Gaussian short-rate model with a deterministic shift ($r_t=g(t)+\sigma W_t^Q$)

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.

To match a prescribed initial instantaneous forward rate curve $F_0(T)$ in the Gaussian short-rate model with a deterministic shift, chooseSince $W_0^Q=0$, its model forward rate is then $f_0(T)=g(T)-{\sigma }^2{T}^2/2=F_0(T)$. Moreover,so the initial zero-coupon bond curve matches as well. A continuous initial forward curve gives pointwise calibration; a locally integrable curve gives the corresponding almost-everywhere interpretation of the instantaneous forward rate.