- 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

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