pa allel
s ep
and
i
he
asse
p ice
is
a
he
node
(i
−
1,
i
−
1
−
(j
−
2))
and
he
i h
s ep
is
an
up
s ep.
Thus
he
e en
(i,
j,
k)
occu s
when
he
e en
(i
−
1,
j,
k)
occu s
and
he
i h
s ep
is
a
down
s ep
o
when
he
e en
(i
−
1,
j
−
1,
k)
akes
place
and
he
i h
s ep
is
pa allel
s ep
o
when
he
e en
(i
−
1,
j
−
2,
k)
akes
place
and
he
i h
s ep
is
an
up
s ep,
i.e.
(i,
j,
k)=(i
−
1,
j,
k)∩
Ld∪(i
−
1,
j
−
1,
k)∩
Lm
∪(i
−
1,
j
−
2,
k)∩
Lu(10)
whe e
Lu ep esen s
he
e en
ha
he
las
s ep
aken
by
he
unde -
lying
asse
p ice
is
an
up
s ep.
Hence,
he
p obabili y
o
he
s a e
o
na u e
(i,
j,
k)
is
p ob (i,
j,
k)=
p ob
(i
−
1,
j,
k)∩
Ld∪(i
−
1,
j
−
1,
k)∩
Lm
∪(i
−
1,
j
−
2,
k)∩
Lu
=
p ob (i
−
1,
j,
k)×
qj
+(−1)i +
p ob (i
−
1,
j
−
1,
k)
×1
−
pj−1
+(i−1) −
qj−1
+(i−1) 
+p ob (i
−
1,
j
−
2,
k)×
pj−2
+(i−1)
(11)
The
case
o
he
p obabili y
o
each
s a e
o
na u e
ep esen ed
by
he
iple
(i,
j,
h)
in
he
inomial
ee
can
be
compu ed
as
be o e.
We
mus
dis inguish
be ween
wo
cases,
he
fi s
is
when
he
cu -
en
p ice
o
he
unde lying
asse ,
Sj
+i is
equal
o
he
maximum
alue
o
he
asse
p ice,
i.e.
h=
j-.i.
In
his
case
he
las
s ep
o
he
asse
p ice
mus
be
an
up
s ep
o
pa allel
s ep
and
maximum
may
be
eached
wi h
he
las
ime
s ep
o
du ing
he
fi s
i-1
ime
s eps.
Hence,
(i,
j,
h)=(i
−
1,
j
−
2,
h
−
1)∩
Lu∪(i
−
1,
j
−
2,
h)∩
Lu
∪(i
−
1,
j
−
1,
h)∩
Lm(12)
and
p ob (i,
j,
h)=
p ob (i
−
1,
j
−
2,
h
−
1)×
pj−2
+(−1)i
+
p ob (i
−
1,
j
−
2,
h)×
pj−2
+(i−1)
+p ob (i
−
1,
j
−
1,
h)×1
−
pj−1
+(i−1) −
qj−1
+(i−1) 
(13)
On
he
con a y,
when
he
maximum
unde lying
asse
p ice
is
g ea e
han
he
cu en
asse
p ice,
i.e.
h
>
j
−
.i,
we
ha e
(i,
j,
h)=(i
−
1,
j,
h)∩
Ld∪(i
−
1,
j
−
2,
h)∩
Lu
∪(i
−
1,
j
−
1,
h)∩
Lm(14)
and
p ob (i,
j,
h)=
p ob (i
−
1,
j,
h)×
qj
+(−1)i
+
p ob (i
−
1,
j
−
2,
h)×
pj−2
+(i−1)
+p ob (i
−
1,
j
−
1,
h)×1
−
pj−1
+(i−1) −
qj−1
+(i−1) 
(15)
I
is
wo h
no ing
ha
he
ecu si e
algo i hm
o
compu ing
he
isk
neu al
p obabili y
o
each
s a e
o
na u e
a
he
beginning
o
he
ee,
assigns
he
isk-neu al
p obabili y
1
o
he
s a e
o
na u e
(0,0,0)
and
a
isk-neu al
p obabili y
0
o
he
s a es
o
na u es
ha
can
ne e
occu ,
i.e,
when
k
<
0
o
k
>
i
−
j/2i
−
j/2and
h
<
0
o
h
>j/2.
To
cla i y
how
he
p ocedu e
wo ks
we
conside
asse
wi h
p ice
S
=
100
ha
mo es
in
he
inomial
la ice
illus a ed
in
Sec ion
2.
We
conside
he
numbe
o
ime
s eps
n
=
3,
he
isk
=
ee
in e es
a e
=
10%,
he
ime
o
ma u i y
T
=
6
mon hs
and
elas ici y
ac o
a
=
05.
he
ola ili y
pa ame e
is
adjus ed
in
such
a
way
ha
he
ins an aneous
ola ili y
o
he
asse
p ice
a
incep ion
is
he
same
o
all
he
alues
o
he
pa ame e
a
conside ed.
We
se
he
ini ial
