Information Technology Reference
In-Depth Information
Fig. 7.1
Yield curves in the CIR model with parameters
α
=
0
.
03,
β
=
0
.
5,
σ
=
0
.
5 and different
initial conditions
r
0
has to be made to produce an amount of one unit of currency at maturity, starting
from
B(t,T,r)
units of currency at time
t
, when accruing occurs proportionally to
the investment time, i.e.
1
−
B(t,T,r)
L(t, T , r)
:=
t)B(t,T,r)
.
(T
−
We now turn to the definition of
forward rates
. Forward rates are characterized by
three time instants, namely the time
t
at which the rate is considered, its expiry
T
and
its maturity
T
1
, with
t
T
1
. Forward rates are interest rates that can be locked
in today for an investment in a future time period, and are set consistently with
the current structure of zero coupon bonds. We can define a forward rate through a
forward rate agreement. This contract gives its holder an interest rate payment for
the period between
T
and
T
1
. At the maturity
T
1
a fixed payment
K
is exchanged
for a variable payment
L(T , T
1
,r)
. The value of the contract in
T
1
is given as
(T
1
−
T )(K
−
L(T , T
1
,r
T
))
. Recalling the definition of
L(T , T
1
,r
T
)
the value
V(t,r)
of
the contract at
t
is given as
≤
T
≤
B(t,T
1
,r).
The value of
K
that renders the contract fair at time
t
,i.e.
V(t,r)
=
V(t,r)
=
B(t,T
1
, r)(T
1
−
T)K
−
B(t,T,r)
+
0, is called
simply compounded forward interest rate
.
Definition 7.2.2
The simply compounded forward interest rate at time
t
for the
expiry
T>t
with maturity
T
1
>T
at short rate
r
is denoted by
F(t,T,T
1
)
and
defined by
B(t,T,r)
B(t,T
1
,r)
−
1
.
1
T
1
−
F(t,T,T
1
,r)
:=
T
Search WWH ::
Custom Search