J. Wissel
Financial Engineering with Stochastic Calculus I
Fall 2010
Assignment Sheet 7
1.
Let
Z
≥
0 be a random variable on a space (Ω
,
F
,P
) with
E
[
Z
] = 1.
a)
Show that
˜
P
[
A
] =
E
[
ZI
A
]
,
A
∈ F
deﬁnes a probability measure
˜
P
on (Ω
,
F
).
b)
Suppose that
Z >
0
P
a.s. Show that
P
and
˜
P
are equivalent, and that
1
Z
is a
RadonNikod´ym derivative of
P
w.r.t.
˜
P
.
2.
Consider the BlackScholes model for a stock price
S
(
t
) with interest rate
r
and vola
tility
σ
. Use the riskneutral pricing formula to compute the price at time
t
∈
[0
,T
] of
a
digital option
which pays 1 unit of cash at time
T
if
S
(
T
)
≥
K
, and zero otherwise.
3.
In the BlackScholes model with timevarying nonrandom coeﬃcients, the bank ac
count and stock price are given by
dB
(
t
) =
B
(
t
)
r
(
t
)
dt,
B
(0) = 1
,
dS
(
t
) =
S
(
t
)
r
(
t
)
dt
+
S
(
t
)
σ
(
t
)
d
f
W
(
t
)
,
S
(0) =
S
0
for a Brownian motion
f
W
(
t
) under the riskneutral measure
˜
P
, and
r
(
t
)
,σ
(
t
) are
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 '09
 J.WISSEL
 Probability theory, bank account, Mathematical finance, riskneutral measure, quadratic hedging error

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