Pricing Basket of Credit Default Swaps and Collateralised Debt Obligation by Lévy Linearly Correlated, Stochastically Correlated, and Randomly Loaded Factor Copula Models and Evaluated by the Fast and Very Fast Fourier Transform

摘要

In the last decade, a considerable growth has been added to the volume of the credit riskudderivatives market. This growth has been followed by the current financial marketudturbulence. These two periods have outlined how significant and important are theudcredit derivatives market and its products. Modelling-wise, this growth has parallelisedudby more complicated and assembled credit derivatives products such as mth to defaultudCredit Default Swaps (CDS), m out of n (CDS) and collateralised debt obligationud(CDO).udIn this thesis, the Lévy process has been proposed to generalise and overcome the CreditudRisk derivatives standard pricing model's limitations, i.e. Gaussian Factor CopulaudModel. One of the most important drawbacks is that it has a lack of tail dependence or,udin other words, it needs more skewed correlation. However, by the Lévy Factor CopulaudModel, the microscopic approach of exploring this factor copula models has beenuddeveloped and standardised to incorporate an endless number of distribution alternativesudthose admits the Lévy process. Since the Lévy process could include a variety ofudprocesses structural assumptions from pure jumps to continuous stochastic, then thoseuddistributions who admit this process could represent asymmetry and fat tails as theyudcould characterise symmetry and normal tails. As a consequence they could captureudboth high and low events¿ probabilities.udSubsequently, other techniques those could enhance the skewness of its correlation andudbe incorporated within the Lévy Factor Copula Model has been proposed, i.e. theud'Stochastic Correlated Lévy Factor Copula Model' and 'Lévy Random Factor LoadingudCopula Model'. Then the Lévy process has been applied through a number of proposedudPricing Basket CDS&CDO by Lévy Factor Copula and its skewed versions and evaluated by V-FFT limiting and mixture cases of the Lévy Skew Alpha-Stable distribution and GeneralizedudHyperbolic distribution.udNumerically, the characteristic functions of the mth to default CDS's and udud(n/m) th touddefault CDS's number of defaults, the CDO's cumulative loss, and loss given defaultudare evaluated by semi-explicit techniques, i.e. via the DFT's Fast form (FFT) and theudproposed Very Fast form (VFFT). This technique through its fast and very fast formsudreduce the computational complexity from O(N2) to, respectively, O(N log2 N ) andud O(N ).