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Copulae and correlation products

Picone, D. (2005). Copulae and correlation products. (Unpublished Doctoral thesis, City, University of London)


This thesis studies copula applications to correlation products. There are six self-contained but related projects in this research, with the following objectives: 1) to review reduced-form approaches to model the default process of a single-name obligor and their extension to model the joint distribution of defaults in a portfolio of obligors; 2) to set up the CDO market; 3) to introduce copulae and to provide a justification of copulae as modelling tools; 4) to provide with our view regarding the most suitable copula when modelling complex correlation products; 5) to prepare a time-inhomogeneous intensity model for valuing cash-flow CDOs, which explicitly incorporates the credit rating of the firms in the collateral portfolio as the indicator of the likelihood of default; 6) to prepare a pricing model for CDOs of EDSs.

We found strong evidence that the Clayton copula is a suitable tool when modelling correlation products with Li’s Survival model. The Clayton copula had some important consequences: we noticed a redistribution of losses from the Junior note to the Mezzanine and Senior notes; in addition, it picked up some extra risk in the Senior note, and finally, when compared with the Normal copula, it overestimated the fair compensation of the Senior and the Mezzanine notes and underestimated the fair compensation of the Junior note. We also found the Clayton copula was very adaptable into the dynamic copula framework of Schonbucher and Schubert.

Modelling the notes of cash-flow CDOs with copulae and time-inhomogeneous transition matrices has not been an easy task. This is because the computation of the transition matrices for arbitrary periods of time was based on an annual transition matrix. In addition, this matrix, as most of the empirical annual transition matrices, was not compatible with a continuous Markov process since it did not admit a valid generator. Therefore, we computed a modified version of a true generator. Following this, we successfully applied one method, originally advanced by JLT (1997), to calibrate the adjusted matrix to the S&P’s probabilities of default. Finally, we described how to simulate the credit rating migration of one single credit, and how to join n -credit rating migrations via the Normal copula. Modelling the collateral credit risk in this way is very powerful, since it allowed us to take into account quality trigger linked to the rating-performance of the collateral and to keep the model of the joint credit rating migrations, totally separate with copulae. For example, when there are performance triggers linked to the collateral average rating, our Rating Transition Copula model perfectly captures the diversion of cash from the interest waterfall to the principal waterfall for the benefit of the Senior and Mezzanine notes.

To price single-name Equity Default Swaps and CDOs of Equity Default Swaps, we extended the GARCH option framework of Duan (1995). Volatility of the underlying equity price is the critical factor affecting option prices, and in our EDS model, the variance of the equity return followed a nonlinear GARCH in mean. When pricing single-name EDS, we proposed two nonlinear GARCH in mean (NGARCH-M): normal and /-Student NGARCH-M model. As a benchmark, we assumed that the equity returns moved accordingly to the standard homoskedastic lognormal process of Black & Scholes and priced the single-name EDS with the Rubinstein and Reiner model for binary barrier options. The problem we found with this approach was that the implied volatilities for very deep out-of-the-money put options were not available. When the volatility was modelled as a GARCH process, it was not possible to derive the future distribution of the underlying equity. Therefore, our model relied on Monte Carlo simulations. To ensure that the simulated option price did not violate rational option pricing bounds, we used the empirical martingale simulation originally advanced by Duan and Simonato, coupled with the standard variance reduction technique. To address the issue of how to price a basket of EDSs, we resorted to the concept of copula. With copulae, we were able to decouple the pricing problem: keeping the aspect of modelling the marginal distribution of the equity returns via NGARCH-M, totally separate from addressing the dependence problem.

Publication Type: Thesis (Doctoral)
Subjects: H Social Sciences > HG Finance
Departments: Bayes Business School > Bayes Business School Doctoral Theses
Bayes Business School > Finance
Doctoral Theses
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