ACTSC/STAT 446/846 – Assignment 2 – Solutions
1. Binomial Model
[10pts]
Using a binomial tree with
n
= 8 time steps,
S
(0) = 100, expiration of 1 year (and therefore
h
= 1
/
8),
r
= 5%,
σ
= 0
.
3 and
δ
= 3
.
5%. (a) Compute European and American put option prices for
K
= 90
,
110.
(b) Compute European and American call option prices for
K
= 90
,
110.
See the calculations in CalculA2.xls. We first build the tree of stock prices using
u
= exp((
r

δ
)
h
+
σ
√
h
) = 1
.
114 and
d
= exp((
r

δ
)
h

σ
√
h
) = 0
.
9011. We then compute the option value at time 0
recursively, using
H
(
i, j
) =
e

rh
(
qV
(
i
+ 1
, j
+ 1) + (1

q
)
V
(
i
+ 1
, j
))
,
where
q
= (
e
(
r

δ
)
h

d
)
/
(
u

d
) = 0
.
4735 and in the case of the American option,
V
(
i, j
) = max(
E
(
i, j
)
, H
(
i, j
),
where
E
(
i, j
) is the exercise value in node (
i, j
), while in the European case,
V
(
i, j
) =
H
(
i, j
). In both
cases, the recursion is initilaized using
V
(
N, j
) =
E
(
N, j
).
call
put
K
Euro
Amer
Euro
Amer
90
17.48
17.52
6.53
6.65
110
8.68
8.69
16.76
17.22
(c) Repeat (a) and (b) but assuming that
δ
= 0. Comment on what you observe.
call
put
K
Euro
Amer
Euro
Amer
90
19.87
19.87
5.48
5.74
110
10.13
10.13
14.77
15.76
As we know, when
δ
= 0 the American and European calls have the same value. With no dividends,
the stock prices are higher, and this makes the call option more valuable while the put options become
less valuable.
(d) Repeat (a) and (b) but with
r
= 0 and
δ
= 3
.
5%. Comment on what you observe.
call
put
K
Euro
Amer
Euro
Amer
90
14.95
15.55
8.39
8.39
110
7.03
7.23
20.47
20.47
Now it is the American and European put that have the same value. When
r
is 0, there is no advantage
in exercising earlier based on the timevalue of money for the put option, and since
δ >
0, it is best to
hold on to the stock as long as possible.
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 Spring '09
 Adam
 Option style, Euro Amer

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