Cox–Ingersoll–Ross model

In mathematical finance, the Cox–Ingersoll–Ross (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[1] by John C. Cox, Jonathan E. Ingersoll and Stephen A. Ross as an extension of the Vasicek model, itself an Ornstein–Uhlenbeck process.

The CIR model describes the instantaneous interest rate

with a Feller square-root process, whose stochastic differential equation is where

is a Wiener process (modelling the random market risk factor) and

It ensures mean reversion of the interest rate towards the long run value

, with speed of adjustment governed by the strictly positive parameter

, avoids the possibility of negative interest rates for all positive values of

An interest rate of zero is also precluded if the condition is met.

) also becomes very small, which dampens the effect of the random shock on the rate.

Consequently, when the rate gets close to zero, its evolution becomes dominated by the drift factor, which pushes the rate upwards (towards equilibrium).

for the CIR model, we must use the Fokker-Planck equation: Our interest is in the particular case when

and rearranging terms leads to the equation: Integrating shows us that: Over the range

, this density describes a gamma distribution.

Stochastic simulation of the CIR process can be achieved using two variants: Under the no-arbitrage assumption, a bond may be priced using this interest rate process.

The bond price is exponential affine in the interest rate: where The CIR model uses a special case of a basic affine jump diffusion, which still permits a closed-form expression for bond prices.

Time varying functions replacing coefficients can be introduced in the model in order to make it consistent with a pre-assigned term structure of interest rates and possibly volatilities.

[3] A more tractable approach is in Brigo and Mercurio (2001b)[4] where an external time-dependent shift is added to the model for consistency with an input term structure of rates.

A significant extension of the CIR model to the case of stochastic mean and stochastic volatility is given by Lin Chen (1996) and is known as Chen model.

A more recent extension for handling cluster volatility, negative interest rates and different distributions is the so-called "CIR #" by Orlando, Mininni and Bufalo (2018,[5] 2019,[6][7] 2020,[8] 2021,[9] 2023[10]) and a simpler extension focussing on negative interest rates was proposed by Di Francesco and Kamm (2021,[11] 2022[12]), which are referred to as the CIR- and CIR-- models.