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2
σ
2
r
1
+
2
μ
∂
r
φ∂
r
ϕ
d
r
+
σ
2
1
2
+
μ
R
R
1
a
CIR
(φ, ϕ)
:=
r
2
μ
∂
r
φϕ
d
r
0
0
R
R
βr)r
2
μ
∂
r
φϕ
d
r
r
2
μ
+
1
φϕ
d
r.
−
(α
−
+
0
0
The spaces
W
1
/
2
,μ
(G)
and
H
μ
(G)
are defined in (4.29) and (4.33) with
ρ
=
1
/
2
and
μ
∈
(
−
1
/
2
,
0
)
. Applying Theorem 3.2.2, one can show the well-posedness of
the pricing problem.
Remark 7.1.4
The localization of the pricing problem (
7.2
) to a bounded domain
cannot be justified as in Theorem 4.3.1 as [143, Theorem 25.18] is not applicable.
Instead, we can use a different approach as in Theorem 9.4.1 to rigorously justify
the use of homogeneous Dirichlet boundary conditions for large
r
. Alternatively, we
could use the approach described in [136] using local times and leading to homoge-
neous Neumann conditions for large
r
.
A quantity that is often considered next to the bond price is the
yield
, which is
for a given maturity
T
given by
Y(t,r)
1
T
−
t
log
B(t,r)
. It is the constant rate
of continuously compounding interest which an investment has to be made starting
from
B(t,r)
units of currency at time
t
to obtain one unit of currency at maturity.
The corresponding graph for different maturities of
T>t
is called a yield curve
and is often considered in the financial context. Typical shapes of yield curves are
normal, i.e. monotonously increasing in
T
, inverse, i.e. monotonously decreasing
in
T
, and humped, i.e. having exactly one local maximum and no local minimum
in
T
.
:= −
Example 7.1.5
We consider the CIR model with parameters
α
=
0
.
03,
β
=
0
.
5 and
σ
0
.
5. The bond prices are computed solving PDE (
7.6
) and the yield obtained
postprocessing the bond prices. We plot in Fig.
7.1
the yield curve for different
initial values of
r
0
. It can be seen that the three types of yield curved mentioned
above (normal, inverse and humped) can be reproduced. We refer to [102] for further
details.
=
Remark 7.1.6
For other types of interest rate models, such as the Vasicek model,
the well-posedness results can be obtained as in Sect. 4.5.2.
7.2 Interest Rate Derivatives
Derivatives in interest rate markets are usually not written on the bonds directly, but
on
simply compounded interest rates
which are defined as follows. To emphasize
the dependency on the maturity, we now denote the bond price with
B(t,T,r)
.
Definition 7.2.1
The simply compounded interest rate prevailing at time
t
for the
maturity
T
is denoted by
L(t, T , r)
and is the constant rate at which an investment
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