A Reference List of One-Factor and Multifactor Short-Rate Models
Summary
The document surveys named stochastic short-rate models, grouping them into one-factor and multifactor families. The one-factor list spans models with constant or time-varying drift and volatility, mean-reverting Gaussian dynamics, square-root volatility, and log-rate specifications. The multifactor examples describe rates driven by several evolving state variables.
The equations serve as a compact taxonomy rather than a comparative analysis. The answer adds the two-factor Hull–White model as another entry. There is no discussion of calibration, bond pricing, empirical performance, or how to choose among the models, and the equations are presented without checking conventions or assumptions. Readers can use the list to identify model families, but should consult dedicated references before implementing any specification.
Key ideas
- Short-rate models can be organized into one-factor and multifactor specifications.
- One-factor models differ in their drift, volatility structure, and whether rates or log-rates are modeled.
- Multifactor models represent interest-rate movements through several stochastic state variables.
- The list is an introductory catalog and does not compare calibration or pricing properties.
- The answer identifies two-factor Hull–White as an additional model family.
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Full text
# Short rate models (stochastic)
# 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)$$\begin{align} & d{{r}_{t}}=({{\theta }_{t}}-{{\alpha }_{t}})dt+\sqrt{{{r}_{t}}}{{\sigma }_{t}}d{{W}_{t}} \\ & d{{\alpha }_{t}}=({{\zeta }_{t}}-{{\alpha }_{t}})dt+\sqrt{{{\alpha }_{t}}}{{\sigma }_{t}}d{{W}_{t}} \\ & d{{\sigma }_{t}}=({{\beta }_{t}}-{{\sigma }_{t}})dt+\sqrt{{{\sigma }_{t}}}{{\eta }_{t}}d{{W}_{t}} \\ \end{align} $$
## Answer by wsw (score 1)
https://quant.stackexchange.com/a/30494
There is the two-factor Hull-White model:
https://en.wikipedia.org/wiki/Hull%E2%80%93White_model#Two-factor_modelShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.