## The Cox-Ingersoll-Ross model

Like the Vasicek model, the Cox-Ingersoll-Ross model (*Cox at al., 1985*), which is often cited as the CIR model, is a continuous, affine, one-factor stochastic interest rate model. In this model, the instantaneous interest rate dynamics are given by the following stochastic differential equation:

Here, *α*, *β*, and *σ* are positive constants, *r _{t}* is the interest rate,

*t*is the time, and

*W*denotes the standard Wiener process. It is easy to see that the drift component is the same as in the Vasicek model; hence, the interest rate follows a mean-reverting process again,

_{t}*β*is the long-run average, and

*α*is the rate of adjustment. The difference is that the volatility term is not constant but is proportional to the square root of the interest rate level. This 'small' difference has dramatic consequences regarding the probability distribution of the future short rates. In the CIR model, the interest rate has non-central chi-squared distribution, with the following density...