Overview
The forward rate curve is a curve (function of time, {% T %}) that indicates overnight (instantaneous) rate that one could receive today, for a forward rate agreement starting at time {% T %}.
The curve itself is a function of what the current time {% t %} is, hence it is often written as a function of two time points, the current time, and the forward time.
{% f(t,T) %}
or sometimes simplified to
{% f_t(T) %}
The Forward Price
The forward price is given by (see forward rate agreements)
{% f(t_1, t_2) = \frac{(r(t_2)t_2 - r(t_1)t_1)} {(t_2 - t_1)} %}
which is equivalent to
{% f(t_1, t_2) = r(t_2) + t_1 \times \frac{(r(t_2) - r(t_1))}{(t_2 - t_1)} %}
The Instantaneous Forward Rate
The instantaneous forward rate curve at time {% t %} is defined to be
{% f(t,T) = \lim_{S \rightarrow T} F(t,T,S) = -\frac{\partial}{\partial T} ln P(t,T) = -\frac{\partial P(t,T)/\partial T}{P(t,T)} %}
where here {% P(t,T) %} is the price of a zero coupon bond (paying {% $1 %}) at time {% T %}, valued today (time {% t %}).
This implies that:
{% P(t,T) = exp[-\int_t ^T f(t,u) du] %}