Short-rate bond pricing equation ($P_t+{\mu }_Q{P}_r+\frac{1}{2}{\eta }^2{P}_{rr}-rP=0$)

For pricing drift ${\mu }_Q$, a unit zero-coupon bond price $P(t,r;T)$ satisfies $P_t+{\mu }_Q{P}_r+{\eta }^2{P}_{rr}/2-rP=0$, with $P(T,r;T)=1$. Itô formula makes its bank-account-discounted value a local martingale; suitable integrability gives $P(t,r;T)={\mathbb{E}}^Q[\exp (-{\int }_t^T{r}_{s}ds)\mid r_t=r]$. The rate kills the diffusion semigroup, as in the Feynman-Kac formula.

In a one-factor short-rate model, a derivative $V(t,r)$ can be hedged locally by a nondegenerate traded zero-coupon bond $P(t,r;T_1)$. Match Brownian exposures by holding $V_r/P_r$ bonds and putting the remaining value into the bank account. Substitution into the two pricing equations proves the gain identity. This requires $\eta P_r\ne 0$; completeness at a degenerate boundary cannot be inferred from the dimension count alone.

When ${\mu }_Q=a(b-r)$ and $\eta (r{)}^2={\eta }_0^2+{\eta }_1^{2}r$, substituting $P=A(\tau )e^{-B(\tau )r}$ gives $B^{\prime }=1-aB-{\eta }_1^2{B}^2/2$ and $(\log A{)}^{\prime }=-abB+{\eta }_0^2{B}^2/2$, with $B(0)=0$, $A(0)=1$. The Vasicek model uses ${\eta }_1=0$, and the CIR model uses ${\eta }_0=0$.