= Gaussian short-rate model with constant coefficients
{c}
{title2=$dr_t=a_0dt+b_0dW_t$}
= Constant-coefficient Ho-Lee model
{c}
{synonym}
For $dr_t=a_0dt+b_0dW_t$ under a <risk-neutral measure>, the <instantaneous forward rate> and <zero-coupon bond> prices are
$$
f_t(T)=r_t+a_0(T-t)-\frac12b_0^2(T-t)^2,\qquad P(t,T)=\exp\left(-(T-t)r_t-\frac{a_0}2(T-t)^2+\frac{b_0^2}6(T-t)^3\right).
$$
Indeed, conditional on current information, $\int_t^T r_sds$ 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 $A'=0$, $A(0)=1$, and $B'=a_0-b_0^2\theta$, $B(0)=0$. This is the constant-drift instance of the Ho-Lee Gaussian <short rate> model; the model permits negative short rates.
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