Short-rate model
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
However, no-arbitrage arguments show that, under some fairly relaxed technical conditions, if we model the evolution of
Exogenous short rate models are models where such term structure is an input, as the model involves some time dependent functions or shifts that allow for inputing a given market term structure, so that the term structure comes from outside (exogenous).
represents a standard Brownian motion under a risk-neutral probability measure and
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 are endogenous models and 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 a few observed market prices ("calibration") of zero coupon bonds or linear products such as forward rate agreements or swaps, typically, or a best fit is done to these linear products to find the endogenous short rate models parameters that are closest to the market prices.
[5] In this way, exogenous models such as 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, and the remaining parameters can be used for options calibration.
The implementation is usually via a (binomial) short rate tree [6] or simulation; see Lattice model (finance) § Interest rate derivatives and Monte Carlo methods for option pricing, although some short rate models have closed form solutions for zero coupon bonds, and even caps or floors, easing the calibration task considerably.
We now list a number of exogenous short rate models.
The idea of a deterministic shift can be applied also to other models that have desirable properties in their endogenous form.
to the Vasicek model, but due to linearity of the Ornstein-Uhlenbeck process, this is equivalent to making
a time dependent function, and would thus coincide with the Hull-White 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".
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.
