Economy – Quantitative Finance – Pricing of Securities
22 pages, 9 figures, initial draft: Feb 26, 2009, presented to 2009 Quant Congress USA
This paper describes a flexible and tractable bottom-up dynamic correlation modelling framework with a consistent stochastic recovery specification. The stochastic recovery specification only models the first two moments of the spot recovery rate as its higher moments have almost no contribution to the loss distribution and CDO tranche pricing. Observing that only the joint distribution of default indicators is needed to build the portfolio loss distribution, we propose a generic class of default indicator copulas to model CDO tranches, which can be easily calibrated to index tranche prices across multiple maturities. This correlation modelling framework has the unique advantage that the joint distribution of default time and other dynamic properties of the model can be changed separately from the loss distribution and tranche prices. After calibrating the model to index tranche prices, existing top-down methods can be applied to the common factor process to construct very flexible systemic dynamics without changing the already calibrated tranche prices. This modelling framework therefore combines the best features of the bottom-up and top-down models: it is fully consistent with all the single name market information and it admits very rich and flexible spread dynamics. Numerical results from a non-parametric implementation of this modelling framework are also presented. The non-parametric implementation achieved fast and accurate calibration to the index tranches across multiple maturities even under extreme market conditions. A conditional Markov chain method is also proposed to construct the systemic dynamics, which supports an efficient lattice pricing method for dynamic spread instruments. We also showed how to price tranche options as an example of this fast lattice method.
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