Math 238, Financial Mathematics
Problem Set 4 Solutions
February 28, 2009
Problem 1
Consider the Vasicek (or OrnsteinUhlenbeck) model for the observed short rate (money market
rate)
r
t
which is
dr
t
=
a
(
r
∞

r
t
)
dt
+
σdW
t
Here
r
∞
is the observed equilibrium level,
α
is the rate of mean reversion and
σ
the volatility.
We want to price a zero coupon bond by constructing a replicating, selfFnancing portfolio using a
bond with a longer maturity and a money market account. Let
P
(
t, r
;
T
) be the price of the zero
coupon bond to be priced, maturing at time
T
, with
t
and
r
the current time and short interest rate,
respectively. Let
P
l
(
t, r
;
T
l
) be the bond with the longer maturity
T
l
> T
. Let
I
(
t
) = exp(
R
t
0
r
s
ds
)
be the short rate growth factor, with
dI
(
t
) =
r
t
I
(
t
)
dt
. The portfolio for
P
is
P
= Δ
P
l
+
bI,
with
dP
= Δ
dP
l
+
bdI
(a) Explain why selfFnancing implies the second condition above.
In discrete time this condition is
P
n
+1

P
n
= Δ
n
(
P
l
n
+1

P
l
n
) +
b
n
(
I
n
+1

I
n
). In this form we
see clearly that the instantaneous change in the value of the portfolio is due to the change in the
price of the bond and in the accumulation of interest. After the new value
P
n
+1
is calculated then
we go to the replication relation and calculate the new hedge ratio Δ
n
+1
. No new funds are added
to the portfolio.
(b) Use Ito’s formula in
dP
= Δ
dP
l
+
bdI
and then choose Δ so that the coe±cients of the
dW
terms cancel out. Show that this gives
Δ =
∂P
∂r
∂P
l
∂r
and then equate the
dt
terms and rearrange them to get
1
∂P
∂r
±
∂P
∂t
+
1
2
σ
2
∂
2
P
∂r
2

rP
²
=
1
∂P
l
∂r
±
∂P
l
∂t
+
1
2
σ
2
∂
2
P
l
∂r
2

rP
l
²
This is done by direct calculation with Ito’s formula and use of the SDE.
(c) Explain what the essential conclusion from the above relation is. Show that if the relevant
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 Winter '08
 Papanicolaou
 Math, Ito, dRT

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