ins aneous
ola ili y
o
he
unde lying
asse
a e
o
e u n
equal
o
0.25
and
selec
he
alue
o
␴
such
ha
ıSa/2−1
=
0.25.
Figu e
1
illus a es
how
he
o wa d
induc ion
based
p ocedu e
wo ks
in
compu ing
he
isk
neu al
p obabili y
o
each
minimum
and
maximum
in
he
inomial
ee.
In
he
Figu e
1,
we
depic
he
inomial
ee
ha
desc ibes
he
e olu ion
o
he
unde lying
asse
p ice
and
we
epo ed
he
isk-neu al
p obabili y
o
each
up
s ep
and
o
each
down
s ep
and
middle
s ep.
Fo
example,
0.4572
is
he
p obabili y,
p0
+1/3 ha
he
unde lying
asse
p ice
jumps
om
he
alue
S0
T+1/3 o
he
p ice
S2
+1/2 ep esen ed
by
he
node
(3,2).
The
le
hand
side
o
Figu e
1
illus a es
he
isk-neu al
p obabili y
o
each
possible
minimum
eached
by
he
unde lying
asse
ha
cu en ly
is
loca ed
a
he
node
(i,
i-j).
Fo
example,
a
he
node
(3,2),
he
unde lying
asse
p ice
has
a
cu en
p ice
S2
+1/2.
The
minimum
p ice
eached
by
he
unde lying
asse
p ice
i
he
fi s
o
second
s ep
was
an
down
s ep
is
S0
+1/6,
hence
k
=
1
and
he
s a e
o
na u e
occu ed
is
(3,2,1)
because
i
−
j
=
k,
p ob(3,
2,
1)
=
[p ob (2,
2,
0)+
p ob (2,
2,
1)]
×
q2
+1/3
+p ob (2,
1,
1)×1
−
p1
+1/3−
q1
+1/3
=(0.1889
+
0.1953)×
0.3186
+
0.1150
×
0.2196
=
0.1479
Con e sely,
in
he
case
o
wo
consecu i e
down
s eps
o
he
unde lying
asse
p ice
a
incep ion,
he
minimum
p ice
eached
by
he
unde lying
asse
p ice
is
S0
+1/3,
k
=
2,
and
he
s a e
o
na u e
occu ed
is
(3,2,2).
Because
k
>
i
−
j
p ob (3,
2,
2)=
p ob (2,
2,
2)×
q2
+1/3+
p ob (2,
1,
2)
×1
−
p1
+1/3−
q1
+(1/3
+p ob (2,
0,
2)×
p0
+1/3=
0.1067
×
0.4512
=
0.0734
No e
ha
he
s a e
o
na u e
(2,2,2)
and
(2,1,2)
can
ne e
occu .
The
igh
hand
side
o
Figu e
1
illus a es
he
isk-neu al
p obabili y
o
each
possible
maximum
eached
by
he
unde -
lying
asse
ha
cu en ly
is
loca ed
a
he
node
(i,
i
−
j).
Fo
ins ance,
a
he
node
(3,4),
he
unde lying
asse
p ice
has
a
cu -
en
p ice
S4
+1/2.
The
maximum
p ice
eached
by
he
unde lying
asse
p ice
i
he
fi s
o
second
s ep
was
an
up
s ep
is
S2
+1/6,
hence
h
=
1
and
he
s a e
o
na u e
occu ed
is
(3,2,1)
because
j
−
i
=
h,
p ob (3,
4,
1)=p ob (2,
2,
0)+
p ob (2,
2,
1)×
p2
+1/3
+p ob (2,
3,
1)×1
−
p3
+1/3−
q3
+1/3
=
(0.2151
+
0.1425)
×
0.4631
+
0.2005
×
0.2313
=
0.2114
On
he
con a y,
in
he
case
o
wo
consecu i e
up
s eps
o
he
unde lying
asse
p ice
a
incep ion,
he
maximum
p ice
eached
by
he
unde lying
asse
p ice
is
S4
+1/3,
h
=
2,
and
he
s a e
o
na u e
occu ed
is
(3,4,2).
Because
h
>
j
−
i
p ob(3,
4,
2)
=
p ob 2,
4,
2×
q4
+1/3+
p ob (2,
2,
2)×
p2
+1/3
+p ob (2,
3,
2)×1
−
p3
+1/3−
q3
+1/3=
0.2205
×
0.2875
=
0.0634
No e
ha
he
s a e
o
na u e
(2,2,2)
and
(2,3,2)
can
ne e
occu .

