Cox-Ingersoll-Ross model

In this chapter, we have considered both equilibrium and arbitrage-free interest-rate models. These are one-factor Gaussian models of the term structure of interest rates. We saw that in order to specify a term structure model, the respective authors described the dynamics of the price process, and that this was then used to price a zero-coupon bond. The short-rate that is modelled is assumed to be a risk-free interest rate, and once this is modelled, we can derive the forward rate and the yield of a zero-coupon bond, as well as its price. So, it is possible to model the entire forward rate curve as a function of the current short-rate only, in the Vasicek and Cox-Ingersoll-Ross models, among others. Both the Vasicek and Merton models assume constant parameters, and because of equal probabilities of forward rates and the assumption of a normal distribution, they can, xmder certain conditions relating to the level of the standard deviation, produce negative forward rates. 

Ren-Raw Chen and Louis Scott, Pricing Interest Rate Options in a Two-Factor Cox-Ingersoll-Ross Model of the Term Structure, The Review of Financial Studies 5, no. 4 (1992), pp. 613-636. 

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). 

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) ... 

The Vasicek, Cox-Ingersoll-Ross, Hull-White and other models incorporate mean reversion. As the time to maturity increases and as it approaches infinity, the forward rates converge to a point at the long-run mean reversion level of the current short-rate. This is the limiting level of the forward rate and is a function of the volatility of the current short-rate. As the time to maturity approaches zero, the short-term forward rate converges to the same level as the instantaneous short-rate. In the Merton and Vasicek models, the mean of the short-rate over the maturity period T is assumed to be constant. The same constant for the mean, or the drift of the interest rate, is described in the Ho-Lee model, but not the extended Vasicek or Hull-White model. 

This chapter considers some of the techniques used to fit the model-derived term structure to the observed one. The Vasicek, Brennan-Schwartz, Cox-Ingersoll-Ross, and other models discussed in chapter 4 made various assumptions about the nature of the stochastic process that drives interest rates in defining the term structure. The zero-coupon curves derived by those models differ from those constructed from observed market rates or the spot rates implied by market yields. In general, market yield curves have more-variable shapes than those derived by term-structure models. The interest rate models described in chapter 4 must thus be calibrated to market yield curves. This is done in two ways either the model is calibrated to market instruments, such as money market products and interest rate swaps, which are used to construct a yield curve, or it is calibrated to a curve constructed from market-instrument rates. The latter approach may be implemented through a number of non-parametric methods. 

Model consistency, is the fitting method consistent with a model such as Vasicek or Cox-ingersoll-Ross ... 

Cox, Ingersoll and Ross [22] and Jamshidian [42] demonstrate that closed-form solutions for zero-coupon bond options can be derived for single-factor square root and Gaussian models. More generally, Duffie, Pan and Singleton... 

The first generation of term structure models started with a finite factor modeling of the process dynamics with constant coefficients (e.g. Vasicek [73], Brennan and Schwartz [10], Cox, Ingersoll, and Ross [22]). Due to the fact that this type of models are inconsistent with the current term structure, the second generation of models exhibits time dependent coefficients (e.g. Hull and White [41]). A completely different approaeh starts from the direct modeling of the forward rate dynamies, by using the initial term strueture as an input (e.g. Ho, and Lee [39], Heath, Jarrow, and Morton [35]). 

Brown, R., Schaefer, S., 1994. The term structure of real interest rates and the Cox, Ingersoll and Ross model. J. Financ. Econ. 35, 2-42. 

When calcnlating option prices in a one-factor model, a frequently made assnmption is that the process is driven by the short rate often with a mean reversion featnre linked to the short rate. There are several popnlar models which fall into this category, for example, the Vasicek model, and the Cox, Ingersoll, and Ross model both of which will be discussed in more detail later. Calculating option prices in a two-factor model involves both the short- and long-term rates linked by a mean reversion process. 

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. 

Another approach, which was used by Cox, Ingersoll, and Ross to model the term structure in a general equilibrium environment, consists of a model of the dynamics of interest rates. This process provides a... 

Brown, R., and S. Schaefer. 1994. The Term Structure of Real Interest Rates and the Cox, Ingersoll and Ross Model. Jourrutl of Financial Economics 35(1) 3 2. 

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