Equivalence of interest rate models and lattice gases

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 to lattice gases with attractive two-body interaction, V (t(1), t(2)) = -Cov [x(t(1)),x(t(2))]. We consider in some detail the Black-Karasinski model with x(t) as an Ornstein-Uhlenbeck process, and show that it is similar to a lattice gas model considered by Kac and Helfand, with attractive long-range two-body interactions, V (x, y) = -alpha(e(-gamma vertical bar x-y vertical bar)-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.