Discrete Choice Models and Behavioral Response to Congestion Pricing Strategies
Discrete space time option pricing forum fsr
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18
• D
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ng
fsrforum
• ja
arga
ng 1
2 • ed
itie
#5
Dis
cret
e Sp
ace-
Tim
e O
ptio
ns P
rici
ng •
19
Dis
cret
e Sp
ace-
Tim
e O
ptio
ns P
ricin
g
Ilya
Gik
hm
an
This
pap
er
pre
sents
a f
orm
al a
p-
pro
ach t
o t
he d
eri
vati
ves
pri
cing. In
this
pap
er
we w
ill s
tudy
deri
vati
ves
pri
cing in a
dis
crete
spac
e-t
ime
appro
xim
atio
n. Th
e p
rim
ary
pri
nci
-
ple
of
the p
rici
ng t
heory
we intr
o-
duce
in t
he p
aper
is t
he n
oti
on o
f
equal
ity
of
inve
stm
ent
whic
h b
ased
on t
he inve
stors
goal
: ‘in
vest
ing in
a gre
ater
retu
rn’.
We
say
that
tw
o in
vest
men
ts a
re e
qual
at
a m
omen
t of
tim
e if
th
eir
inst
anta
neo
us
rate
s of
retu
rn a
t th
is m
omen
t at
are
equ
al. I
f eq
ual
ity
of t
wo
inve
stm
ents
hol
ds a
ny
mom
ent
of t
ime
over
[ t
, T
] t
hen
th
ese
inve
stm
ents
are
equ
al o
n [
t ,
T ]
. Th
is d
efin
itio
n r
epre
sen
ts in
vest
men
t
equ
alit
y, I
E l
aw o
r pr
inci
ple
wh
ich
wil
l be
app
lied
th
rou
ghou
t of
th
e pa
per
for
defi
nit
ion
of
a
deri
vati
ve p
rice
. Nex
t w
e u
se c
ash
flow
not
ion
th
at im
plie
s a
seri
es o
f tra
nsa
ctio
ns
as a
spe
cifi
-
cati
on o
f som
ewh
at b
road
not
ion
of i
nve
stm
ent.
It is
not
diff
icul
t to
see
that
this
con
cept
of t
he in
vest
men
t equ
alit
y is
a p
erfe
ct a
nd m
ore
accu
rate
than
the
pres
ent v
alue
, PV
con
cept
. Ind
eed,
if tw
o in
vest
men
ts a
re e
qual
in I
E s
ense
then
they
are
equa
l for
any
pos
sibl
e sc
enar
io o
r si
mpl
y to
say
alw
ays.
On
the
othe
r ha
nd, i
t is
cle
ar t
hat
if tw
o
inve
stm
ents
are
equ
al in
IE
sen
se t
hen
they
are
equ
al in
the
PV.
The
inve
rse
stat
emen
t is
inco
r-
rect
. Mor
e ac
cura
tely
, if t
wo
inve
stm
ents
are
equ
al in
the
PV
sen
se th
en it
is e
asy
to p
rese
nt e
xam
-
ple
of a
sce
nari
o th
at d
emon
stra
tes
arbi
trag
e op
port
unit
y. I
n th
is e
xam
ple,
the
pre
sent
val
ue o
f
two
inve
stm
ents
are
equ
al w
hile
one
inve
stm
ent
has
a hi
gher
rat
e of
ret
urn
over
a t
ime
subi
nter
-
val a
nd lo
wer
ove
r an
othe
r su
bint
erva
l tha
n ot
her
inve
stm
ent.
As
far
as P
V s
ugge
sts
equa
l pri
ce
for
two
inve
stm
ents
one
can
sel
l a lo
wer
rat
e in
stru
men
t ov
er t
he c
orre
spon
dent
sub
inte
rval
and
buy
the
inst
rum
ent w
ith
high
er r
ate
of r
etur
n in
stru
men
t. T
hen
at th
e en
d of
this
per
iod
inve
stor
wou
ld s
ell
shor
t hi
gher
pri
ced
inst
rum
ent
whi
ch p
rom
ises
low
er r
ate
of r
etur
n ov
er t
he n
ext
peri
od a
nd b
uy fo
r lo
wer
pri
ce o
ther
inst
rum
ent w
hich
pro
mis
es h
ighe
r ra
te o
f ret
urn.
At t
he e
nd
of t
he p
erio
d th
e in
vest
or h
as p
ure
prof
it fo
r th
e sc
enar
io t
houg
h tw
o in
vest
men
ts h
ave
equa
l PV.
This
type
of t
he e
xam
ple
illus
trat
es th
e fa
ct th
at P
V r
educ
tion
of c
ash
flow
s in
suffi
cien
t to
be u
sed
as a
def
init
ion
of t
he e
qual
ity
of t
wo
cash
flow
s. N
ever
thel
ess,
PV
red
ucti
on m
ight
be
help
ful f
or
cons
truc
tion
of t
he m
arke
t es
tim
ates
of t
he s
pot
or fu
ture
pri
ces.
Bea
ring
in m
ind
that
pri
ce d
ef-
init
ion
depe
nds
on a
sce
nari
o w
e sh
ould
be
awar
e th
at a
ny s
pot
pric
e ca
lcul
ated
wit
h th
e he
lp o
f
PV
or
othe
r ru
le im
plie
s ri
sk. T
his
risk
for
buye
r is
mea
sure
d by
the
pro
babi
lity
of t
he e
vent
s fo
r
whi
ch s
cena
rio’
s pr
ice
is b
ello
w t
han
spot
pri
ce.
1. P
lain
Van
illa
opti
on
s va
luat
ion
.L
et u
s in
trod
uce
th
e de
fin
itio
n o
f th
e pl
ain
van
illa
opt
ion
con
trac
ts w
hic
h is
a c
lass
opt
ion
cov
-
ered
Eu
rope
an a
nd
Am
eric
an t
ypes
. An
opt
ion
is a
rig
ht
to b
uy
or s
ell a
n a
sset
at
a kn
own
pri
ce,
wit
hin
a g
iven
per
iod
of ti
me.
Th
e kn
own
pri
ce, K
is c
alle
d ex
erci
se o
r st
rike
pri
ce. T
he
last
dat
e,
T o
f th
e li
feti
me
of t
he
opti
on i
s ca
lled
mat
uri
ty. T
he
righ
t to
bu
y is
kn
own
as
the
call
opt
ion
,
wh
ile
the
righ
t to
sel
l is
th
e pu
t op
tion
. T
he
pric
e of
th
e op
tion
als
o re
ferr
ed t
o as
pre
miu
m.
Eu
rope
an o
ptio
ns
can
on
ly b
e ex
erci
sed
at m
atu
rity
, wh
erea
s A
mer
ican
typ
e of
th
e op
tion
s ca
n
be e
xerc
ised
at
any
mom
ent
up
to m
atu
rity
.
Let
S (
t )
den
ote
an a
sset
spo
t pr
ice
at d
ate
t, t
0. F
or-
mal
ly a
n E
uro
pean
opt
ion
con
trac
t is
def
ined
by
its
payo
ff
at e
xpir
atio
n.
Th
e ca
ll a
nd
put
valu
es a
t ex
pira
tion
T a
re
defi
ned
by
form
ula
s
C (
T ,
S (
T )
) =
max
{ S
( T
) –
K ,
0 }
(1
.1)
P (
T ,
S (
T )
) =
max
{ K
– S
( T
) ,
0 }
Th
us,
a b
uye
r of
th
e ca
ll o
ptio
n w
ould
agr
ee t
o ex
erci
se t
he
righ
t to
bu
y th
e u
nde
rlyi
ng
asse
t in
cas
e if
th
e va
lue
of t
he
call
opt
ion
at
mat
uri
ty T
is
posi
tive
, i.e
. if
C (
T ,
S (
T )
) >
0. I
t is
cle
ar t
hat
th
ere
is n
o se
nse
in r
eali
zati
on o
f th
e ri
ght
wh
en C
( T
, S
( T
) )
0
. O
n t
he
oth
er h
and
a bu
yer
of t
he
put
opti
on w
ould
exe
rcis
e th
e ri
ght
to s
ell
the
un
derl
yin
g
asse
t in
cas
e if
S (
T )
< K
, i.e
. a p
ut
hol
der
is in
tere
sted
to
sell
ass
et w
ith
pri
ce S
( T
) f
or K
wh
en S
( T
) <
K. T
hat
is
a h
olde
r of
th
e pu
t op
tion
can
exe
rcis
e th
e ri
ght
to s
ell
the
opti
on w
hen
P (
T ,
S (
T )
) >
0. O
ther
wis
e, i
f P
( T
, S
( T
)
) 0
th
e ri
ght
wil
l n
ot b
e ex
erci
sed.
Th
e op
tion
pri
cin
g pr
oble
m i
s to
det
erm
ine
the
call
( p
ut
)
opti
on p
rice
at
any
mom
ent
of t
ime
t be
fore
th
e ex
pira
tion
date
T.
To i
llu
stra
te p
rici
ng
met
hod
olog
y w
e be
gin
wit
h a
sim
ple
exam
ple.
Nex
t w
e u
se t
he
term
s as
set,
sto
ck, o
r se
cu-
rity
as
syn
onym
s.
Exa
mp
le 1
. Let
a s
tock
pri
ce a
t t
= 0
be
S (
0 )
= 2
an
d at
T
=
1 st
ock
take
th
e va
lues
S (
1 )
=
{
5 , 1
}. I
ntr
odu
ce t
he
prob
abil
ity
spac
e of
sce
nar
ios
of t
he
prob
lem
. Den
ote ω =
{
u ,
d },
wh
ere
u d
enot
es t
he
scen
ario
{ S
( 0
) =
2, S
( 1
) =
5 }
, an
d d
= {
S (
0 )
= 2
, S (
1 )
= 1
}. P
utt
ing
stri
ke p
rice
K =
2 w
e en
able
to
defi
ne
the
call
opt
ion
pri
ce
for
each
sce
nar
io.
Den
ote
C (
t ,
x ;
T ,
K ,
)
th
e va
lue
of
the
call
opt
ion
for
fixe
d s
cen
ario
a
t th
e m
omen
t t
, giv
en
S (
t )
= x
. Her
e t
, x a
re v
aria
bles
of t
he
fun
ctio
n C
( )
, wh
ile
T, K
, a
re i
nte
rpre
ted
as p
aram
eter
s. T
he
valu
e of
par
am-
eter
s T
an
d K
are
ass
um
ed t
o be
fix
ed a
nd
for
the
wri
tin
g
sim
plic
ity
we
wil
l om
it t
hem
nex
t. L
et u
s sp
ecif
y th
e va
lue
of t
he
opti
on a
lon
g th
e sc
enar
io
u .
App
lyin
g IE
pri
nci
ple
we
arri
ve a
t th
e eq
uat
ion
wit
h r
espe
ct t
o u
nkn
own
C (
t =
0
, S (
0 )
= 2
; u
)
Th
e so
luti
on o
f th
e eq
uat
ion
is C
( 0
, 2
; u
) =
1.2
. Th
en
as f
ar a
s th
e op
tion
pay
off
for
the
scen
ario
u
is
max
{ 1
-
2 , 0
} =
0 w
e pu
t by
def
init
ion
C (
0 ,
2 ;
d )
= 0
. Th
ere
is n
o se
nse
to
pay
for
the
opti
on a
su
m if
opt
ion
val
ue
in t
he
futu
re m
omen
t is
0. T
her
efor
e, b
y de
fin
itio
n w
e pu
t C
( 0
,
2 ;
d )
=
0.
Th
e se
curi
ty d
istr
ibu
tion
is
the
prob
abil
itie
s
of t
he
two
scen
ario
s: P
u
=
P (
u
), P
d =
P
(
d )
. Th
is
dist
ribu
tion
is
assi
gned
th
en t
o th
e op
tion
pre
miu
m. T
hu
s
Let
us
con
side
r tw
o st
ocks
th
at h
ave
prob
abil
itie
s P
1 (
u )
= P
2 (
d )
= 0
.99
an
d
P 1
( d
) =
P
2 (
u )
=
0.
01 c
orre
spon
din
gly.
Th
e ca
ll
opti
on p
rice
is
the
ran
dom
var
iabl
e ta
kin
g va
lues
: C
i ( 0
,
2 ;
u )
= 1
, C
i ( 0
, 2
; d
) =
0 ,
i =
1, 2
. Th
en
*)
Th
e av
erag
e ra
te o
f re
turn
on
1st s
tock
is
equ
al t
o 1.
48
and
– 0
.43
on 2
nd
sto
ck.
**)
Th
e av
erag
e ra
te o
f re
turn
on
cal
l op
tion
wri
tten
on
1st
stoc
k
wh
ile
the
aver
age
rate
of
retu
rn o
n c
all
opti
on w
ritt
en o
n
the
2nd
stoc
k is
equ
al t
o 1.
5 %
. Ass
um
e th
at m
arke
t pr
ice
at
t =
0
is e
qual
to
the
mea
n o
f th
e op
tion
at
this
mom
ent.
Th
e ex
pect
atio
ns
of t
he
opti
ons
pric
e on
th
e 1st
an
d th
e 2n
d
stoc
ks a
t t
= 0
are
c 1
( 0
, 2 )
= E
C 1
( 0
, 2 ;
) =
1.2
× 0
.99
= 1
.188
c 2
( 0
, 2 )
= E
C 2
( 0
, 2 ;
) =
1.2
× 0
.01
= 0
.012
Oth
er p
ossi
ble
esti
mat
e of
th
e sp
ot o
ptio
n p
rice
s ca
n b
e
base
d on
th
e P
V c
once
pt. A
ssu
min
g th
at t
he
risk
fre
e in
ter-
est
is e
qual
to
0 w
e se
e th
at
»
As
far
as P
V s
ugg
ests
equ
al p
rice
for
two in
vest
men
ts
on
e c
an
se
ll a
lo
we
r ra
te i
nst
rum
en
t o
ver
the
co
rres
ponden
t su
bin
terv
al
an
d b
uy t
he
in
stru
me
nt
wit
h h
igh
er
rate
of
retu
rn i
nst
rum
en
t.
We
say t
hat
two i
nve
stm
ents
are
e
qu
al
at
a m
om
en
t o
f ti
me
if
the
ir
inst
anta
neo
us
rate
s o
f re
turn
at
this
m
om
en
t a
re e
qu
al.
