Evaluation of tranche in securitization and long-range Ising model

Abstract

This econophysics work studies the long-range Ising model of a finite system with N spins and the exchange interaction J / N and the external field H as a model for homogeneous credit portfolio of assets with default probability P d and default correlation ρ d . Based on the discussion on the ( J , H ) phase diagram, we develop a perturbative calculation method for the model and obtain explicit expressions for P d , ρ d and the normalization factor Z in terms of the model parameters N and J , H . The effect of the default correlation ρ d on the probabilities P ( N d , ρ d ) for N d defaults and on the cumulative distribution function D ( i , ρ d ) are discussed. The latter means the average loss rate of the“tranche” (layered structure) of the securities (e.g. CDO), which are synthesized from a pool of many assets. We show that the expected loss rate of the subordinated tranche decreases with ρ d and that of the senior tranche increases linearly, which are important in their pricing and ratings.