Totouom Tangho, Daniel (2007) Copules dynamiques: applications en finance & en économie. PhD thesis Economie et finance, CERNA - Centre d'économie industrielle, ENSMP p.130.

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Abstract

In this thesis, we show that with the growth of credit derivatives markets, new products are continually being created and market liquidity is increasing. After reviewing these products starting out from the credit default swap, CDS, and describing their evolution since their inception in the early 90s, we demonstrate that this development has been market driven, with the mathematical models used for pricing lagging behind. As the market developed, the weak points of the models became apparent and improved models had to be developed. In October 2003 when the work on this thesis started, CDOs (Collateralised Debt Obligations) were becoming standard products. A new generation of products which we will refer to as third generation credit derivatives were starting to come on line: these include forward-starting CDS, forward-starting CDOs, options on CDOs, CPDO (in full) and so forth. In contrast to early products, these derivatives require a dynamic model of the evolution of the “correlation” between the names over time, something which base correlation was not designed to do.

Our objective was to develop a family of multivariate copula processes with different types of upper and lower tail dependence so as to be able to reproduce the correlation smiles/skews observed in credit derivatives in practice. We chose to work with a dynamic version of Archimedean copulas because unlike many other copulas found in the literature, they are mathematically consistent multivariate models. Chapter 2 presents two different approaches for developing these processes. The first model developed is a non-additive jump process based on a background gamma process; the second approach is based on time changed spectrally positive Levy process. The first approach is very convenient for simulations; the second approach is based on additive building blocks and hence is a more general. Two applications of these models to credit risk derivatives were carried out. The first one on pricing synthetic CDOs at different maturities (Chapter 5) was presented at the 5th Annual Advances in Econometrics Conference in Baton Rouge, Louisiane, November 3-5 2006 and has been submitted for publication. The second one which presents a comparison of the pricing given by these dynamic copulas with five well-known copula models, has been submitted to the Journal of Derivatives (see Chapter 6).

Having tested the basic dynamic copula models in a credit derivative context, we went on to combine this framework with matrix migration approach (Chapter 3). In order to market structured credit derivatives, banks have to get them rated by rating agencies such as S&P, Moody’s and Fitch. A key question is the evolution of the rating over time (i.e. its migration).

As the latest innovations in the credit derivatives markets such as Constant Proportion Debt Obligation (CPDO) require being able to model credit migration and correlation in order to handle substitutions on the index during the roll, we propose a model for the joint dynamics of credit ratings of several firms.

We then proposed a mathematical framework were individual credit ratings are modelled by a continuous time Markov chain, and their joint dynamics are modelled using a copula process. Copulas allow us to incorporate our knowledge of single name credit migration processes, into a multivariate framework.

This is further extended with the multi-factor and time changed approach. A multifactor approach is developed within the new formulated dynamic copula processes, and a time changed Levy process is used to introduce dependency on spread dynamics.

References

Albanese C., O. Chen & A Dalessandro (2005): Dynamic Credit Correlation Model, working paper

Andersen , P., and Borgan, O. (1995): Counting Process Models for Life History Data: A Review, Scandinavian Journal of Statistics, 12, 97-158.

Andersen, L. & J. Sidenius, (2004), Extensions to the Gaussian Copula: Random Recovery and Random Factor Loadings, Journal of Credit Risk, 1(1) 29-70. Available on www.defaultrisk.com

Andersen, L., (2006) Portfolio Losses in Factor Models: Term Structures and Intertemporal Loss Dependence www.defaultrisk.com

Andersen, L., J. Sidenius & S. Basu, (2003), All your Hedges in One Basket, RISK, November, 67-72

Anonymous (2005a), BCs 5yr CDX 22 July 05.xls from website www.wilmott.com

Anonymous (2005b), 5yrCDX22July2005HedgeRatios.xls, from website www.wilmott.com

Barndorff-Nielsen, O.E. (1977a): Infinite divisibility of the hyperbolic and generalized inverse gaussian distributions. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete,38:309–312

Barndorff-Nielsen, O.E. (1977b): Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society London, A(353):151–157

Barndorff-Nielsen O.E.(1977): Normal inverse gaussian distributions and stochastic volatility modelling. Scandinavian Journal of Statistics, 24(1):1–13, 1997.

