# CIR process

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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

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### Look at other dictionaries:

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