A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written
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The short rate
Under a short rate model, the stochastic state variable is taken to be the instantaneous spot rate. The short rate,
where
Particular short-rate models
Throughout this section
One-factor short-rate models
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. 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. Here, the implementation is usually via a (binomial) short rate tree; see Lattice model (finance)#Interest rate derivatives.
- Merton's model (1973) explains the short rate as
r t = r 0 + a t + σ W t ∗ W t ∗ - The Vasicek model (1977) models the short rate as
d r t = ( θ − α r t ) d t + σ d W t d r t = a ( b − r t ) d t + σ d W t - The Rendleman–Bartter model (1980) explains the short rate as
d r t = θ r t d t + σ r t d W t - The Cox–Ingersoll–Ross model (1985) supposes
d r t = ( θ − α r t ) d t + r t σ d W t d r t = a ( b − r t ) d t + r t σ d W t σ r t - The Ho–Lee model (1986) models the short rate as
d r t = θ t d t + σ d W t - The Hull–White model (1990)—also called the extended Vasicek model—posits
d r t = ( θ t − α r t ) d t + σ t d W t θ , α andσ are not time-dependent. The model may also be applied as lognormal. Lattice-based implementation is usually trinomial. - The Black–Derman–Toy model (1990) has
d ln ( r ) = [ θ t + σ t ′ σ t ln ( r ) ] d t + σ t d W t d ln ( r ) = θ t d t + σ d W t - The Black–Karasinski model (1991), which is lognormal, has
d ln ( r ) = [ θ t − ϕ t ln ( r ) ] d t + σ t d W t - The Kalotay–Williams–Fabozzi model (1993) has the short rate as
d ln ( r t ) = θ t d t + σ d W t
Multi-factor short-rate models
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".
Other interest rate models
The other major framework for interest rate modelling is the Heath–Jarrow–Morton framework (HJM). Unlike the short rate models described above, this class of models is generally non-Markovian. This makes general HJM models computationally intractable for most purposes. The great advantage of HJM models is that they give an analytical description of the entire yield curve, rather than just the short rate. For some purposes (e.g., valuation of mortgage backed securities), this can be a big simplification. The Cox–Ingersoll–Ross and Hull–White models in one or more dimensions can both be straightforwardly expressed in the HJM framework. Other short rate models do not have any simple dual HJM representation.
The HJM framework with multiple sources of randomness, including as it does the Brace–Gatarek–Musiela model and market models, is often preferred for models of higher dimension.