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 ncp_t = 2c_tr₀e ^{− θt}, it can be shown that 2c_tr_t follows a noncentral chi-squared distribution with degree of freedom df and non-centrality parameter ncp_t. 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[r_t | 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

• Ordinary least squares

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.

• Martingale estimation
• Maximum likelihood

Simulation

• Discretization
• Exact

References

• Cox JC, Ingersoll JE and Ross SA (1985). "A Theory of the Term Structure of Interest Rates". Econometrica 53: 385–407. doi:10.2307/1911242.