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Statistical Physics of Correlated Failures and Defaults

, and . Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)


Systemic failure problems are hot topics in econophysics, financial engineering and computer engineering. For a long time, this area lacks good data and it was difficult to examine their modelings. Recently, empirical data analysis of correlated failures in Wide-area network systems and the researches on market quotes of credit derivatives have clarified their statistical properties. The probability function for the number of failures in storage systems has a long tail, which means that the failures are highly correlated. The implied loss functions of credit portfolios, which are estimated based on the market quotes of iTraxx-CJ, CDX-IG and iTraxx-Europe, also have the same nature. Credit markets expect that the defaults of the assets in the portfolio do not occur independently and their correlations are relatively strong. In this paper, we would like to review some results on a method to study the correlated failure data. In addition, we compare some theoretical models and understand their behaviors. At first, we show the general method to construct correlated binomial models. We introduce conditional correlations $\rho_ij$ and expectations $p_ij$, where the suffix $_ij$ means the condition that $i (j)$ of $N$ variables take $1 (resp. 0)$. Based on them, we show how to construct joint probability function and derive recursive relations, which are necessary conditions to ensure the probability conservation of the model. Next, by using the recursive relations we show how to calibrate the correlation structures ($\rho_ij$ and $p_ij$) based on the probability function. By the method, we compare the correlation structures of several probabilistic models. In addition, we also compare those of empirical probability functions and discuss the physical mechanism to induce them. References:\\ 1) M.Hisakado, K.Kistukawa and S.Mori, J.Phys.A:Math.Gen.39(2006) 15365-15378. \\ 2) S. Mori, K. Kitsukawa and M. Hisakado, Default Distribution and Credit Market Implications (arXiv:physics0609093).\\ 3) S.Mori, K.Kitsukawa, M.Hisakado, Moody's Correlated Binomial Default Distributions for Inhomogeneous Portfolios (arXiv:physics0603036).

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