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Continuous Time Finance, Spring 2004 – Homework 3
Distributed 2/27/04, due 3/10/04
(1) Assume the Vasicek model
dr
= (
θ

ar
)
dt
+
σ dw
for the riskneutral short rate process.
Consider a call option with maturity
T
and strike
K
, on a zerocoupon bond with maturity
S > T
. Its payoﬀ at time
T
is (
P
(
T, S
)

K
)
+
. Show using Black’s formula that the value
of this option at time
t
is
P
(
t, S
)
N
(
d
1
)

KP
(
t, T
)
N
(
d
2
)
where
d
1
=
1
σ
p
log
P
(
t, S
)
P
(
t, T
)
K
+
1
2
σ
p
,
d
2
=
1
σ
p
log
P
(
t, S
)
P
(
t, T
)
K

1
2
σ
p
with
σ
p
=
σ
±
1

e

2
a
(
T

t
)
2
a
!
1
/
2
B
(
T, S
)
.
(The function
B
(
T, S
) is the one from the representation
P
(
t, T
) =
A
(
t, T
)
e

B
(
t,T
)
r
(
t
)
.)
(2) Since Vasicek is a onefactor model, the call option of Problem 1 can be replicated by a
selfﬁnancing trading strategy using any pair of tradeables.
(a) What trading strategy produces a replicating portfolio using tradeables
P
(
t, T
) and
P
(
t, S
)?
(b) What trading strategy produces a replicating portfolio using tradeables
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.
 Fall '11
 Bayou
 Finance

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