MATH503
Homework3
Solution
Note: In this solution, a strategy
h
= (
x,y
1
,
···
,y
n
), where
x
is the riskless asset,
y
i
is the number
of shares of the risky asset.
1)
a)
i. ”
⇒
”. If 1 +
R > u
, consider the strategy
h
= (
S
0
,

1). Clearly,
V
h
0
= 0, and
V
h
1
=
±
S
0
(1 +
R
)

uS
0
, w.p. P
u
S
0
(1 +
R
)

dS
0
, w.p. P
d
Since 1 +
R > u
, we have
V
h
1
>
0, which implies
V
is dominant. This contradicts
with the no dominant strategy condition, so 1 +
R
≤
u
.
If 1 +
R < d
, consider the strategy
h
= (

S
0
,
1). Again, we have
V
h
0
= 0, and
V
h
1
=
±
uS
0

S
0
(1 +
R
)
, w.p. P
u
dS
0

S
0
(1 +
R
)
, w.p. P
d
Since 1+
R < d
, we have
V
h
1
>
0, which leads to the contradiction. Thus, 1+
R
≥
d
.
Finally, we have
d
≤
1 +
R
≤
u
.
ii. ”
⇒
”. Suppose there is a dominant strategy
H
= (
x,y
). Then we must have
x
=

yS
0
and
V
H
1
=
±
yS
0
[
u

(1 +
R
)]
, w.p. P
u
yS
0
[
d

(1 +
R
)]
, w.p. P
d
H
is dominant, so we must have
±
d >
1 +
R, if y >
0
u <
1 +
R, if y <
0
Clearly, it’s a contradiction. So there is no dominant strategy.