Price estimation of basket credit default swaps using numerical and quasianalytical methods

PhD (Risk Analysis), North-West University, Potchefstroom CampusThe introduction of the credit derivatives in finance has facilitated the concept of credit risk transfer, together with its analysis and management. The reason is obviously due to the fact that these derivative instruments can be pre-defined and tailor-made to conform to the needs of its investors, thereby expediting the hedging and diversification of credit risk. This research focuses and gives a broader overview of the multi-name credit derivatives, which have received less attention as compared to the single-name credit derivatives. The basket credit default swaps, as multi-name derivatives have appealing features to the financial investors owing to their substantial leveraging benefits, as well as their less expensive nature. Hence, the overall theme of this thesis is to price the basket credit default swaps (BCDS) using numerical and quasi-analytical techniques. These methods are targeted towards estimating the joint default probability distributions or the joint dependent defaults which characterizes the basket pricing concepts. This investigation was channelled into three major phases, as evident in the main three-part chapters presented in this work. Phase one of this thesis focused on pricing the BCDS using the stochastic default intensity models. Here, we modelled the hazard rate or the intensity default process using the one-factor Vasicek and the Cox-Ingersoll-Ross (CIR) models. Next, we approximated the joint survival probability distribution functions which describes the intensity models under the risk-neutral pricing measure, for both the homogeneous and the heterogeneous portfolios. We next utilized the Monte-Carlo method, under the Gaussian copula model to numerically approximate the default time distribution function. The nth-to-default basket credit swaps, in which the spreads depend on the nth default time, were successfully priced using the above methodologies, as well as the consideration of the effects of different swap parameters to various nth-to-default swaps. Furthermore, this phase equally considered the estimation of the survival probabilities, the swap spreads and the equal-weighted portfolio values defined within the context of the Vasicek and the CIR model. Results obtained showed that the CIR is more applicable in modelling the hazard rate process, and this, in turn, resulted in more efficient pricing of the basket swaps, as compared to the Vasicek counterpart. Also, from the results, we observed that the nth-to-default swap prices behave differently with respect to varying in the intensity rate and the default correlation, as the rank of default protection increases. Hence, we can recommend that investors who wish to trade first-to-default swaps with highest swap premium should sell protections on entities with low correlations. Phase two of the thesis estimated the valuation of the portfolio of credit derivatives (specif-ically the nth-to-default basket swap), as well as the comparative analysis on the effect of using the one-factor elliptical and the Archimedean copula models to swap pricing. Here, we used the Gaussian and the Student-t as elliptical models, as well as the Clayton, the Frank, and the Gumbel as Archimedean copulas, to model the corresponding default times. For the numerical computation, we employed the Monte-Carlo simulations as the benchmark of the estimation process. The break-even swap premium valuation was made viable through the estimation of the default times and the payment leg streams of the contingent claims. Finally, from our results, we investigated the choice of copula models from existing works of literature, and then made our appropriate choice of copulas for BCDS pricing based on their computation time. Furthermore, the corresponding numerical experiments which were pre-sented clearly showed that the selection of the copula model hugely affects the quantitative risk analysis of the portfolio. Finally, the quasi-analytic techniques were employed in Phase three of the thesis to the valuation of the BCDS premiums. Here, the main focus was on the one-factor copula model, which was applied to minimize the dimensionality issues resulting from the basket default swap pricing. The one-factor Gaussian, student-t and the Clayton copula models were used to estimate the conditional default probability. Furthermore, the conditional characteristic function for the corresponding portfolio loss distribution using the Fast Fourier transform was obtained, and then, we retrieved the unconditional characteristic function with the aid of the inverse fast Fourier transform using numerical integration. Hence, the quasi-analytical expressions for the computation of the premium payment leg, the default payment leg and then the nth-to-default swap formulas were derived by incorporating the concept of the Fourier transform, together with the distribution function of a counting process. From our findings, we observed that in the absence of the trending simulation method, a semi-analytic method which involves the applications of the discrete Fourier transform could be utilized to price the basket credit default swaps effectively.Doctora