Short-rate model

Following are the one-factor models, where a single stochastic factor – the short rate – determines the future evolution of all interest rates. Other than Rendleman–Bartter and Ho–Lee, which do not capture the mean reversion of interest rates, these models can be thought of as specific cases of Ornstein–Uhlenbeck processes. The Vasicek, Rendleman–Bartter and CIR models have only a finite number of free parameters and so it is not possible to specify these parameter values in such a way that the model coincides with observed market prices ("calibration"). This problem is overcome by allowing the parameters to vary deterministically with time.[2][3] In this way, Ho-Lee and subsequent models can be calibrated to market data, meaning that these can exactly return the price of bonds comprising the yield curve. The implementation is usually via a (binomial) short rate tree [4] or simulation; see Lattice model (finance) § Interest rate derivatives and Monte Carlo methods for option pricing.

1. Merton's model (1973) explains the short rate as ${\displaystyle r_{t}=r_{0}+at+\sigma W_{t}^{*}}$: where ${\displaystyle W_{t}^{*}}$ is a one-dimensional Brownian motion under the spot martingale measure.[5]
2. The Vasicek model (1977) models the short rate as ${\displaystyle dr_{t}=(\theta -\alpha r_{t})\,dt+\sigma \,dW_{t}}$; it is often written ${\displaystyle dr_{t}=a(b-r_{t})\,dt+\sigma \,dW_{t}}$.[6]
3. The Rendleman–Bartter model (1980) explains the short rate as ${\displaystyle dr_{t}=\theta r_{t}\,dt+\sigma r_{t}\,dW_{t}}$.[7]
4. The Cox–Ingersoll–Ross model (1985) supposes ${\displaystyle dr_{t}=(\theta -\alpha r_{t})\,dt+{\sqrt {r_{t}}}\,\sigma \,dW_{t}}$, it is often written ${\displaystyle dr_{t}=a(b-r_{t})\,dt+{\sqrt {r_{t}}}\,\sigma \,dW_{t}}$. The ${\displaystyle \sigma {\sqrt {r_{t}}}}$ factor precludes (generally) the possibility of negative interest rates.[8]
5. The Ho–Lee model (1986) models the short rate as ${\displaystyle dr_{t}=\theta _{t}\,dt+\sigma \,dW_{t}}$.[9]
6. The Hull–White model (1990)—also called the extended Vasicek model—posits ${\displaystyle dr_{t}=(\theta _{t}-\alpha r_{t})\,dt+\sigma _{t}\,dW_{t}}$. In many presentations one or more of the parameters ${\displaystyle \theta ,\alpha }$ and ${\displaystyle \sigma }$ are not time-dependent. The model may also be applied as lognormal. Lattice-based implementation is usually trinomial.[10][11]
7. The Black–Derman–Toy model (1990) has ${\displaystyle d\ln(r)=[\theta _{t}+{\frac {\sigma '_{t}}{\sigma _{t}}}\ln(r)]dt+\sigma _{t}\,dW_{t}}$ for time-dependent short rate volatility and ${\displaystyle d\ln(r)=\theta _{t}\,dt+\sigma \,dW_{t}}$ otherwise; the model is lognormal.[12]
8. The Black–Karasinski model (1991), which is lognormal, has ${\displaystyle d\ln(r)=[\theta _{t}-\phi _{t}\ln(r)]\,dt+\sigma _{t}\,dW_{t}}$.[13] The model may be seen as the lognormal application of Hull–White;[14] its lattice-based implementation is similarly trinomial (binomial requiring varying time-steps).[4]
9. The Kalotay–Williams–Fabozzi model (1993) has the short rate as ${\displaystyle d\ln(r_{t})=\theta _{t}\,dt+\sigma \,dW_{t}}$, a lognormal analogue to the Ho–Lee model, and a special case of the Black–Derman–Toy model.[15] This approach is effectively similar to “the original Salomon Brothers model" (1987),[16] also a lognormal variant on Ho-Lee.[17]

Besides the above one-factor models, there are also multi-factor models of the short rate, among them the best known are the Longstaff and Schwartz two factor model and the Chen three factor model (also called "stochastic mean and stochastic volatility model"). Note that for the purposes of risk management, "to create realistic interest rate simulations", these multi-factor short-rate models are sometimes preferred over One-factor models, as they produce scenarios which are, in general, better "consistent with actual yield curve movements".[18]

• The Longstaff–Schwartz model (1992) supposes the short rate dynamics are given by

${\displaystyle {\begin{aligned}dX_{t}&=(a_{t}-bX_{t})\,dt+{\sqrt {X_{t}}}\,c_{t}\,dW_{1t},\\[3pt]dY_{t}&=(d_{t}-eY_{t})\,dt+{\sqrt {Y_{t}}}\,f_{t}\,dW_{2t},\end{aligned}}}$

where the short rate is defined as

${\displaystyle dr_{t}=(\mu X+\theta Y)\,dt+\sigma _{t}{\sqrt {Y}}\,dW_{3t}.}$[19]

• The Chen model (1996) which has a stochastic mean and volatility of the short rate, is given by

${\displaystyle {\begin{aligned}dr_{t}&=(\theta _{t}-\alpha _{t})\,dt+{\sqrt {r_{t}}}\,\sigma _{t}\,dW_{t},\\[3pt]d\alpha _{t}&=(\zeta _{t}-\alpha _{t})\,dt+{\sqrt {\alpha _{t}}}\,\sigma _{t}\,dW_{t},\\[3pt]d\sigma _{t}&=(\beta _{t}-\sigma _{t})\,dt+{\sqrt {\sigma _{t}}}\,\eta _{t}\,dW_{t}.\end{aligned}}}$[20]