Overview
Forward
A forward (or forward rate agreement) is a contract between two parties that specifies the interest rate to be used in fixed income contract that begins in the future. That is, two parties at time {% t_0 %} agree that the borrower will borrow a given principal at time {% t_1 %} to be repaid at time {% t_2 %} at the agreed upon rate, {% r %}.
Using the notation, {% P(t_1, t_2) %} as the price at time {% t_1 %} for {% $1 %} received at time {% t_2 %}, the arbitrage free value of the iterest rate for the forward is given by
{% \frac{P(t, u)}{P(t, u+ \Delta t)} = 1 + r \Delta t %}
see back
That is, the loan provider could meet the forward obligation by investing {% 1/P(t_0, t_1) %} dollars at time {% t_0 %}. That investment matures at {% $1 %} at time {% t_1 %}. That dollar is then lent to the borrower, which pays the loan back at time {% t_2 %}. However, the loan provider could have just invested the dollar at time {% t_0 %} for maturity at time {% t_2 %}. In order to prevent arbitrage, the loan provider should set the forward rate such that at {% t_2 %} she receives the same amount from the forward, as should would from just investing the money.