Understanding the complexities of the fine structure of interest rates: a Wasserstein barycenter learning approach

A novel methodology to investigate the fine structure of interest rates based on Machine Learning techniques is discussed. The aim is to capture in an unsupervised way the common stochastic structure that drives the dynamics of interest rates of different maturities. The proposed approach is based on the Wasserstein barycenter, a powerful tool of analysis that allows us to construct, from a set of assigned probability distributions, a single probability distribution that captures the essential features of the whole set. To identify common stochastic factors, a Gaussian Mixture Model is fitted to the Wasserstein barycenter by maximum likelihood using the Expectation-Maximization algorithm with an initialization strategy based on Graph Machine Learning techniques. A fine-tuning of single-maturity interest rates is discussed in an attempt to capture maturity-specific stochastic factors. The proposed analysis also gives us the opportunity to test the hypothesis of a market segmentation into a short-term segment, the money market, and a long-term segment, the capital market, each with its own segment-specific stochastic factors. The methodology is tested on the US zero-coupon Treasury yield curve. The results obtained seem to show that most of the stochastic nature of the dynamics of the US zero-coupon yield curve can be captured by a three-component Gaussian Mixture Model describing the Wasserstein barycenter of the short-term segment of the yield curve.