CDOs

Optimisation des cash flows liés à des opérations d'achat ou de vente sur des tranches de CDOs.

Les Collateralized Debt Obligations, ou CDOs, sont des dérivés exotiques obtenus par le principe de titrisation. Les CDOs ne sont plus autant utilisés qu'avant la crise des subprimes de 2007. Au vue de la complexité de ces produits, il semble pertinent d'optimiser les cash flows liés à des opérations d'achat ou de vente sur des tranches de CDOs.

Dans notre article, nous traitons uniquement le cas des CDO synthétiques. Ce dérivé est composé en totalité de Credit Default Swaps et découpé en différentes tranches. La tranche choisie par l'investisseur définit le nombre de CDS concernés par l'opération d'achat ou de vente. Plus l'investissement est risqué, plus le nombre de CDS concernés par les opérations est faible et plus les montants des jambes de paiement seront élevés.

Nous utilisons un modèle binomial (considéré gaussien en raison de la taille de notre population, égale à 125 CDS) dans lequel la corrélation et la probabilité inconditionnelle de défaut sont mises en évidence. Des milliers de simulations sont ensuite effectuées sur la base de ce modèle et pour différents scénarios, afin d'évaluer les cash flows pour un nombre de défauts donné à chaque instant de paiement. Les cash flows ne sont pas uniquement calculés sur une seule tranche achetée ou vendue, mais plutôt sur une combinaison d'achat et de vente des tranches. Par l'intermédiaire d’opérateurs booléens et d'un filtrage logique, on déduit l'ensemble des stratégies et l'évaluation du P&L devient possible. Nous utilisons ensuite l'Analyse en Composante Principale (ACP) pour visualiser les résultats obtenus et mettre alors en évidence les stratégies les plus pertinentes pour chaque scénario. Pour optimiser la totalité des stratégies obtenues, nous avons été amenés à mettre en place la méthode du Simplexe. Pour ce faire, la probabilité inconditionnelle et la corrélation de défaut doivent être déterminées. Dans un premier temps, nous avons utilisé la méthode du bootstrap sur Matlab pour déterminer la probabilité inconditionnelle, puis nous avons fait varier la seconde valeur sur son domaine de définition (0-0,3). Ainsi, pour chaque valeur de corrélation de défaut, nous pouvons calculer l'apparition de défaut et donc déterminer les cash flows associés à chaque tranche. La méthode du Simplexe nous permet donc de déterminer la meilleure combinaison d'achat et de vente des tranches pour chaque échéance.

La modélisation des résultats obtenus fournit une surface qui dépend de la maturité et de la corrélation de défaut. La meilleure combinaison de maturité et de corrélation de défaut est représentée par le rayon de courbure de la plus grande pente de cette surface. Il faut noter que le modèle gaussien que nous utilisons n'est pas réaliste dans les situations de crise. D'autre part, notre système ne permet pas l'achat ou la vente partielle d'une tranche. On considère en effet que l'investisseur achète ou vend la tranche en totalité. Notre travail apporte cependant un support pertinent de conseil pour l'investisseur qui souhaite effectuer des opérations liées aux CDOs synthétiques.

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