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J. Wissel
Financial Engineering with Stochastic Calculus I
Fall 2010
Assignment Sheet 4
1.
Consider the
N
period binomial model with
d <
1 +
r < u
. For each
T
= 0
,...,N
and
K
≥
0 let
C
0
(
K,T
) be the fair price at time 0 of a European call option with payoﬀ
(
S
T

K
)
+
at time
T
. Show that for
T
1
≤
T
2
and
K
≥
0 we have
C
0
(
K,T
2
)
≥
C
0
±
K
(1+
r
)
T
2

T
1
,T
1
²
.
Hint: Use assignment 3 problem 2b), and Corollary 2.44 of the lecture notes.
2.
We revisit the oneperiod trinomial model on a space Ω =
{
ω
1
,ω
2
,ω
3
}
as discussed in
the recitations. We assume
B
0
=
B
1
= 1 (zero interest rate),
S
0
= 1, and
S
1
and the
realworld probability measure
P
is deﬁned by
S
(
ω
3
) = 2
, P
[
ω
3
] =
1
4
,
S
(
ω
2
) = 1
, P
[
ω
2
] =
1
2
,
S
(
ω
1
) =
1
2
, P
[
ω
1
] =
1
4
.
By the FTAP, the market
B,S
is arbitragefree if and only if
E
˜
P
[
S
1
] =
S
0
where
˜
P
is a probability measure on Ω which is equivalent to
P
. Any such
˜
P
is called
a riskneutral measure. Let ˜
p
=
˜
P
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This note was uploaded on 01/25/2011 for the course ORIE 5600 at Cornell University (Engineering School).
 '09
 J.WISSEL

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