# CIR process

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):

$dr_t = \theta (\mu-r_t)\,dt + \sigma\, \sqrt r_t dW_t\,$

where Wt is a standard Wiener process and $\theta\,$, $\mu\,$ and $\sigma\,$ are the parameters. The parameter $\theta\,$ corresponds to the speed of adjustment, $\mu\,$ to the mean and $\sigma\,$ to volatility.

This process can be defined as a sum of squared Ornstein–Uhlenbeck process. The CIR is an ergodic process, and possesses a stationary distribution, which is a gamma.

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 r0 and defining $c_t=\frac{2 \theta}{\sigma^2(1-e^{-\theta t})}$, $df=\frac{4\theta \mu}{\sigma^2}$ and ncpt = 2ctr0e − θt, it can be shown that 2ctrt follows a noncentral chi-squared distribution with degree of freedom df and non-centrality parameter ncpt. Note that df is constant.

• Stationary distribution

Provided that 2θμ > σ2, the process has a stationary gamma distribution with shape parameter df / 2 and scale parameter $\frac{\sigma^2}{2\theta}$.

## Properties

• Mean reversion,
• Level dependent volatility ($\sigma \sqrt{r_t}$),
• For given positive r0 the process will never touch zero, if $2\theta\mu\geq\sigma^2$; otherwise it can occasionally touch the zero point,
• E[rt | r0] = r0e − θt + μ(1 − e − θt), so long term mean is μ,
• $Var[r_t|r_0]=r_0 \frac{\sigma^2}{\theta} (e^{-\theta t}-e^{-2\theta t}) + \frac{\mu\sigma^2}{2\theta}(1-e^{-\theta t})^2$.

## Calibration

The continuous SDE can be discretized as follows

$r_{t+\Delta t}-r_t =\theta (\mu-r_t)\,\Delta t + \sigma\, \sqrt r_t \epsilon_t$,

which is equivalent to

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

## Simulation

• Discretization
• Exact

## References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• cir|ca´di|an|ly — cir|ca|di|an «su KAY dee uhn», adjective. of or having to do with a biological or behavioral process that recurs in an innate daily rhythm, such as the 24 hour cycle of sleep and wakefulness in man: »the circadian activity of sparrows. ╂[<… …   Useful english dictionary

• cir|ca|di|an — «su KAY dee uhn», adjective. of or having to do with a biological or behavioral process that recurs in an innate daily rhythm, such as the 24 hour cycle of sleep and wakefulness in man: »the circadian activity of sparrows. ╂[< Latin circā… …   Useful english dictionary

• cir|can|ni|an — «sur KAN ee uhn», adjective. of or having to do with a biological or behavioral process that recurs in an innate yearly rhythm, as the annual cycle of hibernation and activity in animals. ╂[< Latin circā around + annus year + English an] …   Useful english dictionary

• cir|cum|am|bu|la|tion — «SUR kuhm AM byuh LAY shuhn», noun. 1. a walking around or about. 2. Figurative. an indirect process; a beating about the bush …   Useful english dictionary

• cir|cum|ben|di|bus — «SUR kuhm BEHN duh buhs», noun. a roundabout process or method; twist; turn; circumlocution. ╂[pretended classical derivation < circum + bend + Latin ablative plural ending ibus] …   Useful english dictionary

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• Heston model — In finance, the Heston model is a mathematical model describing the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even… …   Wikipedia

• Cox-Ingersoll-Ross model — The Cox Ingersoll Ross model in finance is a mathematical model describing the evolution of interest rates. It is a type of one factor model (Short rate model) as describes interest rate movements as driven by only one source of market risk. The… …   Wikipedia