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| {{Merge|CIR process|date=September 2010}}
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| [[Image:SQRTDiffusion.png|thumb|Three trajectories of CIR Processes]]
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| In [[mathematical finance]], the '''Cox–Ingersoll–Ross model''' (or '''CIR model''') describes the evolution of [[interest rate]]s. 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 can be used in the valuation of [[interest rate derivative]]s. It was introduced in 1985 by [[John C. Cox]], [[Jonathan E. Ingersoll]] and [[Stephen Ross (economist)|Stephen A. Ross]] as an extension of the [[Vasicek model]].
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| == The model ==
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| The CIR model specifies that the [[instantaneous interest rate]] follows the [[stochastic differential equation]], also named the [[CIR process]]:
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| :<math>dr_t = a(b-r_t)\, dt + \sigma\sqrt{r_t}\, dW_t</math>
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| where ''W<sub>t</sub>'' is a [[Wiener process]] modelling the random market risk factor.
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| The drift factor, ''a''(''b'' − ''r''<sub>''t''</sub>), is exactly the same as in the Vasicek model. It ensures [[mean reversion (finance)|mean reversion]] of the interest rate towards the long run value ''b'', with speed of adjustment governed by the strictly positive parameter ''a''.
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| The [[standard deviation]] factor, <math>\sigma \sqrt{r_t}</math>, avoids the possibility of negative interest rates for all positive values of a and b. An interest rate of zero is also precluded if the condition
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| :<math>2 a b \geq \sigma^2 \,</math>
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| is met. More generally, when the rate is at a low level (close to zero), the standard deviation also becomes very small, which dampens the effect of the random shock on the rate. Consequently, when the rate gets close to zero, its evolution becomes dominated by the drift factor, which pushes the rate upwards (towards [[steady state|equilibrium]]).
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| The same process is used in the [[Heston model]] to model stochastic volatility.
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| ===Future distribution===
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| The distribution of future values of a CIR process can be computed in closed form:
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| :<math>r_{t+T} = cY</math>,
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| where <math>c=\frac{(1 - e^{-aT})\sigma^2}{2a}</math>, and ''Y'' is a non-central Chi-Squared distribution with <math>\frac{4ab}{\sigma^2}</math> degrees of freedom and non-centrality parameter <math>2cr_te^{-aT}</math>.
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| ==Bond pricing==
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| Under the no-arbitrage assumption, a bond may be priced using this interest rate process. The bond price is exponential affine in the interest rate:
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| :<math>P(t,T) = A(t,T) \exp(-B(t,T) r_t)\!</math>
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| where
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| <math>
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| A(t,T) = \left(\frac{2h \exp((a+h)(T-t)/2)}{2h + (a+h)(\exp((T-t)h) -1)}\right)^{2ab/\sigma^2}</math>
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| <math>
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| B(t,T) = \frac{2(\exp((T-t)h)-1)}{2h+(a+h)(\exp((T-t)h)-1)}</math>
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| <math> | |
| h = \sqrt{a^2+2\sigma^2}
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| </math> | |
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| ==Extensions==
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| Time varying functions replacing coefficients can be introduced in the model in order to make it consistent with a pre-assigned term structure of interest rates and possibly volatilities. The most general approach is in Maghsoodi (1996). A more tractable approach is in Brigo and Mercurio (2001b) where an external time-dependent shift is added to the model for consistency with an input term structure of rates. A significant extension of the CIR model to the case of stochastic mean and stochastic volatility is given by [[Lin Chen]] (1996) and is known as [[Chen model]]. A CIR process is a special case of a [[basic affine jump diffusion]], which still permits a [[closed-form expression]] for bond prices.
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| ==See also==
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| *[[Hull-White model]]
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| *[[Vasicek model]]
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| *[[Chen model]]
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| *[[CIR process]]
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| ==References==
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| *{{Cite book | author=Hull, John C. | title=Options, Futures and Other Derivatives| year=2003 | publisher = Upper Saddle River, NJ: [[Prentice Hall]] | isbn = 0-13-009056-5}}
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| *{{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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| *{{Cite journal | author=Maghsoodi, Y.| title=Solution of the extended CIR Term Structure and Bond Option Valuation | journal=Mathematical Finance| year=1996 | issue=6 | pages=89–109}}
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| *{{Cite book | title = Interest Rate Models — Theory and Practice with Smile, Inflation and Credit| author = Damiano Brigo, Fabio Mercurio | publisher = Springer Verlag | year = 2001 | edition = 2nd ed. 2006 | isbn = 978-3-540-22149-4}}
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| *{{Cite journal | author=Brigo, Damiano and Fabio Mercurio| title=A deterministic-shift extension of analytically tractable and time-homogeneous short rate models| journal=Finance & Stochastics| year=2001b |volume = 5 |issue=3 | pages=369–388|url=http://ideas.repec.org/a/spr/finsto/v5y2001i3p369-387.html}}
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| {{Bond market}}
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| {{Stochastic processes}}
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| {{DEFAULTSORT:Cox-Ingersoll-Ross Model}}
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| [[Category:Finance theories]]
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| [[Category:Interest rates]]
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| [[Category:Mathematical finance]]
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| [[Category:Fixed income analysis]]
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| [[Category:Stochastic processes]]
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| [[Category:Short-rate models]]
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| [[de:Wurzel-Diffusionsprozess#Cox-Ingersoll-Ross-Modell]]
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