Chapter 3
American Options in Discrete
Time
In this short chapter we’ll brieﬂy discuss the pricing and hedging of American
options and the relationship with the “optimal stopping problem.” We will
continue to work in a finite market model and we will assume that this is
complete so that every ECC can be hedged.
By the second fundamental
theorem there is a unique martingale measure which is denoted by
P
∗
.
3.1
The Price Process for an American Op
tion
In Chapter 2, we have considered the pricing and hedging of a general ECC
X
in discrete time. So
X
is the payoff (or value) of an ECC at the terminal
date
T
and it is an
F
T
measurable random variable. Whereas an ECC can
only be exercised at time
T
, a general ACC (
American contingent claim
)
may be exercised at any time
n
= 1
,
2
, . . . , T
. To model this we construct an
adapted process (
X
(
n
)
, n
∈ T
) whereby each
X
(
n
) is an ECC with terminal
date
n
, e.g.
for an American put option with exercise price
k
,
X
(
n
) =
max
{
k
−
S
(
n
)
,
0
}
. We call (
X
(
n
)
, n
∈ T
) the
payoff process
.
We want to price the option at any time
n
and so we want to construct
a process (
U
(
n
)
, n
∈
N
) which we will call the
price process
. The trick with
American options is to argue
backwards
in time starting at time
T
. Clearly
we must have
U
(
T
) =
X
(
T
)
.
What is the price of the process at time
T
−
1? At this time, the holder has
two options  either exercise the option to earn
X
(
T
−
1) or wait until time
19
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T
to earn
X
(
T
). In the second case, the option is equivalent to an ECC held
over the time period
T
−
1 to
T
and so the writer of the option needs to
invest
β
(
T
−
1)
−
1
E
∗
(
β
(
T
)
X
(
T
)
F
T
−
1
) in a replicating portfolio in order to
generate
X
(
T
) at time
T
. Hence we can argue that the price of the option
at time
T
−
1 should be
U
(
T
−
1) = max
{
X
(
T
−
1)
, β
(
T
−
1)
−
1
E
∗
(
β
(
T
)
X
(
T
)
F
T
−
1
)
}
.
Iterating the argument, we deduce that the price at time
n
−
1 is
U
(
n
−
1) = max
{
X
(
n
−
1)
, β
(
n
−
1)
−
1
E
∗
(
β
(
n
)
U
(
n
)
F
n
−
1
)
}
,
and if we discount both payoff and price processes by defining
U
(
n
) =
β
(
n
)
U
(
n
) and
X
(
n
) =
β
(
n
)
X
(
n
) we find that for
n
= 1
,
2
, . . . , T
U
(
n
−
1) = max
{
X
(
n
−
1)
,
E
∗
(
U
(
n
)
F
n
−
1
)
}
.
(3.1.1)
This ensures that at each time
n
−
1, the writer has enough funds to
either cover the cost of the option being exercised at time
n
−
1 or generate
enough funding to cover all eventualities at time
n
.
Mathematically, the structure obtained in (3.1.1) has been previously
studied by probabilists who are interested in modelling
optimal stopping prob
lems
. We will look at this in the next section. Before we do that notice that
there is a nice pattern in (3.1.1). In particular, if
X
(
n
−
1)
≤
E
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 Spring '08
 mikson
 Snell Envelope

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