Solutions 293
8.9The delta of a European call becomes equal to 1 at the first stepnsuch that
the option will be in the money independently of whether the stock goes up
or down at the next step, that is,S(0)(1 +u)n(1 +d)>X(in this case
S(0)(1 +u)n+1>Xas well). This givesn>lnX−lnln(1 +S(0)−u)ln(1 +d).8.10The stock values are
n 0123
79. 86
72. 60 <
66. 00 < 68. 97
S(n) 60. 00 < 62. 70 <
57. 00 < 59. 57
54. 15 <
51. 44
The American put prices are
n 01 23
0. 00
0. 00 <
0. 50 < 0. 00
PA(n) 2. 52 < 1. 10 <
5. 00 < 2. 43
7. 85 <
10. 56
The option will be exercised early in the d node at time 1 or in the dd node
at time 2 (bold figures).
8.11The values of the European and American calls are the same,
n 012
52. 80
34. 91 <
CE(n)=CA(n) 22. 92 < 9. 60
5. 82 <
0. 00
At time 2 both options have, of course, the same payoff. At time 1 the American
call will not be exercised in the up state, as this would bring only 24 dollars,
which less than the value of holding the option until expiry. In the down state
the American call will be out of the money and will not be exercised either.
Similarly, it will not be exercised at time 0. As a result, the American call is
equivalent to its European counterpart.
8.12The ex-dividend stock prices are
n 01 2
12. 32
11. 20 <
/ 10. 64
S(n) 12. 00
ex-div \ 10. 34
9. 40 <
8. 93