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 ${\displaystyle r_{t}\,}$.

Under a short rate model, the state variable is taken to be the instantaneous spot rate. The short rate, ${\displaystyle r_{t}\,}$, then, is the (continuously compounded, annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time ${\displaystyle t}$. 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 ${\displaystyle r_{t}\,}$ as a under a risk-neutral measure ${\displaystyle Q}$ then the price at time ${\displaystyle t}$ of a zero-coupon bond maturing at time ${\displaystyle T}$ with a payoff of 1 is given by

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