Equivalence of interest rate models and lattice gases

TL;DR

This paper demonstrates the equivalence between certain short rate interest rate models and lattice gas systems with specific attractive interactions, providing new insights into their phase transition behavior.

Contribution

It establishes a formal equivalence between interest rate models and lattice gases, including explicit solutions and phase transition analysis for the Black-Karasinski model.

Findings

Interest rate models are equivalent to lattice gases with attractive two-body interactions.

The Black-Karasinski model exhibits a phase transition similar to lattice gas systems.

Explicit solutions relate interest rate models to lattice gas state sums.

Abstract

We consider the class of short rate interest rate models for which the short rate is proportional to the exponential of a Gaussian Markov process x(t) in the terminal measure r(t) = a(t) exp(x(t)). These models include the Black, Derman, Toy and Black, Karasinski models in the terminal measure. We show that such interest rate models are equivalent with lattice gases with attractive two-body interaction V(t1,t2)= -Cov(x(t1),x(t2)). We consider in some detail the Black, Karasinski model with x(t) an Ornstein, Uhlenbeck process, and show that it is similar with a lattice gas model considered by Kac and Helfand, with attractive long-range two-body interactions V(x,y) = -\alpha (e^{-\gamma |x - y|} - e^{-\gamma (x + y)}). An explicit solution for the model is given as a sum over the states of the lattice gas, which is used to show that the model has a phase transition similar to that found previously in the Black, Derman, Toy model in the terminal measure.