Interest Rates and Information Geometry

TL;DR

This paper explores the geometric structure of probability distributions related to interest rates, using information geometry to analyze yield curve dynamics and their representation in Hilbert space.

Contribution

It introduces a geometric framework for interest rate modeling by mapping yield curves to probability densities in Hilbert space, linking arbitrage-free dynamics with information geometry.

Findings

Yield curves can be represented as density functions in Hilbert space.

Interest rate dynamics correspond to processes in the space of smooth densities.

Derived dynamics for moments of the distribution related to yield curves.

Abstract

The space of probability distributions on a given sample space possesses natural geometric properties. For example, in the case of a smooth parametric family of probability distributions on the real line, the parameter space has a Riemannian structure induced by the embedding of the family into the Hilbert space of square-integrable functions, and is characterised by the Fisher-Rao metric. In the nonparametric case the relevant geometry is determined by the spherical distance function of Bhattacharyya. In the context of term structure modelling, we show that minus the derivative of the discount function with respect to the maturity date gives rise to a probability density. This follows as a consequence of the positivity of interest rates. Therefore, by mapping the density functions associated with a given family of term structures to Hilbert space, the resulting metrical geometry can be used to analyse the relationship of yield curves to one another. We show that the general arbitrage-free yield curve dynamics can be represented as a process taking values in the convex space of smooth density functions on the positive real line. It follows that the theory of interest rate dynamics can be represented by a class of processes in Hilbert space. We also derive the dynamics for the central moments associated with the distribution determined by the yield curve.