Svenja Hager Pricing Portfolio Credit Derivatives by Means of Evolutionary Algorithms GABLER EDITION WISSENSCHAFT Svenja Hager Pricing Portfolio Credit Derivatives by Means of Evolutionary Algorithms With a foreword by Prof Dr.-Ing Rainer Schöbel GABLER EDITION WISSENSCHAFT Bibliographic information published by Die Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at Dissertation Universität Tübingen, 2007 1st Edition 2008 All rights reserved © Betriebswirtschaftlicher Verlag Dr Th Gabler | GWV Fachverlage GmbH, Wiesbaden 2008 Editorial Office: Frauke Schindler / Anita Wilke Gabler-Verlag is a company of Springer Science+Business Media www.gabler.de No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior permission of the copyright holder Registered and/or industrial names, trade names, trade descriptions etc cited in this publication are part of the law for trade-mark protection and may not be used free in any form or by any means even if this is not specifically marked Cover design: Regine Zimmer, Dipl.-Designerin, Frankfurt/Main Printed on acid-free paper Printed in Germany ISBN 978-3-8349-0915-2 Dedicated to my parents and my beloved husband Foreword Collateralized Debt Obligations (CDOs) are the most prominent example of portfoliorelated credit derivatives They make it possible to diversify and transfer credit risk by pooling and redistributing the risks of an underlying portfolio of defaultable assets It comes as no surprise that the dependence structure of portfolio assets is crucial for the valuation of CDO tranches The standard market model is the Gaussian copula model, which uses only one parameter to summarize the correlations of default times in the underlying credit portfolio Comparable with the volatility smile from option pricing, this simplification leads to an implied correlation smile when the model is confronted with market data There is a growing interest in literature searching for solutions of this problem Dr Svenja Hager contributes to this literature by extending the Gaussian copula model, allowing for a heterogeneous specification of the dependence structure of the underlying portfolio She shows that heterogeneous correlation matrices are able to explain the correlation smile Based on this discovery, she develops a method to find the implied correlation matrix which optimally reproduces the observed tranche spreads of a CDO structure To overcome the complexity of the resulting optimization problems, Evolutionary Algorithms are applied successfully This monograph puts a new complexion on the standard market model and should therefore be recognized for its substantial contribution in this fascinating field of research on credit derivatives Rainer Schöbel Acknowledgements First and foremost, I would like to express gratitude and appreciation for my advisor Prof Dr.-Ing Rainer Schöbel Prof Schöbel gave me valuable academic advice and support during my three years at the University of Tübingen I am thankful that he supervised my dissertation and that he provided me the freedom to research in the exciting area of credit risk modeling I would also like to thank the second referee of this thesis, Prof Dr Joachim Grammig, for his comments and suggestions in our internal seminars Next, I record my gratitude to Dr Axel Hager-Fingerle His enthusiasm for science has always encouraged me to investigate new areas of research Numerous discussions with him significantly improved the ideas expressed in this dissertation I also thank Dr Markus Bouziane for his companionship and for providing plenty of useful information, and Vera Klöckner for her organizational assistance I am much obliged to Ute Hager for her literature search and to Clemens Dangelmayr for his computational assistance I wish to acknowledge financial assistance of the Deutsche Forschungsgemeinschaft by funding my research at the University of Tübingen, and of the Stiftung Landesbank Baden-Württemberg by supporting the publication of this dissertation Moreover, I thank JPMorgan Chase for providing the data sets used in this study