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2013-12-23T06:48:51Z

122.178.112.241: /* Properties */

{{Merge|Cox–Ingersoll–Ross model|date=September 2010}}

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):
:<math>dr_t = \theta (\mu-r_t)\,dt + \sigma\, \sqrt r_t dW_t\,</math>

where Wt is a standard [[Wiener process]] and <math> \theta\, </math>, <math> \mu\, </math> and <math> \sigma\, </math> are the [[parameter]]s. The parameter <math> \theta\, </math> corresponds to the speed of adjustment, <math> \mu\, </math> to the mean and <math> \sigma\, </math> to volatility.
[[File:CIR Process.png|thumb|right|CIR process]]
This process can be defined as a sum of squared [[Ornstein–Uhlenbeck process]]. The CIR is an [[ergodic]] process, and possesses a stationary distribution.

This process is widely used in [[finance]] to model short term [[interest rate]] (see [[Cox–Ingersoll–Ross model]]). It is also used to model [[stochastic volatility]] in the [[Heston model]].

==Distribution==
*Conditional distribution
Given <math>r_0</math> and defining <math>c_t=\frac{2 \theta}{\sigma^2(1-e^{-\theta t})}</math>, <math>df=\frac{4\theta \mu}{\sigma^2}</math> and <math>ncp_t=2c_t r_0 e^{-\theta t}</math>, it can be shown that <math> 2c_t r_t </math> follows a [[noncentral chi-squared distribution]] with degree of freedom <math>df</math> and non-centrality parameter <math>ncp_t</math>. Note that <math>df</math> is constant.

*Stationary distribution
Provided that <math>2\theta \mu >\sigma^2</math>, the process has a stationary [[gamma distribution]] with shape parameter <math>df/2</math> and scale parameter <math>\frac{\sigma^2}{2\theta}</math>.

==Properties==
*[[Mean reversion]],
*Level dependent volatility (<math>\sigma \sqrt{r_t}</math>),
*For given positive <math>r_0</math> the process will never touch zero, if <math>2\theta\mu\geq\sigma^2</math>; otherwise it can occasionally touch the zero point,
*<math>E[r_t|r_0]=r_0 e^{-\theta t} + \mu (1-e^{-\theta t})</math>, so long term mean is <math>\mu</math>,
*<math>Var[r_t|r_0]=r_0 \frac{\sigma^2}{\theta} (e^{-\theta t}-e^{-2\theta t}) + \frac{\mu\sigma^2}{2\theta}(1-e^{-\theta t})^2</math>.

==Calibration==
*[[Ordinary least squares]]
The continuous SDE can be discretized as follows

<math> r_{t+\Delta t}-r_t =\theta (\mu-r_t)\,\Delta t + \sigma\, \sqrt r_t \epsilon_t </math>,

which is equivalent to

<math> \frac{r_{t+\Delta t}-r_t}{\sqrt r_t} =\frac{\theta\mu\Delta t}{\sqrt r_t}-\theta \sqrt r_t\Delta t + \sigma\, \epsilon_t </math>.This equation can be used for a linear regression.

*Martingale estimation
*[[Maximum likelihood]]

==Simulation==
[[Stochastic simulation]] of the CIR process can be achieved using two variants:
*[[Discretization]]
*Exact

==References==
*{{Cite journal | author=Cox JC, Ingersoll JE and Ross SA | title=A Theory of the Term Structure of Interest Rates | journal=[[Econometrica]]| year=1985 | volume=53 | pages=385–407 | doi=10.2307/1911242}}

{{DEFAULTSORT:Cir Process}}
[[Category:Stochastic processes]]

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