Polynomial Models of Yield Term Structure

Abstract

We consider the possibility of representing yield term structures in the form of polynomials and power series in models with short-term interest rate processes described by stochastic differential equations with polynomial drift and diffusion functions. It is shown that such a representation occurs only in the case when the drift and diffusion functions are polynomials of degree not higher than one. The problem of representing term structures by power series is connected with the solution of an infinite system of first-order ordinary differential equations for the coefficients of the series. This system of equations has singularities that do not allow one to obtain its solution in an analytic form in the general case. In those particular cases, when this was done, the representation of the term structure in the form of a power series does not exist, since the coefficients of the series do not satisfy the required properties. Solutions are constructed for the equation for a family of term structure models that are based on short-term rate processes in which the square of the volatility is proportional to the third power of the short-term rate. The solution of the equation is sought in the form of a definite functional series and, as a result, is reduced to a confluent hypergeometric function. Three versions of the underlying stochastic differential equations for short-term rate processes are considered: with zero drift, linear drift, and quadratic drift. Numerical examples are given for the yield curve and the forward rate curve for each case. Some conditions for the existence of nontrivial solutions of the equation of time structure in the family of processes under consideration are formulated. The results are illustrated for the known Ahn–Gao and CIR (1980) models of the processes of the short-term interest rate.