利率模型与格子气体的等价性
Equivalence of interest rate models and lattice gases
摘要：
考虑一类短期利率与高斯马尔可夫过程x（t）在终端测度r（t）= a（t）exp（x（t））中的指数成比例的短期利率模型.这些型号包括黑色，德曼，玩具和黑色，卡拉辛斯基型号的终端措施。我们证明了这样的利率模型与具有吸引两体相互作用的晶格气体V（t1，t2）= -Cov（x（t1），x（t2））是等价的。我们详细地研究了x（t）为Ornstein，Uhlenbeck过程的Black，Karasinski模型，证明了它与Kac和Helfand考虑的具有吸引长程两体相互作用V（x，y）= -α（e^{-γ| X - y|} - e^{-γ（x + y）}）。一个明确的解决方案，该模型给出了作为一个总和以上的状态的晶格气，这是用来表明，该模型有一个相变类似于以前发现的黑色，德曼，玩具模型中的终端措施。
英文摘要：
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) = -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.
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