54
B.
Peng,
F.
Peng
/
Jou nal
o
Economics,
Finance
and
Adminis a i e
Science
21
(2016)
50–55
(0,0,0)
(1,2,1)
0.4665
(1,1,0)
0.2098
(1,0,0)
0.3237
(2,4,0)
0.2205
0.4728
0.4793
0.2216.
0.3278
0.3056.
0.3318
0.4632
0.4655
0.4602
0.4572
0.3186.
0.3206.
0.3156.
0.2110
0.4598
0.2875.
02332.
02972.
02373
0.2212
0.2120
0.2196.
0.4631.
0.2183.
(3,6,0)
0.1057
(2,2,0)
0.1889
(2,2,1)
0.1953
(3,0,3)
0.0354
(3,1,2)
0.0225
(2,0,2)
0.1067
(3,4,0)
0.1097
(3,4,1)
0.1368
(3,5,0)
0.1447
(3,3,0)
0.0997
(3,3,1)
0.0956
(2,1,1)
0.1150
(3,2,1)
0.1479
(3,2,2)
0.0734
(2,3,0)
0.2005
(0,0,0)
(1,2,1)
0.4665
(1,1,0)
0.2098
(1,0,0)
0.3237
(2,4,2)
0.2205
0.4728
0.4793
0.2216.
0.3278
0.3056.
0.3318
0.4632
0.4655
0.4602
0.4572
0.3206.
0.3156.
0.2110
0.4598
0.2875.
02332.
02972.
02373
0.2212
0.2120
0.2196.
0.4631
0.2183.
(3,6,3)
0.1057
(2,2,0)
0.2151
(3,0,0)
0.0354
(3,1,0)
0.0225
(2,0,0)
0.1067
(3,5,2)
0.1447
(2,1,0)
0.1150
(2,3,1)
0.2005
(3,4,1)
0.2114
(3,4,2)
0.0634
(3,3,0)
0.0998
(3,3,1)
0.0907
0.3186
(2,2,1)
0.1425
(3,2,0)
0.0740
(3,2,1)
0.0454
Figu e
1.
The
ansi ion
p obabili y
o
each
s a e
o
na u e
(i,
j,
h)
and
(i,
j,
h).
Once
we
compu e
he
isk-neu al
p obabili y
o
eaxh
s a e
o
na u e
a
ma u i y,
(n,
j,
k)
and
(n,
j,
h),
j
=
0,1,.
.
..2n,
k
=
max (i
−
j,
0),
....... i
−
j/2h
=
max (j
−
i,
0),
........ j/2;
we
a e
eady
o
e alua e
he
bidi ec ional
hindsigh
op ions
since
o
each
possible
s a e
o
na u e
a
ma u i y
(n,j,k)
and
(n,
j,
h)
a e
espec-
i ely
associa ed
he
op ion
payo
Sj
+n −S0
+k and S2h
+h −
Sj
+n ;
hence,
he
p ice
a
ime
o
a
floa ing
s ike
op ion
is
FOP
=
2n

