The credit valuation adjustment (CVA) on an over-the-counter derivative transaction is the price of the risk associated with the potential default of the counterparties to the trade. This dissertation provides an introduction to the concept of CVA, beginning with the required backdrop of counterparty risk and the basics of default risk modelling. Right and wrong way risks are central themes of the dissertation. A model for the pricing of both the unilateral and the bilateral CVA on a credit default swap (CDS) is implemented. Each step of this process is explained thoroughly. Results are reported and discussed for a range of parameters. The trends observed in the CDS CVA numbers produced by the model are all justified and the right and wrong way nature of the exposures captured. In addition, the convergence and stability of the numerical schemes utilised are shown to be appropriate. A case study, in which the model is applied to a set of market scenarios, concludes the dissertation. Since the field is far from established, a number of areas are suggested for further research. Copyright / Dissertation (MSc)--University of Pretoria, 2012. / Mathematics and Applied Mathematics / unrestricted

Submitted by Eitan Chernizon (eitan.chernizon@sgcib.com) on 2013-02-15T17:46:24Z No. of bitstreams: 1 MODELAGEM DA DEPENDÊNCIA ENTRE FATORES DE CRÉDITO E MERCADO.pdf: 1474762 bytes, checksum: 19b13b065762c89e556619042eaf016d (MD5) / Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2013-02-18T12:58:52Z (GMT) No. of bitstreams: 1 MODELAGEM DA DEPENDÊNCIA ENTRE FATORES DE CRÉDITO E MERCADO.pdf: 1474762 bytes, checksum: 19b13b065762c89e556619042eaf016d (MD5) / Made available in DSpace on 2013-02-18T13:20:44Z (GMT). No. of bitstreams: 1 MODELAGEM DA DEPENDÊNCIA ENTRE FATORES DE CRÉDITO E MERCADO.pdf: 1474762 bytes, checksum: 19b13b065762c89e556619042eaf016d (MD5) Previous issue date: 2013-02-01 / Apesar das recentes turbulências nos mercados, a utilização de derivativos negociados fora de uma câmara de compensação tem apresentado rápido crescimento, constituindo um dos maiores componentes do mercado financeiro global. A correta inclusão da estrutura de dependência entre fatores de crédito e mercado é de suma importância no apreçamento do risco de crédito adjacente a exposições geradas por derivativos. Este é o apreçamento, envolvendo simulações de Monte Carlo, feito por uma instituição negociante para determinar a redução no valor do seu portfólio de derivativos devido a possibilidade de falência da contraparte. Este trabalho apresenta um modelo com abordagem paramétrica para lidar com a estrutura de dependência, intuitivo e de fácil implementação. Ao mesmo tempo, os números são contrastados com os resultados obtidos através de uma abordagem neutra ao risco para um portfólio replicante, sob o mesmo processo estocástico. O modelo é aplicado sobre um contrato a termo de câmbio, e diferentes cópulas e fatores de correlação são utilizados no processo estocástico. / Despite recent turmoils, the use of derivatives traded outside of a clearinghouse has shown rapid growth and is a major component of the global financial market. The correct inclusion of the dependence structure between market and credit factors is of high importance in the pricing of credit risk exposures generated by the adjacent derivatives. This pricing, involving Monte Carlo simulations, is done by a dealer to determine the reduction in the value of its derivatives portfolio because of the bankruptcy of the counterparty. This paper presents a model with parametric approach to deal with the dependence structure, intuitive and easily implemented. Meanwhile, the numbers are contrasted with results obtained using a risk neutral approach for a replicating portfolio under the same stochastic process. The model is applied on a forward exchange contract, and different copulas and correlation factors are used in the stochastic process.

