The CIR process is a Markov process

Markov process

In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...

 with continuous paths defined by the following stochastic differential equation

Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....

 (SDE):

where Wt is a standard Wiener process

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

 and , and are the parameter

Parameter

Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

s. 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

Finance

"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...

 to model short term interest rate

Interest rate

An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...

 (see Cox–Ingersoll–Ross model). It is also used to model stochastic volatility

Stochastic volatility

Stochastic volatility models are used in the field of mathematical finance to evaluate derivative securities, such as options. The name derives from the models' treatment of the underlying security's volatility as a random process, governed by state variables such as the price level of the...

 in the Heston model

Heston model

In finance, the Heston model, named after Steven Heston, is a mathematical model describing the evolution of the volatility of an underlying asset...

.

Distribution

• Conditional distribution

Given and defining , and , it can be shown that follows a noncentral chi-squared distribution with degree of freedom and non-centrality parameter . Note that is constant.

• Stationary distribution

Provided that , the process has a stationary gamma distribution with shape parameter and scale parameter .

Properties

• Mean reversion
  Mean reversion
  Mean reversion is a mathematical concept sometimes used for stock investing, but it can be applied to other assets. In general terms, the essence of the concept is the assumption that both a stock's high and low prices are temporary and that a stock's price will tend to move to the average price...
  
  ,
• Level dependent volatility (),
• For given positive the process will never touch zero, if ; otherwise it can occasionally touch the zero point,
• , so long term mean is ,
• .

Calibration

• Ordinary least squares
  Ordinary least squares
  In statistics, ordinary least squares or linear least squares is a method for estimating the unknown parameters in a linear regression model. This method minimizes the sum of squared vertical distances between the observed responses in the dataset and the responses predicted by the linear...
  
  

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
  Maximum likelihood
  In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....