184 Mathematics for Finance
induction:
DA(N)=f(S(N)),DA(N−1) = max{
f(S(N−1)),1
1+r[p∗f(S(N−1)(1 +u))
+(1−p∗)f(S(N−1)(1 +d))]}
=:fN− 1 (S(N−1)),DA(N−2) = max{
f(S(N−2)),^1
1+r[p∗fN− 1 (S(N−2)(1 +u))+(1−p∗)fN− 1 (S(N−2)(1 +d))]}
=:fN− 2 (S(N−2)),
..
.
DA(1) = max{
f(S(2)),1
1+r[p∗f 2 (S(1)(1 +u))+(1−p∗)f 2 (S(1)(1 +d))]}
=:f 1 (S(1)),DA(0) = max{
f(S(0)),1
1+r[p∗f 1 (S(0)(1 +u))+(1−p∗)f 1 (S(0)(1 +d))]}
.
Exercise 8.10
Compute the value of an American put expiring at time 3 with strike
priceX= 62 dollars on a stock with initial priceS(0) = 60 dollars in a
binomial model withu=0.1,d=− 0 .05 andr=0.03.Exercise 8.11
Compare the prices of an American call and a European call with strike
priceX = 120 dollars expiring at time 2 on a stock with initial price
S(0) = 120 dollars in a binomial model withu=0.2,d=− 0 .1and
r=0.1.Example 8.2
The last exercise can be modified to show that the equality of European and
American call prices may not hold if a dividend is paid. Suppose that a dividend
of 14 dollars is paid at time 2. Otherwise, we shall use the same data as in