The
dif
fere
nce
be
twe
en
po
ssib
le
spo
t p
rice
s is
the
valu
e of
risk
ta
ke
n b
y i
nve
sto
rs.
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fsrforum
• ja
arga
ng 1
2 • ed
itie
#5
20
• D
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ngD
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ng •
21
Her
e, t
he
last
row
C (
0 ,
S (
0 )
) =
C (
0 ,
2 )
repr
esen
ts t
he
opti
on p
rice
at
tim
e 0.
Eac
h e
ntr
y in
th
e th
ird
row
has
pro
b-
abil
ity
of 1
/ 6
. Th
e si
tuat
ion
rep
rese
nte
d by
th
e Ta
ble
is t
he
sim
ples
t in
sen
se t
hat
th
e op
tion
’s r
etu
rn p
erfe
ctly
rep
lica
tes
the
stoc
k re
turn
. To
illu
stra
te m
ore
a ge
ner
al c
ase
in w
hic
h
the
poss
ibil
ity
perf
ectl
y re
plic
ates
the
stoc
k re
turn
by
the
call
opti
on is
impo
ssib
le w
e as
sum
e, fo
r in
stan
ce, t
hat
K =
$ 2
.5.
Th
e co
rres
pon
den
t op
tion
pay
off a
t m
atu
rity
is t
he
row
C (
1,
S (
1 )
) an
d th
e pr
ice
of t
he
opti
on is
def
ined
by
the
thir
d ro
w.
Its
valu
es c
an b
e ca
lcu
late
d ap
plyi
ng
IE c
once
pt. T
hu
s
S (
1 )
12
34
56
C
( 1, S
( 1 )
)0
00.5
1.5
2.5
3.5
C
( 0, S
( 0 )
)0
01
/3
3/
41
7/
6
Ris
k m
anag
emen
t. M
ean
of
the
opti
on p
rice
at
t =
0
is
0.54
17. H
ence
, if t
he
mar
ket
pric
e of
th
e op
tion
is A
= 0
.541
7
then
if
outc
ome
is 1
or
2 in
vest
or’s
los
ses
are
its
prem
ium
,
i.e. 0
.541
7. T
her
efor
e th
e lo
ss o
f 0.5
417
occu
rs w
ith
pro
babi
l-
ity
1/3.
Th
en t
he
prob
abil
ity
1/6
is a
ssig
n t
o ea
ch o
f th
e n
ext
prof
it-l
oss
outc
omes
: [ 0
.5
– 1
/3 ]
=
0.
17, [
1.5
–
3/4
]
=
0.75
, [ 2
.5 –
1 ]
= 1
.5,
[ 3.
5 –
7/6
] =
2.3
th
at c
orre
spon
d
to s
tock
val
ues
3, 4
, 5, 6
. We
intr
odu
ce s
ome
use
ful r
isk
char
-
acte
rist
ics.
Th
ese
are
aver
age
prof
it o
f th
e op
tion
def
ined
by
the
form
ula
< P
rofit
( 0
, T ;
K )
> =
E
C (
0, S
( 0
)) χ
{ C
( 1,
S (
1 ))
> K
}
and
aver
age
loss
es
< L
oss
( 0 ,
T ; K
) >
=
E
C (
0, S
( 0
)) χ
{ C
( 1,
S (
1 ))
K
}
The
prof
it-l
oss
rati
o is
< P
rofit
( 0
, T ;
K )
> /
< L
oss
( 0 ,
T ; K
)
>. T
hese
are
pri
mar
y ri
sk c
hara
cter
isti
cs o
f the
opt
ion.
Let u
s hi
ghlig
ht th
e di
ffere
nce
betw
een
Eur
opea
n an
d Am
eric
an
opti
on p
rice
s. C
onsi
der
Amer
ican
opt
ion
wri
tten
on
stoc
k. T
he
IE r
ule
appl
ying
for
the
Eur
opea
n ca
ll or
put
opt
ions
at t
, t <
T
in e
ithe
r di
scre
te o
r co
ntin
uous
tim
e is
a s
olut
ion
of th
e eq
uati
on
0 t
T
. Her
e
c 1
( 0
, 2 )
= E
C 1
( t
= 1
; )
= 3
× 0
.99
=
2.97
c 2
( 0
, 2 )
= E
C 2
( t
= 1
; )
= 3
× 0
.01
=
0.03
We
can
see
th
at t
her
e n
o u
niq
ue
rule
to
defi
ne
a ‘fa
ir’ p
rice
to t
he
opti
on a
s fa
r as
an
y n
um
ber
use
d as
a s
pot
pric
e
impl
ies
the
risk
. Th
e di
ffer
ence
bet
wee
n p
ossi
ble
spot
pri
ces
is t
he
valu
e of
ris
k ta
ken
by
inve
stor
s. T
his
ris
k is
th
e
mar
ket
risk
wh
ich
spe
cifi
ed b
y th
e fu
ture
beh
avio
r of
th
e
un
derl
yin
g as
set.
Th
e ri
sk m
anag
emen
t pr
oble
m is
a c
alcu
-
lati
on o
f th
e m
arke
t ri
sk.
Ris
k m
an
ag
em
en
t.*)
Con
side
r fo
r ex
ampl
e ca
ll o
ptio
n w
ritt
en o
n s
tock
1.
An
inve
stor
pay
s pr
emiu
m A
for
th
e op
tion
at
t =
0 t
akes
ris
k
asso
ciat
ed w
ith
th
e sc
enar
ios
Th
is i
s th
e se
t of
sce
nar
ios
for
wh
ich
rat
e of
ret
urn
on
cal
l
opti
on w
ill
be l
ower
th
an t
he
rate
of
retu
rn o
n u
nde
rlyi
ng
stoc
k. I
nde
ed, b
uyi
ng
call
opt
ion
for
A a
nd
rece
ivin
g S
( T
) -
K a
t T
impl
ies
rate
of r
etu
rn e
qual
to
the
left
han
d si
de o
f th
e
latt
er i
neq
ual
ity
wh
ile
the
righ
t h
and
side
is
the
rate
of
retu
rn o
n s
tock
ove
r th
e sa
me
peri
od.
In o
ther
wor
ds t
his
risk
set
of
scen
ario
s fo
rms
buye
r ri
sk w
hen
in
vest
ors
pay
hig
her
pri
ce f
or s
tock
th
at i
mpl
ies
by t
he
mar
ket.
Th
us
the
prob
abil
ity
P { ω r
isk-
buye
r (
A )
} i
s a
mea
sure
of
the
buye
r
risk
. L
et u
s co
nsi
der
are
mor
e co
mpl
ex c
ase
wh
en r
ando
m
stoc
k ad
mit
s m
ult
iple
val
ues
.
Exa
mple
2. L
et u
s st
udy
th
e ro
llin
g di
ce e
xam
ple
to il
lust
rate
the
mu
ltip
le v
alu
es s
toch
asti
c se
curi
ty i
n t
he
opti
on p
rici
ng
prob
lem
. Let
aga
in a
ssu
me
that
tim
e ta
kes
two
valu
e t
= 0
, 1
wh
ich
are
th
e in
itia
l an
d ex
pira
tion
dat
es o
f th
e op
tion
. Th
e
set
1, 2
..., 6
rep
rese
nts
pos
sibl
e va
lues
of t
he
stoc
k an
d pr
ob-
abil
itie
s of
th
e ev
ents
{ S
( 1
) =
j }
, j =
1, 2
, ...
6 a
re e
qual
to 1
/ 6.
Th
e pa
yoff
at
the
mat
uri
ty is
def
ined
C (
1 ,
S (
1 )
) =
max
{ S
( 1
) –
K ,
0 }
an
d le
t K
= $
0.8
. Th
e va
lue
S (
0 )
=
$2 c
an b
e in
terp
rete
d as
a p
rice
to
roll
th
e di
ce .
App
lyin
g th
e
IE c
once
pt w
e ar
rive
at
the
defi
nit
ion
of t
he
call
opt
ion
pri
ce.
Th
e op
tion
pri
ce is
a r
ando
m v
aria
ble
taki
ng
diff
eren
t va
lues
j -
0. 8
, j =
1, 2
, ...
6. W
e ex
pres
s th
e th
eore
tica
l pri
ce o
f th
e
gam
e w
ith
th
e h
elp
of t
he
tabl
e
S (
1 )
12
34
56
C
( 1, S
( 1 )
)0.2
1.2
2.2
3.2
4.2
5.2
C
( 0, S
( 0 )
) 0
.41
.21
.47
1.6
1.6
81
.73
»
are
payo
ffs
on c
all a
nd
put
opti
ons
at m
atu
rity
T. T
he
Am
eri-
can
opt
ion
can
be
exer
cise
d at
an
y ti
me
up
to m
atu
rity
T.
Th
eref
ore,
its
pay
off
depe
nds
on
tim
e in
terv
al d
uri
ng
wh
ich
the
opti
on c
an b
e ex
erci
sed.
Ass
um
ing
for
sim
plic
ity
that
ris
k
free
rat
e eq
ual
to
0 it
loo
ks r
easo
nab
le t
o ex
erci
se o
ptio
n a
t
the
date
wh
en p
ayof
f rea
ches
its
max
imu
m. H
ence
, th
e ex
er-
cise
pri
ce o
f th
e A
mer
ican
opt
ion
is
{ S
( t
)
- K
, 0
}.
App
lyin
g IE
for
th
e A
mer
ican
cal
l pr
icin
g le
ads
to t
he
equ
a-
tion
for
Am
eric
an c
all o
ptio
n v
alu
e at
t =
0
wh
ere τ ( ω )
= {
t
T :
= m
ax }
. Giv
en d
istr
ibu
tion
S (
t )
an in
vest
or c
an e
stab
lish
th
e le
vel L
su
ch t
hat
th
e A
mer
i-
can
opt
ion
wou
ld b
e ex
erci
sed
befo
re T
if
= L
for
t T
.
Oth
erw
ise
the
opti
on w
ould
be
exer
cise
d at
T if
S (
T )
> K
.
Den
ote
C A
( t
, S
( t
) ;
T )
Am
eric
an o
ptio
n p
rice
at
t. T
hen
C A
( 0
, S (
0 )
; T
= 2
) =
C
E (
0 ,
S (
0 )
; T =
1 )
) χ {
S (
1 )
3
S (
2 )
} +
+ C
E (
0 ,
S (
0 )
; T =
2 ) χ {
S (
2 )
> S
( 1
) }
Not
e, f
or e
xam
ple,
th
at w
hen
th
e ev
ent
{ S
( 1
) 3
S (
2 )
} i
s
tru
e w
hen
an
nu
aliz
ed r
ate
of r
etu
rn o
n s
tock
ove
r th
e pe
riod
[0, 1
] is
hig
her
th
an o
ver
the
peri
od [
0, 2
]. T
he
latt
er fo
rmu
la
expr
esse
s A
mer
ican
opt
ion
pri
ce t
hro
ugh
Eu
rope
an o
ptio
n
pric
e an
d th
e co
ncl
usi
on t
hat
foll
ows
from
th
is fo
rmu
la d
oes
not
coi
nci
de w
ith
a w
ell-
know
n s
tate
men
t th
at t
he
curr
ent
pric
es o
f A
mer
ican
an
d E
uro
pean
opt
ion
s on
no
divi
den
d
asse
t are
iden
tica
l. T
he
last
form
ula
can
be
easi
ly e
xten
ded
on
mu
ltip
le s
teps
eco
nom
y.
In o
ne-
step
eco
nom
y, l
et u
s br
iefl
y ou
tlin
e th
e co
nst
ruct
ion
of t
he
call
opt
ion
pri
ce.
Let
t a
nd
T d
enot
e in
itia
l m
omen
t
and
opti
on m
atu
rity
an
d le
t u
nde
rlyi
ng
valu
es a
t T
an
d st
rike
pric
e sa
tisf
y in
equ
alit
ies:
S 1
< S
2 <
…. <
S p
K
S
p
+ 1
<
… <
S
n
. T
hen
cal
l op
tion
pre
miu
m i
s de
fin
ed a
s a
ran
dom
var
iabl
e
j =
p
+ 1
, …
, n
. If
c 0
is a
mar
ket
pric
e of
th
e op
tion
th
en
the
risk
con
nec
ted
to t
he
pric
e is
a c
han
ce t
hat
rea
lize
d sc
e-
nar
io b
elon
gs t
o th
e se
t
wh
ere
is a
sol
uti
on o
f th
e eq
uat
ion
. Th
is e
quat
ion
spe
cifi
es o
ne-
to-
on
e
corr
espo
nde
nce
bet
wee
n S
to
the
opti
on p
rice
c. I
f th
e va
lue
of th
e u
nde
rlyi
ng
at m
atu
rity
T w
ill b
e be
low
than
S th
en th
is
scen
ario
is
an e
lem
ent
of t
he
risk
y se
t ri
sk (
c
) as
soci
ated
wit
h t
he
inve
stor
’s m
arke
t ri
sk.
We
con
side
r n
ow o
ptio
ns
wri
tten
on
exc
han
ge r
ate.
Th
is
prob
lem
is s
imil
ar t
o th
e pr
oble
ms
stu
died
abo
ve n
ever
the-
less
som
e pe
culi
arit
ies
are
nee
ded
to b
e sp
ecif
ied.
We
wil
l
use
cro
ss c
urr
ency
exc
han
ge a
s u
nde
rlyi
ng
of t
he
opti
on
con
trac
ts. L
et K
den
ote
stri
ke p
rice
mea
sure
d in
$ /
£ an
d q
( t
) de
not
es $
/ £
- ex
chan
ge r
ate
at t
ime
t. T
hat
is £
1 (
t )
=
$ q
( t
) an
d th
eref
ore
a £1
can
be
inte
rpre
ted
as a
por
tion
of
asse
t th
at c
an b
e so
ld o
r bo
ugh
t on
$-m
arke
t. A
ll c
ontr
acts
are
sett
led
by d
eliv
ery
of t
he
un
derl
yin
g cu
rren
cy. B
y de
fin
i-
tion
, th
e co
ntr
act
payo
ff a
t m
atu
rity
T is
N m
ax {
Q (
T )
– K
, 0 }
, wh
ere
N d
enot
es a
con
trac
t si
ze. F
or in
stan
ce, t
he
size
of a
Bri
tish
pou
nd
call
opt
ion
con
trac
t tr
aded
on
PL
HX
is N
= £
31,
250.
