Continuous Time Finance, Spring 2004 – Homework 1
Distributed 1/28/04, due 2/4/04
(1) In the Section 1 notes, we proved that if
V
solves the BlackScholes PDE with finalvalue
f
, then
V
(
S
0
,
0) =
e

rT
E
[
f
(
S
T
)] where
S
solves the SDE
dS
=
rS dt
+
σS dw
with initial
value
S
(0) =
S
0
. Let’s do something similar for a stochastic interest rate. Suppose the spot
rate
r
t
solves a diffusion of the form
dr
=
α dt
+
β dw
with
r
(0) =
r
0
, where
α
=
α
(
r, t
) and
β
=
β
(
r, t
) are fixed functions of
r
and
t
. Consider the function
U
(
r, t
) defined by solving
U
t
+
αU
r
+
1
2
β
2
U
rr

rU
= 0 with final value
U
(
r, T
) = 1. Show that
U
(
r
0
,
0) =
E
e

T
0
r
(
s
)
ds
.
[Comment: if the SDE for
r
is the riskneutral process, then
U
(
r
0
,
0) is the value of a zero
coupon bond that pays one dollar at time
T
. Hint: show that
U
(
r
(
t
)
, t
) exp

t
0
r
(
s
)
ds
is a martingale.]
(2) Consider a nondividendpaying stock whose share price satisfies
dS
=
μS dt
+
σS dw
,
and assume for simplicity that the riskfree rate
r
is constant. Consider a European option
with maturity
T
and payoff
f
(
S
T
).
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 Fall '11
 Bayou
 Finance, Mathematical finance, BlackScholes PDE, Martingale Representation Theorem, St /Bt

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