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Concept
Short-rate model
Summary
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 .
Under a short rate model, the stochastic state variable is taken to be the instantaneous spot rate. The short rate, , then, is the (continuously compounded, annualized) interest rate at which an entity can borrow money for an infinitesimally short period of time from time . 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 as a stochastic process under a risk-neutral measure , then the price at time of a zero-coupon bond maturing at time with a payoff of 1 is given by
where is the natural filtration for the process. The interest rates implied by the zero coupon bonds form a yield curve, or more precisely, a zero curve. Thus, specifying a model for the short rate specifies future bond prices. This means that instantaneous forward rates are also specified by the usual formula
Short rate models are often classified as endogenous and exogenous. Endogenous short rate models are short rate models where the term structure of interest rates, or of zero-coupon bond prices , is an output of the model, so it is "inside the model" (endogenous) and is determined by the model parameters. 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).
Throughout this section represents a standard Brownian motion under a risk-neutral probability measure and its differential. Where the model is lognormal, a variable is assumed to follow an Ornstein–Uhlenbeck process and is assumed to follow .
Following are the one-factor models, where a single stochastic factor – the short rate – determines the future evolution of all interest rates.
Official source
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