Th
e ca
ll o
ptio
n e
quat
ion
(1.
2) c
an b
e re
wri
tten
in t
he
form
Th
en t
he
$-va
lue
of t
he
call
opt
ion
con
trac
t at
dat
e t
is
Th
is f
orm
ula
hol
ds r
egar
dles
s w
het
her
th
e ex
chan
ge r
ate
q
( t
) i
s su
ppos
ed t
o be
sto
chas
tic
or d
eter
min
isti
c. F
or
inst
ance
, let
N =
£ 3
1,25
0, K
= $
/ £ 1
.50,
q (
T )
= $
/ £1
.55.
Th
en p
ayof
f at
mat
uri
ty T
is e
qual
to
N m
ax {
q (
T )
-
K ,
0 }
=
£ 31
,250
× $
/ £
( 1.
55 –
1.5
) =
$ 1,
562.
5
Now
we
appl
y fo
r m
ore
com
plex
opt
ion
pro
blem
that
invo
lves
inte
rmed
iate
mom
ent
of t
ime
wit
h m
ore
than
2 s
tate
s at
expi
rati
on. A
ssu
me
that
th
e va
lue
of 1
00 B
riti
sh p
oun
ds o
ver
thre
e da
tes
0, 1
, 2 a
re g
iven
as
foll
ow
t = 0
t = 1
t = 2
q(2
) = 1
86 p
(185, 186 )
= 1
/4
q(1
) = 1
85, p
(180, 185)
= 2
/3
q(2
) = 1
82 p
(178, 182 )
= 1
/8
q(0
) = 1
80
q(2
) = 1
81 p
(178, 181 )
= 1
/4
q(1
) = 1
78, p
(180, 178)
= 1
/3
q(2
) = 1
79 p(1
85, 179 )
= 3
/4
q(2
) = 1
76 p
(178, 176 )
= 5
/8
wh
ere
p (
a, b
) d
enot
es t
ran
siti
on p
roba
bili
ty f
rom
th
e st
ate
‘a’ t
o st
ate
‘b’.
Ass
um
e th
at a
ll t
ran
siti
ons
are
mu
tual
ly in
de-
pen
den
t. C
onsi
der
Eu
rope
an c
all o
ptio
n w
ith
th
e st
rike
pri
ce
K =
180
. W
e be
gin
wit
h c
alcu
lati
ons
of t
he
opti
on p
rice
by
mov
ing
back
war
d in
tim
e. A
pply
ing
the
met
hod
th
at w
e u
sed
abov
e ov
er p
erio
d it
is e
asy
to s
ee t
hat
and
Th
en
Her
e, p
(a,
b, c
) =
P
{ q
(0)
= a
, q (
1) =
b, q
(2)
= c
}an
d {a
}
{b} i
s th
e u
nio
n o
f tw
o st
ates
‘a’ a
nd
‘b’.
We
sum
mar
ize
cal-
cula
tion
s in
th
e ta
ble
C( 0
, 180 )
C (
1, ω )
C (
2, ω )
p (ω
)
5.8
07
5.9
68
61/
6
1.9
78
1.9
56
21/
24
0.9
94
0.9
83
11/
12
00
017/
24
Th
e pr
obab
ilit
ies
in t
he
fou
rth
col
um
n r
elat
ed t
o th
e ev
ents
in e
ach
cel
l in
th
e ro
w.
Now
let
us
inve
stig
ate
a po
ssib
le
inve
stor
’s s
trat
egy.
Th
e av
erag
e re
turn
on
th
e ex
chan
ge r
ate
over
An
inve
stor
wh
o m
igh
t in
tere
sted
in c
alcu
lati
on o
f th
e va
lue
of th
e op
tion
pri
ce w
hic
h e
xpec
ted
retu
rn w
ould
be
not
wor
se
then
1.0
148.
Th
is p
rice
is a
sol
uti
on o
f th
e eq
uat
ion
E C
( 1
, ω )
/ x =
1.0
148.
Sol
vin
g th
is e
quat
ion
for
x yi
elds
Hen
ce,
the
prem
ium
of
1.14
on
cal
l op
tion
wit
h s
trik
e K
=
180
appr
oxim
atel
y in
ave
rage
pro
mis
es t
he
retu
rn o
f 1.4
8% .
Th
e ri
sk o
f bu
yin
g op
tion
for
$1.1
4 is
th
e pr
obab
ilit
y
The A
meri
can
op
tion
can
be e
xerc
ised
at
any
tim
e u
p
to m
atu
rity
T. T
here
fore
, it
s p
ayoff
dep
ends
on t
ime
inte
rval
du
rin
g w
hic
h t
he o
pti
on
can
be e
xerc
ised
.A
ssu
min
g f
or
sim
plici
ty t
hat ri
sk f
ree
rate
equ
al t
o 0
it l
oo
ks
rea
son
ab
le t
o e
xerc
ise
op
tio
n a
t th
e d
ate
w
he
n p
ayo
ff r
ea
che
s it
s m
axim
um
.
(1.2
)
(1.3
)
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#5
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• D
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
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iscr
ete
Spac
e-Ti
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ions
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23
(2.1
)
(2.2
)
(2.3
)
(2.4
)
C cn (
0, 180 )
C cn (
1, ω )
C cn (
2, ω )
p (ω
)
0.9
677
0.9
946
11/
6
0.9
89
0.9
78
11/
24
0.9
945
0.9
834
11/
12
00
017/
24
Eac
h r
aw in
th
is t
able
rep
rese
nts
a p
ath
of t
he
call
opt
ion
for
a fi
xed
scen
ario
ω 0
=
{ q
( 0, ω 0 )
, q (
1, ω 0 )
, q (
2, ω 0 )
} a
nd
ther
efor
e fo
r th
e fi
xed
scen
ario ω 0 t
he
opti
on’s
rat
es o
f ret
urn
coin
cide
wit
h t
he
corr
espo
nde
nt
rate
s of
ret
urn
of t
he
un
der-
lyin
g ex
chan
ge r
ate.
Sim
ilar
cla
ss o
f exo
tics
is a
sset
s-or
-not
h-
ing
call
an
d pu
t op
tion
s pa
yoff
at
mat
uri
ty a
re d
efin
ed a
s
Can
( T
, q
( T
))
= q
( T
) χ {
q (
T )
> K
}
Pan
( T
, q
( T
))
= q
( T
) χ {
q (
T )
< K
}
Th
e pr
icin
g fo
rmu
las
can
be
deri
ved
from
the
gen
eral
pri
cin
g
form
ula
s
Can
( t
, q
( t
))
= N
q (
t ) χ {
q (
T )
> K
}
Pan
( t
, q
( t
))
=
N q
( t
) χ {
q (
T )
< K
}
Gap
opt
ion
s ar
e co
nta
cts
for
wh
ich
Eu
rope
an c
all p
ayof
f is
be
wri
tten
in t
he
form
Cg (
T ,
q (
T )
) =
( q
( T
) –
R
) χ {
q (
T )
> K
}
wh
ere
K, R
are
kn
own
con
stan
ts a
nd
K >
R. T
he
valu
e of
th
e
con
trac
ts c
an b
e re
pres
ente
d by
th
e ca
sh-o
r-n
oth
ing
opti
on
solu
tion
wh
ere
X =
q (
T )
– R
. Th
e ga
p-pu
t pa
yoff
is
Pg (
T ,
q (
T )
) =
( R
- q
( T
)) χ {
q (
T )
< K
}
wh
ere
K <
R. T
hen
th
e ga
p-pu
t pr
icin
g fo
rmu
la c
an b
e pe
r-
form
by
the
seco
nd
form
ula
(2.
1) w
her
e X
= R
– q
( T
).
Pay
late
r op
tion
s ca
ll a
nd
put
payo
ffs
are
defi
ned
by
form
ula
s
Cpl (
T ,
q (
T ))
= [
q (
T )
- K
- C
pl (
t ,
q (
t ))
] χ {
q (
T )
> K
}
P pl (
T ,
q (
T ))
= [
K -
q (
T )
- P
pl (
t ,
q (
t ))
] χ {
q (
T )
< K
}
wh
ere
Cpl (
t ,
q (
t )
) ,
Ppl (
t ,
q (
t )
) a
re t
he
valu
es o
f th
e
opti
ons
at th
eir
date
of o
rigi
nat
ion
dat
e t a
nd
paid
on
ly o
n th
e
exer
cise
of
the
opti
ons.
Th
ese
are
up-
fron
t pa
ymen
ts p
aid
at
date
t.
We
show
th
at t
he
payl
ater
pay
off
can
be
neg
ativ
e. T
o
prod
uce
th
e va
luat
ion
of
the
prob
lem
on
e n
eeds
to
use
th
e
ben
chm
ark
form
ula
(1.
3).
Th
e so
luti
on o
f th
is e
quat
ion
wh
en N
= 1
can
be
pres
ente
d in
th
e fo
rm
Cpl (
t ,
q (
t ))
=
C
pl (
T ,
q (
T )
)
Bea
rin
g in
min
d fo
rmu
la (1
.3) t
he
abov
e eq
uat
ion
can
be
rep-
rese
nte
d in
th
e fo
rm
Cpl (
t , q
( t )
) =
[ q
( T
) -
K -
Cpl (
t , q
( t )
) ] χ {
q ( T
) >
K }
Sol
vin
g th
e eq
uat
ion
for
Cpl (
t ,
q (
T )
) w
e ar
rive
at
the
call
payl
ater
opt
ion
pri
ce
Cpl (
t , q
( t )
) =
(
q ( T
) -
K )
χ {
q (
T )
> K
} =
=
(
q (
T )
-
K )
χ {
q (
T )
> K
}
Th
is m
igh
t be
a h
igh
ris
k fo
r an
in
vest
or. N
ote
that
on
e ca
n
reac
h a
n a
rbit
rary
hig
h a
vera
ge r
etu
rn b
y ch
osen
th
e op
tion
pric
e su
ffic
ien
tly
smal
l bu
t th
e ri
sk o
f an
y pr
ice
wil
l be
not
less
th
an 1
7/24
. We
can
use
dat
a pr
ovid
ed b
y th
e la
tter
Tab
le
to p
rese
nt c
alcu
lati
on fo
r th
e p
ut o
ptio
n. C
onsi
der
Eu
rope
an
put
opti
on w
ith
th
e st
rike
K =
182
. Th
en
and
Th
en
Th
e m
ean
an
d st
anda
rd d
evia
tion
of
the
put
prem
ium
are
2.86
97,
2.05
98 c
orre
spon
din
gly.
Let
for
exa
mpl
e, i
nve
stor
pays
$1
prem
ium
for
th
e pu
t op
tion
th
en t
he
risk
to
rece
ive
at e
xpir
atio
n l
ess
retu
rn t
han
in
vest
ed i
s 7/
24.
Th
is l
oss
is
asso
ciat
ed w
ith
th
e sc
enar
io
{ q
(2)
= 1
86, 1
82, o
r 18
1 }
If t
he
put
prem
ium
is $
4 th
en t
he
risk
is
P [
q (
2 )
= {
186
, 182
, 181
, 179
}]
= 1
9/24
.
2. Ex
oti
cs o
pti
on
s.
In t
his
sec
tion
, w
e in
trod
uce
opt
ion
pri
cin
g fo
rmu
las
for
som
e po
pula
r ex
otic
cla
sses
. T
he
exot
ics
or n
on-s
tan
dard
opti
ons
are
thos
e w
hic
h p
ayof
f ca
nn
ot b
e re
duce
d to
Am
eric
an o
r E
uro
pean
opt
ion
s. T
hey
are
div
ided
on
to t
wo
prim
ary
clas
ses
refe
rred
to
as
to
pa
th-d
epen
den
t an
d
path
-in
depe
nde
nt.
Exo
tic
opti
ons
are
gen
eric
nam
e of
thes
e de
riva
tive
s. E
xoti
c op
tion
s ar
e re
ferr
ed t
o as
pat
h-
inde
pen
den
t if
th
eir
payo
ff d
oes
not
dep
end
on t
he
path
duri
ng
the
life
tim
e of
th
e op
tion
.
Cas
h-o
r-n
oth
ing o
ptio
ns
also
kn
own
as
digi
tal
or b
inar
y
opti
ons.
Th
e ca
ll a
nd
put
digi
tal
opti
ons
are
defi
ned
by
thei
r pa
yoff
at
mat
uri
ty a
s
Ccn
( T
, q
( T
))
= X
χ {
q (
T )
> K
}
Pcn
( T
, q
( T
))
= X
χ {
q (
T )
< K
}
wh
ere
X is
a p
rede
term
ined
con
stan
t an
d q
( t
) ca
n b
e in
ter-
pret
ed a
s a
spot
exc
han
ge r
ate
in d
olla
rs p
er u
nit
of
fore
ign
curr
ency
at
tim
e t,
t
T.
Not
e, t
hat
in
con
tras
t to
th
e co
n-
tin
uou
s pa
yoff
of t
he
Eu
rope
an o
r A
mer
ican
opt
ion
s th
e di
g-
ital
opt
ion
s h
ave
disc
onti
nu
ous
payo
ff.
