One-factor short-rate model ($d{r}_t=\mu (t,r_t)dt+\eta (t,r_t)d{W}_t$)

A one-factor short-rate model specifies $d{r}_t=\mu (t,r_t)dt+\eta (t,r_t)d{W}_t$ with deterministic coefficient functions. All zero-coupon bond prices are functions of time, maturity and this single scalar state. The physical drift and pricing drift differ by $\eta \lambda$, where $\lambda$ is the market price of risk.

For smooth derivative prices depending on a Markov short rate, Itô formula gives physical drift $m=V_t+b{V}_r+{\sigma }^2{V}_{rr}/2$ and exposure $s=\sigma V_r$. The common short-rate market price of risk gives $m-rV=\theta s$, yielding this equation. Under the risk-neutral measure, the rate drift is $b-\sigma \theta$ and the bank-discounted price is a martingale under suitable integrability. A unit zero-coupon bond has terminal value one and price ${\mathbb{E}}^Q[\exp (-{\int }_t^T{r}_{s}ds)\mid r_t=r]$.

If all traded zero-coupon bonds depend on one scalar rate driven by one Brownian motion, their diffusion exposures form one vector $v_i=\eta P_{i,r}$. Their instantaneous covariance matrix is $v{v}^T$, of rank at most one. Nondegenerate maturity changes are therefore instantaneously perfectly correlated, with correlation $+1$ or $-1$, though finite-horizon bond-return correlations need not be one.

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 ${\eta }_0=0$.