Polynomial Models of Yield Term Structure

Abstract

The possibility of representation of yield term structures in the form of polynomials or power series in models where short-term interest rate processes are described by stochastic differential equations is considered. In most diffusion models of short-term interest rate processes the functions of the drift and diffusion are polynomials. Among the well-known analytical form of yield term structures, corresponding to these models, there is a class of affine models in which term structure is also described by polynomials. The question therefore arises whether there are more such models short-term interest rate processes, for which the term structure are polynomials on the values of the interest rate. The paper shows that the answer to this question in the general case is negative. Such a representation takes place only in the case when the functions of drift and diffusion are polynomials not higher than of first order. Somewhat more complex to analyze is the assumption that in the present case, the term structure could be described by a power series on values of the interest rate. The problem of representation of the term structures by power series is connected with the solution of the infinite system of ordinary differential equations of first order for the coefficients of the series. This system of equations has features that do not allow to obtain the solution in analytical form in the general case. The conditions under which a power series could be a description of the term structure of yields are discussed. It is shown that there are models of the short rate for which the equations for the coefficients of the series are such that the subsequent coefficients are determined only by the previous one. Such models are, in particular, diffusion models of short-term interest rates CIR (1980) and Ahn–Gao. For them the system of differential equations is solved analytically and it is shown that a power series on values of the interest rate cannot be used as a model of term structure for these models. Unfortunately, the proof of this in the general case has not yet been found.