4 Interest Rate Derivative Securities
251
×
ˆ
£
2
α
+ (
γ
+
ψ
)
2
/
e
ψ
(
T

t
)
(
γ
+
ψ
)
£
(
γ
+
ψ
)
e
ψ
(
T

t
)

(
γ

ψ
)
/
!
μ
(
γ
+
ψ
)
/αψ
=
2
ψ
(
γ
+
ψ
)
e
ψ
(
T

t
)

(
γ

ψ
)
¶
μ
(
ψ

γ
)
/αψ
×
2
ψ e
ψ
(
T

t
)
(
γ
+
ψ
)
e
ψ
(
T

t
)

(
γ

ψ
)
¶
μ
(
γ
+
ψ
)
/αψ
=
2
ψ
(
γ
+
ψ
)
e
ψ
(
T

t
)

(
γ

ψ
)
¶
2
μ/α
e
μ
(
γ
+
ψ
)(
T

t
)
/α
=
2
ψ e
(
γ
+
ψ
)(
T

t
)
/
2
(
γ
+
ψ
)
e
ψ
(
T

t
)

(
γ

ψ
)
¶
2
μ/α
,
the two expressions are identical.
10. Describe a way to determine the market price of risk for the spot interest
rate.
Solution
:
Let
dr
=
u
(
r, t
)
dt
+
w
(
r, t
)
dX,
where
r
is the spot interest rate. Suppose that the price of any interest
rate derivative satisfies
∂V
∂t
+
1
2
w
2
∂
2
V
∂r
2
+ (
u

λ
(
r, t
)
w
)
∂V
∂r

rV
= 0
,
where
λ
(
r, t
) is the market price of risk for the spot interest rate. Because
λ
(
r, t
) is unknown, in order to use this equation to price derivatives, we
need to find
λ
. Suppose that
λ
is a function of
t
, i.e.,
λ
=
λ
(
t
). Then
this function, as the solution of the following inverse problem, can be
determined by the term structure of interest rates or, equivalently, by the
zerocoupon bond price curve. Suppose that
t
= 0 corresponds to today
and today’s spot interest rate is
r
*
.
Let
V
(
r, t
;
T
*
) be the solutions of the
problem
∂V
∂t
+
1
2
w
2
∂
2
V
∂r
2
+ (
u

λ
(
t
)
w
)
∂V
∂r

rV
= 0
,
r
l
≤
r
≤
r
u,
0
≤
t
≤
T
*
,
V
(
r, T
*
;
T
*
) = 1
, r
l
≤
r
≤
r
u
.
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 Spring '10
 Zhu
 Interest Rates, Rate Derivative Securities, ∂2V ∂V

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