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 $\frac{J}{N}$ and the external field $H$ as a modely for homogeneous credit portfolio of assets with default probability $P_d$ and default correlation ${\rho }_d$. Based on the discussion on the $(J,H)$ phase diagram, we develop a perturbative calculation method for the model and obtain explicit expressions for $P_d,{\rho }_d$ and the normalization factor $Z$ in terms of the model parameters $N$ and $J,H$. The effect of the default correlation ${\rho }_d$ on the probabilities $P(N_d,{\rho }_d)$ for $N_d$ defaults and on the cumulative distribution function $D(i,{\rho }_d)$ 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 ${\rho }_d$ and that of the senior tranche increases linearly, which are important in their pricing and ratings.

21 pages, 9 figures

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