Th
e co
nst
ant
X i
s
usu
ally
ass
um
ed e
qual
to 1
. Th
e va
luat
ion
of t
he
opti
ons
con
-
trac
ts c
an b
e re
pres
ente
d by
th
e fo
rmu
la
Ccn
( T
, q
( T
))
=
N
X χ {
q (
T )
> K
}
P cn
( T
, q
( T
))
=
N
X χ {
q (
T )
< K
}
Her
e N
is
the
con
trac
t si
ze e
xpre
ssed
in
for
eign
cu
rren
cy, K
is t
he
stri
ke p
rice
, q
( T
)
is t
he
curr
ency
exc
han
ge r
ate
at
date
T. L
et u
s th
e n
um
eric
exa
mpl
e. A
ssu
me
that
th
e u
nde
r-
lyin
g se
curi
ty d
ata
is g
iven
by
the
Tabl
e on
pag
e 7
and
N =
X
= 1
. Th
en u
sin
g th
e sa
me
alge
bra
one
arri
ves
at t
he
tabl
e
χ {
q (
T )
K
1 } =
1 - χ {
q (
T )
> K
1 }
χ {
q (
T )
( K
1 , K
2 ]
} =
χ {
q (
T )
> K
1 } - χ {
q (
T )
> K
2 }
one
can
see
that
pay
off o
f the
col
lar
can
be p
rese
nted
as
follo
win
g
I (
T )
= K
1 -
K1 χ {
q (
T )
> K
1 } +
q (
T ) χ {
q (
T )
> K
1 } -
- q
( T
) χ {
q (
T )
> K
2 } +
K
2 χ
{ q
( T
) >
K2 }
= K
1 +
+ [
q (
T )
- K
1 ] χ {
q ( T
) >
K1 }
- [
q ( T
) -
K2 ] χ {
q ( T
) >
K2
}
Th
e ri
ght
han
d si
de o
f th
is e
qual
ity
is e
qual
to
a po
rtfo
lio
hol
din
g $K
1 ca
sh,
lon
g E
uro
pean
cal
l w
ith
th
e st
rike
pri
ce
K1
, an
d sh
ort
Eu
rope
an c
all
wit
h t
he
stri
ke p
rice
K2.
T
his
deco
mpo
siti
on o
f th
e co
llar
pay
off i
s n
ot u
niq
ue.
In
deed
, on
e
can
be
easi
ly v
erif
y ot
her
pay
off’s
rep
rese
nta
tion
I (
T )
= K
1 +
K2
- q
( T
) +
[ q
( T
) -
K
1 ] χ {
q (
T )
> K
1 } -
- [
K2 -
q (
T )
] χ {
q (
T )
< K
2 }
Thus
col
lar
payo
ff is
equ
ival
ent
now
to
the
valu
e of
the
por
tfol
io
that
con
tain
s $(
K1
+ K
2 ) c
ash
, sho
rt s
tock
, lo
ng E
urop
ean
call,
and
shor
t E
urop
ean
put.
The
pric
e of
a c
olla
r co
ntra
ct a
t an
y
tim
e pr
ior
expi
rati
on c
oinc
ides
wit
h th
e va
lue
of th
e po
rtfo
lio.
We
intr
oduc
e di
rect
eva
luat
ion
of t
he c
olla
r co
ntra
ct a
pply
ing
form
ula
(2.4
). It
follo
ws
that
the
col
lar
payo
ff (2
.4)
is t
he b
aske
t
of th
e th
ree
hypo
thet
ical
fina
ncia
l ins
trum
ents
wit
h pa
yoffs
at
I 1 ( T
) =
K 1 χ
{ q
( T
)
K1
}
I 2 ( T
) =
q (
T ) χ {
q (
T )
(
K 1 ,
K 2
] }
I 3 ( T
) =
K 2 χ
{ q
( T
) >
K 2
}
wit
h th
e sa
me
mat
urit
y T.
The
n th
e co
llar
cont
ract
pri
ce a
t t is
I (
t )
= I
1 ( t
) +
I 2 (
t )
+ I
3 ( t
) ,
wh
ere
Sim
ilar
ly,
P pl (
t ,
q (
t ))
=
(
K -
q (
T )
) χ {
q (
T )
> K
}
In t
he
nex
t Ta
ble
we
encl
ose
the
valu
atio
n o
f th
e pa
ylat
er c
all
opti
on w
hen
un
derl
yin
g is
th
e va
lue
of fo
reig
n c
urr
ency
un
it
wh
ich
val
ue
give
n b
y th
e Ta
ble
on p
age
7.
C pl (
0, 180 )
C pl (
1, ω )
C pl (
2, ω )
p (ω
)
0.2
.911
2.9
919
3.0
89
1/
6
0.9
944
0.9
889
1.0
056
1/
24
0.4
978
0.4
958
0.5
022
1/
12
00
017/
24
Inde
ed, a
pply
ing
form
ula
(2.
3) w
e se
e th
at
Not
e th
at w
e om
itte
d fo
r w
riti
ng
sim
plic
ity
inde
x ‘p
l’ th
at
spec
ifie
s pa
ylat
er o
ptio
n. O
ne
mig
ht
not
e th
at t
he
risk
ch
ar-
acte
rist
ics
of t
he
payl
ater
cal
l op
tion
as
wel
l as
oth
er e
xoti
cs
call
opt
ion
wit
h t
he
sam
e st
rike
pri
ce h
ave
been
in
trod
uce
d
abov
e co
inci
de w
ith
th
e co
rres
pon
den
t ri
sk c
har
acte
rist
ics
of
the
stan
dard
Eu
rope
an o
ptio
n w
ith
th
e sa
me
stri
ke p
rice
. All
thes
e op
tion
s of
fere
d th
e sa
me
retu
rn t
hou
gh t
hei
r pr
emi-
um
s an
d pa
yoff
s ar
e di
ffer
ent.
A c
oll
ar c
on
trac
t pa
yoff
at
mat
uri
ty T
is d
efin
ed b
y a
form
ula
I (
T )
= m
in {
max
{ q
( T
) ,
K1
} ,
K 2
}}.
Not
e th
at t
his
pay
off c
an b
e re
wri
tten
in a
mor
e co
mpr
ehen
-
sive
form
I ( T
) =
K1 χ
{ q (
T )
K1 }
+ q
( T
) χ {
q ( T
) (
K1 ,
K2
] } +
+
K2 χ {
q (
T )
> K
2 }
Bel
ow w
e w
ill
intr
odu
ce s
tan
dard
arg
um
ents
th
at p
erfo
rm
the
valu
atio
n o
f th
e co
llar
con
trac
t. U
sin
g id
enti
ties
The
exo
tics
or
non-s
tandard
opti
ons
are
th
ose
w
hic
h p
ayo
ff c
an
no
t b
e r
educe
d t
o A
mer
ican o
r E
uro
pea
n o
pti
ons .
Th
ey a
re d
ivid
ed
in
to t
wo
p
rim
ary
cla
sse
s re
ferr
ed
to
as
path
-de
pe
nd
en
t
an
d p
ath
-in
de
pe
nd
en
t.
»
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fsrforum
• ja
arga
ng 1
2 • ed
itie
#5
24
• D
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ngD
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ng •
25
Th
eref
ore,
th
e re
turn
to
the
choo
ser
opti
on c
an b
e re
pre-
sen
ted
in t
he
form
Rec
all t
hat
exc
han
ge r
ate
q (
* )
is t
he
un
derl
yin
g pr
oces
s fo
r
adm
issi
ble
scen
ario
s. T
her
efor
e th
e eq
uat
ion
for
co
( t
, q
)
cou
ld b
e pr
esen
ted
in t
he
form
Sol
vin
g th
is e
quat
ion
, w
e fi
gure
ou
t th
at t
he
valu
e of
th
e
choo
ser
opti
on is
A c
liqu
et o
r ra
tch
et o
ptio
n is
a s
erie
s of
at
the
mon
ey o
ptio
ns,
wit
h p
erio
dic
sett
lem
ent,
res
etti
ng
of t
he
stri
ke p
rice
at
the
rese
t da
te s
pot
pric
e le
vel,
at w
hic
h t
he
opti
on l
ocks
in
th
e
diff
eren
ce b
etw
een
th
e ol
d an
d n
ew s
trik
e pr
ices
an
d pa
ys
that
dif
fere
nce
ou
t as
th
e pr
ofit
. Th
is p
rofi
t m
igh
t be
pai
d ou
t
at e
ach
res
et d
ate
or c
ould
be
accu
mu
late
d u
nti
l m
atu
rity
.
Th
us,
a c
liqu
et o
ptio
n c
an b
e th
ough
t, a
s a
seri
es o
f op
tion
s
that
set
tles
per
iodi
call
y at
th
e re
set
date
s is
an
exa
mpl
e of
th
e
path
-dep
ende
nt
clas
s of
opt
ion
s.
Let
us
intr
odu
ce a
n-y
ears
cli
quet
opt
ion
wit
h k
-res
ets
ann
u-
ally
. Let
t j i
be
rese
t m
omen
ts o
f tim
e j =
0, 1
, … ,
n ;
i = 0
, 1,
…k
– 1
and
T d
enot
es m
atu
rity
. Th
e pa
yoff
ove
r th
e pe
riod
[ t j
i , t
j i +
1 ]
th
at is
du
e to
pai
d at
th
e t
j i +
1 is
max
{ q
( t
j i +
1 )
- q
( t
j i )
, 0
}
Th
is f
orm
ula
cor
resp
onds
to
the
case
wh
en o
ptio
n w
rite
r
pays
ou
t pe
riod
ical
ly a
t th
e re
set
date
s. D
enot
e th
e u
nde
rly-
ing
of t
he
cliq
uet
opt
ion
q (
s )
= q
( s
; t
, x )
, s
t a
nd
C (
t ,
x ; T
, Q
) t
he
valu
e of
th
e E
uro
pean
cli
quet
cal
l opt
ion
at d
ate
t wit
h s
trik
e pr
ice
Q a
nd
expi
rati
on d
ate
T. A
pply
ing
IE
valu
atio
n w
e ar
rive
at
the
pric
ing
equ
atio
n
A ch
oo
ser
or a
s-yo
u-l
ike
op
tio
n i
s ot
her
exo
tic
opti
on
type
. A
hol
der
of t
his
opt
ion
can
ch
oose
wh
eth
er t
he
opti
on i
s a
call
or
put
afte
r sp
ecif
ied
peri
od o
f ti
me.
An
inte
rest
ing
poin
t is
th
at t
he
choo
ser
opti
on p
ayof
f do
es
not
spe
cify
it
as c
all
or p
ut
opti
ons.
Mor
e ac
cura
tely
, th
is
type
of
deri
vati
ves
cou
ld b
e n
amed
as
a fo
rwar
d-ch
oice
opti
ons
con
trac
t. C
onsi
der
a ch
oose
r op
tion
th
at m
atu
res
at m
omen
t T ch
, t
he
mat
uri
ty o
f th
e u
nde
rlyi
ng
call
an
d
put
den
ote
T c ,
Tp
res
pect
ivel
y m
in (
T c ,
Tp
) >
Tch
. T
hu
s,
the
valu
es o
f u
nde
rlyi
ng
call
an
d pu
t at
th
e da
te T
ch
are
C (
Tch
, q (
T ch )
; T c ,
Kc
) a
nd
P (
Tch
, q
(Tch
) ;
Tp,
Kp
)
corr
espo
ndi
ngl
y, q
( t
) i
s th
e u
nde
rlyi
ng
secu
rity
of
the
call
and
put
opti
ons,
an
d K
c ,
Kp
are
the
corr
espo
nde
nt
stri
ke
pric
es. T
he
payo
ff t
o th
e ch
oose
r op
tion
at
mat
uri
ty T
ch is
co (
T ch ,
q (T
ch )
) =
m
ax {
C (
Tch
, q
(Tch
) ;
T c , K
c )
, P
( T
ch ,
q (T
ch )
; T
p, K
p )
}
Not
e th
at t
he
payo
ff c
an b
e ex
pres
sed
in t
he
form
co (
T ch ,
q (T
ch )
) =
C (
Tch
, q
(Tch
) ;
T c , K
c )
×
× χ
{ C
( T
ch ,
q (
T ch )
; T c ,
Kc
)
P (
Tch
, q
(Tch
) ;
T p , K
p )
} +
+ P
( T
ch ,
q (
T ch )
; T
p , K
p ) χ
{C
( T
ch ,
q (
T ch )
; T c ,
Kc )
< P
(
T ch ,
q (T
ch )
; T
p , K
p )}
Usi
ng
expl
icit
rep
rese
nta
tion
of t
he
call
an
d pu
t pr
ices
giv
en
by (
1.2.
1) it
is e
asy
to v
erif
y eq
ual
itie
s
exch
ange
rat
es q
( s
) ,
s
t. A
n i
nve
stor
bu
ys t
he
ladd
er
opti
on w
ith
a s
trik
e pr
ice
Q =
Q 0.
Th
us
a la
dder
sta
rt w
ith
the
hei
ght
Q a
nd
goin
g u
pwar
ds in
th
e st
ep in
terv
al o
f ε >
0 u
nti
l
the
max
imu
m r
un
g of
Q N
, Q
j =
Q +
j ε
, j =
0, 1
, … ,
N. A
t
mat
uri
ty,
T b
uye
r of
th
e la
dder
cal
l op
tion
wou
ld r
ecei
ve
payo
ff
Fro
m t
his
for
mu
la,
foll
ows
that
th
e la
dder
pay
off
take
s in
to
acco
un
t th
e m
axim
um
val
ue
of t
he
un
derl
yin
g pr
ice
over
life
tim
e of
th
e op
tion
. To
con
stru
ct l
adde
r ca
ll o
ptio
n p
rice
assu
me
that ω
{ ω :
Q j
< Q
j +
1 }
for
som
e j .
Th
en l
adde
r ca
ll o
ptio
n p
ayof
f re
aliz
ed f
or t
his
sce
nar
io w
ill
be e
qual
to
C la
d (
T ,
q (
T )
) =
max
{ q
( T
) -
Q ,
Q j
- Q
}
Th
eref
ore
the
IE r
ule
bri
ngs
us
to t
he
valu
atio
n e
quat
ion
Th
us
Rem
ark.
Oth
er m
odif
icat
ion
of
the
ladd
er c
all
opti
on c
an b
e
intr
odu
ced
by a
ssu
min
g th
at c
all
opti
on p
ayof
f is
def
ined
as
foll
owin
g
In t
his
cas
e in
wh
ich
th
e pa
yoff
is s
imil
ar t
o th
e la
dder
wh
ich
adm
its
a fi
nit
e n
um
ber
of v
alu
es 0
, Q 1 -
Q, …
, Q
N –
Q w
ith
prob
abil
itie
s P
j =
P{
Q j
Q j +
1 }
,
j =
0,
1,
… ,
N -
1,
and
P N
=
P
{ >
Q N
}.
Th
e
valu
atio
n f
orm
ula
in
th
is c
ase
can
be
obta
ined
fro
m t
he
abov
e fo
rmu
la b
y re
plac
ing
payo
ff i
n t
he
brac
kets
by
its
mod
ific
atio
n.
