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. An interest rate is the price a borrower pays for the use of money he does not own, and the return a lender receives for deferring his consumption, by lending to the borrower. ... A derivative is a financial contract whose payoffs over a period of time are derived from the performance of assets, interest rates, exchange rates, or indices. ... A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... An interest rate is the price a borrower pays for the use of money he does not own, and the return a lender receives for deferring his consumption, by lending to the borrower. ...

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 the price at time t of a zero-coupon bond maturing at time T is given by The US dollar yield curve as at 9 February 2005. ... In economics, arbitrage is the practice of taking advantage of a state of imbalance between two or more markets: a combination of matching deals are struck that exploit the imbalance, the profit being the difference between the market prices. ... In mathematical finance, a risk-neutral measure is a probability measure in which todays fair (i. ... In finance, a bond is a debt security, in which the issuer owes the holders a debt and is obliged to repay the principal and interest (the coupon). ...

where 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 In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with an index set I that is a totally ordered set, subject only to the condition that if i ≤ j in I then Si is contained in Sj. ...

And it's third euivalent, the yields are given as well.

Particular short-rate models

Throughout this section W_t represents a standard Brownian motion and dW_t its differential. An example of 1000 simulated steps of Brownian motion in two dimensions. ... In mathematics, the word differential has various meanings: In calculus, a differential is an infinitesimal change in the value of a function. ...

The Ho-Lee model models the short rate as

The Hull-White model (also called the Vasicek model almost interchangeably) posits . In many presentations one or more of the parameters θ,α and σ are not time-dependent. The process is called an Ornstein-Uhlenbeck process.

The Cox-Ingersoll-Ross model supposes

In the Black-Karasinski model a variable X_t is assumed to follow an Ornstein-Uhlenbeck process and r_t is assumed to follow r_t = expX_t.

In financial mathematics, the Ho-Lee model is a Short rate model of future interest rates. ... The Hull-White model is a mathematical model of future interest rates. ... Oldrich Vasicek (1942-) a Czech mathematician, received his masters degree in math from the Czech Technical Institute, 1964, and a doctorate from Charles University four years later, at the time the tanks of the Soviet Union arrived in Prague to enforce the Brezhnev doctrine. ... In mathematics, the Ornstein-Uhlenbeck process (also known as the Mean reverting process in probability) is a stochastic process given by the following stochastic differential equation where, θ, μ and σ are parameters. ...

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. Heath-Jarrow-Morton framework is a set of techniques to price interest-rate derivatives that stems from the work of D. Heath, R.A. Jarrow and A. Morton in the late 1980s, especially Bond pricing and the term structure of interest rates: a new methodology (1987) -- working paper, Cornell University...

References

• Martin Baxter and Andrew Rennie (1996). Financial Calculus, Cambridge University Press. ISBN 9780521552899.
• Riccardo Rebonato (2002). Modern Pricing of Interest-Rate Derivatives, Princeton University Press. ISBN 0691089736.