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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
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