- 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):
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.
- Conditional distribution
Given r0 and defining , 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 .
- Mean reversion,
- Level dependent volatility (),
- For given positive r0 the process will never touch zero, if ; otherwise it can occasionally touch the zero point,
- E[rt | r0] = r0e − θt + μ(1 − e − θt), so long term mean is μ,
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
Wikimedia Foundation. 2010.