Lecture 5
American Options
An American option is a contract between two parties made at a certain time
t
such that the
buyer of the contract has the right, but not the obligation, to exercise the option at any time
τ
with
t
≤
τ
≤
T
. If the option is exercised at
τ
, then the seller pays the buyer an amount
Y
(
τ
)
≥
0. For
instance,
Y
(
τ
) = (
S
(
τ
)

c
)
+
for an American call option and
Y
(
τ
) = (
c

S
(
τ
))
+
for an American
put option based on a single stock.
One can identify an American option by its payoff process
Y
A
=
{
Y
(
t
)
, t
= 0
,
1
, . . . , T
}
. American options enjoy the additional flexibility — possibility of
exercising earlier than
T
— compared to their European option counterparts. What is the value
V
A
(
t
) of an American option?
5.1
A special case: “American
=
European”
Since the holder of an American option can always choose not to exercise the option until time
T
,
V
A
(
t
)
≥
V
(
t
) where
V
(
t
) is the time
t
value of the European option with the payoff
Y
=
Y
(
T
).
Nevertheless, there are situations where the two value processes coincide.
Proposition 5.1
Consider an American option
Y
A
and the corresponding European option with
time
T
value
Y
=
Y
(
T
)
. If
V
(
t
)
≥
Y
(
t
)
for all
t
, then
V
(
t
) =
V
A
(
t
)
for all
t
, and it is optimal to
wait until time
T
to exercise.
Proof
For the holder of an American option, exercising at
t
only ends up with payoff
Y
(
t
), while
selling the corresponding European option (or shorting the portfolio which replicates the European
option) would guarantee you a time
t
payoff
V
(
t
). Hence the option should not be exercised at
t
.
Since
t
is arbitrary, it is optimal to wait until
T
to decide whether to exercise.
Consider the American call option with
Y
(
t
) = (
S
(
t
)

c
)
+
at each
t
where
c
= 2
.
05 in the
example given in Lecture 2. Proposition 5.1 applies to this case. See Figure 5.1.
Note:
“American calls = European calls”
Yes, the quote is really true, i.e.
American calls have the same values as their European
counterparts in the simple setup given in this section, thus there should be no earlier exercises.
This result will be proved in Section 5.2 following Theorem 5.1. Briefly, it is based on the fact that
{
Y
*
(
τ
)
}
is a submartingale.
A stochastic process
X
=
{
X
(
t
)
, t
= 0
,
1
, . . . , T
}
is called a
Q

supermartingale
under a proba
bility measure
Q
on Ω and with respect to
FF
, if the conditional expectation
E
Q
(
X
(
t
)
 F
t

1
)
≤
X
(
t

1)
∀
t
= 1
, . . . , T
;
1
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V
(0) = 0
.
28
Y
(0) = 0
.
00
'
'
'
'
'
'
'*
H
H
H
H
H
H
Hj
V
(1) = 0
.
31
Y
(1) = 0
.
09
'
'
'
'
'
'
'*
H
H
H
H
H
H
Hj
V
(1) = 0
.
04
Y
(1) = 0
.
00
'
'
'
'
'
'
'*
H
H
H
H
H
H
Hj
V
(2) = 0
.
36
Y
(2) = 0
.
24
'
'
'
'
'
'
'*
H
H
H
H
H
H
Hj
V
(2) = 0
.
05
Y
(2) = 0
.
00
'
'
'
'
'
'
'*
H
H
H
H
H
H
Hj
V
(2) = 0
.
00
Y
(2) = 0
.
00
'
'
'
'
'
'
'*
H
H
H
H
H
H
Hj
V
(3) = 0
.
40
Y
(3) = 0
.
40
V
(3) = 0
.
06
Y
(3) = 0
.
06
V
(3) = 0
.
00
Y
(3) = 0
.
00
V
(3) = 0
.
00
Y
(3) = 0
.
00
Figure 5.1: Exercise at
t
or
T
?
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 Spring '08
 Staff
 Option style, American Option

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