forward interest rate
A forward interest rate is always in relation to some specific underlying spot interest rater.
Assume the existence of a lending/borrowing contract C that starts at time T.
Further assume this contract C stipulates either directly or indirectly a corresponding interest rate r(T) that is agreed between the parties entering into the contract at time T.
Then the forward interest rate at time t < T is the number f(t,T) with the following meaning:
No two counterparties would incur an immediate gain or loss by deciding to enter into a binding agreement at time t to hold the contract C at the latter time T under the provision that the contract's underlying rate at T will equal f(t,T).
As an example, let r(T) be a certain ibor rate prevailing at time T.
In simplifing terms, the associated contract C is a lending/borrowing contract between two banks over a term that starts at T and ends a certain time interval later, for example at T + 6 months.
Then the forward interest rate f(t,T) is the number agreed between two parties at time t, such that they will enter into the contract C at the latter time T using the rate f(t,T) even if that rate is not fair any more.
It follows that f(t,T) -> r(T) as t → T.
Similar to the spot interest rate function r(t), we may also speak of a function f(t,T) that - keeping T fixed - maps each time t to the respective forward interest rate value f(t,T).
Assuming t = 0 designates the time now, the value f(t,T), t > 0 and t < T is not a simple number but rather a random variable, since it is not possible to know with certainty the forward interest rate that is going to prevail at the future tinme t.
It follows, the function f(t,T) represents a mapping from t to some random variable, and therefore is a stochastic process.