A T-maturity zero-coupon bond (ZCB) is a bond that pays the holder one unit of the currency at time T, with no intermediate payments.

The contract value at time t, t<T, is denoted as P(t,T)

Clearly, P(T,T) = 1 for all T

The question that now arises is what is the relationship between the discount factor D(t,T) and the ZCB price P(t,T)

If the rates, r, are deterministic, then D(t,T) is also deterministic and therefore D(t,T) = P(t,T)

However, if rates are stochastic, D(t,T) is a random quantity at time t depending on the future evolution of rates r between time t and T. The ZCB price P(t,T) is however know (deterministic) at time t.

It can be shown that the ZCB price P(t,T) can be viewed as the expectation of the randon variable D(t,T) under a particular probability measure.