Continuous Compounding

The continuously compounded spot interest rate, denoted as R(t,T), is the rate prevailing at time t for the maturity T.

It is the constant rate at which an investment P(t,T) at time t accrues continuously to yield a unit amount of curency at maturity T

As we have seen before, the zero coupon bond price for a deterministic rate R(t,T) can be expressed as

If we set [T-t] as 

and given the fact that P(T,T) = 1, taking logs of both sides

giving

Re-arranging, the continuously compounded spot interest rate is given by

The Zero Coupn Bond price expressed in terms of the continuously compounded spot interest rate is