Short rate model - Definition and Overview

In the context of interest rate derivatives, a short rate model is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate.

The short rate

The short rate, usually written r_t is the (annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time t. Specifying the current short rate does not specify the entire yield curve. However no-arbitrage arguments show that, under some fairly relaxed technical conditions, if we model the evolution of r_t as a stochastic process under a risk-neutral measure Q then bond prices are given by

<math> P(t,T) = \mathbb{E}[ \exp{(-\int_t^T r_s ds) } | \mathcal{F}_t ] <math>

where <math>\mathcal{F}<math> is the natural filtration for the process. Thus specifying a model for the short rate specifies future bond prices. This means that future instantaneous forward rates are also specified by the usual formula

<math> f(t,T) = - \frac{\partial}{\partial T} P(t,T) <math>

Particular short-rate models

Throughout this section <math>W_t<math> represents a standard Brownian motion and <math>dW_t<math> its differential.

The Ho-Lee model models the short rate as <math>r_t = \theta_t dt + \sigma dW_t<math>

The Hull-White model (also called the Vasicek model almost interchangeably) posits <math>r_t = (\theta_t-\alpha r_t)dt + \sigma_t dW_t<math>. In many presentations one or more of the parameters <math>\theta, \alpha<math> and <math>\sigma<math> are not time-dependent. The process is called a Ornstein-Uhlenbeck or OU process.

The Cox-Ingersoll-Ross model supposes <math>r_t = (\theta_t-\alpha r_t)dt + \sqrt(r_t)\sigma_t dW_t<math>

In the Black-Karasinski model a variable X_t is assumed to follow an OU process and r_t is assumed to follow <math>r_t = \exp{X_t}<math>

Other interest rate models

The other major framework for interest rate modelling is the Heath-Jarrow-Morton framework. Whilst the two frameworks are actually equivalent in scope for modelling interest rates with one source of uncertainty (one driving Brownian motion), the latter, including as it does the Brace-Gatarek-Musiela model and market models, are often preferred for models of higher dimension.

References

• Martin Baxter and Andrew Rennie (1996). Financial Calculus. Cambridge University Press. ISBN 9780521552899.