Gaussian short-rate model with constant coefficients ($d{r}_t=a_{0}dt+b_{0}d{W}_t$)

For $d{r}_t=a_{0}dt+b_{0}d{W}_t$ under a risk-neutral measure, the instantaneous forward rate and zero-coupon bond prices areIndeed, conditional on current information, ${\int }_t^T{r}_{s}ds$ is Gaussian with mean $(T-t)r_t+a_0(T-t{)}^2/2$ and variance $b_0^2(T-t{)}^3/3$. Its negative exponential expectation gives the bond formula, and differentiating its logarithm gives the instantaneous forward rate. Equivalently the affine forward-rate equation has $A^{\prime }=0$, $A(0)=1$, and $B^{\prime }=a_0-b_0^2\theta$, $B(0)=0$. This is the constant-drift instance of the Ho-Lee Gaussian short rate model; the model permits negative short rates.