- 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
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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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