On term structure of yield rates. 2. The Cox - Ingersoll - Ross model

Historically the first popular model of dynamics of the interest rate is the Vasicek model that was proposed in 1977. It had been considered in previous paper [1]. In this model the interest rate has the normal distribution that obviously is economically inconsistent because in terms of the interest rate cannot take negative values. At the same time this model was often used for the reason that in many cases the relation between the expectations and the variances of real interest rates is that the probability of occurrence of their negative values is very small. However the analysis of the Vasicek model and the prices of the assets, based on it, is very simple, as leads to linear problems. Later in 1985 Cox, Ingersoll and Ross had offered other model named else "model with a square root" in which the interest rate takes only nonnegative values and has the gamma distribution. The analysis of interest rates and the prices of the assets based on this model, also supposes analytical results, but they are essentially more complicated, as assume a solution of nonlinear problems. A possibility of deriving of analytical results is principal advantage of affine models. Analytical results are important, because otherwise yield should be calculated either by methods of Monte Karlo, or by methods of solution of the equations with partial derivatives. Both these approach are in the computing ratio labour-intensive, especially when the model parameters need to be estimated, using the sample data of bond yields. Therefore the literature by definition of the prices of bonds, starting papers of Vasicek and Cox, Ingersoll and Ross, concentrated attention on solutions in the closed form. From the practical point of view it is interesting to consider a problem, how much can differ the results obtained by means of these models. Main objective of the present paper is deriving the analytical solutions by the analysis of term structure of interest rates of yield of zero-coupon bonds, using the Cox-Ingersoll-Ross model in one-factor and multifactor versions. Also comparison of the yield curves and the forward curves implying from mentioned above models of behavior of the short-term interest rate is represented.