Equivalence of interest rate models and lattice gases

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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 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)=-α(e -γ |x -y |-e -γ (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.

Original languageEnglish
Article number046116
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume85
Issue number4
DOIs
StatePublished - 25 Apr 2012

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