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Derivative Securities, Fall 2007 – Homework 4.
Distributed 10/17/07, due 10/31/07.
A solution sheet to HW3 will be posted 10/25; a solution sheet to HW4 will be posted 11/8;
no late HW’s will be accepted once the corresponding solution sheet has been posted.
1. Let
F
solve the SDE
dF
=
μF dt
+
σF dw
, where
μ
and
σ
are constant and
w
is
Brownian motion. Find the SDE solved by
(a)
V
(
F
) =
AF
, where
A
is constant
(b)
V
(
F
) =
√
F
(c)
V
(
F
) = cos
F
(d)
V
(
F,t
) =
F
3
t
2
.
2. We continue to assume that
F
solves
dF
=
μF dt
+
σF dw
, where
μ
and
σ
are constant
and
w
is Brownian motion. Find a function
V
(
F
) such that the process
t
7→
V
(
F
(
t
))
is a martingale (i.e. the SDE describing
V
(
F
) has no
dt
term).
3. This problem should help you understand Ito’s formula. If
w
is Brownian motion,
then Ito’s formula tells us that
z
=
w
2
satisﬁes the stochastic diﬀerential equation
dz
= 2
wdw
+
dt
. Let’s see this directly:
(a) Suppose
a
=
t
0
< t
1
< .. . < t
N

1
< t
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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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