I am deeply grateful to my parents for being there for me every single day of my life All the confidence and joy that I possess is born of your love and belief in me I am wholeheartedly thankful for having a wonderful sister who has always been my long-distance confidante Above all, I want to thank my beloved husband I am grateful for your caring love and friendship You make my life complete Svenja Hager Table of Contents List of Tables xvii List of Figures xix List of Notations xxiii Introduction Collateralized Debt Obligations: Structure and Valuation 2.1 Introduction 2.2 Credit Risk Transfer Instruments 2.2.1 Credit Default Swaps 2.2.2 CDS Indices 2.2.3 Collateralized Debt Obligations 10 2.2.3.1 Arbitrage and Balance Sheet CDOs 11 2.2.3.2 Cash Flow and Market Value CDOs 12 2.2.3.3 Static Structures and Managed Structures 13 2.2.3.4 Cash Structures and Synthetic Structures 13 2.2.3.5 Single-Tranche Deals 15 2.2.3.6 Effect of Correlation 16 CDS Index Tranches 17 2.2.4 2.3 Credit Risk Modeling 18 2.3.1 18 Single-Name Credit Risk: Intensity-Based Models 2.3.1.1 Stopping Times and the Hazard Rate Function 19 2.3.1.2 Homogeneous Poisson Processes 22 2.3.1.3 Inhomogeneous Poisson Processes 23 2.3.1.4 Cox Processes 24 2.3.2 Multi-Name Credit Risk: Copula Models 25 2.3.3 Valuation of Synthetic CDOs 29 Table of Contents xii 2.3.3.1 Joint Distribution of Default Times in the Gaussian Copula Approach 30 2.3.3.2 Joint Distribution of Default Times in the Gaussian OneFactor Copula Approach 32 2.3.3.3 Pricing the Default Leg and the Margin Leg of a CDO 33 2.3.3.3.1 The Default Leg 33 2.3.3.3.2 The Margin Leg 34 2.3.3.4 proach 35 2.3.3.5 2.4 Distribution of the Portfolio Loss in the One-Factor ApMonte-Carlo Simulation of CDO Tranche Spreads 35 Valuation of CDOs: Literature 36 Explaining the Implied Correlation Smile 41 3.1 Introduction 41 3.2 Sensitivity of the Tranche Price to the Level of Correlation 42 3.3 The Implied Tranche Correlation 44 3.4 The Implied Correlation Smile 45 3.5 The Implied Base Correlation 47 3.6 Evolution of the Implied Correlation Smile 49 3.7 Modeling the Correlation Smile: Literature 56 3.8 Heterogeneous Dependence Structures 58 3.8.1 Heterogeneous Dependence Structures Can Cause Implied Correlation Smiles 60 3.8.1.1 The Existence Problem 60 3.8.1.2 The Uniqueness Problem 61 3.8.1.3 Exemplary Heterogeneous Matrices 62 3.8.2 Different Dependence Structures Can Lead to Identical Implied 3.8.3 Heterogeneous Dependence Structures Do Not Necessarily Lead to Tranche Correlations Implied Correlation Smiles 3.8.4 63 64 Heterogeneous Dependence Structures Allow for Flexible Portfolio Loss Distributions 3.9 65 Conclusion 65 Optimization by Means of Evolutionary Algorithms 73 4.1 Introduction 73 4.2 Evolutionary Algorithms 74 Chapter Summary and Outlook 145 Even if a relatively liquid and transparent market for tranched credit risk evolved in recent years, we point out that the credit derivatives market is still not yet fully developed At the current stage it is supposable that market imperfections contribute biased effects which aggravate the fit to observed data Irrespective of these misbalances, it is important that pricing models are developed further In this study we illustrated the tremendous influence of heterogeneous default dependence structures on the risk content of portfolio credit derivatives In this regard, frameworks have to be enhanced so as to realistically capture credit risk correlations The development of stochastic correlation models incorporating the dynamics of tranche prices beyond purely spread-driven dynamics will be a major challenge References [1] Ahluwalia R, McGinty L, Beinstein E (2004) A relative value framework for credit correlation JP Morgan, Credit Derivatives Strategy, April [2] Alemdar NM, Ozyildirim S (1998) A genetic game of trade, growth, and externalities Journal of Economic Dynamics and Control 22(6):811-832 [3] Alga E, Tomassini M (2002) Parallelism and evolutionary algorithms IEEE Transactions on Evolutionary Computation 6(5):443-462 [4] Amato JD, Gyntelberg