| Summary: | The thesis is an investigation into the pricing of credit risk under the intensity framework with a copula generating default dependence between obligors. The challenge of quantifying credit risk and the derivatives that are associated with the asset class has seen an explosion of mathematical research into the topic. As credit markets developed the modelling of credit risk on a portfolio level, under the intensity framework, was unsatisfactory in that either: 1. The state variables of the intensities were driven by diffusion processes and so could not generate the observed level of default correlation (see Schönbucher (2003a)) or, 2. When a jump component was added to the state variables, it solved the problem of low default correlation, but the model became intractable with a high number of parameters to calibrate to (see Chapovsky and Tevaras (2006)) or, 3. Use was made of the conditional independence framework (see Duffie and Garleanu (2001)). Here, conditional on a common factor, obligors’ intensities are independent. However the framework does not produce the observed level of default correlation, especially for portfolios with obligors that are dispersed in terms of credit quality. Practitioners seeking to have interpretable parameters, tractability and to reproduce observed default correlations shifted away from generating default dependence with intensities and applied copula technology to credit portfolio pricing. The one factor Gaussian copula and some natural extensions, all falling under the factor framework, became standard approaches. The factor framework is an efficient means of generating dependence between obligors. The problem with the factor framework is that it does not give a representation to the dynamics of credit risk, which arise because credit spreads evolve with time. A comprehensive framework which seeks to address these issues is developed in the thesis. The framework has four stages: 1. Choose an intensity model and calibrate the initial term structure. 2. Calibrate the variance parameter of the chosen state variable of the intensity model. 3. When extended to a portfolio of obligors choose a copula and calibrate to standard market portfolio products. 4. Combine the two modelling frameworks, copula and intensity, to produce a dynamic model that generates dependence amongst obligors. The thesis contributes to the literature in the following way: • It finds explicit analytical formula for the pricing of credit default swaptions with an intensity process that is driven by the extended Vasicek model. From this an efficient calibration routine is developed. Many works (Jamshidian (2002), Morini and Brigo (2007) and Schönbucher (2003b)) have focused on modelling credit swap spreads directly with modified versions of the Black and Scholes option formula. The drawback of using a modified Black and Scholes approach is that pricing of more exotic structures whose value depend on the term structure of credit spreads is not feasible. In addition, directly modelling credit spreads, which is required under these approaches, offers no explicit way of simulating default times. In contrast, with intensity models, there is a direct mechanism to simulate default times and a representation of the term structure of credit spreads is given. Brigo and Alfonsi (2005) and Bielecki et al. (2008) also consider intensity modelling for the purposes of pricing credit default swaptions. In their works the dynamics of the intensity process is driven by the Cox Ingersoll and Ross (CIR) model. Both works are constrained because the parameters of the CIR model they consider are constant. This means that when there is more than one tradeable credit default swaption exact calibration of the model is usually not possible. This restriction is not in place in our methodology. • The thesis develops a new method, called the loss algorithm, in order to construct the loss distribution of a portfolio of obligors. The current standard approach developed by Turc et al. (2004) requires differentiation of an interpolated curve (see Hagan and West (2006) for the difficulties of such an approach) and assumes the existence of a base correlation curve. The loss algorithm does not require the existence of a base correlation curve or differentiation of an interpolated curve to imply the portfolio loss distribution. • Schubert and Schönbucher (2001) show theoretically how to combine copula models and stochastic intensity models. In the thesis the Schubert and Schönbucher (2001)framework is implemented by combining the extended Vasicek model and the Gaussian copula model. An analysis of the impact of the parameters of the combined models and how they interact is given. This is as follows: – The analysis is performed by considering two products, securitised loans with embedded triggers and leverage credit linked notes with recourse. The two products both have dependence on two obligors, a counterparty and a reference obligor. – Default correlation is shown to impact significantly on pricing. – We establish that having large volatilities in the spread dynamics of the reference obligor or counterparty creates a de-correlating impact: the higher the volatility the lower the impact of default correlation. – The analysis is new because, classically, spread dynamics are not considered when modelling dependence between obligors. • The thesis introduces a notion called the stochastic liquidity threshold which illustrates a new way to induce intensity dynamics into the factor framework. • Finally the thesis shows that the valuation results for single obligor credit default swaptions can be extended to portfolio index swaptions after assuming losses on the portfolio occur on a discretised set and independently to the index spread level.
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