Ingersoll-Ross CIR Model

Although published officially in 1985, the Cox-Ingersoll-Ross model was described in academic circles in 1977, or perhaps even earlier, which would make it the first interest rate model. Like Vasiceks it is a one-factor model that defines interest rate movements in terms of the dynamics of the short rate. It differs, however, in incorporating an additional feature, which relates the variation of the short rate to the level of interest rates. This feature precludes negative interest rates. It also reflects the fact that interest rate volatility rises when rates are high and correspondingly decreases when rates are low. The Cox-lngersoll-Ross model is expressed by equation (4.11). 

Note that the CIR model is often stated with k used to denote the speed of mean reversion as a was earlier. 

Deriving the zero-coupon bond price given this model is formalized in equation (4.12), from Ingersoll (1987), chapter 18. 

Again we use k to denote the speed of mean reversion, and A is a user-defined parameter to adjust the level of this if necessary, connected with the risk premium associated with long-dated bond yields. 

Some researchers— Van Deventer and Imai (1997) cite Fleseker (1993), for example—have stated that the difficulties in determining parameters for the CIR model have limited its use among market practitioners. Van Deventer and Imai, however, conclude that it deserves further empirical analysis and remains worthwhile for practical applications. 

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The traditional one-, two- and multi-factor equilibrium models, known as ajfine term structure models (see James and Webber, 2000 or Duffie, 1996, p. 136). These include Gaussian affine models such as Vasicek, Hull-White and Steeley, where the model describes a process with constant volatility and models that have a square-root volatility such as Cox-Ingersoll-Ross (CIR) ... 

Formulas for bond options were found by Cox, Ingersoll, and Ross using the CIR model (square root process) for short rates, and by Jam-shidian, Rabinovitch, and by Chaplin using the Vasicek model for the short rate process. 

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