 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):where Wt is a standard Wiener process and , and are the parameters. The parameter corresponds to the speed of adjustment, to the mean and 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.
Contents
Distribution
 Conditional distribution
Given r_{0} and defining , and ncp_{t} = 2c_{t}r_{0}e ^{− θt}, it can be shown that 2c_{t}r_{t} follows a noncentral chisquared distribution with degree of freedom df and noncentrality parameter ncp_{t}. Note that df is constant.
 Stationary distribution
Provided that 2θμ > σ^{2}, the process has a stationary gamma distribution with shape parameter df / 2 and scale parameter .
Properties
 Mean reversion,
 Level dependent volatility (),
 For given positive r_{0} the process will never touch zero, if ; otherwise it can occasionally touch the zero point,
 E[r_{t}  r_{0}] = r_{0}e ^{− θt} + μ(1 − e ^{− θt}), so long term mean is μ,
 .
Calibration
The continuous SDE can be discretized as follows
,
which is equivalent to
.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.
Categories: Stochastic processes
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