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Portfolio Losses and the Term Structure of Loss Transition Rates: A new methodology for the pricing of portfolio credit derivatives

by Philipp J. Schönbucher of ETH Zürich

June 2006

Abstract: In this paper, we present a model for the joint stochastic evolution of the cumulative loss process of a credit portfolio and of its probability distribution. At any given time, the loss distribution of the portfolio is represented using forward transition rates, i.e. the transition rates of an auxiliary time-inhomogeneous Markov chain which reproduces the desired transition probability distribution. This approach allows a straightforward calibration of the model (e.g. to a full initial term- and strike structure of synthetic CDOs including the correlation smile) and every arbitrage-free loss distribution admits such a representation with forward transition rates. Next, the stochastic evolution of the loss distribution is modelled by equipping the transition rates with stochastic dynamics. Martingale / drift restrictions on these dynamics are derived which ensure absence of arbitrage in the model. Like in HJM-type interest-rate models, the arbitrage-free dynamics are then fully determined by the specification of the volatilities of the transition rates. We present various volatility specifications which easily capture index spread volatility, correlation moves and relative price moves of STCDOs in the framework. Finally, extending the framework to allow for jumps in the transition rates, we show that this modelling framework encompasses the class of multivariate intensity-based models while being more general than e.g. multivariate Cox processes alone.

JEL Classification: G13.

Keywords: Default Correlation, Stochastic Correlation, CDO Pricing, HJM-Models.

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