On Finite-dimensional Term Structure models

TL;DR

This paper characterizes all finite-dimensional Heath-Jarrow-Morton models capable of fitting any initial yield curve, establishing the uniqueness of affine models with time-dependent coefficients and exploring their properties.

Contribution

It proves that affine models with time-dependent coefficients are the only finite-factor models fitting arbitrary initial yield curves and analyzes their geometric and invariant properties.

Findings

Affine models are uniquely capable of fitting any initial yield curve.

Time-homogeneous affine models have an invariant singular set of initial curves.

Non-affine, functional-dependent volatility structures cannot produce finite-dimensional realizations.

Abstract

In this paper we provide the characterization of all finite-dimensional Heath--Jarrow--Morton models that admit arbitrary initial yield curves. It is well known that affine term structure models with time-dependent coefficients (such as the Hull--White extension of the Vasicek short rate model) perfectly fit any initial term structure. We find that such affine models are in fact the only finite-factor term structure models with this property. We also show that there is usually an invariant singular set of initial yield curves where the affine term structure model becomes time-homogeneous. We also argue that other than functional dependent volatility structures -- such as local state dependent volatility structures -- cannot lead to finite-dimensional realizations. Finally, our geometric point of view is illustrated by several examples.