Phase transition in a log-normal Markov functional model

TL;DR

This paper presents an exact solution for a one-dimensional log-normal Markov functional model of interest rates, revealing a phase transition between low and high volatility regimes linked to the zeros of a generating function.

Contribution

It introduces an exact analytical solution for the model and uncovers a phase transition phenomenon related to the distribution of interest rates at different volatility levels.

Findings

Identifies two distinct volatility regimes in the model.

Shows the phase transition is connected to zeros of a generating function.

Provides conditions under which the phase transition occurs.

Abstract

We derive the exact solution of a one-dimensional Markov functional model with log-normally distributed interest rates in discrete time. The model is shown to have two distinct limiting states, corresponding to small and asymptotically large volatilities, respectively. These volatility regimes are separated by a phase transition at some critical value of the volatility. We investigate the conditions under which this phase transition occurs, and show that it is related to the position of the zeros of an appropriately defined generating function in the complex plane, in analogy with the Lee-Yang theory of the phase transitions in condensed matter physics.