182 Mathematics for Finance
To begin with, we shall analyse an American option expiring after 2 time
steps. Unless the option has already been exercised, at expiry it will be worth
DA(2) =f(S(2)),where we have three values depending on the values ofS(2). At time 1 the op-
tion holder will have the choice to exercise immediately, with payofff(S(1)), or
to wait until time 2, when the value of the American option will becomef(S(2)).
The value of waiting can be computed by treatingf(S(2)) as a one-step Euro-
pean contingent claim to be priced at time 1, which gives the value
1
1+r[p∗f(S(1)(1 +u)) + (1−p∗)f(S(1)(1 +d))]at time 1. In effect, the option holder has the choice between the latter value or
the immediate payofff(S(1)). The American option at time 1 will, therefore,
be worth the higher of the two,
DA(1) = max{
f(S(1)),^1
1+r[p∗f(S(1)(1 +u)) + (1−p∗)f(S(1)(1 +d))]}
=f 1 (S(1))(a random variable with two values), where
f 1 (x)=max{
f(x),1
1+r[p∗f(x(1 +u)) + (1−p∗)f(x(1 +d))]}
.
A similar argument gives the American option value at time 0,
DA(0) = max{
f(S(0)),1
1+r[p∗f 1 (S(0)(1 +u)) + (1−p∗)f 1 (S(0)(1 +d))]}
.
Example 8.1
To illustrate the above procedure we consider an American put option with
strike priceX= 80 dollars expiring at time 2 on a stock with initial price
S(0) = 80 dollars in a binomial model withu=0. 1 ,d=− 0 .05 andr=0.05.
(We consider a put, as we know that there is no difference between American
and European call options, see Theorem 7.4.) The stock values are
n 012
96. 80
88. 00 <
S(n) 80. 00 < 83. 60
76. 00 <
72. 20