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

Historically the first popular model of dynamics of the interest rate is the Vasiček model thatwas proposed in 1977. It had been considered in previous paper [1]. In this model the interest ratehas the normal distribution that obviously is economically inconsistent because in terms of theinterest rate cannot take negative values. At the same time this model was often used for the reasonthat in many cases the relation between the expectations and the variances of real interestrates is that the probability of occurrence of their negative values is very small. However theanalysis of the Vasiček model and the prices of the assets, based on it, is very simple, as leads tolinear 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 thegamma 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 solutionof nonlinear problems. A possibility of deriving of analytical results is principal advantage of affinemodels. Analytical results are important, because otherwise yield should be calculated eitherby 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 parametersneed to be estimated, using the sample data of bond yields. Therefore the literature bydefinition of the prices of bonds, starting papers of Vasiček and Cox, Ingersoll and Ross, concentratedattention on solutions in the closed form. From the practical point of view it is interestingto 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 termstructure of interest rates of yield of zero-coupon bonds, using the Cox- Ingersoll-Ross model inone-factor and multifactor versions. Also comparison of the yield curves and the forward curvesimplying from mentioned above models of behavior of the short-term interest rate is represented.

Keywords

Authors

References