• There are a number of short rate models that give $r(t)$.
  • How can those be used to construct the whole of the yield curve $y(t,T)$ (where $y(t, 0) = r(t)$)?

closed as off-topic by noob2, LocalVolatility, Helin, Alex C, amdopt Nov 17 '17 at 13:50

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  • $\begingroup$ Under the "expectations hypothesis" (which assumes term premia are zero), the prices of ZCB are given by the following expectation $P(0,T)=E^Q [\exp(-\int_0^T r(t)dt)]$ where $r(t)$ is the stochastic process driving the short rate. From the prices of ZCB you can find the yields $y(0,T)$ $\endgroup$ – Alex C Nov 11 '17 at 21:21
  • $\begingroup$ @Alex C Is this not a closed loop? You calculate ZCB $P(0,T) = e^{-T \cdot r(t)}$, then derive $y(0,T)$ by $y(0,T) = \frac{ln\bigg( P(0,T) \bigg)}{-T}$ getting back the original $r(t)$. So the yield curve is always $y(0,T) = r(t) \; \forall \; T$? $\endgroup$ – A.L. Verminburger Nov 13 '17 at 12:00