Th
e pu
rch
aser
of
the
ladd
er p
ut
wil
l re
ceiv
e at
mat
uri
ty
payo
ff o
f
Hen
ce,
C (
t j
i , q
( t
j i )
; t
j i +
1 ,
q (
t j i )
)
=
Usi
ng
this
form
ula
, we
can
cal
cula
te c
liqu
et c
all v
alu
e re
cur-
sive
ly.
On
th
e ot
her
han
d, t
he
pres
ent
valu
e at
t o
f th
e st
o-
chas
tic
cash
flo
ws
gen
erat
ed b
y th
e se
ries
of
1) t
he
init
ial
valu
es o
f th
e fo
rwar
d st
art
opti
ons
and
2) p
ayof
f of
th
ese
opti
ons
are
equ
al t
o
corr
espo
ndi
ngl
y.
If c
is
spot
pri
ce o
f th
e op
tion
at
t th
en
buye
r an
d se
ller
ris
k ca
n b
e es
tim
ated
by
prob
abil
itie
s P
{ C
(
t , x
, ω )
< c
} ,
P {
PC
( t
, x
, ω )
< c
} w
hic
h a
re t
he
mea
s-
ure
of
scen
ario
s w
hen
th
ese
cou
nte
rpar
ties
pay
s m
ore
than
the
scen
ario
s pr
ovid
e fo
r.
A C
ou
ple
opt
ion
is a
sim
ilar
typ
e of
th
e cl
iqu
et o
ptio
ns.
As
for
cliq
uet
opt
ion
, pa
yoff
to
a h
olde
r co
uld
tak
e pl
ace
eith
er a
t
spec
ifie
d re
set
date
s or
at
mat
uri
ty.
Th
e on
ly d
isti
nct
ion
betw
een
cou
ple
and
cliq
uet
is th
at t
he
cou
ple
opti
ons
at r
eset
date
s sw
itch
its
valu
e to
th
e sm
alle
r of
th
e cu
rren
t sp
ot le
vel
and
the
init
ial
stri
ke p
rice
. T
he
cash
flo
w g
ener
ated
by
the
cou
ple
call
opt
ion
is
wh
ere
t =
t 0
< t
1 < …
< t
N =
T a
re r
eset
dat
es, a
nd
min
[ q
(
t j )
, K
] i
s re
set
stri
ke p
rice
. T
he
pric
e of
th
e ca
ll a
nd
put
cou
ple
opti
ons
are
C cp
( t
j , q
( t
j ) ;
t j +
1 ,
min
[ q
( t
j ) ,
K ]
)
=
P cp
( t
j , q
( t
j ) ;
t j +
1 ,
min
[ q
( t
j ) ,
K ]
)
=
A la
dder
opt
ion
pay
off
is a
lso
sim
ilar
to
a cl
iqu
et p
ayof
f w
ith
exce
ptio
n t
hat
th
e ga
ins
are
lock
ed i
n w
hen
th
e as
set
pric
e
brea
ks t
hro
ugh
cer
tain
pre
dete
rmin
e ru
ng.
Th
e st
rike
pri
ce
is t
hen
in
term
itte
ntl
y re
set.
Con
side
r a
ladd
er o
ptio
n o
n
A c
liquet
or
ratc
het
opti
on
is
a s
eri
es
of
at
the
mo
ne
y o
pti
on
s,
wit
h p
eri
od
ic s
ett
lem
en
t, r
ese
ttin
g o
f th
e s
trik
e p
rice
at
th
e r
ese
t d
ate
sp
ot
pri
ce l
eve
l, a
t w
hic
h t
he
op
tio
n l
ock
s in
th
e d
iffe
rence
bet
wee
n t
he
old
and n
ew s
trik
e p
rice
s a
nd
p
ays
that
dif
fere
nce
ou
t a
s th
e p
rofi
t.
»
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fsrforum
• ja
arga
ng 1
2 • ed
itie
#5
26
• D
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ngD
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ng •
27
exte
ndi
ble
expi
rati
on d
ate.
Th
e ad
diti
onal
pre
miu
m o
f $d
is
paid
by
the
hol
der
in c
ase
wh
en e
xten
sion
fea
ture
is
chos
en
to e
xerc
ise
at T
e . T
her
e ar
e n
ew fa
ctor
s in
volv
ed t
o th
e pr
ob-
lem
. Va
luat
ion
equ
atio
n o
f th
e ca
ll e
xten
dibl
e ca
n b
e re
pre-
sen
ted
in t
he
form
Th
e in
dica
tor
on t
he
righ
t h
and
side
of t
he
equ
alit
y co
nta
ins
un
ion
of
two
even
ts,
wh
ich
sig
nif
y th
at a
t le
ast
one
of t
he
poss
ibil
itie
s at
Te
shou
ld b
e st
rict
ly p
osit
ive.
Oth
erw
ise,
th
e
valu
e of
C eh
( T
e , q
( T
e ))
and
C eh
( t
, q
( t
)) fo
r th
is p
arti
cu-
lar
scen
ario
is 0
.
Taki
ng
this
in
to a
ccou
nt
and
solv
ing
call
opt
ion
pri
ce e
qua-
tion
we
arri
ve a
t th
e pr
emiu
m fo
rmu
la
Th
e fo
rmu
la fo
r th
e h
olde
r ex
ten
dibl
e pu
t op
tion
can
be
per-
form
in t
he
sim
ilar
way
A r
ecip
roca
l pro
blem
giv
en o
ptio
n p
rice
to e
stim
ate
the
valu
e
of t
he
prem
ium
d is
impo
rtan
t to
o.
A w
rite
r ex
ten
dibl
e op
tion
all
ows
a se
ller
of
the
opti
on,
opti
on w
rite
r to
ext
end
the
opti
on e
ith
er c
all o
r pu
t wit
h z
ero
cost
at
the
mat
uri
ty T
e if
th
e op
tion
is
out-
of-m
oney
. R
ecal
l
that
opt
ion
cal
l ( p
ut
) is
ou
t-of
-mon
ey a
t da
te t
if it
s va
lue
at
this
mom
ent
is l
ess
( la
rger
) or
equ
al t
o th
e st
rike
pri
ce.
Th
us,
if o
ptio
n h
ave
a n
egat
ive
valu
e it
s ca
n b
e ex
erci
se la
ter
at a
dat
e T.
Th
eref
ore,
wri
ter
exte
ndi
ble
payo
ff o
f th
e ca
ll a
nd
put
are
equ
al t
o
C ew
( T
e , q
( T
e ))
= [
q (
Te )
- Q
] χ {
q (
T e )
Q }
+ [
q (
T )
- Q
] χ {
q (
T e ) <
Q }
Her
e, Q
– M
<
Q –
M +
1
< …
<
Q –
1 <
Q
is
a ru
ng
sequ
ence
.
Th
e pr
icin
g eq
uat
ion
for
the
ladd
er p
ut
is
if ω
{ ω :
< Q
– M }
. Th
e va
lue
of th
e pu
t lad
der
opti
on
is t
hen
Ext
endib
le o
ptio
ns
hav
e be
com
e po
pula
r ov
er r
ecen
t ti
me
for
vola
tile
un
derl
yin
g. T
her
e ar
e tw
o ty
pes
of t
he
exte
ndi
ble
opti
ons:
hol
der
and
wri
ter
exte
ndi
ble.
A h
olde
r ex
ten
dibl
e
opti
on i
s an
opt
ion
th
at c
an b
e ex
ten
ded
by t
he
hol
der
at
opti
on m
atu
rity
Te.
Th
is p
ossi
bili
ty i
s re
quir
ed a
n a
ddit
ion
al
prem
ium
. Th
e h
olde
r of
th
e ex
ten
dibl
e op
tion
on
cal
l or
pu
t
has
a c
hoi
ce t
o ge
t an
ord
inar
y ca
ll o
ptio
n p
ayof
f or
by p
ayin
g
a pr
edet
erm
ine
prem
ium
$d
to t
he
wri
ter
at t
ime
T e to
get
call
opt
ion
wit
h e
xten
ded
mat
uri
ty.
Ass
um
e th
at t
he
hol
der’
s ch
oice
is
base
d on
th
e m
axim
um
valu
e of
th
e op
tion
pay
offs
at
T e . T
hat
is
C eh
( T e ,
q (
T e ))
=
max
{{ q
( T e )
- Q
, 0
} , C
( T e ,
q (
T e ) ;
T ,
K )
- d
} =
=
max
{ q
( T
e ) -
Q ,
C (
Te ,
q (
Te )
; T
, K
) -
d ,
0 }
,
P eh
( T
e , q
( T
e ))
=
max
{ Q
- q
( T
e ) ,
P (
Te ,
q (
Te )
; T
, K
) -
d ,
0 }
Th
ough
for
exam
ple
opti
on b
uye
r at
Te m
ay e
xpec
t ov
er [
Te ,
T ]
to
get
hig
her
ove
rall
ret
urn
by
exer
cise
ext
endi
bili
ty t
han
to g
et c
all o
ptio
n p
ayof
f at
T e an
d in
vest
ing
it a
t ri
sk fr
ee r
ate.
We
do n
ot a
nal
yse
such
pos
sibi
lity
.
In a
bove
form
ula
s C
( T
e , x
; K
, T
) ,
P (
Te ,
x ;
K ,
T )
den
ote
the
pric
e of
th
e E
uro
pean
cal
l or
pu
t op
tion
s at
dat
e Te
wit
h
a st
rike
pri
ce K
th
at m
igh
t be
equ
al t
o Q
, an
d T
, T
> T
e is
the
Th
ese
deri
vati
ve c
ontr
acts
can
be
inte
rpre
ted
as d
eriv
ativ
es
hav
ing
vari
able
str
ikes
in
con
tras
t to
a c
onst
ant
stri
ke u
sed
in t
he
prev
iou
s ex
ampl
es.
Th
e pr
icin
g fo
rmu
las
to t
he
con
-
trac
ts c
an b
e ob
tain
ed u
sin
g st
anda
rd I
E p
rici
ng
rule
. In
deed
,
only
tw
o po
ssib
ilit
ies
are
avai
labl
e u
nde
rlyi
ng
exch
ange
an
d
deri
vati
ves.
If a
sce
nar
io ω is
su
ch th
at C
e ( T
; Δ
,T 0 )
= 0
then
ther
e is
no
sen
se t
o in
vest
in
ext
rem
e ca
ll. I
f C
e ( T
; Δ
,T 0
)
> 0
th
en t
her
e is
th
e u
niq
ue
pric
e to
avo
id a
rbit
rage
. T
his
pric
e is
def
ined
as
a so
luti
on o
f th
e eq
uat
ion
Hen
ce,
Sim
ilar
ly,
Oth
er p
ath
-dep
ende
nt
opti
on c
lass
is
Lookbac
k o
ptio
ns.
Th
e
extr
eme
exot
ic o
ptio
ns
intr
odu
ced
abov
e so
met
imes
are
con
-
side
red
as s
ubc
lass
of l
ookb
ack
opti
ons
and
call
ed it
ext
rem
a
look
back
opt
ion
s. T
wo
prim
ary
form
s of
the
look
back
opt
ion
s
exis
t ba
sed
on s
trik
e pr
ice
defi
nit
ion
. Fir
st fo
rm is
def
ined
as
look
back
opt
ion
s w
ith
fix
ed s
trik
e pr
ice.
Th
e pa
yoff
s of
th
e
call
an
d pu
t op
tion
s ar
e
resp
ecti
vely
. A
pply
ing
the
sam
e ar
gum
ents
as
for
extr
eme
opti
ons
pric
ing
we
arri
ve a
t th
e fo
rmu
las
Th
e lo
okba
ck o
ptio
ns
wit
h fl
oati
ng
stri
ke p
rice
can
be
sett
led
in c
ash
or
asse
ts i
n c
ontr
ast
wit
h t
he
fixe
d st
rike
opt
ion
s in
wh
ich
cas
h s
ettl
emen
t is
on
ly a
dmit
ted.
Th
e pa
yoff
of
the
look
back
cal
l an
d pu
t op
tion
s w
ith
flo
atin
g st
rike
pri
ce a
re
defi
ned
as
foll
owin
g
P ew
( T
e , q
( T
e ))
= [
Q -
q (
Te )
] χ {
Q
q (
Te )
} +
[ Q
- q
(
T )
] χ {
Q
q (
Te )
}
corr
espo
ndi
ngl
y. T
hes
e pa
yoff
typ
es g
ive
addi
tion
al b
enef
it t
o
buye
rs o
f th
e ca
ll o
r pu
t op
tion
s. T
he
firs
t te
rm o
n t
he
righ
t
han
d si
de o
f th
e ca
ll a
nd
put
opti
on p
ayof
fs,
corr
espo
nd
to
“in
-th
e-m
oney
” sc
enar
ios
at T
e wh
ile
the
seco
nd
term
impl
ies
“ou
t-of
-th
e-m
oney
” sc
enar
ios.
B
y u
sin
g ex
ten
ded
feat
ure
does
not
cos
t or
im
ply
mor
e lo
sses
for
cou
nte
rpar
ties
. T
he
valu
atio
n e
quat
ion
of
wri
ter
exte
ndi
ble
call
an
d pu
t op
tion
s
can
be
pres
ente
d in
th
e fo
rm
= Th
ese
pric
ing
equ
atio
ns
brin
g u
s to
th
e va
luat
ion
form
ula
s
Not
e th
at E
uro
pean
typ
e of
th
e u
nde
rlyi
ng
opti
ons
can
als
o
be r
epla
ced
by A
mer
ican
opt
ion
s.
Th
e E
xtre
me
or
Rev
erse
Ext
rem
e ex
oti
c opti
on
s w
as i
ntr
o-
duce
d in
199
6. C
all
extr
eme
opti
ons
payo
ff a
t m
atu
rity
T i
s
dete
rmin
ed b
y th
e di
ffer
ence
bet
wee
n m
axim
um
val
ues
on
com
plim
ent
subi
nte
rval
s co
nst
itu
ted
the
life
tim
e of
an
un
derl
yin
g as
set.
