Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 1 / c

Let $q(T)$ denote the original zero-coupon bond price $P(0,T)$ computed above. A deterministic shift $h_t$ of the short rate, with $h_0=0$, gives ${\hat{r}}_t=r_t+h_t$. Subtracting the two recursions shows that the required added drift isSince the shift is nonrandom, the new discount factor differs by $\exp (-{\sum }_{s=1}^T{h}_s)$, and hence $\hat{P}(0,T)=q(T)\exp (-{\sum }_{s=1}^T{h}_s)$.

For deterministic calibration of an autoregressive short rate, defineAll these quantities are finite. The telescoping sum ${\sum }_{s=1}^T{h}_s=d_T$ gives $\hat{P}(0,T)=p(T)$. Explicitly,These constants fit every maturity simultaneously. Each successive maturity fixes $h_T$ uniquely, so the deterministic calibration is also unique with the prescribed $r_0$.