18 • Discrete Space-Time Options Pricing fsrforum • jaargang 12 • editie #5 Discrete Space-Time Options Pricing • 19 Discrete Space-Time Options Pricing Ilya Gikhman This paper presents a formal ap- proach to the derivatives pricing. In this paper we will study derivatives pricing in a discrete space-time approximation. The primary princi- ple of the pricing theory we intro- duce in the paper is the notion of equality of investment which based on the investors goal : ‘investing in a greater return’. We say that two investments are equal at a moment of time if their instantaneous rates of return at this moment at are equal. If equality of two investments holds any moment of time over [ t , T ] then these investments are equal on [ t , T ]. This definition represents investment equality, IE law or principle which will be applied throughout of the paper for definition of a derivative price. Next we use cash flow notion that implies a series of transactions as a specifi- cation of somewhat broad notion of investment. It is not difficult to see that this concept of the investment equality is a perfect and more accurate than the present value, PV concept. Indeed, if two investments are equal in IE sense then they are equal for any possible scenario or simply to say always. On the other hand, it is clear that if two investments are equal in IE sense then they are equal in the PV. The inverse statement is incor- rect. More accurately, if two investments are equal in the PV sense then it is easy to present exam- ple of a scenario that demonstrates arbitrage opportunity. In this example, the present value of two investments are equal while one investment has a higher rate of return over a time subinter- val and lower over another subinterval than other investment. As far as PV suggests equal price for two investments one can sell a lower rate instrument over the correspondent subinterval and buy the instrument with higher rate of return instrument. Then at the end of this period investor would sell short higher priced instrument which promises lower rate of return over the next period and buy for lower price other instrument which promises higher rate of return. At the end of the period the investor has pure profit for the scenario though two investments have equal PV. This type of the example illustrates the fact that PV reduction of cash flows insufficient to be used as a definition of the equality of two cash flows. Nevertheless, PV reduction might be helpful for construction of the market estimates of the spot or future prices. Bearing in mind that price def- inition depends on a scenario we should be aware that any spot price calculated with the help of PV or other rule implies risk. This risk for buyer is measured by the probability of the events for which scenario’s price is bellow than spot price. 1. Plain Vanilla options valuation. Let us introduce the definition of the plain vanilla option contracts which is a class option cov- ered European and American types. An option is a right to buy or sell an asset at a known price, within a given period of time. The known price, K is called exercise or strike price. The last date, T of the lifetime of the option is called maturity. The right to buy is known as the call option, while the right to sell is the put option. The price of the option also referred to as premium. European options can only be exercised at maturity, whereas American type of the options can be exercised at any moment up to maturity. Let S ( t ) denote an asset spot price at date t, t 0. For- mally an European option contract is defined by its payoff at expiration. The call and put values at expiration T are defined by formulas C ( T , S ( T ) ) = max { S ( T ) – K , 0 } (1.1) P ( T , S ( T ) ) = max { K – S ( T ) , 0 } Thus, a buyer of the call option would agree to exercise the right to buy the underlying asset in case if the value of the call option at maturity T is positive, i.e. if C ( T , S ( T ) ) > 0. It is clear that there is no sense in realization of the right when C ( T , S ( T ) ) 0. On the other hand a buyer of the put option would exercise the right to sell the underlying asset in case if S ( T ) < K, i.e. a put holder is interested to sell asset with price S ( T ) for K when S ( T ) < K. That is a holder of the put option can exercise the right to sell the option when P ( T , S ( T ) ) > 0. Otherwise, if P ( T , S ( T ) ) 0 the right will not be exercised. The option pricing problem is to determine the call ( put ) option price at any moment of time t before the expiration date T. To illustrate pricing methodology we begin with a simple example. Next we use the terms asset, stock, or secu- rity as synonyms. Example 1. Let a stock price at t = 0 be S ( 0 ) = 2 and at T = 1 stock take the values S ( 1 ) = { 5 , 1 }. Introduce the probability space of scenarios of the problem. Denote ω = { u , d }, where u denotes the scenario { S ( 0 ) = 2, S ( 1 ) = 5 }, and d = { S ( 0 ) = 2, S ( 1 ) = 1 }. Putting strike price K = 2 we enable to define the call option price for each scenario. Denote C ( t , x ; T , K , ) the value of the call option for fixed scenario at the moment t , given S ( t ) = x. Here t , x are variables of the function C ( ), while T, K, are interpreted as parameters. The value of param- eters T and K are assumed to be fixed and for the writing simplicity we will omit them next. Let us specify the value of the option along the scenario u . Applying IE principle we arrive at the equation with respect to unknown C ( t = 0 , S ( 0 ) = 2 ; u ) The solution of the equation is C ( 0 , 2 ; u ) = 1.2. Then as far as the option payoff for the scenario u is max { 1 - 2 , 0 } = 0 we put by definition C ( 0 , 2 ; d ) = 0. There is no sense to pay for the option a sum if option value in the future moment is 0. Therefore, by definition we put C ( 0 , 2 ; d ) = 0. The security distribution is the probabilities of the two scenarios: P u = P ( u ), P d = P ( d ). This distribution is assigned then to the option premium. Thus Let us consider two stocks that have probabilities P 1 ( u ) = P 2 ( d ) = 0.99 and P 1 ( d ) = P 2 ( u ) = 0.01 correspondingly. The call option price is the random variable taking values : C i ( 0 , 2 ; u ) = 1, C i ( 0 , 2 ; d ) = 0 , i = 1, 2. Then *) The average rate of return on 1 st stock is equal to 1.48 and – 0.43 on 2 nd stock. **) The average rate of return on call option written on 1 st stock while the average rate of return on call option written on the 2 nd stock is equal to 1.5 %. Assume that market price at t = 0 is equal to the mean of the option at this moment. The expectations of the options price on the 1 st and the 2 nd stocks at 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 Other possible estimate of the spot option prices can be based on the PV concept. Assuming that the risk free inter- est is equal to 0 we see that » As far as PV suggests equal price for two investments one can sell a lower rate instrument over the correspondent subinterval and buy the instrument with higher rate of return instrument. We say that two investments are equal at a moment of time if their instantaneous rates of return at this moment are equal. The difference between possible spot prices is the value of risk taken by investors.
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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.
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
)
fsrforum
• ja
arga
ng 1
2 • ed
itie
#5
22
• D
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ngD
iscr
ete
Spac
e-Ti
me
Opt
ions
Prici
ng •
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.
»
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.
»
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 }
»
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
»
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
»
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.
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