Heath-Jarrow-Morton (HJM) models provide an important framework for understanding the dynamics of the term structure of interest rates. In this article, we focus on exploring the concepts of the long bond, long forward measure, and long-term factorization within these models, based on the function space framework of Filipović (2001).

The Role of the Stochastic Discount Factor

The stochastic discount factor (SDF) is a crucial tool in financial modeling and asset pricing. It provides a mechanism for assigning today's prices to risky future payoffs at different investment horizons. This is accomplished by simultaneously discounting future payoffs and adjusting for risk.

Here, we first need to understand that there is a standard representation of the SDF in which it is divided into two factors: one that discounts at the short-term risk-free interest rate, and a risk-adjusting martingale component. This martingale carries out the change of probabilities from the data-generating (physical) measure P to the risk-neutral measure Q.

The Long-term Factorization of the SDF

However, recent studies have considered another approach – decomposing the pricing kernel (PK) into the long-term discount rate, a process characterizing gross holding period returns on the long bond net of the long-term discount rate, and a positive martingale that defines a long-term forward measure L. This is known as the long-term factorization of the SDF and plays a critical role in the pricing of long-life assets and the investigation of the term structure of risk-return trade-off.

The long-term factorization was originally introduced by Alvarez and Jermann (2005) in discrete-time ergodic economies. Later, Hansen and Scheinkman (2009) extended the long-term factorization to continuous-time Markovian economies, expressing it in terms of the Perron-Frobenius principal eigenfunction of the pricing operator. Qin and Linetsky (2017) further extended the long-term factorization to general semimartingale economies.

Applying Long-term Factorization in Heath-Jarrow-Morton Models

While the existing literature has mainly focused on Markovian model specifications, the goal of this article is to construct the long-term factorization in Heath-Jarrow-Morton term structure models, thereby illustrating how the long-term factorization operates in non-Markovian models. To achieve this, we adopt the viewpoint of Filipović (2001) and view forward curves as elements of an appropriately specified function space.

We provide a sufficient condition on the asymptotic behavior of forward rate volatility that ensures the existence of the long bond process, the long forward measure, and long-term factorization in HJM models. This condition is natural from the interest rate modeling point of view and confirms the existence of the volatility process for the long bond.

In summary, the explicit construction of the long-term factorization in HJM models offers insight into how long-term factorizations arise and furnishes an alternative mechanism for the same, relative to the original theory of Hansen and Scheinkman (2009). Understanding the detailed mechanisms of how this intricate facet of financial economics works is essential for financial modeling and risk management in the bond market and beyond.