The main ingredient for pricing is the zero curve r(t) which assigns an interest rate to any given maturity t > 0. It tells us what the value of 1 currency unit will be at time t if invested at the risk-free rate. For most theoretical applications, the zero rate is expressed as continuously compounding, so the value at time t will be given by

Other conventions are also common. Linear compounding is typically used for short-term interest (less than one year):

Simple compounding takes interest on interest into account, in particular for maturities beyond one year:

Conversely, today’s value of one currency unit paid in t years is given by

P(0,t) is the price of a risk-free zero bond with maturity t, as seen today (at time 0). It is also referred to as the (deterministic) discount factor for time t, df_t.

The most important building block in interest rate modelling is the forward rate agreement, or FRA for short. This is a contract by which two parties agree today (at t = 0) on an interest rate f(0;t₁,t₂) to be paid in t₂ for a loan spanning a future period t₁ to t₂. If the market (i.e. LIBOR) rate L(t₁,t₂) which is fixed in t₁ for that period exceeds f(0;t₁,t₂), the payer of the rate has made a profit. Otherwise, the receiver gains more than the market rate.

Market practice is that the payment is actually paid in t₁ by computing the cash flow in t₂ and discounting it to t₁ with the fixed LIBOR rate. For pricing purposes, this is virtually irrelevant (see [117]), so we ignore this distinction.

Note that, in general, the period lengths are not exactly a quarter or half a year but rather depend on the day count fraction of the rates used. In Euroland, this would be ACT/360, for example.

In other words

The present value of the forward rate paid on a notional of 1 unit is therefore

which also implies that

Forward rates are used as expected values for the LIBOR fixing for a future time period. Most importantly, this is done in interest rate swaps.

An interest rate swap, or swap for short, is a contract by which two parties agree to exchange interest payments on a predetermined notional on a regular basis. One party pays the fixed rate with the frequency which is standard in the chosen currency. For EUR, for instance, this is annually; for USD, on the other hand, this is semi-annually. The other party pays a floating rate linked to LIBOR of some given frequency (1, 3, 6 or 12 months), possibly with a spread. There is also a standard frequency for floating legs in most currencies: in EUR, this is six months, and in USD, it is three months, for instance.

At inception, the v...