Evaluation of Tranche in Securitization and Long-range Ising Model

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.