Assume that a zero-coupon bond that pays 1 at time \(T\) has price \(p(t,T)\) contracted at time \(t\). Naturally, for all \(t\), \(p(t,t)=1\) and \(p(t,T)\) is well-defined for all \(t\in \mathbb{R}\). Assume that \(p(t,T)\) is also well-defined for all \(T\in \mathbb{R}\) and is differentiable w.r.t \(T\).
If the accumulation of interest can be described by a constant continuous-time interest rate on \([t,T]\) denoted \(y(t,T)\), then \[ p(t,T)e^{y(t,T)(T-t)}=1, \] solves \[ y(t,T)=- \frac{1}{T-t}\log p(t,T), \] called the yield curve viewed as a function in \(T\) after fixing \(t\).