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J. Wissel
Financial Engineering with Stochastic Calculus I
Fall 2010
Assignment Sheet 6
1.
Let
c
(
t,S
(
t
)
)
be the price of a European call option with maturity
T
and strike
K
in
the BlackScholes model, and let Δ(
t
) be the stock position in the replication portfolio.
a)
Using the formulas for
c
(
t,x
) and
d
±
(
t,x
) in the lecture notes (equation (4.20)
and following), show that
c
x
(
t,x
) = Φ
(
d
+
(
t,x
)
)
.
b)
Let
X
(
t
) =
η
(
t
)
B
(
t
)+Δ(
t
)
S
(
t
) be the value at time
t
of the replication portfolio.
Compute the bond position
η
(
t
). In particular, you will ﬁnd that
η
(
t
)
<
0.
c)
The function
c
(
t,x
) is given by equation (4.20) only for
t < T
. For
t
=
T
we have
c
(
T,x
) = (
x

K
)
+
. Show that
c
(
t,x
) is continuous at
t
=
T
, i.e.,
lim
t
→
T
c
(
t,x
) = (
x

K
)
+
for all
x >
0
.
2.
Let us denote the BlackScholes model price of a European call option with maturity
T
and strike
K
by
c
(
t,S
(
t
)
,T,K,r,σ
)
to express the dependence on parameters.
a)
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 '09
 J.WISSEL

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