Cox–Ingersoll–Ross model

In mathematical finance, the Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. It is a type of "one factor model" (short rate model) as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives. It was introduced in 1985 by John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross as an extension of the Vasicek model.

The CIR model specifies that the instantaneous interest rate follows the , also named the CIR Process:

where ${\displaystyle W_{t}}$ is a Wiener process (modelling the random market risk factor) and ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle \sigma \,}$ are the parameters. The parameter ${\displaystyle a}$ corresponds to the speed of adjustment, ${\displaystyle b}$ to the mean and ${\displaystyle \sigma \,}$ to volatility. The drift factor, ${\displaystyle a(b-r_{t})}$, is exactly the same as in the Vasicek model. It ensures mean reversion of the interest rate towards the long run value ${\displaystyle b}$, with speed of adjustment governed by the strictly positive parameter ${\displaystyle a}$.

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