Barndorff-Nielsen, O.E. and P. Blæsild (1981): Hyperbolic distributions and ramifications: Contributions to theory and application. Statistical Distributions in Scientific Work, 4:19–44

Barndorff-Nielsen, O.E. and Shephard, N. (2001): Non-Gaussian Ornstein: Uhlenbeck-based models and Some of Their Uses in Financial Economics, Journal of the Royal Statistical Society, Series B 63, 167-241

Black and Cox (1976): “Valuing Corporate Securities: Some Effects of Bonds Indenture Provisions,” Journal of Finance, 31,351-367

Black, F. and M. Scholes (1973) "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81: 637-659.



Bohn, J. R. (2000a). "A Survey of Contingent-Claims Approaches to Risky Debt Valuation." Journal of Risk Finance 1(3): 53-78.

Burtschell, X., J. Gregory & J.-P. Laurent (2005a) A comparative analysis of CDO pricing models , working paper, ISFA Actuarial School, University of Lyon & BNPParibas

Burtschell, X., J. Gregory & J.-P. Laurent (2005b) Beyond the Gaussian copula: stochastic & local correlation, Petit Déjeuner de la Finance 12 Octobre 2005, available at http://laurent.jeanpaul.free.fr/Petit_dejeuner_de_la_finance_12_octobre_2005.pdf

Cariboni and W. Schoutens (2006): Jumps in Intensity Models, working paper available at www.defaultrisk.com

Carr, P., Geman, H., Madan, D.H. and Yor, M. (2001): Stochastic Volatility for Levy Processes, Prepublications du Laboratoire de Probabilités et Modèles Aléatoires 645, Universités de Paris 6 & Paris 7, Paris

Cont R. and P. Tankov (2004), Financial Modelling with Jump Processes, Book: Chapman & Hall / CRC Financial Mathematics Series

Brigo D. & al (2007): Calibration of CDO Tranches with the Dynamical Generalized-Poisson Loss Model

Tasche D. & al (2002): Calculating Value-at-Risk contributions in CreditRisk+

Fouque & al (2006): “Multiscale Stochastic Volatility Asymptotics,” SIAM Journal of Multiscale Modelling and Simulation, 2(1), 2003 (22-42)

Fouque & al (2006): Modelling correlated Defaults: First Passage Model under Stochastic Volatility, www.defaultrisk.com

Gordy, M. B. (2001) Calculation of Higher Moments in CreditRisk+ with Applications, Working paper. http://mgordy.tripod.com/

Gregory J. & C. Donald (2005), Structured credit related value strategy, Internal Report, BNPParibas, Sept 2005

Gregory J. & J-P Laurent (2003) I Will Survive, RISK, June 103-107

Hawkes, A.G. & D. Oakes (1974), `A cluster process representation of a self-exciting process', Journal of Applied Probability 11, 493-503.

Hawkes, A.G. (1971), `Spectra of some self-exciting and mutually exciting point processes', Biometrika 58(1), 83-90.

Hull J & A. White (2006) Dynamic models of portfolio credit risk: a simplified approach. Working paper, University of Toronto. 29pp



Hull J. & A. White 2004, Valuation of a CDO and an nth-to-default CDS without Monte Carlo simulation, J Derivatives 2, 8-23

Hull J. and A. White (2006) Dynamic Models of Portfolio Credit Risk: A Simplified Approach, Working paper, University of Toronto. 29pp

Jarrow and Turnbull (1995): Pricing derivatives on financial securities subject to credit risk Journal of Finance 50, pages 53–85

Jarrow, A., Lando D., and Turnbull, M.(1997): A Markov Model for the Term Structure of Credit Risk Spreads, The Review of Financial Studies, 10 (2), 481-523

Joe, H. 1997. Multivariate Models and Dependence Concepts, vol. 37 of Monographs on Statistics and Applied Probability. Chapman and Hall, London, Weinheim, New York.

