# Structural Model For Credit Default In One And Higher Dimensions

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## Abstract

In this thesis, we provide a new structural model for default of a single name which is an extension in several directions of Merton's seminal work [41] and also propose a new hierarchical model in higher dimensions in a heterogeneous setting. Our new model takes advantage of the fact that currently much more data is readily available about the equity (stock) markets, and through our analysis, can be translated to the much less transparent credit markets. We show how this can be used to provide volatilities for the default indices in structural models for these same stocks. More importantly, we use the equity data to obtain an implied probability distribution for the firms' liabilities, a quantity that is only reported quarterly, and often with questionable reliability. This completes the structural model for a single firm by specifying (probabilistically) the absorbing default barrier. In particular, we can then obtain the default probability of this firm and capture its Credit Default Swap(CDS) spreads. For several companies selected from different industry sectors, the values that our model obtain are in good agreement with the credit market data. Furthermore, we are able to extend this approach to higher dimensional models (e.g., with 125 firms) where the correlations among the firms are essential. Specifically, we use hierarchical models for which each firm’s default boundary a linear combination of a systematic factor (e.g, the Dow Jones Industrial Average) and an idiosyncratic factor, with firm-to-firm correlations obtained through their correlations with the systemic factor. Once again the parameters for these high dimensional structural models are obtained from equity data and the resulting values for the tranche spreads for the CDX: NAIG Series 17 Collateralized Debt Obligations (CDO) compare favorably with actual market data. In the course of this work we also provide results for the probabilistic inverse first passage problem for a Brownian motion default index: given a default probability, find the probability distribution for linear default barriers (equivalently initial distributions) that reproduce the given default probability.

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