j=9
n−j/2

k=max(n−j,0)
p ob (n,
j,
k)Sj
+n −S0
+k 
+
2n

j=9
j/2

h=max(j−n,0)
p ob (n,
j,
h)S2h
+h −
Sj
+n (16)
Subs i u ing
he
s iking
p ice
K
in o
Sj
+n we
can
ob ain
a
fixed
s ike
op ion
as
ollow
OP
=
2n

j=9
n−j/2

k=max(n−j,0)
p ob (n,
j,
k)max 0,K
−
S0
+k 
+
2n

j=9
j/2

h=max(j−n,0)
p ob (n,
j,
h)max 0,S2h
+h −
K (17)
Finally
we
u n
o
he
e alua ion
o
he
compu a ional
cos
o
he
ecu si e
algo i hm
o
p icing
maximum-minimum
bidi ec-
ional
op ions
in
inomial
CEV
model.
we
de e mine
he
numbe
o
possible
minima
o
maxima
eached
by
he
unde lying
asse
a
each
node
(i,
j)
(i
=
0,1,.
.
..,n;
j
=
0,.
.
..
.
..,2i)
o
he
ee.
Wi h-
ou
loss
o
gene ali y,
we
conside
a
inomial
ee
wi h
an
e en
numbe ,
n,
o
ime
s eps.
We
need
dis inguish
wo
cases.
The
fi s
conside s
he
s a es
o
na u e
cha ac e ized
by
he
iple
(i,j,k)
and
(i,j,h)
such
ha
i
≥
j,
ha
I
o
say
j
=
0,1,.
.
..,i.
In
Figu e
1,
we
can
easily
obse e
ha
each
node
eached
by
a
ajec o y
wi h
j/2
up
s eps
o
i-j/2
up
s eps
is
cha ac e ized
by
j/2
+
1
minima
o
maxima
o
j
=
2
m.
and
(j
+
1)/2
minima
o
maxima
o
j
=
2m
+
1
(m
=
0,1,.
.
..,i/2).
Since
in
he
ee
he e
exis
n
−
j
+
1
nodes
cha ac-
e ized
by
j/2
+
1
and
(j
+
1)/2
di e en
minima
o
maxima.
These
nodes
co espond
o
(n
−
j
+
1)*(j/2
+
1)
and
(n
−
j
+
1)*(j
+
1)/2
min-
ima
o
maxima.
The
second
case
conside s
he
s a es
o
na u e
cha ac e ized
by
he
iple
(i,j,k)
and
(i,j,h)
such
ha
i
<
j
i.e.
j
=
i
+
1,.
.
..,2i,
in
he
ee
he e
exis
j
−
n
nodes
cha ac e ized
byn-
j/2
+
1
and
n-(j
+
1)/2
+
1di e en
minima
o
maxima
espec i ely
o
j
=
2w
and
j
=
2w-1(w
=
i/2
+
1,.
.
..,i).
These
nodes
co espond
o
(j
−
n)*(n
−
j/2
+
1)
and
(j
−
n)*(n
−
(j
+
1)/2
+
1)
minima
o
maxima.
Thus
he
o al
numbe
o
minima
o
maxima
in
a
inomial
ee
wi h
n
ime
s eps
is
equal
o:
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩

