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.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 MAJIN
 Math, Trigraph, Dominant strategy, Fundamental theorem of linear algebra, v1h

Click to edit the document details