= Gaussian short-rate model with a deterministic shift
{c}
{title2=$r_t=g(t)+\sigma W_t^Q$}
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 is
$$
P_t(T)=\exp\left[-\int_t^Tg(s)ds-\sigma(T-t)W_t^Q+\frac{\sigma^2(T-t)^3}{6}\right].
$$
Indeed, conditional on $\mathcal F_t$, the random part of the integrated future <short rate> is $\sigma\int_t^T(T-u)dW_u^Q$, 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> is
$$
f_t(T)=-\partial_T\log P_t(T)=g(T)+\sigma W_t^Q-\tfrac12\sigma^2(T-t)^2.
$$
For merely locally integrable $g$, the same identity holds almost everywhere. The model allows negative <short rates> and gives explicit bond prices and forward curves.
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