J (2005) CDS index tranches and the pricing of credit risk correlations Bank for International Settlements, BIS Quarterly Review, March [5] Andersen LBG, Sidenius J (2005) CDO pricing with factor models: survey and comments Journal of Credit Risk 1(3):71-88 [6] Andersen LBG, Sidenius J (2004) Extensions to the Gaussian copula: random recovery and random factor loadings Journal of Credit Risk 1(1):29-70 [7] Andersen LBG, Sidenius J, Basu S (2003) All your hedges in one basket Risk, November, 67-72 [8] Arifovic J (1996) The behavior of the exchange rate in the genetic algorithm and experimental economies Journal of Political Economy 104(3):510-541 [9] Arifovic J (1995) Genetic algorithms and inflationary economies Journal of Monetary Economics 36(1):219-243 [10] Arifovic J, Genỗay R (2000) Statistical properties of genetic learning in a model of exchange rate Journal of Economic Dynamics and Control 24(5-7):981-1005 References 148 [11] Axelrod R (1987) The evolution of strategies in the iterated prisoner’s dilemma In: Davis L (ed) Genetic algorithms and simulated annealing Pitman, London, 32-41 [12] Baglioni S, Sorbello D, da Costa Pereira C, Tettamanzi AGB (2000) Evolutionary multiperiod asset allocation In: Whitley D, Goldberg D, Cantu-Paz E, Spector L, Parmee I, Beyer H (eds) Proceedings of the genetic and evolutionary computation conference Morgan Kaufmann, San Francisco, California, 597-604 [13] Bäck T (1996) Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms Oxford University Press, Oxford [14] Baxter M (2006) Dynamic modelling of single-name credits and CDO tranches Working paper, Nomura [15] Bearse PM, Bozdogan H, Schlottmann AM (1997) Empirical econometric modelling of food consumption using a new informational complexity approach Journal of Applied Econometrics 12(5):563-592 [16] Belsham T, Vause N, Wells S (2005) Credit correlation: interpretation and risks Bank of England, Financial Stability Review, Vol 19, December [17] Bielecki TR, Rutkowski M (2002) Credit risk: modeling, valuation, and hedging Springer, Berlin [18] Birchenhall CR (1995) Modular technical change and genetic algorithms Computational Economics 8(3):233-253 [19] Black F, Cox J (1976) Valuing corporate securities: some effects of bond indenture provisions Journal of Finance 31(2):351-367 [20] Black F, Scholes M (1973) The pricing of options and corporate liabilities Journal of Political Economy 81(3):637-654 [21] Blanco R, Brennan S, Marsh IW (2005) An empirical analysis of the dynamic relation between investment-grade bonds and credit default swaps Journal of Finance 60(5):2255-2281 [22] Blochinger W, Dangelmayr C, Schulz S (2006) Aspect-oriented parallel discrete optimization on the Cohesion Desktop Grid Platform IEEE International Symposium on Cluster Computing and the Grid 1(16-19):49-56 References 149 [23] Bluhm C, Overbeck L, Wagner C (2003) An introduction to credit risk modeling Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, Florida [24] Briys E, de Varenne F (1997) Valuing risky fixed rate debt: an extension Journal of Financial and Quantitative Analysis 32(2): 239-248 [25] Bullard J, Duffy J (1998) Learning and the stability of cycles Macroeconomic Dynamics 2(1):22-48 [26] Burtschell X, Gregory J, Laurent J-P (2005a) A comparative analysis of CDO pricing models Working paper, BNP Paribas [27] Burtschell X, Gregory J, Laurent J-P (2005b) Beyond the Gaussian copula: stochastic and local correlation Working paper, BNP Paribas [28] Cantú-Paz E (2000) Markov chain models of parallel genetic algorithms IEEE Transactions on Evolutionary Computation 4(3):216-226 [29] Chen L, Filipović D (2004) Credit derivatives in an affine framework Working paper, Lehman Brothers and University of Munich [30] Chen S-H (ed) (2002) Evolutionary computation in economics and finance Studies in Fuzziness and Soft Computing, Physica Verlag, Heidelberg [31] Chen S-H (2000) Toward an agent-based computational modeling of bargaining strategies in double auction markets with genetic programming Lecture Notes in Computer Sciences 