Zero Coupon Rates

We can now develop the model for zero coupon bonds using the principles discussed in the continuous compounding section.

Let us define the terms for the model.

We define R(t,T) to be the spot interest rate prevailing at time t for a zero coupon bond P maturing at time T, denoted by P(t,T).

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 ∀ T

Assuming that R(t,T) is deterministic, the zero coupon bond price can be expressed as

If we set [T-t] to be 𝝉, and given the fact that P(T,T) = 1 (it is the bonds face value at maturity), equation (1) can be written as

Taking logs of both sides

Equation (5) is the zero coupon bond price expressed in terms of the continuously compounded spot interest rate.

Let D(t,T) denote the discount factor. What is the relationship between D(t,T) and 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(t,T) 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 random variable D(t,T) under a particular probability measure.