j=2m(n
−
j
+
1)j
+
1
2+
j=2m(n
−
j
+
1)j
2+
1=
i/2

m=0
(2n
−
4m
+
1)(m
+
1)
=n3
12 +5n2
8+17n
12 +
1;
i
≥
j

j=2w(j
−
n)n
−j
2+
1+
j=2w−1(j
−
n)i
−j
+
1
2+
1=
i

w=i/2+1
(4w
−
2n
−
1)(n
−
m
+
1)
=n3
12 +2n2
8+5n
12;
i
<
j
To
compu e
he
isk-neu al
p obabili y
o
each
minimum
o
maximum
in
he
ee,
acco ding
o
he
o wa d
induc ion
p oce-
du e
de eloped
abo e,
we
ha e
o
implemen
wo
mul iplica ions
and
one
addi ion,
hence,
n3/6
+
7n2/4
+
22n/6
mul iplica ions
and
n3/12
+
7n2/8
+
22n/12
addi ions
( he
node
(0.0.0)
has
a
isk-neu al
p obabili y
equal
o
1).
Mo eo e ,
o
he
isk-neu al
p obabili y
o
each
possible
minimum
p ice
a
he
las
ime
s ep,
n,
we
ha e
o
associa e
he
co esponding
op ion
payo
a
ma u i y
and
hen
sum
hem
up.
Table
1
illus a e
he
nume ical
esul s
o
he
app oach
desc ibed
abo e
o
e alua ing
floa ing
s ike
and
fixed
s ike
maximum-minimum
bidi ec ional
op ions
in
he
inomial
CEV
model.
The
ocus
o
he
able
is
o
check
he
app oach
pe o mance
and
con e gence.
.in
o de
o
allow
compa isons
wi h
o he
al-
ua ion
me hods,
we
choose
he
same
op ion
pa ame e s
as
hese
used
in
Boyle
and
Tian
(1999)
illus a ed
in
Du ydo
and
Line sky
(2001).
The
cu en
asse
p ice
S may
be
90
100
110,
he
ime
o
ma u i y
is
T
=
6
mon hs,
he
annualized
isk- ee
in e es
a e
equals
10%
and
he
ola ili y
o
he
unde lying
asse
a e
o
e u n
B.
Peng,
F.
Peng
/
Jou nal
o
Economics,
Finance
and
Adminis a i e
Science
21
(2016)
50–55
55
Table
1
Valua ion
o
he
maximum-minimum
bidi ec ional
op ions
in
he
inomial
CEV
model.
nFixed
s ike
op ion
floa ing
s ike
op ion
S a
=
0.5
a
=
1
a
=
1.5
a
=
0.5
a
=
1
a
=
1.5
25
90
20.3697
20.3937
20.4647
30.1459
30.2299
30.3291
100
19.0725
19.2022
19.3697
28.6135
28.7147
28.8511
110
18.1462
18.1375
18.4731
29.1786
29.2291
29.3975
100
90
20.3734
20.3954
20.4681
30.1452
29.7188
30.3438
100
19.0827
19.2009
19.3722
28.6128
28.7143
28.8548
110
18.1439
18.1299
18.4724
29.1399
29.2511
29.4146
250
90
20.3723
20.3945
20.4644
30.1437
30.2185
30.3107
100
18.9898
19.1925
19.3707
28.6078
28.7071
28.8355
110
18.1405
18.1285
18.4699
29.1232
29.2486
29.4063
Close- o m
solu ion
90
20.3727
20.3942
20.4644
30.1435
30.2185
30.3255
100
19.0799
19.1923
19.3707
28.6078
28.7075
28.8353
110
18.1405
18.1285
18.4699
29.1233
29.2485
29.4065
Own
elabo a ion.
is
25%
pe
annum.
The
s ike
p ice
is
100.
The
numbe
o
ime
s eps
n
used
anges
among
25,
100,
250.
The
elas ici y
ac o
a
=
0.5,
1
o
1.5.
Fu he mo e,
a
close- o m
solu ion
is
compu ed
wi h
Da ydo
and
Line sky
me hod
expanded
o
compa ison.
As
wi h
o he
nume ical
me hod
applica ions,
p ices
om
ou
inomial
CEV
model
is
con e ge
apidly
wi h
all
a
alue
o
he
closed- o m
solu ion
as
he
numbe
o
s eps
n
inc eases.
When
n
=
25,
on
a e age,
he
alue
om
ou
inomial
op ion
p icing
app oach
is
close
o