1983:517-531, Springer, Berlin [32] Chen S-H, Kuo TW (1999) Towards an agent-based foundation of financial econometrics: an approach based on genetic-programming artificial markets In: Banzhaf W, Daida J, Eiben A, Garzon M, Honovae V, Jakiela M, Smith R (eds) Proceedings of the Genetic and Evolutionary Computation Conference, Vol 2, Morgan Kaufmann, San Francisco, California, 966-973 [33] Chen S-H, Yeh C-H, (1997a) Trading restrictions, speculative trades and price volatility: an application of genetic programming In: Proceedings of the 3rd International Mendel Conference on Genetic Algorithms, Optimization Problems, Fuzzy Logic, Neural Networks, Rough Sets, 31-37 [34] Chen S-H, Yeh C-H (1997b) Using Genetic Programming to model volatility in financial time series: the case of Nikkei 225 and S&P 500 In: Proceedings of the 4th JAFEE International Conference on Investments and Derivatives, 288-306 References 150 [35] Chidambaran NK, Jevons Lee C-W, Trigueros JR (1998) An adaptive evolutionary approach to option pricing via genetic programming In: Koza J, Banzhaf W, Chellapilla K, Deb K, Dorigo M, Fogel D, Garzon M, Goldberg D, Iba H, Riolo R (eds) Proceedings of the Third Annual Conference on Genetic Programming, Morgan Kaufmann, San Francisco, California, 187-192 [36] Cousseran O, Rahmouni I (2005) The CDO market: functioning and implications in terms of financial stability Banque de France, Financial Stability Review, Vol 6, June [37] Crouhy M, Galai D, Mark R (2000) A comparative analysis of current credit risk models Journal of Banking and Finance 24(1-2):59-117 [38] Dangelmayr C (2006) Evolutionäre Algorithmen und ihre Parallelisierung bei der Bewertung von Kreditderivaten Diplomarbeit, Universität Tübingen, Wirtschaftswissenschaftliche Fakultät [39] Dangelmayr C (2005) Eine aspektorientierte Desktop Grid Platform zur Diskreten Optimierung Diplomarbeit, Universität Tübingen, Fakultät für Informations- und Kognitionswissenschaften [40] Davis M, Lo V (2001a) Infectious defaults Quantitative Finance 1(4):382-387 [41] Davis M, Lo V (2001b) Modelling default correlation in bond portfolios In: Alexander C (ed), Mastering Risk, Vol 2, Financial Times Prentice Hall [42] Dawid H (1999) On the convergence of genetic learning in a double auction market Journal of Economic Dynamics and Control 23(9-10):1545-1567 [43] Demarta S, McNeil AJ (2005) The t copula and related copulas International Statistical Review 73(1):111-129 [44] Deutsche Bundesbank (2004a) Credit Default Swaps: Funktionen, Bedeutung und Informationsgehalt Monatsbericht, Dezember [45] Deutsche Bundesbank (2004b) Instrumente zum Kreditrisikotransfer: Einsatz bei deutschen Banken und Aspekte der Finanzstabilität Monatsbericht, April [46] Duffie D (2004) Time to adapt copula methods for modelling credit risk correlation Risk, April, p 77 References 151 [47] Duffie D, Gˆrleanu N (2001) Risk and valuation of collateralized debt obligations a Financial Analysts Journal 57(1):41-59 [48] Duffie D, Lando D (2001) Term structures of credit spreads with incomplete accounting information Econometrica 69(3):633-664 [49] Duffie D, Pan J, Singleton K (2000) Transform analysis and asset pricing for affine jump diffusions Econometrica 68(6):1343-1376 [50] Duffie D, Singleton KJ (2003) Credit risk: pricing, measurement, and management Princeton University Press, Princeton [51] Duffie D, Singleton KJ (1998) Simulating correlated defaults Working paper, Stanford University [52] Eiben AE, Smith JE (2003) Introduction to Evolutionary Computing Springer, Berlin [53] Elouerkhaoui Y (2003a) Credit derivatives: basket asymptotics Working paper, UBS Warburg [54] Elouerkhaoui Y (2003b) Credit risk: correlation with a difference Working paper, UBS Warburg [55] Embrechts P, Lindskog F, McNeil A (2003) Modelling dependence with copulas and applications to risk management In: Rachev S (ed) Handbook of heavy tailed distributions in finance, Elsevier [56] Fabozzi FJ, Choudhry M, Anson MJP, Chen RR (2004) Credit derivatives: instruments, applications and pricing Wiley, Hoboken, New Jersey [57] Fairbrother S, McCarthy P, Kehn D (2004) The Java Developer’s Guide to Eclipse Addison-Wesley, Boston, Massachusetts [58] Finger C (2004) Issues in the pricing of synthetic CDOs Journal of Credit Risk 1(1):113-124 [59] Flanagan D (2005) Java in a Nutshell O’Reilly, Cambridge, Massachusetts [60] Folino G, Pizzuti C, Spezzano G (2001) Parallel hybrid method for SAT that couples genetic algorithms and local search IEEE Transactions on Evolutionary Computation 5(4):323-334 References 152 [61] Franzen D (2000) Design of Master Agreements for OTC Derivatives Springer, Berlin [62] Frey R, Backhaus J (2004) Portfolio credit risk models with interacting default intensities: a Markovian approach Working paper, University of Leipzig [63] Frey R, McNeil A (2003) Dependent defaults in models of portfolio credit risk Journal of Risk 6(1):59-92 [64] Friend A, Rogge E (2004) Correlation at first sight Economic Notes: Review of Banking, Finance and Monetary Economics 34(2):155-183 [65] Galiani SS (2003) Copula functions and their application in pricing and risk managing multiname credit derivative products Master Thesis, University of London [66] Geske R (1977) The valuation of corporate liabilities as compound options Journal of Financial and Quantitative Analysis 12(4):541-552 [67] Gibson MS (2004) Understanding the risk of synthetic CDOs Finance and Economics Discussion Series, Vol 36 [68] Giesecke K (2003) A simple exponential model for dependent defaults Journal of Fixed Income 13(3):74-83 [69] Giesecke K, Goldberg L (2005) A top down approach to multi-name credit Working paper, Stanford University [70] Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning Addison-Wesley, Reading, Massachusetts [71] Goodman LS (2002) Synthetic CDOs: an introduction Journal of Derivatives 9(3):60-72 [72] Goodman LS, Fabozzi FJ (2002) Collateralized debt obligations: structures and analysis Wiley, Hoboken, New Jersey [73] Gordy MB (2003) A risk-factor model foundation for ratings-based bank capital rules Journal of Financial Intermediation 12(3):199-232 [74] Gordy MB, Jones D (2003) Random tranches Risk, March, 78-83 [75] Grama A, Gupta A, Karypis G, Kumar V (2003) Introduction to Parallel Computing Benjamin-Cummings, San Francisco, California References 153 [76] Greenberg A, Mashal R, Naldi M, Schlögl L (2004) Tuning correlation and tail risk to the market prices of liquid tranches Quantitative Research Quarterly, Lehman Brothers, March [77] Gregory J, Laurent J-P (2004) In the core of correlation Risk, October, 87-91 [78] Gregory J, Laurent J-P (2003) I will survive Risk, June, 103-107 [79] Guégan D, Houdain J (2005) Collateralized debt obligations pricing and factor models: a new methodology using normal inverse gaussian distributions Working paper, Ecole Normale Superieure, Cachan [80] Gupton G, Finger C, Bhatia M (1997) CreditMetrics: the benchmark for understanding credit risk Technical Document, JP Morgan [81] Hager S, Schöbel R (2006) Deriving the dependence structure of portfolio credit derivatives using evolutionary algorithms Lecture Notes in Computer Science, Vol 3994, 340-347, Springer, Berlin [82] Hager S, Schöbel R (2005) A Note on the Correlation Smile Tübinger Diskussionsbeitrag, Vol 297 [83] Hamida SB, Cont R (2005) Recovering volatility from option prices by evolutionary optimization Journal of Computational Finance 8(4):43-76 [84] Haupt RL, Haupt SE (2004) Practical Genetic Algorithms 2nd ed., Wiley, Hoboken, New Jersey [85] Höfling S (2006) Credit risk modeling and valuation: the reduced form approach and copula models Master Thesis, TU München [86] Holland JH (1975) Adaption in natural and artificial systems University of Michigan Press, Ann Arbor, Michigan [87] Hull J, Predescu M, White A (2005) The valuation of correlation-dependent credit derivatives using a structural model Working paper, University of Toronto [88] Hull J, White A (2006) Valuing credit derivatives using an implied copula approach Forthcoming in: Journal of Derivatives [89] Hull J, White A (2004) Valuation of a CDO and an nth to default CDS