Dynamic copulas: applications to finance & economics

In this thesis, we show that with the growth of credit derivatives markets, new products are continually being created and market liquidity is increasing. After reviewing these products starting out from the credit default swap, CDS, and describing their evolution since their inception in the early 90s, we demonstrate that this development has been market driven, with the mathematical models used for pricing lagging behind. As the market developed, the weak points of the models became apparent and improved models had to be developed. In October 2003 when the work on this thesis started, CDOs (Collateralised Debt Obligations) were becoming standard products. A new generation of products which we will refer to as third generation credit derivatives were starting to come on line: these include forward-starting CDS, forward-starting CDOs, options on CDOs, CPDO (in full) and so forth. In contrast to early products, these derivatives require a dynamic model of the evolution of the “correlation” between the names over time, something which base correlation was not designed to do. Our objective was to develop a family of multivariate copula processes with different types of upper and lower tail dependence so as to be able to reproduce the correlation smiles/skews observed in credit derivatives in practice. We chose to work with a dynamic version of Archimedean copulas because unlike many other copulas found in the literature, they are mathematically consistent multivariate models. Chapter 2 presents two different approaches for developing these processes. The first model developed is a non-additive jump process based on a background gamma process; the second approach is based on time changed spectrally positive Levy process. The first approach is very convenient for simulations; the second approach is based on additive building blocks and hence is a more general. Two applications of these models to credit risk derivatives were carried out. The first one on pricing synthetic CDOs at different maturities (Chapter 5) was presented at the 5th Annual Advances in Econometrics Conference in Baton Rouge, Louisiane, November 3-5 2006 and has been submitted for publication. The second one which presents a comparison of the pricing given by these dynamic copulas with five well-known copula models, has been submitted to the Journal of Derivatives (see Chapter 6). Having tested the basic dynamic copula models in a credit derivative context, we went on to combine this framework with matrix migration approach (Chapter 3). In order to market structured credit derivatives, banks have to get them rated by rating agencies such as S&P, Moody's and Fitch. A key question is the evolution of the rating over time (i.e. its migration). As the latest innovations in the credit derivatives markets such as Constant Proportion Debt Obligation (CPDO) require being able to model credit migration and correlation in order to handle substitutions on the index during the roll, we propose a model for the joint dynamics of credit ratings of several firms. We then proposed a mathematical framework were individual credit ratings are modelled by a continuous time Markov chain, and their joint dynamics are modelled using a copula process. Copulas allow us to incorporate our knowledge of single name credit migration processes, into a multivariate framework. This is further extended with the multi-factor and time changed approach. A multifactor approach is developed within the new formulated dynamic copula processes, and a time changed Levy process is used to introduce dependency on spread dynamics.