Dynamic polynomial models for the term structure of interest rates

In this paper we discuss a problem quite debated in the literature, namely how to obtain dynamic versions of cross-sectional models for the term structure of interest rates which satisfies some desirable requirements: (a) providing a good reconstruction of market data; (b) having theoretical control of the dynamic no-arbitrage conditions. These requirements are often conflicting: many empirical models in the literature satisfy (a) but not (b), and many no-arbitrage models which control (b) are not satisfactory with respect to (a). We consider a cross-sectional polynomial model for the logarithm of the discount function, expressed in time-to-maturity, and we study two kinds of extensions: a dynamic, arbitrage-free polynomial model and a dynamic, quasi-arbitrage-free polynomial model. We argue that the quasi-arbitrage-free model can give a better reconstruction of the market data (addressing requirement (a)), while allowing to control the approximation error (addressing (b)). The reasons for this behavior are twofold. One one hand, the model might suffer of a specification error if the true curve is not a polynomial. On the other hand, even if the model were correctly specified, the observed prices might only approximately follow an arbitrage-free evolution, due to faulty observations or imperfection in the markets. Therefore, the (polynomial) statistical model has to be regarded as an approximation of the theoretical, arbitrage-free curve, and the problem becomes to study how the arbitrage-free evolution of the theoretical curve reflects on its polynomial approximation. We are able to give a simple answer to this question by using constructive polynomials, namely Bernstein polynomials, and we show two properties of quasi-no-arbitrage for the proposed model. A discrete-time version of the model can be written in a state-space form, and estimation can be obtained by filtering procedures. The performance of our model is illustrated in a study with real data. 1This work was partially supported by grants from Università Bocconi, Ricerca di Base. Francesco Corielli was supported by the Progetto Strategico C.N.R. ”Mathematical Modelling of Economic Phenomena”