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Credit Derivatives – Pricing and Bootstrapping Credit Derivatives – Pricing and Bootstrapping Author: Mi Sun University College London MSc Financial Computing Project 2009-2010 Project Supervisor: Christopher Clack Submission Date: 15 September 2010 Disclaimer: This report is submitted as part requirement for the MSc Degree in Financial Computing at University College London. It is substantially the result of my own work except where explicitly indicated in the text. The report may be freely copied and distributed provided the source is explicitly acknowledged. MSc Financial Computing Project 2009-2010 by Mi Sun Abstract This project explores different pricing models for credit derivatives and implements the bootstrapping method for the survival curve and base correlation. The credit derivatives discussed in this report include credit default swap (CDS) and collateralized debt obligation (CDO). The main aim of the project is to make sure that there is no arbitrage opportunity between quoted (CDS and index CDO tranches) instruments and unquoted (bespoke CDO tranches) instruments. In doing so, the objectives include obtaining the CDS survival curve from market spreads. Then together with the standardized index CDO tranche spreads, the CDS survival curve is an input parameter for obtaining the base correlation. Finally using the base correlation, the bespoke (i.e. customized) CDO tranche can be priced consistently. There are seven chapters. The first one introduces the background and underlying logic of the whole project. The second chapter starts to introduce the pricing of credit default swaps (CDSs), and discusses the bootstrapping and interpolation of the CDS survival curve. The third chapter moves on to introduce the pricing of collateralized debt obligations (CDOs), and investigates two models for building the CDO tranche survival curve. The fourth chapter discusses the latest development in credit derivative pricing in the market, namely the stochastic recovery rate. The fifth chapter looks into the correlation between credits and implements the bootstrapping method for base correlation. The sixth chapter briefly discusses Matlab programming techniques. The last chapter provides a summary and critical evaluation of the project. 2 MSc Financial Computing Project 2009-2010 by Mi Sun Acknowledgements First of all, I am very grateful to Afzal Tarar at PwC China for providing me the opportunity to meet Colin Lawrence at the Financial Service Authority. I want to particularly thank Colin Lawrence for introducing me to the Quantitative Modeling Team (formerly the Analytics and Risk Technology Team) at the FSA. This team is definitely the perfect place to hold the project for my degree in financial computing. I would like to thank all the people from the Quantitative Modeling Team. Each of them has provided me with a lot of help and advice. I also want to say millions of thanks to my dear friend Fatema and my parents for providing me immense support at the most difficult time of my personal life. Finally, I would like to reserve my biggest thanks for Igor Tsatskis, Don Stewart, and Dan Oprescu at the Quantitative Modeling Team for their very generous support and great encouragement during the project. 3 MSc Financial Computing Project 2009-2010 by Mi Sun Table of Contents Chapter 1. Background ...........................................................................................................5 1.1 Credit Derivatives ........................................................................................................................................................ 5 1.2 The Pricing Logical Flow.......................................................................................................................................... 7 Chapter 2 The CDS Pricing Box ............................................................................................9 2.1 The Direct Problem ..................................................................................................................................................... 9 2.2 The Inverse Problem and the Bootstrap..............................................................................................................12 2.3 Interpolation.................................................................................................................................................................15 Chapter 3 The CDO Pricing Box..........................................................................................20 3.1 The Tranche Loss and Tranche Survival Curve...............................................................................................20 3.2 The Legs........................................................................................................................................................................23 3.3 The Portfolio Loss Distribution and the Gaussian Copula Model.............................................................25 3.4 The Tranche Survival Curve under the FHP Model.......................................................................................27 3.5 The Tranche Survival Curve under the LHP Model.......................................................................................29 Chapter 4 The Stochastic Recovery Rate ............................................................................36 Chapter 5 A Discussion on Default Correlation .................................................................43 5.1 Compound Correlation and Correlation Smile.................................................................................................43 5.2 Base Correlation Bootstrap.....................................................................................................................................45 Chapter 6 Programming with Matlab .................................................................................47 Chapter 7 Summary and Evaluation ...................................................................................50 7.1 Summary .......................................................................................................................................................................50 7.2 Critical Evaluation .....................................................................................................................................................51 Appendix A User Manual......................................................................................................53 Appendix B Code Listing ......................................................................................................56 Appendix C Published M-Files.............................................................................................78 Bibliography...........................................................................................................................86 4 MSc Financial Computing Project 2009-2010 by Mi Sun Chapter 1. Background This project explores different pricing models for credit derivatives and implements the bootstrapping method for the survival curve and base correlation. The credit derivatives discussed in this report include credit default swap (CDS) and collateralized debt obligation (CDO). The main aim of the project is to make sure that there is no arbitrage opportunity between quoted (CDS and index CDO tranches) instruments and unquoted (bespoke CDO tranches) instruments. In doing so, the objectives include obtaining the CDS survival curve from market spreads. Then together with the standardized index CDO tranche spreads, the CDS survival curve is an input parameter for obtaining the base correlation. Finally using the base correlation, the bespoke (i.e. customized) CDO tranche can be priced consistently. All models and algorithms in this project implemented in Matlab1. It is a programming language developed by MathWorks that is designed specifically for technical computing. It has very strong abilities in handling numerical calculations, data visualization, and algorithm development. Hence it is very suitable for this project. 1.1 Credit Derivatives Credit derivatives were introduced to the market in the mid-1990s. Since then, the size of the credit derivative market has been dramatically growing. Initially, it was primarily used by banks to hedge their credit risk of bonds or loans. Nowadays, the uses of credit derivatives include increasing asset liquidity, diversifying credit risk, and diversifying investment portfolios (O’Kane 2008, p. 2). It is important to price all the instruments consistently so that the market is fair and arbitrage-free. The credit default swap (CDS) is the simplest yet the most important single-name credit derivative. The expression ‘single-name’ means that one CDS contract is only exposed to the default risk of one credit. A CDS is a bilateral over-the-counter contract between the protection seller and the protection buyer. During the contract, the protection buyer will make periodical payments (referred to as the premium leg), usually quarterly, to the protection 1 Matlab stands for Matrix Laboratory 5 MSc Financial Computing Project 2009-2010 by Mi Sun seller until the occurrence of a credit event2 or the contract maturity date, whichever comes sooner. In return for the stream of payments, the protection seller will compensate the protection buyer (the payment is referred to as the protection leg) in the case of a credit event. Following the single-name credit derivatives come the multi-name credit derivatives, where there are more than one credit
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