Hull-White model

The short rate $r_t$ follows $dr_t = (\theta(t) - a r_t)\,dt + \sigma\,dW_t$. Here $a$ is the speed of mean reversion, $\sigma$ is the volatility, and $\theta(t)$ is a function chosen for each point in time. Hull and White presented it in 1990 as an extension of the Vasicek model.

Consider a hypothetical example in which the model with $a = 0.1$ and $\sigma = 1\%$ is fitted to a flat 2% curve (continuously compounded). Then $\theta(t)$ rises from 0.002 toward 0.0025, and the standard deviation of the short rate after 10 years is 2.08%.

Because the short rate is normally distributed, it can take negative values. The model is driven by one factor and produces only a flat smile. The article page has the formulas and the worked examples.