Understanding the Complexities of the Fine Structure of Interest Rates: A Wasserstein Barycenter Learning Approach

https://link.springer.com/article/10.1007/s00521-024-10202-5

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Published: July 11, 2024

Interest rates are among the most closely watched indicators in economics. They influence investment decisions, shape monetary policy, and signal the health of an economy. Yet, the way interest rates of different maturities evolve over time — what economists call the term structure or yield curve — remains one of the most intricate puzzles in financial analysis.

A recent study by Carlo Mari and Cristiano Baldassari , published in Neural Computing and Applications (Springer, 2024), introduces an innovative framework to investigate these complexities. By combining concepts from machine learning, graph theory, and optimal transport, the authors propose a novel methodology that uncovers the common stochastic structure underlying interest rate movements.

From traditional models to unsupervised learning

Classical term-structure models, such as those by Vasicek or Cox–Ingersoll–Ross, describe interest rates through one or more latent factors. While mathematically elegant, these models often fall short in capturing the nonlinearities and extreme events that characterize modern financial markets.

Mari and Baldassari take a fundamentally different approach. Instead of assuming the number or form of underlying factors, they apply unsupervised machine learning to discover them directly from data. Their goal is to identify whether a shared stochastic pattern drives the evolution of interest rates across maturities — and if so, to describe it in a data-driven and interpretable way.

The Wasserstein barycenter as a unifying structure

At the core of the method lies the Wasserstein barycenter, a mathematical construct rooted in optimal transport theory. Given several probability distributions — in this case, the empirical distributions of daily changes in yields across maturities — the barycenter represents a single “central” distribution that minimizes the cumulative Wasserstein distance to all of them.This allows the researchers to synthesize a collective probabilistic representation of the entire yield curve, effectively summarizing its shared dynamics.The barycenter is then modeled using a Gaussian Mixture Model (GMM) estimated via the Expectation–Maximization (EM) algorithm. To ensure robust convergence, the initialization is performed using Graph Machine Learning: time series are converted into visibility graphs, embedded into low-dimensional spaces, and clustered using topological data analysis techniques. This hybrid strategy enhances the stability and interpretability of the estimation process.

Capturing maturity-specific dynamics

After identifying the global structure, the authors extend the model by introducing a term-specific Gaussian component for each maturity. This allows them to fine-tune the representation and capture localized stochastic effects that are not shared across the yield curve. The result is a hierarchical model that simultaneously captures both common and maturity-specific sources of randomness.

Empirical application and findings

The methodology was applied to the U.S. zero-coupon Treasury yield curve between 2018 and 2022, using high-frequency data from the Liu–Wu dataset. Three experimental configurations were tested:

Overall mode – including all maturities from 1 to 360 months,
Short mode – focusing on the 1–12 month segment (money market), and
Long mode – covering maturities from 13 to 360 months (capital market).

The results reveal that a three-component Gaussian Mixture Model estimated on the short-term segment is sufficient to explain most of the stochastic variability observed in the entire yield curve. This suggests that short-term interest rates play a dominant role in shaping the broader term structure, consistent with the view that monetary policy primarily transmits through the short end of the market.

The study also supports the hypothesis of a segmented market, in which short-term and long-term maturities are governed by partially distinct stochastic factors.

Implications and outlook

The proposed framework represents a significant methodological advancement in the modeling of interest rate dynamics. By integrating optimal transport, mixture modeling, and graph-based machine learning, it provides a flexible and fully data-driven tool for studying the fine structure of the yield curve — without imposing restrictive parametric assumptions.

Beyond its immediate application to finance, the approach holds promise for other domains characterized by complex, multi-scale stochastic systems, such as energy markets or macroeconomic indicators.

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