- Cox-Ingersoll-Ross model
The

**Cox-Ingersoll-Ross model**in finance is amathematical model describing the evolution ofinterest rate s. It is a type of "one factor model" (Short rate model ) as describes interest rate movements as driven by only one source ofmarket risk . The model can be used in the valuation ofinterest rate derivative s. It was introduced in1985 byJohn C. Cox ,Jonathan E. Ingersoll and Stephen A. Ross as an extension of theVasicek model .The model specifies that the

instantaneous interest rate follows thestochastic differential equation , also named theCIR process ::$dr\_t\; =\; a(b-r\_t),\; dt\; +\; sigmasqrt\{r\_t\},\; dW\_t$

where "W

_{t}" is aWiener process modelling the random market risk factor.The drift factor, $a(b-r\_t)$, is exactly the same as in the Vasicek model. It ensures

mean reversion of the interest rate towards the long run value "b", with speed of adjustment governed by the strictly positive parameter "a".The

standard deviation factor, $sigma\; sqrt\{r\_t\}$, corrects the main drawback of Vasicek's model, ensuring that the interest rate cannot become negative. Thus, at low values of the interest rate, the standard deviation becomes close to zero, cancelling the effect of the random shock on the interest rate. Consequently, when the interest rate gets close to zero, its evolution becomes dominated by the drift factor, which pushes the rate upwards (towards equilibrium).**Bond Pricing**An arbitrage-free bond may be priced using this interest rate process. The bond price is exponential affine in the interest rate:

:$B(t,T)\; =\; exp(A(t,T)\; +\; r(t)\; C(t,T))!$

**ee also***

Hull-White model

*Vasicek model **References***cite book | author=Hull, John C. | title=Options, Futures and Other Derivatives| year=2003 | publisher = Upper Saddle River, NJ:

Prentice Hall | id = ISBN 0-13-009056-5

*cite journal | author=Cox, J.C., J.E. Ingersoll and S.A. Ross | title=A Theory of the Term Structure of Interest Rates | journal=Econometrica | year=1985 | volume=53 | pages=385–407 | doi=10.2307/1911242

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Cox–Ingersoll–Ross model**— Three trajectories of CIR Processes In mathematical finance, the Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. It is a type of one factor model (short rate model) as it describes interest rate movements as… … Wikipedia**Short rate model**— 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 rateThe short rate, usually written r t… … Wikipedia**Vasicek model**— In finance, the Vasicek model is a mathematical model describing the evolution of interest rates. It is a type of one factor model (short rate model) as it describes interest rate movements as driven by only one source of market risk. The model… … Wikipedia**Stephen Ross (economist)**— Stephen Alan Steve Ross is the inaugural Franco Modigliani Professor of Financial Economics at the MIT Sloan School of Management. He is known for initiating several important theories and models in financial economics. He is a widely published… … Wikipedia**John Carrington Cox**— is the Nomura Professor of Finance at the MIT Sloan School of Management. He is one of the world s leading experts on options theory and one of the inventors of the Cox Ross Rubinstein model for option pricing, as well as of the Cox Ingersoll… … Wikipedia**Jonathan E. Ingersoll**— Jonathan Edwards Jon Ingersoll, Jr. is an American economist. He is currently the Adrian C. Israel Professor at Yale School of Management, having previously taught at the University of Chicago.Jonathan Ingersoll s research area is finance,… … Wikipedia**Stephen Ross**— Stephen Alan Ross (* 1944) ist ein US amerikanischer Wirtschaftswissenschaftler und Finanzmathematiker. Sein Name ist eng verbunden mit der Entwicklung der Arbitrage und der Optionspreistheorie. Werdegang, Forschung und Lehre Ross studierte… … Deutsch Wikipedia**CIR process**— The CIR process (named after its creators John C. Cox, Jonathan E. Ingersoll, and Stephen A. Ross) is a Markov process with continuous paths defined by the following stochastic differential equation (SDE): where Wt is a standard Wiener process… … Wikipedia**List of finance topics**— Topics in finance include:Fundamental financial concepts* Finance an overview ** Arbitrage ** Capital (economics) ** Capital asset pricing model ** Cash flow ** Cash flow matching ** Debt *** Default *** Consumer debt *** Debt consolidation ***… … Wikipedia**Outline of finance**— The following outline is provided as an overview of and topical guide to finance: Finance – addresses the ways in which individuals, businesses and organizations raise, allocate and use monetary resources over time, taking into account the risks… … Wikipedia