Markovian Quadratic Term Structure Models For Risk-free And Defaultable Rates

Abstract. In this paper, a class of regular quadratic Gaussian processes is defined to characterize quadratic term structure models (QTSMs) in a general Markovian setting. The primary motivation for this definition is to provide a more general model for the quadratic term structure of the forward curve, while maintaining the analytical tractability of the traditional QTSMs. It is demonstrated that the tractability of QTSMs does not necessarily rely on the Ornstein-Uhlenbeck state processes used in their traditional definition. Rather, the crucial element that provides analytical solutions for the prices of zero-coupon bonds and their options is a so-called quadratic Gaussian property as defined in this paper. In order to retain this property for a general Markov process, it is shown that, under the regularity conditions, no jumps are allowed in the infinitesimal generator of the process. It is further shown that the coefficient functions defined in the quadratic Gaussian property can be determined by multi-variate Riccati equations with a unique admissible parameter set. The implications of this result for modeling the term structure of risk-free rates and defaultable rates are discussed. Key words and phrases. Quadratic term structure models, option pricing, defaultable rates, timehomogenous Markov processes. We are indebted to Damir Filipović not only for initiating the study of affine processes but also for his comments and suggestions on some proofs in this paper.