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Continuous Time Finance, Spring 2004 – Homework 4
Posted 3/19/04, due 3/31/04
(1) To solve Problem 5 of HW3 you needed to know that if
dr
= (
θ

ar
)
dt
+
σ dw
then
the function
v
(
x, t
) deﬁned by
v
(
x, t
) =
E
r
(
t
)=
x
±
e

R
T
t
r
(
s
)
ds
f
(
r
(
T
))
²
(1)
solves
v
t
+ (
θ

ax
)
v
x
+
1
2
σ
2
v
xx

xv
= 0
for
t < T
, with ﬁnaltime condition
v
(
x, T
) =
f
(
x
). This is a special case of the Feynman
Kac formula. Give a selfcontained proof, using the method of HW1, problem 1. (You
should assume that the PDE has a unique solution with this ﬁnaltime condition; your task
is to prove that the solution of the PDE satisﬁes (1).)
(2) The Section 6 notes explain how a trinomial tree can be used to approximate the
random walk
dx
=
σ dw
, and how working backward in this tree amounts to a standard
ﬁnitediﬀerence scheme for solving the backward Kolmogorov equation
u
t
+
1
2
σ
2
u
xx
= 0.
Let’s try to do something similar for the “geometric Brownian motion with drift” process
dy
=
μy dt
+
σy dw
, whose backward Kolmogorov equation is
v
t
+
μyv
y
+
1
2
σ
2
y
2
v
yy
= 0.
Assume the time interval is Δ
t
, and at time
t
=
n
Δ
t
the tree has nodes at

n
Δ
y, .
. . , n
Δ
y
.
The process on the tree goes from (
y, t
) to (
y
+Δ
y, t
+Δ
t
) with probability
p
u
, to (
y, t
+Δ
t
)
with probability
p
m
, and to (
y

Δ
y, t
+ Δ
t
) with probability
p
d
.
(a) How must
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 Fall '11
 Bayou
 Finance

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