Wishart Quadratic Term Structure Models

Abstract

This paper reveals that the class of affine term structure models introduced by Duffie and Kan (1996) is much larger than it has been usually considered in the literature. We study fundamental risk factors, which represent multivariate risk aversion of the consumer or the volatility matrix of the technological activity returns, and argue that they can be defined as symmetric positive matrices. For such matrices we introduce a dynamic affine process called the Wishart autoregressive (WAR) process; this process is used to reveal the associated term structure. In this framework: i) we derive very simple restrictions on the parameters to ensure positive yields at all maturities; ii) we observe that the usual constraint that the volatility matrix of an affine process be diagonal up to a path independent linear invertible transformation can be considerably relaxed. The Wishart Quadratic Term Structure Model is the natural extension of the one-dimensional Cox-Ingersoll-Ross model and of the quadratic models introduced in the literature.