Math 238, Financial Mathematics
Final Problem set
March 3, 2009
These problems are due on Tuesday March 10. You can give them to me in class, drop them in
my office, or put them in my mailbox outside the mathematics department.
Problem 1
Discuss the pricing of options on bonds, that is, using the Vasicek short rate model let
P
(
t, r
t
, T
2
)
be the price at time
t < T
2
of a zero coupon bond maturing at time
T
2
. Price a call option on this
bond maturing at time
T
1
with
t < T
1
< T
2
, with strike price
K
. (Bjork Chapter 22.) You may
use the framework of Chapter 24 in Bjork, sections 24.5, 24.6 and 24.7, if you wish, provided you
explain how you are using this framework.
Problem 2
In the HJM theory (Bjork Chapter 23) the forward rate
f
(
t, T
) is modeled by a family of SDEs
f
(
t, T
) =
f
(0
, T
) +
integraldisplay
t
0
α
(
s, T
)
ds
+
integraldisplay
t
0
σ
(
s, T
)
dW
(
s
)
for 0
< t < T
. The price of a zero coupon bond is given by
p
(0
, T
) = exp
{
integraldisplay
T
0
f
(0
, s
)
ds
}
and the short rate
r
(
t
) =
f
(
t, t
).
(a) Using Ito’s formula formally (without worrying about differentiability conditions) derive the
consistency relation
α
(
t, T
) =
σ
(
t, T
)
integraldisplay
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 Papanicolaou
 Math, zero coupon bond

Click to edit the document details