# Short rate models (stochastic)

I want to make a quick reference or some pages, that contains short rate models . I know some models but I am not sure that ,this list is complete ...please help me to $\textbf{improve}$ this list .thanks in advanced.

$$\textbf{One-factor short-rate models}$$

Merton's model (1973) $${{r}_{t}}={{r}_{0}}+at+\sigma W_{t}^{*}$$ Vasicek model (1977) $$d{{r}_{t}}=(\theta -\alpha {{r}_{t}})dt+\sigma d{{W}_{t}}$$ Rendleman–Bartter model (1980)$$d{{r}_{t}}=\theta {{r}_{t}}dt+\sigma {{r}_{t}}d{{W}_{t}}$$ Cox–Ingersoll–Ross model (1985) $$d{{r}_{t}}=(\theta -\alpha {{r}_{t}})dt+\sqrt{{{r}_{t}}}\sigma d{{W}_{t}}$$ Ho–Lee model (1986)$$d{{r}_{t}}={{\theta }_{t}}dt+\sigma d{{W}_{t}}$$ Hull–White model (1990)—also called the extended Vasicek model $$d{{r}_{t}}=({{\theta }_{t}}-\alpha {{r}_{t}})dt+{{\sigma }_{t}}d{{W}_{t}}$$ Black–Derman–Toy model (1990) $$d\ln (r)=[{{\theta }_{t}}+\frac{{{{{\sigma }'}}_{t}}}{{{\sigma }_{t}}}\ln (r)]dt+{{\sigma }_{t}}d{{W}_{t}}$$ Black–Karasinski model (1991) $$d\ln (r)=[{{\theta }_{t}}-{{\phi }_{t}}\ln (r)]dt+{{\sigma }_{t}}d{{W}_{t}}$$ Kalotay–Williams–Fabozzi model (1993) $$d\ln ({{r}_{t}})={{\theta }_{t}}dt+\sigma d{{W}_{t}}$$

$$\textbf{Multi-factor short-rate models}$$

Longstaff–Schwartz model (1992)\begin{align} & d{{X}_{t}}=({{a}_{t}}-b{{X}_{t}})dt+\sqrt{{{X}_{t}}}{{c}_{t}}d{{W}_{1t}} \\ & d{{Y}_{t}}=({{d}_{t}}-e{{Y}_{t}})dt+\sqrt{{{Y}_{t}}}{{f}_{t}}d{{W}_{2t}} \\ & d{{r}_{t}}=(\mu X+\theta Y)dt+{{\sigma }_{t}}\sqrt{Y}d{{W}_{3t}} \\ \end{align} Chen model (1996)\begin{align} & d{{r}_{t}}=({{\theta }_{t}}-{{\alpha }_{t}})dt+\sqrt{{{r}_{t}}}{{\sigma }_{t}}d{{W}_{t}} \\ & d{{\alpha }_{t}}=({{\zeta }_{t}}-{{\alpha }_{t}})dt+\sqrt{{{\alpha }_{t}}}{{\sigma }_{t}}d{{W}_{t}} \\ & d{{\sigma }_{t}}=({{\beta }_{t}}-{{\sigma }_{t}})dt+\sqrt{{{\sigma }_{t}}}{{\eta }_{t}}d{{W}_{t}} \\ \end{align}

There is the two-factor Hull-White model:

https://en.wikipedia.org/wiki/Hull%E2%80%93White_model#Two-factor_model