without Monte-Carlo simulation Journal of Derivatives 12(2):8-23 References 154 [90] Hull J, White A (1995) The impact of default risk on the prices of options and other derivative securities Journal of Banking and Finance 19(2):299-322 [91] Hyder IU (2002) The Barclays Capital guide to cash flow collateralized debt obligations Barclays Capital, March [92] Izumi K, Ueda K (2000) Learning of virtual dealers in an artificial market: comparison with interview data Lecture Notes in Computer Sciences, Vol 1983, 511-516, Springer, Berlin [93] Izumi K, Ueda K (1998) Emergent phenomena in a foreign exchange market: analysis based on an artificial market approach In: Artificial Life VI, MIT Press, Cambridge, Massachusetts, 398-402 [94] Jarrow RA, Turnbull SM (1995) Pricing derivatives on financial securities subject to credit risk Journal of Finance 50(1):53-85 [95] Jarrow RA, Yu F (2001) Counterparty risk and the pricing of defaultable securities Journal of Finance 56(5):1765-1799 [96] Jay White A (1998) A genetic adaptive neural network approach to pricing options: a simulation analysis Journal of Computational Intelligence in Finance 6(5):13-23 [97] Joshi S, Bedau MA (1998) An explanation of generic behavior in an evolving financial market In: Standish R, Henry B, Watt S, Marks R, Stocker R, Green D, Keen S, Bossomaier T (eds) Complex Systems, Complexity between the Ecos: From Ecology to Economics, 327-335 [98] Joshi S, Parker J, Bedau MA (1999) Technical trading creates a prisoner’s dilemma: results from an agent-based model Computational Finance, MIT Press, Cambridge, Massachusetts, 465-479 [99] Joshi MS, Stacey A (2005) Intensity Gamma: a new approach to pricing portfolio credit derivatives Working paper, Royal Bank of Scotland [100] Kaboudan MA (1999) A measure of time series’ predictability using Genetic Programming applied to stock returns Journal of Forecasting 18(5):345-357 [101] Kalemanova A, Schmid B, Werner R (2005) The normal inverse Gaussian distribution for synthetic CDO pricing Technical Report References 155 [102] Kamat R, Oks M (2004) Full correlation matrices for pricing credit portfolio tranches Working Paper, FEA Research [103] Keber C (1999) Option pricing with the genetic programming approach Journal of Computational Intelligence in Finance 7(6):26-36 [104] Lando D (1998) On Cox processes and credit risky securities Review of Derivatives Research 2(2-3):99-120 [105] Laurent J-P, Gregory J (2005) Basket default swaps, CDOs and factor copulas Journal of Risk 7(4):103-122 [106] LeBaron B (2001) Evolution and time horizons in an agent based stock market Macroeconomic Dynamics 5(2):225-254 [107] Li DX (2000) On default correlation: a copula function approach Journal of Fixed Income 9(4):43-54 [108] Lindholm T, Yellin F (1997) Java - die Spezifikation der virtuellen Maschine Addison-Wesley, Bonn [109] Lindskog F, McNeil A (2003) Common Poisson shock models: applications to insurance and credit risk modelling ASTIN Bulletin 33(2):209-238 [110] Litterman R, Iben T (1991) Corporate bond valuation and the term structure of credit spreads Journal of Portfolio Management 17(3):52-64 [111] Liu Y, Yao X (2001) Evolving neural networks for Hang Seng stock index forecast Proceedings of the Congress on Evolutionary Computation, Vol 1, 256-260 [112] Longstaff FA, Schwartz ES (1995) A simple approach to valuing risky fixed and floating rate debt Journal of Finance 50(3):789-819 [113] Lucas DJ (2001) CDO handbook JPMorgan, Global Structured Finance Research, May [114] Lucas DJ, Goodman LS, Fabozzi FJ (2006) Collateralized Debt Obligations: Structures and Analysis 2nd ed., Wiley, Hoboken, New Jersey [115] Lux T, Schornstein S (2005) Genetic learning as an explanation of stylized facts of foreign exchange markets Journal of Mathematical Economics 41(1-2):169-196 References 156 [116] Madan DB, Konikov M, Marinescu M (2004) Credit and basket default swaps Journal of Credit Risk 2(1) [117] Madan DB, Unal H (1998) Pricing the risks of default Review of Derivatives Research 2(2-3):121-160 [118] Mahfoud S, Mani G (1996) Financial forecasting using genetic algorithms Applied Artificial Intelligence 10(6):543-565 [119] Man KF, Tang KS, Kwong S (1999) Genetic Algorithms Springer, London [120] Markowitz HM (1959) Portfolio selection Wiley, New York [121] Marshall A, Olkin I (1967) A multivariate exponential distribution Journal of the American Statistical Association 62(317):30-44 [122] Mashal R, Naldi M, Tejwani G (2004) The implications of implied correlation Lehman Brothers, Quantitative Credit Research Quarterly, July [123] Mashal R, Naldi M, Zeevi A (2003) On the dependence of equity and asset returns Risk, October, 83-87 [124] Mashal R, Zeevi A (2003) Inferring the dependence structure of financial assets: empirical evidence and implications Working paper, Columbia Business School [125] McGinty L, Beinstein E, Ahluwalia R, Watts M (2004) Introducing base correlations Working paper, JPMorgan [126] Meneguzzo D, Vecchiato W (2004) Copula sensitivity in collateralized debt obligations and basket default swaps Journal of Futures Markets 24(1):37-70 [127] Merino S, Nyfeler M (2002) Calculating portfolio loss Risk, August, 82-86 [128] Merton RC (1977) On the pricing of contingent claims and the Modigliani-Miller theorem Journal of Financial Economics 5(2):241-249 [129] Merton RC (1974) On the pricing of corporate debt: the risk structure of interest rates Journal of Finance 29(2):449-470 [130] Miles R (2005) AspectJ Cookbook O’Reilly, Cambridge, Massachusetts [131] Mitchell M (1996) An introduction to genetic algorithms MIT Press, Cambridge, Massachusetts References 157 [132] Moosbrucker T (2006) Explaining the correlation smile using variance gamma distributions Forthcoming in: Journal of Fixed Income [133] Mortensen A (2006) Semi-analytical valuation of basket credit derivatives in intensity-based models Journal of Derivatives 13(4):8-26 [134] Nelsen R (1999) An introduction to copulas Springer, New York [135] Nomura (2005a) U.S Fixed Income 2006 Outlook/2005 Review Nomura Fixed Income Research, December [136] Nomura (2005b) U.S Fixed Income 2005 Mid-Year Outlook/Review Nomura Fixed Income Research, June [137] Nomura (2004a) U.S Fixed Income 2005 Outlook/2004 Review Nomura Fixed Income Research, December [138] Nomura (2004b) U.S Fixed Income 2004 Mid-Year Outlook/Review Nomura Fixed Income Research, July [139] O’Kane D (2001) Credit derivatives explained: market, products, and regulations Lehman Brothers, Structured Credit Research, March [140] O’Kane D, Livesey M (2004) Base correlation explained Lehman Brothers, Quantitative Credit Research, November [141] O’Kane D, Naldi M, Ganapati S, Berd A, Pedersen C, Schlögl L, Mashal R (2003) The Lehman Brothers guide to exotic credit derivatives Lehman Brothers [142] O’Neill M, Brabazon T, Ryan C, Collins JJ (2001) Developing a market timing system using grammatical evolution In: Proceedings of the Genetic and Evolutionary Computation Conference Morgan Kaufmann, San Francisco, California, 1375-1381 [143] Palmer RG, Arthur WB, Holland JH, LeBaron B, Taylor P (1994) Artificial economic life: a simple model of a stockmarket Physica D 75, 264-274 [144] Pye G (1974) Gauging the default premium Financial Analysts Journal 30(1):4952 [145] Pykhtin M, Dev A (2002) Analytical approach to credit risk modeling Risk, March, 26-32 158 References [146] Ramaswamy K, Sundaresan SM (1986) The valuation of floating-rate instruments, theory and evidence Journal of Financial Economics 17(2):251-272 [147] Rechenberg I (1994) Evolutionsstrategie ’94 Frommann-Holzboog, Stuttgart [148] Rechenberg I (1973) Evolutionsstrategie - Optimierung technischer Systeme nach Prinzipien der biologischen Evolution Frommann-Holzboog, Stuttgart [149] Rogge E, Schönbucher PJ (2003) Modelling dynamic portfolio credit risk Working paper, Imperial College London and ETH Zürich [150] Schlögl L, O’Kane D (2005) A note on the large homogeneous portfolio approximation with the student t copula Finance and Stochastics 9(4):577-584 [151] Schöbel R (1999) A note on the valuation of risky corporate bonds OR Spektrum 21(1-2):35-47 [152] Schönbucher PJ (2006) Portfolio losses and the term structure of loss transition rates: a new methodology for the pricing of portfolio credit derivatives Working paper, ETH Zürich [153] Schönbucher PJ (2003) Credit derivatives pricing models: models, pricing, and implementation Wiley, Chichester [154] Schönbucher PJ (2002) Taken to the limit: simple and not-so-simple loan loss distributions Working paper, University of Bonn [155] Schönbucher PJ (2001) Copula-dependent default risk in intensity models Working Paper, University of Bonn [156] Schönbucher PJ (1998) Term structure modelling of defaultable bonds The Review of Derivatives Studies, Special Issue: Credit Risk and Credit Derivatives 2(2-3):161192 [157] Schönbucher PJ, Schubert D (2001) Copula-dependent default risk in intensity models Working paper, University of Bonn [158] Schwefel HP (1995) Evolution and optimum seeking Wiley, New York [159] Schwefel HP (1975) Evolutionsstrategie und numerische Optimierung Dissertation, Technische Universität Berlin References 159 [160] Sidenius J, Piterbag V, Andersen LBG (2005) A new framework for dynamic credit portfolio loss modelling Working paper, Royal Bank of Scotland and Barclay’s Capital and Bank of America [161] Sklar A (1959) Fonctions de répartition n dimensions et leures marges Publications de l’Institut de Statistique de L’Université de Paris, Vol 8, 229-231 [162] St Pierre M, Rousseau E, Zavattero J, van Eyseren O, Arora A, Pugachevsky D, Fourny M, Reyfman A (2004) Valuing and hedging synthetic CDO tranches using base correlations Technical Paper, Bear Stearns, May [163] Streichert F, Ulmer H (2005) JavaEva: a java based framework for Evolutionary Algorithms - Manual and documentation Technical report, University of Tübingen [164] Streichert F, Ulmer H (2004) Evaluating a hybrid encoding and three crossover operators on the constrained portfolio selection problem Congress on Evolutionary Computation, Proceedings Part I, 932-939 [165] Tanenbaum AS, van Steen M (2002) Distributed systems Prentice Hall [166] Tavares PAC, Nguyen TU, Chapovsky A, Vaysburd I (2004) Composite basket model Working paper, Merrill Lynch Credit, July [167] Tsang E, Li J, Markose S, Er H, Salhi A, Iori G (2000) EDDIE in financial decision making Journal of Management and Economics 4(4) [168] van der Voort M (2005) Factor copulas: totally external defaults Working paper, ABN Amro and University of Rotterdam [169] Vasicek O (1987) Probability of loss on loan portfolio Working paper, KMV Corporation [170] Watkinson L, Roosevelt D (2004) The Layman’s guide to implied correlation Product note, Morgan Stanley, May [171] Weicker K (2002) Evolutionäre Algorithmen Teubner, Stuttgart [172] Willemann S (2005a) An evaluation of the base correlation framework for synthetic CDOs Journal of Credit Risk 1(4):180-190 [173] Willemann S (2005b) Fitting the CDO correlation skew: a tractable structural jump-diffusion model Working paper, Aarhus School of Business References 160 [174] Wise BM (1997) Correlation matrix pseudocolor map with variable grouping Eigenvector Research, Inc., http.//www eigenvector com / MATLABarea.html [175] Wong D (2000) Copula from the limit of a multivariate binary model Working paper, Bank of America Corporation, December [176] Zhou C (1997) A jump-diffusion approach to modeling credit risk and valuing defaultable securities Finance and Economics Discussion Paper Series, Vol 15, Board of Governors of the Federal Reserve System ...Svenja Hager Pricing Portfolio Credit Derivatives by Means of Evolutionary Algorithms GABLER EDITION WISSENSCHAFT Svenja Hager Pricing Portfolio Credit Derivatives by Means of Evolutionary Algorithms. .. we focus on credit derivatives which transfer the credit risk exposure between two parties By means of credit derivatives it is possible to acquire or reduce credit risk exposure Credit risk... prominent example of portfoliorelated credit derivatives They make it possible to diversify and transfer credit risk by pooling and redistributing the risks of an underlying portfolio of defaultable