Let
t <
T 0
< T
an
d de
not
e Δ
= [
t ,
T ]
. Th
en
payo
ffs
to t
he
call
opt
ion
at
mat
uri
ty f
or t
he
extr
eme
and
inve
rse
extr
eme
opti
ons
are
C e (
T ; Δ
,T 0
) =
m
ax {
q
( v
) -
q
( u
) ,
0 }
C i e (
T ; Δ
, T
0 )
=
max
{
q (
u )
-
q (
v )
, 0
}
Th
e pa
yoff
s to
pu
t ex
trem
e an
d pu
t in
vers
e ex
trem
e op
tion
s
at m
atu
rity
get
th
e sp
read
val
ue
betw
een
min
imu
m o
ver
adja
cen
t pe
riod
s, i.
e.
P e (
T ; Δ
, T
0 ) =
m
ax {
q
( v
) -
q
( u
) ,
0 }
P i e (
T ; Δ
, T
0 ) =
m
ax {
q
( u
)
- q
( v
) ,
0 }
»
![Page 6: Discrete space time option pricing forum fsr](https://reader034.fdocuments.in/reader034/viewer/2022052315/55490e61b4c9056b458bc384/html5/thumbnails/6.jpg)
fsrforum
• ja
arga
ng 1
2 • ed
itie
#5
28
• D
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ngD
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ng •
29
In t
he
firs
t li
ne,
th
e ar
ith
met
ic a
vera
ge is
use
d, a
s th
e u
nde
r-
lyin
g w
hil
e in
th
e se
con
d li
ne
form
ula
s th
e ar
ith
met
ic m
ean
is u
sed
as a
str
ike
pric
e. S
omet
imes
, in
cu
rren
cy m
arke
t on
e
appl
ies
the
inve
rse
mea
n a
s u
nde
rlyi
ng.
In th
is c
ase,
cal
l - p
ut
opti
ons
payo
ffs
are
defi
ned
by
form
ula
s
max
{ a
– 1 (
T )
- K
, 0
}
,
m
ax {
K
- a
– 1 (
T )
, 0
}
whe
re a
– 1 (
T )
is
expr
esse
d in
the
sam
e cu
rren
cy a
s th
e a
( T
)
itse
lf. T
he
pric
ing
form
ula
s ar
e
Th
e C
om
po
un
d o
pti
on
s is
a c
lass
of
deri
vati
ves
in w
hic
h
un
derl
yin
g se
curi
ties
are
opt
ion
s or
oth
er t
ype
of c
onti
n-
gen
t cl
aim
s. C
onsi
der
exam
ples
wh
en u
nde
rlyi
ng
inst
ru-
men
ts a
re o
ptio
ns.
Th
is c
lass
is
call
ed c
ompo
un
d or
spl
it
free
opt
ion
s. P
ossi
ble
spec
ific
atio
ns
are
call
opt
ion
s on
a
call
or
put
opti
ons,
an
d pu
t on
cal
l or
pu
t. L
et C
( t
, q
( t
)) =
C (
t ,
q (
t );
T, K
) d
enot
e a
valu
e of
an
Eu
rope
an c
all
opti
on a
t da
te t
wit
h t
he
mat
uri
ty T
an
d t
he
stri
ke p
rice
K w
ritt
en o
n r
ate
q ( *
). C
onsi
der
an o
ptio
n o
n c
all o
ptio
n.
Den
ote
C c
( t
, q
( t
))
=
C c
( t
, C
( t
, q
( t
) ;
T c
, K
c )
;
T, K
) t
he
com
pou
nd
call
opt
ion
pri
ce a
t t
wri
tten
on
th
e
Eu
rope
an c
all
opti
on a
t da
te t
wit
h m
atu
rity
T c ,
T c
T
wit
h s
trik
e pr
ice
K c
. T
hen
th
e pa
yoff
of
the
com
pou
nd
call
opt
ion
is
C c (
T c ,
C (
T c ,
q (
T c )
; T c ,
Kc )
; T,
K )
= m
ax {
C (
T c ,
q (
T c )
; T
, K
) –
K c ,
0 }
Th
e va
luat
ion
form
ula
of t
he
call
on
cal
l opt
ion
is
App
lyin
g tw
ice
the
IE r
ule
we
pres
ent
the
valu
atio
n o
f th
is
equ
atio
n
Not
e th
at t
he
pric
e of
th
e co
mpo
un
d op
tion
at
t de
pen
ds o
n
the
un
derl
yin
g ra
te q
( t
) a
t tw
o fu
ture
dat
es T
c <
T. L
et u
s
con
side
r ot
her
typ
es o
f com
pou
nd
opti
on. T
he
pric
ing
equ
a-
tion
of t
he
put
wri
tten
on
Eu
rope
an c
all o
ptio
n is
An
att
ract
ive
pecu
liar
ity
of t
he
look
back
opt
ion
s w
ith
flo
at-
ing
stri
ke p
rice
is t
hat
th
ey a
re n
ever
ou
t-of
-th
e-m
oney
. Th
e
form
ula
e re
pres
enti
ng
curr
ent
opti
ons
pric
e ar
e
Asi
an o
ptio
ns
is a
pop
ula
r cl
ass
of e
xoti
cs.
Un
derl
yin
g of
an
Asi
an o
ptio
n i
s th
e av
erag
e pr
ice
of a
sset
. In
man
y ca
ses,
un
derl
yin
g of
Asi
an o
ptio
n h
as lo
wer
vol
atil
ity
than
th
e as
set
itse
lf.
Th
ere
are
thre
e m
ain
su
bcla
sses
of
the
Asi
an o
ptio
ns
wh
ich
un
derl
yin
g ar
e fo
rmed
wit
h t
he
hel
p of
ari
thm
etic
,
geom
etri
c, o
r w
eigh
ted
aver
ages
of
asse
t. N
ext
spec
ific
atio
n
of t
he
opti
ons
is t
hat
th
e av
erag
e ca
n b
e u
sed
for
eith
er s
ecu
-
rity
or
as t
he
stri
ke p
rice
. T
hu
s, t
he
payo
ff f
or A
sian
cal
l
opti
ons
can
be
repr
esen
ted
as
Asi
an p
ut
payo
ff u
sin
g th
e ar
ith
met
ic m
ean
as
un
derl
yin
g or
stri
ke p
rice
can
be
pres
ente
d in
th
e fo
rm
corr
espo
ndi
ngl
y. T
he
Am
eric
an s
tyle
of
the
Asi
an o
ptio
ns
is
also
ava
ilab
le fo
r tr
ade.
Th
e pr
icin
g fo
rmu
las
are
wh
ere
q (
t j
) =
q (
t j
; t
, x )
, j
= 0
, 1, …
, n
. For
th
e A
sian
opti
ons
wit
h th
at in
volv
e th
e ge
omet
ric
or w
eigh
ted
aver
ages
to o
btai
n v
alu
atio
n f
orm
ula
e on
e n
eeds
rep
lace
ari
thm
etic
aver
age
in th
e ab
ove
form
ula
e by
thei
r ge
omet
ric
or w
eigh
ted
aver
age
cou
nte
rpar
ts.
Th
e A
sian
opt
ion
s of
th
e E
uro
pean
or
Am
eric
an t
ypes
are
path
dep
ende
nt
clas
s of
exo
tic
opti
ons.
Un
derl
yin
g of
an
Asi
an o
ptio
n i
s an
ave
rage
pri
ce o
f an
ass
et.
Th
e se
curi
ty
aver
age
pric
e ca
n b
e u
sed
as a
str
ike
pric
e to
o. B
y co
mpa
riso
n
wit
h o
ther
opt
ion
s it
s va
lues
are
les
s vo
lati
le d
uri
ng
its
life
and
this
is q
uit
e at
trac
tive
for
inve
stor
s. T
hre
e ty
pes
of m
ean
are
gen
eral
ly a
ppli
ed f
or A
sian
opt
ion
pay
off.
Th
ese
are
pay-
offs
for
m b
y ei
ther
ari
thm
etic
, w
eigh
ted
arit
hm
etic
, or
geo
-
met
ric
aver
ages
Not
e th
at t
hes
e ty
pes
of m
ean
can
be
use
d as
un
derl
yin
g
secu
riti
es a
s w
ell
as a
str
ike
pric
e. F
or e
xam
ple,
Asi
an c
all
and
put
opti
ons
payo
ffs
wit
h a
rith
met
ic m
ean
cou
ld b
e
defi
ned
as
foll
owin
g
max
{ a
( T
) -
K ,
0 }
, m
ax {
q (
T )
- a
( T
) ,
0 }
max
{ K
- a
( T
) ,
0 }
, m
ax {
a (
T )
- S
( T
) ,
0 }
con
tain
s N
× ₤
( t
) a
nd
oth
er q
( t
) N
× $
( t
). A
t a
mom
ent
T ,
T >
t t
he
valu
es o
f th
e po
rtfo
lios
wil
l be
chan
ged.
Th
e in
i-
tial
con
stan
t q
( t
) re
mai
ns
un
chan
ged
wh
ile
N $
( t
) w
ill b
e
tran
sfor
med
in
N $
( T
).
Th
e va
lue
of t
he
seco
nd
port
foli
o
wil
l be
equ
al t
o ₤
N (
T )
. A
s fa
r as
th
ese
port
foli
os a
re n
ot
equ
al a
t T
it
look
s re
ason
able
to
hed
ge t
he
exch
ange
rat
e
wit
h t
he
hel
p of
th
e de
riva
tive
s co
ntr
act
wh
ich
giv
es a
n
inve
stor
th
e op
tion
to
choo
se a
t m
atu
rity
T t
he
max
imu
m
betw
een
$ N
( T
) a
nd
₤ N
( T
) /
q (
t )
=
$ N
( T
). F
or t
his
con
trac
t on
e ca
n
spec
ify
payo
ff a
t T
max
{ 1
, ₤
1 (
T )
/ ₤
1 (
t )
} =
max
{ 1
, }
For
wri
tin
g si
mpl
icit
y pu
t N
= 1
. T
he
pric
e of
th
e ra
inbo
w
con
trac
t ca
n b
e de
rive
d as
foll
owin
g. F
or a
sce
nar
io ω
1
, wh
ere
1
= { ω :
1 }
th
e va
lue
of t
he
payo
ff is
$1
at T
. It
impl
ies
that
at
the
init
ial m
omen
t t
the
con
trac
t va
lue
is B
( t
, T
). O
n t
he
oth
er h
and,
if a
sce
nar
io ω b
elon
gs t
o th
e
com
plim
enta
ry s
et o
f sce
nar
ios
> 1 =
{ ω :
>
1 }
th
e va
lue
of t
he
con
trac
t at
t c
an b
e
deri
ved
from
th
e co
ndi
tion
off
ered
equ
al r
etu
rn o
n U
SA
T-bo
nd
and
the
choo
ser
opti
on, i
.e.
ch (
t )
den
otes
th
e m
ax-c
hoo
ser
opti
on v
alu
e at
t. T
he
solu
-
tion
of t
he
equ
atio
n is
Ch
( t
, ω )
=
. Th
eref
ore,
we
can
pre
sen
t th
e pr
e-
miu
m v
alu
e of
th
e op
tion
con
trac
t
A s
pot
mar
ket
pric
e ch
0 ( t
) im
plie
s ri
sk. B
uye
r ri
sk is
con
-
nec
ted
to s
cen
ario
s, w
hic
h i
mpl
y lo
wer
pri
ce t
han
pai
d by
the
buye
r, i
.e. { ω :
Ch
( t
, ω )
< c
h 0
( t
) },
wh
ile
sell
er r
isk
is t
he
com
plem
enta
ry s
et o
f sc
enar
ios.
Th
eore
tica
lly,
th
e
valu
e ch
0 (
t )
is c
onst
ruct
ed w
ith
“ca
sh-a
nd-
carr
y” s
trat
-
egy.
Th
is s
trat
egy
defi
nes
a s
pot
pric
e as
th
e pr
esen
t va
lue
of
the
face
val
ue
at m
atu
rity
an
d th
eref
ore
ch 0 (
t )
= B
( t
, T
). U
nfo
rtu
nat
ely,
we
shou
ld r
emar
k th
at i
n s
toch
asti
c se
t-
tin
g w
e co
uld
not
ign
ore
mar
ket
flu
ctu
atio
ns
oth
erw
ise
we
wil
l ig
nor
e th
e m
arke
t ri
sk. I
n o
ther
han
d if
mar
ket
has
an
expl
icit
tre
nd
wit
h r
espe
ct t
o ri
sk-f
ree
then
th
is e
stim
ate
of
the
pric
e co
uld
be
bias
ed t
oo.
Th
e ra
inbo
w o
ptio
n c
lass
is
som
ewh
at s
imil
ar t
o th
e ca
ll
opti
on. T
o h
igh
ligh
t th
is s
imil
arit
y on
e ca
n d
efin
e a
vari
atio
n
of t
he
con
trac
t w
hic
h p
ayof
f is
Th
e so
luti
on o
f th
is e
quat
ion
can
be
wri
tten
as
Th
e pr
icin
g of
the
com
pou
nd
call
or
put w
ritt
en o
n E
uro
pean
put
opti
on c
an b
e ob
tain
ed in
a s
imil
ar w
ay a
n c
an b
e re
pre-
sen
ted
in t
he
form
Let
us
con
side
r a
deri
vati
ve c
ontr
act
that
adm
its
a ch
oice
betw
een
tw
o or
mor
e fo
reig
n b
onds
at
a fu
ture
mom
ent
of
tim
e. T
his
type
of t
he
con
trac
t ca
lled
opt
ion
s on
max
imu
m o
r
min
imu
m o
f se
vera
l ri
sky
asse
ts .
Th
is c
lass
of
opti
ons
is
rela
ted
to t
he
rain
bow
or
choo
ser
opti
ons.
Rai
nbo
w o
ptio
ns
get
thei
r n
ame
from
th
e fa
ct t
hat
mor
e th
an o
ne
exch
ange
rate
. Ass
um
e th
at a
t m
atu
rity
T a
hol
der
of t
he
con
trac
t h
as
the
righ
t to
ch
oose
a b
ond
dom
esti
c or
fore
ign
. Ass
um
e th
at
at in
itia
l mom
ent
t, t
he
size
of t
wo
con
trac
ts is
th
e sa
me.
