et
be observation time. We consider an agreement to invest at time
,
,
a fixed amount of cash and collect at
,
,
one unit of reference currency. We denote
the simple compounding rate during the time interval
implied by such a contract. We replicate this contract by selling
units of bond
and purchasing one unit of bond
.
If the implied rate
is fair then the contract should not worth anything at time
:
The cash flow at time
is
.
According to the contract, the investment will grow at the rate
up to the time
when it pays 1 unit of currency.
Therefore
We conclude that the fair rate for the contract is given by the
relationship


(Libor)

By definition, this is the "forward LIBOR". The structure of the last formula
is such that the rate
is a
martingale
with respect to the
forward
probability measure
Prob
Prob
.
