 Gaussian short-rate model with a deterministic shift

Under a risk-neutral measure $Q$, let the short rate be $r_t=g(t)+\sigma W_t^Q$, with deterministic locally integrable $g$ and constant $\sigma >0$. For a zero-coupon bond paying one at $T$, the risk-neutral pricing value isIndeed, conditional on ${\mathcal{F}}_t$, the random part of the integrated future short rate is $\sigma {\int }_t^T(T-u)d{W}_u^Q$, which is Gaussian with mean zero and variance ${\sigma }^2(T-t{)}^3/3$. Its exponential moment gives the formula. At points where $g$ is continuous, the instantaneous forward rate isFor merely locally integrable $g$, the same identity holds almost everywhere. The model allows negative short rates and gives explicit bond prices and forward curves.