Nous entamons ce rapport de thèse par l’évaluation de l’espérance espérée qui représente une des composantes majeures des ajustements XVA. Sous l’hypothèse d’indépendance entre l’exposition et les coûts de financement et de crédit, nous dérivons dans le chapitre 3 une représentation nouvelle de l’exposition espérée comme la solution d’une équation différentielle ordinaire par rapport au temps d’observation du défaut. Nous nous basons, pour le cas unidimensionnel, sur des arguments similaires à ceux de la volatilité locale de Dupire. Et pour le cas multidimensionnel, nous nous référons à la formule de la Co-aire. Cette représentation permet d’expliciter l’impact de la volatilité sur l’exposition espérée : Cette valeur temps fait intervenir la volatilité des sous-jacents ainsi que la sensibilité au premier ordre du prix, évalués sur un ensemble fini de points. Malgré des limitations numériques, cette méthode est une approche précise et rapide pour la valorisation de la XVA unitaire en dimension 1 et 2.Les chapitres suivants sont dédiés aux aspects du risque de corrélations entre les enveloppes d’expositions et les coûts XVA. Nous présentons une modélisation du risque général de corrélation à travers une diffusion stochastique multivariée, comprenant à la fois les sous-jacents des dérivés et les intensités de défaut. Dans ce cadre, nous exposons une nouvelle approche de valorisation par développements asymptotiques, telle que le prix d’un ajustement XVA correspond au prix de l’ajustement à corrélation nulle, auquel s’ajoute une somme explicite de termes correctifs. Le chapitre 4 est consacré à la dérivation technique et à l’étude de l’erreur numérique dans le cadre de la valorisation de dérivés contingents au défaut. La qualité des approximations numériques dépend uniquement de la régularité du processus de diffusion de l’intensité de crédit, et elle est indépendante de la régularité de la fonction payoff. Les formules de valorisation pour CVA et FVA sont présentées dans le chapitre 5. Une généralisation des développements asymptotiques pour le cadre bilatéral de défaut est adressée dans le chapitre 6.Nous terminons ce mémoire en abordant un cas du risque spécifique de corrélation lié aux contrats de migration de rating. Au-delà des formules de valorisation, notre contribution consiste à présenter une approche robuste pour la construction et la calibration d’un modèle de transition de ratings consistant avec les probabilités de défaut implicites de marché / The point of departure of this thesis is the valuation of the expected exposure which represents one of the major components of XVA adjustments. Under independence assumptions with credit and funding costs, we derive in Chapter 3 a new representation of the expected exposure as the solution of an ordinary differential equation w.r.t the default time variable. We rely on PDE arguments in the spirit of Dupire’s local volatility equation for the one dimensional problem. The multidimensional extension is addressed using the co-area formula. This forward representation gives an explicit expression of the exposure’s time value, involving the local volatility of the underlying diffusion process and the first order Greek delta, both evaluated only on finite set of points. From a numerical perspective, dimensionality is the main limitation of this approach. Though, we highlight high accuracy and time efficiency for standalone calculations in dimensions 1 and 2.The remaining chapters are dedicated to aspects of the correlation risk between the exposure and XVA costs. We start with the general correlation risk which is classically modeled in a joint diffusion process for market variables and the credit/funding spreads. We present a novel approach based on asymptotic expansions in a way that the price of an XVA adjustment with correlation risk is given by the classical correlation-free adjustment to which is added a sum of explicit correction terms depending on the exposure Greeks. Chapter 4 is consecrated to the technical derivation and error analysis of the expansion formulas in the context of pricing credit contingent derivatives. The accuracy of the valuation approach is independent of the smoothness of the payoff function, but it is related to the regularity of the credit intensity model. This finding is of special interest for pricing in a real financial context. Pricing formulas for CVA and FVA adjustments are derived in Chapter 5, along with numerical experiments. A generalization of the asymptotic expansions to a bilateral default risk setting is addressed in Chapter 6.Our thesis ends by tackling the problem of modeling the specific Right-Way Risk induced by rating trigger events within the collateral agreements. Our major contribution is the calibration of a rating transition model to market implied default probabilities