The Long Bond, Long Forward Measure and Long-Term Factorization in Heath-Jarrow-Morton Models

TL;DR

This paper establishes the existence of the long bond, long forward measure, and long-term factorization of the stochastic discount factor within Heath-Jarrow-Morton models, extending previous results to a function space framework.

Contribution

It provides a sufficient condition on forward rate volatility curves ensuring the existence of key long-term financial processes in HJM models.

Findings

Existence of the long bond process proven.

Long forward measure characterized as a long-term limit.

Conditions on volatility curves for long-term factorization established.

Abstract

This paper proves existence of the long bond, long forward measure and long-term factorization of the stochastic discount factor (SDF) of Alvarez and Jermann (2005) and Hansen and Scheinkman (2009) in Heath-Jarrow-Morton (HJM) models in the function space framework of Filipovic (2001). A sufficient condition on the weight in the Hilbert space of forward rate volatility curves is given that ensures existence of the long bond volatility process, the long bond process and the long-term factorization of the SDF into discounting at the rate of return on the long bond and a martingale component defining the long forward measure, the long-term limit of T-forward measures.