Let
q (
t )
den
ote
indi
rect
qu
otat
ion
of a
n e
xch
ange
rat
e be
twee
n
two
curr
enci
es a
t t
1 u
nit
of
fore
ign
cu
rren
cy (
at
date
t )
=
q
( t
) u
nit
s of
th
e
dom
esti
c cu
rren
cy (
at
date
t )
Th
e in
dire
ct q
uot
atio
n v
alu
e q
– 1 (
t ) s
how
s a
nu
mbe
r fo
reig
n
curr
ency
un
its
per
dom
esti
c cu
rren
cy. T
he
valu
e of
a g
over
n-
men
t a
0-de
fual
t an
d 0-
cou
pon
bon
d at
t w
ith
un
it fa
ce v
alu
e
is d
efin
ed b
y th
e re
lati
onsh
ip
1 u
nit
cu
rren
cy (
at
date
T )
= B
( t
, T
) 1
un
it c
urr
ency
(at
date
t )
Rec
all,
that
bon
d va
lue
can
als
o be
in
terp
rete
d as
a r
elat
ion
-
ship
bet
wee
n f
utu
re a
nd
curr
ent
valu
es o
f th
e cu
rren
cy. F
or
exam
ple,
let
us
dom
esti
c cu
rren
cy is
US
D a
nd
fore
ign
is G
BP.
Let
at
init
ial
mom
ent
t w
e h
ave
two
equ
al p
ortf
olio
s. O
ne
»
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fsrforum
• ja
arga
ng 1
2 • ed
itie
#5
30
• D
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ngD
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ng •
31
That
is
Q ×
[ S
( T
; t
, S (
t )
) ]
= Q
S (
T ;
t , Q
S (
t )
)
It d
oes
not
dep
end
on a
form
in w
hic
h t
his
low
is g
iven
. Th
e
low
in p
arti
cula
r ca
n b
e re
pres
ente
d by
a s
toch
asti
c or
det
er-
min
isti
c eq
uat
ion
. Th
e so
luti
on t
he
pric
ing
equ
atio
n c
an b
e
perf
orm
in t
he
form
Th
e ra
inbo
w w
ith
min
imu
m o
f n r
isky
ass
ets
payo
ff is
sim
ilar
to t
he
best
of
n a
sset
s st
udi
ed a
bove
. In
ord
er t
o pr
esen
t
form
al v
alu
atio
n o
f a
con
trac
t on
e sh
ould
rep
lace
th
e ‘m
ax’
on ‘m
in’ o
pera
tion
s in
th
e ab
ove
form
ula
s.
Oth
er t
ype
of t
he
exot
ic c
ontr
acts
is
spre
ad o
ptio
ns.
Th
e
payo
ff f
or E
uro
pean
cal
ls a
nd
puts
at
mat
uri
ty T
wit
h t
he
stri
ke p
rice
K c
an b
e w
ritt
en a
s
C sd
( T
, S 1 (
T )
, S 2 (
T )
) =
max
{ S
1 ( T
) -
S 2 (
T )
- K
, 0
}
P sd (
T , S
1 ( T
) , S
2 ( T
) )
= m
ax {
K -
S 1 (
T )
+ S
2 ( T
) ,
0 }
resp
ecti
vely
. Th
e ge
ner
aliz
atio
n o
n n
-ass
ets
in l
ong
posi
tion
and
m-a
sset
s in
sh
ort
is s
trai
ghtf
orw
ard.
For
exa
mpl
e
As
far
as t
he
un
derl
yin
g of
th
e sp
read
opt
ion
is a
lin
ear
com
-
bin
atio
n o
f th
e as
sets
th
e pr
ice
of t
he
call
on
spr
ead
can
be
expr
esse
d by
equ
atio
n
Fro
m w
hic
h it
foll
ows
that
Th
e pr
ice
of t
he
put
spre
ad o
ptio
n c
an b
e de
rive
d si
mil
arly
Now
, we
wil
l lo
ok a
t a
popu
lar
type
of
exot
ics
opti
ons
call
ed
bar
rier
opti
on
. T
his
is
a fa
mil
y of
th
e pa
th-d
epen
den
t. T
he
wh
ich
lead
s to
th
e co
ntr
act
pric
e
Gen
eral
izat
ion
of
the
rain
bow
opt
ion
s on
th
ree
or m
ore
un
derl
yin
g cu
rren
cies
is s
trai
ghtf
orw
ard.
Den
ote
q i
( t
) th
e
dire
ct q
uot
atio
n o
f i-
th c
urr
ency
, i =
1, 2
, … w
ith
res
pect
to
dom
esti
c U
S d
olla
r. I
f th
e pa
yoff
to
the
opti
on a
t m
atu
rity
is
chos
en a
s
Th
en t
he
pric
e in
US
D o
f th
e (
n +
1)-
max
-rai
nbo
w o
ptio
n a
t
date
t is
Let
us
con
side
r th
e ca
se w
hen
n a
sset
s an
d ca
sh a
re in
volv
ed
in p
ayof
f. W
e w
ill
see
that
th
e pr
icin
g fo
rmu
las
for
the
con
-
trac
t th
at d
eals
wit
h t
he
max
imu
m o
f se
vera
l st
ocks
dif
fer
from
th
e on
e pr
esen
ted
abov
e. I
t fo
llow
s fr
om t
he
fact
th
at
fore
ign
exc
han
ge m
arke
t in
stru
men
ts c
an b
e co
mpa
red
if
they
h
ave
the
sam
e cu
rren
cy
form
at
and
ther
efor
e th
e
exch
ange
rat
es p
lay
a si
gnif
ican
t ro
le in
val
uat
ion
s.
Let
SÁ
( t
) ,
…,
S n
( t
) d
enot
e pr
ice
of n
– a
sset
s. A
ssu
me
that
th
e pa
yoff
is g
iven
at
mat
uri
ty T
wh
ere
K 3 0
is a
con
stan
t ca
sh. T
he
reas
onab
le c
hoi
ce a
t T
for
the
con
trac
t is
on
e th
at s
ugg
ests
th
e m
axim
um
ret
urn
.
Th
eref
ore,
the
pric
ing
equ
atio
n c
an b
e pr
esen
ted
as fo
llow
ing
wh
ere
rain
bow
( t
) d
enot
es t
he
con
trac
t pr
ice
at d
ate
t. T
he
righ
t h
and
side
of
the
equ
atio
n c
an b
e si
mpl
ifie
d. I
nde
ed,
putt
ing
S i
( T
) =
S i
( T
; t
, S i
( t
)) a
nd
taki
ng
into
acc
oun
t
the
lin
ear
depe
nde
nce
of
S i
( T
) o
n in
itia
l val
ue
S i
( t
) on
e
can
see
th
at t
he
righ
t h
and
side
of t
he
equ
atio
n a
bove
can
be
rew
ritt
en in
th
e fo
rm
max
{ S
1 ( T
; t
, 1 )
, …
, S
n (
T ;
t , 1
) ,
B –
1 ( t
, T
) }
The
linea
r de
pend
ence
of
an a
sset
on
the
init
ial
valu
e fo
llow
s
from
the
fact
tha
t an
y po
rtio
n Q
of a
sset
s pr
ice
Q S
( *
) o
ver
a
peri
od o
f tim
e is
gov
erne
d by
the
sam
e lo
w a
s th
e si
ngle
ass
et.
valu
e of
th
e ba
rrie
r op
tion
is
spec
ifie
d by
an
eve
nt
wh
eth
er
the
un
derl
yin
g ra
te c
ross
es a
giv
en b
arri
er. T
her
e ar
e tw
o di
f-
fere
nt
way
s of
in
ters
ecti
ons
rega
rded
as
‘in’ o
r ‘o
ut’
an
d tw
o
type
s of
th
e le
vel
‘up’
or
‘dow
n’
wit
h r
espe
ct t
o th
e in
itia
l
valu
e of
th
e sp
ot r
ate.
A d
oubl
e ba
rrie
r op
tion
is
a ba
rrie
r
opti
on w
ith
tw
o ‘u
p’ a
nd
‘dow
n’ b
arri
ers.
The
dow
n-a
nd-o
ut
( kn
ock
-ou
t) o
ptio
n sp
ecifi
es a
low
bar
rier
.
If t
he s
pot
exch
ange
rat
e br
each
es t
his
barr
ier
duri
ng t
he li
fe-
tim
e of
the
opti
on th
en th
e op
tion
pay
off i
s eq
ual t
o 0.
In s
ome
case
s a
reba
te c
an a
lso
be p
rovi
ded
if th
e ba
rrie
r is
cro
ssed
.
Den
ote
d a
barr
ier
leve
l, K
a s
trik
e pr
ice,
and
d <
K. T
he p
ayof
f
to t
he d
own-
and-
out
call
opti
on a
t m
atur
ity
T is
def
ined
as
Let τ d
den
ote
the
firs
t m
omen
t w
hen
pro
cess
q (
l )
, l
t
atta
ins
the
leve
l d. T
hen
wit
h p
roba
bili
ty 1
Th
eref
ore
Th
e do
wn
-an
d-ou
t op
tion
pri
ce c
an b
e co
nst
ruct
ed a
pply
ing
the
stan
dard
IE
equ
atio
n
wh
ich
lead
s to
th
e so
luti
on
Th
e pa
yoff
to
the
dow
n-a
nd-
out
put
opti
on i
s gi
ven
by
the
form
ula
wh
ich
def
ine
its
pric
e at
th
e da
te t
Th
e dow
n-a
nd-i
n (
kn
ock
-in
) c
all
and
put
opti
ons
exer
cise
pric
e at
mat
uri
ty T
are
def
ined
as
foll
owin
g
C di
( T
, q (
T )
) =
max
{ q
( T
) –
K ,
0 } χ ( τ
d T
)
P di
( T
, q (
T )
) =
max
{ K
– q
( T
) ,
0 } χ ( τ
d T
)
Th
ese
payo
uts
impl
y th
e op
tion
s pr
ice
Let
us
pres
ent
for
exam
ple
sim
ple
calc
ula
tion
s, w
hic
h i
llu
s-
trat
e th
e op
tion
s pr
icin
g. A
ssu
me
that
exc
han
ge r
ate
data
is
defi
ned
by
the
Tabl
e on
pag
e 7
and
let
K =
180
, d =
178
.
C do (
0, ω )
C do (
1, ω )
C do (
2, ω )
ω
p(ω
)
5.8
064
5.9
677
6{1
80, 185, 186}
1/
6
00
0{1
80, 185, 179}
1/
2
00
0{1
80, 178, 182}
1/
24
00
0{1
80, 178, 181}
1/
12
00
0{1
80, 178, 176}
5/
24
P d
o (
0, ω )
P d
o (
1, ω )
P d
o (
2, ω )
ω
p(ω
)
00
0{1
80, 185, 186}
1/
6
1.0
056
1.0
335
1{1
80, 185, 179}
1/
2
00
0{1
80, 178, 182}
1/
24
00
0{1
80, 178, 181}
1/
12
00
0{1
80, 178, 176}
5/
24
C di (
0, ω )
C di (
1, ω )
C di (
2, ω )
ω
p(ω
)
00
0{1
80, 185, 186}
1/
6
00
0{1
80, 185, 179}
1/
2
1.9
78
1.9
56
2{1
80
, 1
78
, 1
82
}1
/2
4
0.9
94
50
.98
34
1{1
80
, 1
78
, 1
81
}1
/1
2
00
0{1
80, 178, 176}
5/
24
P d
i (
0, ω )
P d
i (
1, ω )
P d
i (
2, ω )
ω
p(ω
)
00
0{1
80, 185, 186}
1/
6
00
0{1
80, 185, 179}
1/
2
00
0{1
80, 178, 182}
1/
24
00
0{1
80, 178, 181}
1/
12
4.0
909
4.0
455
4{1
80, 178, 176}
5/
24
Let u
s pr
esen
t a r
isk
anal
ysis
of t
he in
vest
men
t in
dow
n-an
d-in
call
opti
on. T
he a
vera
ge o
ptio
n pr
ice
at d
ate
0 is
1.9
78 ×
(1/2
4)+
0.99
45 ×
(1/
12)
= 0
.165
29 m
ulti
plie
d by
a s
ize
of t
he c
ontr
act.
If c
ontr
act s
ize
is 1
000
unit
s of
fore
ign
curr
ency
( £
) , th
en th
e
valu
e of
the
one
cont
ract
is $
165.
29. A
ssum
e th
at in
vest
or c
om-
pare
s tw
o sc
enar
ios
in w
hich
the
pric
es C
di (
0, ω )
are
a $1
00 o
r
$200
. Tha
t is
0.1
or
0.2
per
£-po
und.
If
the
opti
on p
rice
is
0.1
then
the
chan
ce to
exe
rcis
e th
e op
tion
at e
xpir
atio
n im
plie
s on
e
of t
wo
scen
ario
s is
rea
lized
: {18
0, 1
78, 1
82}
or {
180,
178
, 181
}.
The
prob
abili
ty o
f th
e un
ion
of t
hese
tw
o fa
vour
able
eve
nts
is
1/24
+ 1
/12
= 1
/8. T
he e
xpec
ted
rate
of r
etur
n is
equ
al t
o
[ ( 1
/ 24
) ×
1.9
78 +
( 1
/ 12
) ×
0.9
945
– 0
.1 ]
/ 0.
1 =
0.6
529
If th
e op
tion
pri
ce is
0.2
then
the
expe
cted
rat
e of
ret
urn
is a
bout
[ ( 1
/ 24
) ×
1.9
78 +
( 1
/ 12
) ×
0.9
945
– 0
.2 ]
/ 0.
2 =
- 0
.173
54
Th
e m
ean
-val
ue
anal
ysis
doe
s n
ot c
over
ris
k ex
posu
re. G
iven
spot
opt
ion
pri
ce a
t in
itia
l m
omen
t t
risk
is
a ra
ndo
m v
aria
-
ble
th
at r
epre
sen
ts lo
ss o
f th
e in
vest
men
t. T
he
valu
e of
th
e
risk
is m
easu
red
by t
he
cum
ula
tive
loss
-dis
trib
uti
on. I
n c
ase
wh
en t
he
dow
n-a
nd-
in c
all o
ptio
n p
rice
is 0
.1 p
er £
-con
trac
t
»
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fsrforum
• ja
arga
ng 1
2 • ed
itie
#5
32
• D
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ngD
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ng •
33
(3.1
)
Th
e va
luat
ion
of t
he
up-a
nd-i
n b
arri
er o
ptio
n is
sim
ilar
to
the
repr
esen
ted
abov
e.
