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A NOTE ON THE DAI–SINGLETON CANONICAL REPRESENTATION OF AFFINE TERM STRUCTURE MODELS*

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  • *

    An earlier version of this paper was circulated under the title “A Note on the Canonical Representation of Affine Diffusion Processes.”

  • We thank Kenneth Singleton for helpful comments. Damir Filipović thanks Darrell Duffie and George Papanicolaou for their kind hospitality during his stay at Stanford University in September 2006. Patrick Cheridito was supported by NSF Grant DMS-0642361, a Rheinstein Award, and a Peek Fellowship. Damir Filipović was supported by WWTF (Vienna Science and Technology Fund) and Swissquote.

Address correspondence to Patrick Cheridito, ORFE Princeton University, Princeton, NJ 08544, USA; e-mail: dito@princeton.edu.

Abstract

Dai and Singleton (2000) study a class of term structure models for interest rates that specify the short rate as an affine combination of the components of an N-dimensional affine diffusion process. Observable quantities in such models are invariant under regular affine transformations of the underlying diffusion process. In their canonical form, the models in Dai and Singleton (2000) are based on diffusion processes with diagonal diffusion matrices. This motivates the following question: Can the diffusion matrix of an affine diffusion process always be diagonalized by means of a regular affine transformation? We show that if the state space of the diffusion is of the form inline image for integers inline image satisfying inline image or inline image, there exists a regular affine transformation of D onto itself that diagonalizes the diffusion matrix. So in this case, the Dai–Singleton canonical representation is exhaustive. On the other hand, we provide examples of affine diffusion processes with state space inline image whose diffusion matrices cannot be diagonalized through regular affine transformation. This shows that for inline image), the assumption of diagonal diffusion matrices may impose unnecessary restrictions and result in an avoidable loss of generality.

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