QUADRATIC TERM STRUCTURE MODELS FOR RISK‐FREE AND DEFAULTABLE RATES

TLDR

This paper extends the traditional quadratic term structure model based on an Ornstein–Uhlenbeck process by introducing a class of quadratic processes that preserve analytical tractability in a general Markovian setting. The authors show that adding a quadratic potential to these processes enables modeling of default risk, and the model’s exponent functions are derived from a system of Riccati equations. They prove that only OU processes are conservative quadratic processes, and the derived Riccati system yields unique admissible parameters, providing implications for risk‑free and defaultable rate term‑structure modeling.

Abstract

In this paper, quadratic term structure models (QTSMs) are analyzed and characterized in a general Markovian setting. The primary motivation for this work is to find a useful extension of the traditional QTSM, which is based on an Ornstein–Uhlenbeck (OU) state process, while maintaining the analytical tractability of the model. To accomplish this, the class of quadratic processes, consisting of those Markov state processes that yield QTSM, is introduced. The main result states that OU processes are the only conservative quadratic processes. In general, however, a quadratic potential can be added to allow QTSMs to model default risk. It is further shown that the exponent functions that are inherent in the definition of the quadratic property can be determined by a system of Riccati equations with a unique admissible parameter set. The implications of these results for modeling the term structure of risk‐free and defaultable rates are discussed.