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.
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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 .
Les mathématiques financières (aussi nommées finance quantitative) sont une branche des mathématiques appliquées ayant pour but la modélisation, la quantification et la compréhension des phénomènes régissant les opérations financières d'une certaine durée (emprunts et placements / investissements) et notamment les marchés financiers. Elles font jouer le facteur temps et utilisent principalement des outils issus de l'actualisation, de la théorie des probabilités, du calcul stochastique, des statistiques et du calcul différentiel.
In finance, bootstrapping is a method for constructing a (zero-coupon) fixed-income yield curve from the prices of a set of coupon-bearing products, e.g. bonds and swaps. A bootstrapped curve, correspondingly, is one where the prices of the instruments used as an input to the curve, will be an exact output, when these same instruments are valued using this curve. Here, the term structure of spot returns is recovered from the bond yields by solving for them recursively, by forward substitution: this iterative process is called the bootstrap method.
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Explore le cadre Heath-Jarrow-Morton pour les modèles de taux d'intérêt et discute de la dynamique des prix des obligations et d'un modèle de taux court Vasiček.
We give a complete characterization of affine term structure (ATS) models based on a general non-negative Markov short rate process. This applies to the classical Cox—Ingersoll—Ross (CIR) model but in
2002
Changing interest rates constitute one of the major risk sources for banks, insurance companies, and other financial institutions. Modeling the term-structure movements of interest rates is a challeng
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We give a complete characterization of affine term structure models based on a general nonnegative Markov short rate process. This applies to the classical CIR model but includes as well short rate pr