Joshi, M. & A. Stacey (2005) Intensity Gamma: A New Approach to Pricing Portfolio Credit Derivatives, www.defaultrisk.com

Giesecke & al (2007) Pricing Credit from the Top Down with Affine Point Processes, www.defaultrisk.com

Ken-Iti Sato (1999): Levy Processes and Infinitely Divisible Distributions, Cambridge studies in advanced mathematics

Laurent J-P., & J. Gregory 2005, Basket Default Swaps, CDO’s and Factor Copula, Journal of Risk, 7(4), 103-122

Li, D., 2000, On Default Correlation: a Copula Approach, Journal of Fixed Income, 9, 43-54.

Arnsdorf M. 1 I. Halperin (2007): BSLP: Markovian Bivariate Spread-Loss Modelfor Portfolio Credit Derivatives, www.defaultrisk.com

McGinty & Ahulwalia, (2004a) Base Correlation JP Morgan

McGinty & Ahulwalia, (2004b) Base Correlation JP Morgan

McGinty, L., E. Bernstein, R. Ahulwalia & M. Watts, (2004) Introducing base correlation Credit Derivatives Strategy, JP Morgan

Merton, R. C. (1974). "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates." Journal of Finance 29: 449-470.

Moody’s Investors Service, July 1997. “Moody’s Rating Migration and Credit Quality Correlation, 1920-1996”

Nelsen, R.B. 1999 Introduction to Copulas, Springer New York, 220pp

El Karoui, N. (2000): Introduction au calcul stochastique, Ecole Polytechnique, Département de Mathématiques Appliqués



El Karoui N., & J. Neveu (1998): Probabilités, Ecole Polytechnique, Département de Mathématiques Appliqués

Rogge E. & Schonbucher P., (2003), Modelling Dynamic Portfolio Credit Risk, Working Paper, ETH Zurich.

Wilde, T. (1997):CreditRisk+ A Credit Risk Management Framework, CSFB

Totouom D. & M Armstrong, (2007a), Dynamic Copula Processes: A new way of modeling CDO tranches, to appear in Advances in Econometrics, Elsvier

Totouom D. & M Armstrong, (2007b), Dynamic Copula Processes: Comparison with five 1-factor models for pricing CDOs, Working Paper, Cerna, Ecole des Mines de Paris

Totouom D. & M Armstrong, (2007c), Dynamic copulas processes and forward starting credit derivatives, Working Paper, Cerna, Ecole des Mines de Paris

Totouom D. & M. Armstrong (2005) Dynamic copula processes: a new way of modeling CDO tranches, Working paper, www.cerna.ensmp.fr/Documents/DTT-MA-DynamicCopula.pdf

Van der Voort M. (2006) An Implied Loss Model, Working paper, ABN AMRO Bank & Erasmus University Rotterdam

Vasicek, O. A. (1984, 1999). Credit Valuation. KMV Corporation (now Moody's KMV Company) Working Paper

Schoutens, W., E. Simonsy & J. Tistaertz (2003): A Perfect Calibration! Now What ?

Working paper

Table of content

Acknowledgements

Résumé en français

Chapter 1 : Introduction

1.1 Recent subprime crisis

1.2 Structure of the thesis

1.3 Brief history of credit derivatives up to 2000

1.4 Three generations of credit derivatives

1.5 Pricing first generation products

1.6 Archimedean copulas within a credit derivatives framework

Chapter 2 : Dynamic Copula Model

2.1 Levy processes including gamma processes

2.2 Dynamic copulas from a gamma process perspective

2.3 Dynamic copula processes seen from a Levy perspective

2.4 Modelling default times

Chapter 3 Combining Credit Migration and Copulas

3.1 Construction of single name credit migration & default processes

3.2 Construction of risk neutral single name credit migration proceseses

3.3 Copula approach for dependent credit migration & default processes

Chapter 4 Multi-factor & time-changed approach

4.1 Multi-factor approach for dependent default times

4.2 Time-changed Levy process for dependent spread dynamics

Applications

Chapter 5 A New Way of Modelling CDO Tranches

5.1 Dynamic Archimedean copula processes

5.2 Specific dynamic Archimedean copula process

5.3 Pricing a correlation product: CDO

5.4 Conclusions

Chapter 6 Comparison with five 1-factor models

6.1 Dynamic copula process

6.2 Pricing a correlation product

6.3 Conclusions

Statistiques de consultation

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