Let
us
con
side
r a
dou
ble
barr
ier
pric
ing
sch
eme
for
the
case
wh
en K
= 1
80, u
= 1
85,
d =
178
. T
he
payo
ff t
o th
e do
ubl
e-ou
t ba
rrie
r ca
ll a
nd
put
opti
ons
at m
atu
rity
are
def
ined
as
C db
o ( T
, q (
T ))
= m
ax {
q ( T
) –
K ,
0 } χ {
d <
q
( l )
,
q ( l
) <
u }
P db
o ( T
, q (
T ))
= m
ax {
K –
q (
T ) ,
0 } χ
{ d <
q
( l )
,
q
( l )
< u
}
Th
en p
ayof
f to
th
e do
ubl
e-in
bar
rier
cal
l an
d pu
t op
tion
s at
mat
uri
ty is
C db
i ( T
, q (
T ))
= m
ax {
q ( T
) –
K ,
0 } χ {
d
q (
l ) ,
q ( l
) u
}
P db
i ( T
, q (
T ))
= m
ax {
K –
q (
T ) ,
0 } χ
{ d
q
( l )
,
q ( l
) u
}
Pri
cin
g fo
rmu
las
for
thes
e ba
rrie
r op
tion
s ar
e
C db
o ( T
, q
( T ))
=
max
{ q
( T )
– K
, 0
} χ {
d <
q
( l )
,
q ( l
) <
u }
P db
o ( T
, q
( T ))
=
max
{ K
– q
( T
) , 0
} χ { d
<
q ( l
) ,
q ( l
) <
u }
C db
i ( T
, q
( T ))
=
max
{ q
( T )
– K
, 0
} χ {
d
q ( l
) ,
q ( l
)
u }
P db
i ( T
, q
( T ))
=
max
{ K
– q
( T
) , 0
} χ
{ d
q
( l )
,
q ( l
)
u }
Fin
al R
emar
k. H
ere
we
pres
ent
a sh
ort
com
men
t to
th
e bi
no-
mia
l op
tion
pr
icin
g.
Th
is
disc
rete
sp
ace-
tim
e ap
proa
ch
pres
ents
opt
ion
pri
ce u
sin
g a
stan
dard
alg
ebra
ic m
eth
ods.
Tech
nic
ally
, bi
nom
ial
sch
eme
in o
ne
peri
od c
ann
ot p
rese
nt
stra
igh
t fo
rwar
d so
luti
on o
f th
e pr
icin
g pr
oble
m w
ith
arb
i-
trar
y fi
nit
e se
t of s
tate
s. O
n th
e ot
her
han
d th
is fo
rmal
def
ini-
tion
has
obv
iou
s lo
gica
l dra
wba
ck.
Let
sto
ck p
rice
at
date
t =
1 b
e S
( 1
) =
$2
and
by t
he
end
of t
he
sin
gle
peri
od t
he
secu
rity
pri
ce i
s ei
ther
S u
( 2
) =
$4
or S
d ( 2
) =
$1
and
stri
ke p
rice
K =
$2
and
for
sim
plic
ity
let
the
risk
-fre
e in
tere
st r
ate
r =
0. L
et u
s re
call
con
stru
ctio
n o
f
the
bin
omia
l sc
hem
e. I
t de
term
ines
opt
ion
pri
ce w
ith
tw
o
step
s. C
onsi
der
a ca
ll o
ptio
n e
xam
ple.
On
th
e fi
rst
step
th
e
hed
ge r
atio
h o
f th
e h
ypot
het
ical
por
tfol
io in
th
e fo
rm П (
t )
= S
( t
) –
h C
( t
), t
= 1
,2 is
est
abli
shed
. Th
e co
ndi
tion
th
at
use
d fo
r th
e so
luti
on o
f th
e pr
oble
m is
П (
2 )
= S
u (
2 )
– h
C u
( 2
) =
S d (
2 )
– h
C d (
2 )
wh
ere
C u
an
d C
d a
re t
he
call
opt
ion
pay
offs
cor
resp
ondi
ng
two
outc
omes
S u
an
d S
d r
espe
ctiv
ely.
Th
us
C u
( 2
)
=
max
{ 4
– 2
, 0
} =
2
C d
( 2
) =
m
ax {
1 –
2 ,
0 }
= 0
Fro
m (
3.1)
it
foll
ows
that
h =
1.5
. T
he
seco
nd
step
lea
ds t
o
the
opti
on p
rice
. It
foll
ows
from
(3.
1) t
hat
th
e po
rtfo
lio
valu
e
at d
ate
2 is
det
erm
inis
tic
and
equ
al t
o
On
e ca
n t
ran
sfor
m t
his
pri
cin
g re
pres
enta
tion
of
the
risk
into
equ
ival
ent
pres
enta
tion
form
s by
th
e ra
te o
f ret
urn
.
Now
let
us lo
ok a
t th
e ne
xt t
ype
of b
arri
er o
ptio
n in
whi
ch t
he
‘up’
bar
rier
is
spec
ified
. If
the
spo
t ex
chan
ge r
ate
goes
abo
ve
the
‘up’
bar
rier
the
up-a
nd-o
ut
opti
on c
ease
s to
exi
st. A
reb
ate
that
sho
uld
be s
peci
fied
at i
niti
atio
n of
the
con
trac
t m
ay a
lso
be p
rovi
ded
as th
e ba
rrie
r is
cro
ssed
. The
pay
off t
o th
e up
-and
-
out c
all o
r pu
t opt
ions
at m
atur
ity
T is
def
ined
by
the
form
ulas
C u
o ( T
, q
( T
))
= m
ax {
q (
T )
– K
, 0
} χ ( θ
u >
T )
P u
o ( T
, q
( T
))
= m
ax {
K –
q (
T )
, 0
} χ ( θ
u >
T )
resp
ecti
vely
. Th
e ra
ndo
m t
ime θ
u is
def
ined
as
foll
owin
g
θ u
=
m
in {
l : q
( l
) u
, l
[ t
, T
] }
Th
e pr
ice
of t
he
up-
and-
out
call
or
put
opti
ons
at t
are
C u
o ( t
, q
( t
)) =
m
ax {
q (
T )
– K
, 0
} χ ( θ
u >
T )
P u
o ( t
, q
( t
)) =
m
ax {
K –
q (
T )
, 0
} χ ( θ
u >
T )
Th
e pa
yoff
to
the
up-
and-
in c
all,
put
opti
ons
at m
atu
rity
T
can
be
repr
esen
ted
in t
he
form
C u
i ( T
, q (
T )
) =
m
ax {
q (
T )
– K
, 0
} χ ( θ
u
T )
P u
i ( T
, q (
T )
) =
m
ax {
K –
q (
T )
, 0
} χ ( θ
u
T )
Th
eref
ore
C u
i ( t
, q
( t
)) =
m
ax {
q (
T )
– K
, 0
} χ ( θ
u
T )
P u
i ( t
, q
( t
)) =
m
ax {
K –
q (
T )
, 0
} χ ( θ
u
T )
Ass
um
ing
that
th
e u
nde
rlyi
ng
exch
ange
rat
e gi
ven
in T
able
7
and
K =
180
, u =
185
C uo (
0, ω )
C uo (
1, ω )
C uo (
2, ω )
ω
p(ω
)
00
0{1
80, 185, 186}
1/
6
00
0{1
80, 185, 179}
1/
2
1.9
78
1.9
56
2{1
80, 178, 182}
1/
24
0.9
945
0.9
834
1{1
80, 178, 181}
1/
12
00
0{1
80, 178, 176}
5/
24
Hen
ce,
If m
arke
t pri
ce o
f the
opt
ion
is e
qual
to E
C uo
( 0, ω )
= 0
.165
3
then
the
risk
is d
escr
ibed
by
the
set o
f sce
nari
os D
= {1
80, 1
85,
186}
{180
, 185
, 179
}{1
80, 1
78, 1
76}
whi
ch c
orre
spon
d to
0
payo
ff. T
he v
alue
of r
isk
is 2
1/24
, whi
ch is
the
prob
abili
ty o
f the
D. I
n ge
nera
l the
ave
rage
loss
and
ave
rage
pro
fit a
re
E C
uo (
0, ω ) χ
{ C
uo (
0, ω )
< o
ptio
n m
arke
t pr
ice
( 0
) }
E C
uo (
0, ω ) χ
{ C
uo (
0, ω )
> o
ptio
n m
arke
t pr
ice
( 0
) }
Thus
, the
com
plet
e op
tion
pri
ce d
ata
shou
ld b
e su
pplie
d by
the
risk
cha
ract
eris
tics
, whi
ch c
an b
e es
tabl
ishe
d as
sum
ing
a pa
rtic
-
ular
dis
trib
utio
n of
the
und
erly
ing.
The
hig
her
orde
r m
omen
ts
of th
e op
tion
pri
cing
giv
e us
mor
e de
tails
that
are
mor
e ac
cura
te
repr
esen
t ris
k ex
posu
re. A
nalo
gous
ly, o
ne c
an s
ee th
at
P u
o (
0, ω )
P u
o (
1, ω )
P u
o (
2, ω )
ω
p(ω
)
00
0{1
80, 185, 186}
1/
6
1.0
056
1.0
335
1{1
80, 185, 179}
1/
2
00
0{1
80, 178, 182}
1/
24
00
0{1
80, 178, 181}
1/
12
4.0
909
4.0
455
4{1
80, 178, 176}
5/
24
П (
2 )
= 4
– 1
.5 *
2 =
1 –
1.5
* 0
= 1
By
con
stru
ctio
n i
t do
es n
ot d
epen
d on
sce
nar
ios
“up”
or
“dow
n”
at m
atu
rity
. Th
eref
ore,
th
e
chan
ge in
val
ue
of t
he
port
foli
o sh
ould
foll
ow t
he
risk
-fre
e ra
te. T
hat
is a
pply
ing
sim
ple
inte
r-
est
rate
we
hav
e
П (
2 )
= (
1 +
r )
П (
1 )
As
far
as t
he
risk
-fre
e in
tere
st r
ate
r w
as a
ssu
med
to
be e
qual
to
0 th
en
П (
2 )
= П
( 1
) =
1 =
S (
1 )
– 1
.5 C
( 1
)
Hen
ce C
( 1
) =
2/3
. T
his
is
the
theo
reti
cal
opti
on p
rice
su
gges
ted
by t
he
bin
omia
l sc
hem
e.
Let
us
test
th
e th
eore
tica
l so
luti
on a
gain
st p
arti
cula
r sc
enar
ios.
Ass
um
e th
at t
wo
secu
riti
es
diff
er b
y th
e pr
obab
ilit
ies
of t
he
stat
es a
t m
atu
rity
. Let
th
e pr
obab
ilit
y of
th
e st
ate
‘4’ i
s fo
r th
e
firs
t se
curi
ty e
qual
to
0.99
an
d fo
r th
e se
con
d se
curi
ty t
he
stat
e ‘4
’ pr
obab
ilit
y is
equ
al t
o
0.01
. Th
e se
curi
ties
exp
ecte
d ra
te o
f re
turn
s ar
e
[ 4
* 0.
99 +
1*
0.01
– 2
] :
2 =
98.
5%
[ 4
* 0.
01 +
1*
0.99
– 2
] :
2 =
– 4
8.5%
corr
espo
ndi
ngl
y. A
ccor
din
g to
bin
omia
l sc
hem
e th
e u
niq
ue
opti
on p
rice
for
eit
her
of
thes
e
secu
riti
es is
th
e sa
me
2/3.
In
oth
er w
ords
bin
omia
l sch
eme
does
not
sen
siti
ve w
ith
res
pect
to
un
derl
yin
g se
curi
ty r
ates
ret
urn
. O
ne
can
not
e th
at s
elli
ng
opti
on o
n b
ad s
tock
an
d bu
yin
g
opti
on o
n g
ood
stoc
k an
inve
stor
sta
rts
wit
h z
ero
fin
anci
ng.
Th
en a
t th
e en
d of
th
e pe
riod
th
e
inve
stor
has
1 c
han
ce t
o lo
ss p
rem
ium
wh
ile
in 9
9% t
he
inve
stor
rec
eive
s a
prof
it. T
he
inve
st-
men
t of
th
is t
ype
one
can
cal
l st
atis
tica
l ar
bitr
age.
A c
uri
ous
fact
is
that
th
e sa
me
pric
e is
esta
blis
hed
on
tw
o op
tion
-in
vest
men
ts w
ith
dif
fere
nt
expe
cted
rat
es o
f ret
urn
. In
deed
, bu
yin
g
firs
t op
tion
for
2/3
an
d gi
ven
exp
ecte
d ra
te o
f re
turn
is
[ (
4 *
0.99
+ 1
* 0.
01 )
– 2
/3 ]
: 2/
3 =
4.9
25
On
th
e ot
her
han
d, t
he
seco
nd
opti
on s
ugg
ests
exp
ecte
d ra
te o
f re
turn
[ (
4 *
0.01
+ 1
* 0.
99 )
– 2
/3 ]
: 2/
3 =
0.5
37
Th
is o
bser
vati
on c
ontr
adic
ts t
he
com
mon
sen
se a
nd
theo
reti
cal
un
ders
tan
din
g of
th
e pr
ice
in F
inan
ce.
Th
e co
mm
on a
rgu
men
t u
sual
ly u
sed
to j
ust
ify
the
bin
omia
l op
tion
pri
ce i
s it
s
non
-arb
itra
ge p
rici
ng.
Th
is a
rgu
men
t is
a n
eces
sary
con
diti
on o
f th
e co
rrec
t pr
icin
g an
d it
can
als
o be
tru
e if
th
e op
tion
pri
ce i
s de
term
ined
in
corr
ectl
y. T
hat
is
ther
e is
no
arbi
trag
e
betw
een
bin
omia
l pr
icin
g of
th
e op
tion
s an
d ri
sk f
ree
fin
anci
ng.
Th
e n
o-ar
bitr
age
fact
doe
s
not
su
ffic
ien
t to
acc
ept
sugg
este
d co
nst
ruct
ion
as
a th
eore
tica
l def
init
ion
of t
he
pric
e. H
avin
g
theo
reti
call
y co
rrec
t th
e op
tion
pri
ce d
efin
itio
n o
ne
can
app
ly i
t fo
r th
e so
luti
on o
f th
e re
al
wor
ld p
rici
ng
prob
lem
s.