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Equivalence of interest rate models and lattice gases

Dan Pirjol

posted on 05 April 2012

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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.


This paper proves an equivalence between a class of one-factor interest
rate models with a lattice gas (or equivalently an Ising spin chain) with
2-body interactions. Roughly speaking, the volatility corresponds to the
inverse of the temperature. As the volatility in the interest rate model
is increased, the lattice gas cools down and 'condenses' at a critical
temperature. Certain expectation values in the interest rate model which
are important for the calibration of the model are related to the grand
partition function in the lattice gas, and have discontinuous derivatives
at the critical point.