close- o m
solu ion
han
app oxima ion
solu-
ions
using
o he
nume ical
me hod
de eloped
by
Boyle,
Tian
and
Bin.
Fo
n
=
100,
he
esul s
show
ha
he
di e ence
be ween
he
inomial
me hod
and
he
closed- o m
solu ion
is
less
han
o
equal
o
0.01.
The
esul
wi h
ime
s ep
250
is
almos
app oach
o
close-
o m
solu ion.
The
accu acy
o
esul s
is
simila
o
o he
nume ical
me hods
and
he
e ficiency
o
ecu si e
algo i hm
in
he
inomial
CEV
model
is
highe
han
o he
nume ical
algo i hm.
4.
Conclusions
A
inomial
ee
app oxima ion
o
he
CEV
model
was
con-
s uc ed
o
desc ibe
he
e olu ion
o
he
unde lying
asse
wi h
non-cons an
ola ili y.
Once
he
inomial
ee
has
been
buil
up,
we
p oposed
a
ecu si e
algo i hm
based
on
o wa d
induc ion
scheme
o
calcula e
ansi ion
p obabili y
o
possible
minimum
o
maximum
eached
by
he
unde lying
asse
a
each
node
o e
he
i-
nomial
ee
when
ola ili y
a ied
wi h
asse
p ice
le el,
u he
we
use
i
o
alue
di e en
ypes
o
maximum-minimum
bidi ec ional
op ions.
Nume ical
esul s
show
ha
he
inomial
algo i hm
has
sa is ac o y
con e gence
and
p oduces
accu a e
p ices
o
a
wide
ange
o
pa ame e
alues
o
he
CEV
model
compa ed
wi h
o he
nume ical
me hods
such
as
close- o m
solu ion
me hod.
In
addi-
ion,
o
exo ic
pa h-dependen
op ions
ha
depend
on
he
ex eme
o
he
p ocess,
he
p ices
a e
qui e
sensi i e
o
he
specifica ion
o
he
p ocess.
Re e ences
Becke s,
S.
(1980).
The
consan
elas ici y
o
a iance
model
and
i s
implica ions
o
op ion
p icing.
Jou nal
o
Finance,
(35),
661–673.
Black,
F.,
&
Scholes,
M.
(1973).
The
p icing
o
op ion
and
co po a e
liabili ies.
Jou nal
o
Poli ical
Economy,
(81),
637–659.
Boyle,
P.,
&
Tian,
Y.
(1999).
P icing
lookback
and
ba ie
op ions
unde
he
CEV
p ocess.
Jou nal
o
Financial
and
Quan i a i e
Analysis,
34(2),
242–264.
Bin,
P.,
&
Fei,
P.
(2006).
P icing
geome ic
Asian
op ion
unde
he
CEV
p ocess.
In e na ional
Economic
Jou nal,
(4),
515–519.
Cox,
J.
(1996).
No es
on
op ion
p icing
1:
Cons an
elas ici y
o
a iance
di usions.
Unpublished
d a .
Palo
Al o,
CA.
S an o d
Uni e si y
(Sep embe
1975).
Pub-
lished
in
he
Jou nal
o
Po olio
Managemen ,
(23),
5–17.
Cox,
J.
C.,
&
Ross,
S.
A.
(1976).
The
alua ion
o
op ions
o
al e na i e
s ochas ic
p ocesses.
Jou nal
o
Financial
Economics,
(7),
145–166.
Da il,
C.
(1982).
Fu he
esul s
on
he
cons an
elas ici y
o
a iance
call
op ion
p icing
model.
Jou nal
o
Financial
and
Quan i a i e
Analysis,
(4),
533–554.
Du ydo ,
D.,
&
Line sky,
V.
(2001).
P icing
and
hedging
pa h-dependen
op ions
unde
he
CEV
p ocess.
Managemen
Science,
(47),
949–965.
He,
X.
(2001).
P icing
lookback
op ion
unde
CEV
p ocess.
Jou nal
o
Sys ems
Engi-
nee ing,
(4),
296–300.
Macbe h,
J.
D.,
&
Me ille,
L.
J.
(1980).
Tes s
o
he
Black-Scholes
and
Cox
call
op ion
alua ion
models.
Jou nal
o
Finance,
(35),
285–301.