|
|
| Line 1: |
Line 1: |
| {{Merge|Cox–Ingersoll–Ross model|date=September 2010}}
| | Greetings. The author's name is Phebe and she feels comfy when individuals use the complete title. Years ago we moved to North Dakota. His wife doesn't like it the way he does but what he truly likes doing is to do aerobics and he's been doing it for fairly a whilst. My working day job is a meter reader.<br><br>Here is my web site :: std testing at home - [http://www.gaysphere.net/user/Z5147 visit the next web site] - |
| | |
| 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]]
| |
Greetings. The author's name is Phebe and she feels comfy when individuals use the complete title. Years ago we moved to North Dakota. His wife doesn't like it the way he does but what he truly likes doing is to do aerobics and he's been doing it for fairly a whilst. My working day job is a meter reader.
Here is my web site :: std testing at home - visit the next web site -