Correlation Risk Modeling and Management

GUNTER MEISSNER, PH.D., heads Dersoft (www.dersoft.com), the software company behind TradeSmart, a software package that derives futures, options, and swaps prices and risk parameters. In addition, he runs a hedge fund (www. cassandracm.com), and is Adjunct Professor of Mathematical Finance at NYU. Dr. Meissner joined Deutsche Bank in 1990, where he traded interest rate futures, swaps, and options in Frankfurt and New York. He became Head of Product Development in 1994, responsible for originating algorithms for new derivatives products. In 1995/1996 Dr. Meissner became Head of Options at Deutsche Bank Tokyo. From 1997 to 2007, he was Professor of Finance at Hawaii Pacific University. From 2008 to 2013 he was Director of the Master in Financial Engineering program at the Shidler College of Business at the University of Hawaii. The author of numerous published papers on derivatives in international journals, Dr. Meissner also is a frequent speaker at international conferences and seminars and the author of four other books, including The Definitive Guide to CDOs: Application, Pricing, and Risk Management.

Preface xiii


Acknowledgments xvii


About the Author xix


CHAPTER 1
Some Correlation Basics: Properties, Motivation, Terminology 1


1.1 What Are Financial Correlations? 1


1.2 What Is Financial Correlation Risk? 2


1.3 Motivation: Correlations and Correlation Risk Are Everywhere in Finance 5


1.4 How Does Correlation Risk Fit into the Broader Picture of Risks in Finance? 24


1.5 A Word on Terminology 33


1.6 Summary 34


Appendix 1A: Dependence and Correlation 35


Dependence 35


Correlation 36


Independence and Uncorrelatedness 37


Appendix 1B: On Percentage and Logarithmic Changes 38


Practice Questions and Problems 39


References and Suggested Readings 40


CHAPTER 2
Empirical Properties of Correlation: How Do Correlations Behave in the Real World? 43


2.1 How Do Equity Correlations Behave in a Recession, Normal Economic Period, or Strong Expansion? 43


2.2 Do Equity Correlations Exhibit Mean Reversion? 46


2.3 Do Equity Correlations Exhibit Autocorrelation? 50


2.4 How Are Equity Correlations Distributed? 51


2.5 Is Equity Correlation Volatility an Indicator for Future Recessions? 52


2.6 Properties of Bond Correlations and Default Probability Correlations 53


2.7 Summary 54


Practice Questions and Problems 55


References and Suggested Readings 55


CHAPTER 3
Statistical Correlation Models—Can We Apply Them to Finance? 57


3.1 AWord on Financial Models 57


3.2 Statistical Correlation Measures 60


3.3 Should We Apply Spearman’s Rank Correlation and Kendall’s t in Finance? 65


3.4 Summary 66


Practice Questions and Problems 67


References and Suggested Readings 68


CHAPTER 4
Financial Correlation Modeling—Bottom–Up Approaches 69


4.1 Correlating Brownian Motions (Heston 1993) 69


4.2 The Binomial CorrelationMeasure 72


4.3 Copula Correlations 74


4.4 Contagion Correlation Models 88


4.5 Summary 90


Appendix 4A: Cholesky Decomposition 91


Example: Cholesky Decomposition for Three Assets 92


Appendix 4B: A Short Proof of the Gaussian Default


Time Copula 93


Practice Questions and Problems 93


References and Suggested Readings 94


CHAPTER 5
Valuing CDOs with the Gaussian Copula—What Went Wrong? 101


5.1 CDO Basics—What Is a CDO? Why CDOs? Types of CDOs 101


5.2 Valuing CDOs 105


5.3 Conclusion: The Gaussian Copula and CDOs—What Went Wrong? 113


5.4 Summary 115


Practice Questions and Problems 116


References and Suggested Readings 117


CHAPTER 6
The One–Factor Gaussian Copula (OFGC) Model—Too Simplistic? 119


6.1 The Original One–Factor Gaussian Copula (OFGC) Model 121


6.2 Valuing Tranches of a CDO with the OFGC 122


6.3 The Correlation Concept in the OFGC Model 128


6.4 The Relationship between the OFGC and the Standard Copula 131


6.5 Extensions of the OFGC 132


6.6 Conclusion—Is the OFGC Too Simplistic to Evaluate Credit Risk in Portfolios? 135


6.7 Summary 138


Practice Questions and Problems 139


References and Suggested Readings 140


CHAPTER 7
Financial Correlation Models—Top–Down Approaches 143


7.1 Vasicek’s 1987 One–Factor Gaussian Copula (OFGC) Model Revisited 144


7.2 Markov Chain Models 146


7.2.1 Inducing Correlation via Transition Rate Volatilities 146


7.3 Contagion Default Modeling in Top–Down Models 150


7.4 Summary 153


Practice Questions and Problems 154


References and Suggested Readings 154


CHAPTER 8
Stochastic Correlation Models 157


8.1 What Is a Stochastic Process? 157


8.2 Sampling Correlation from a Distribution (Hull and White 2010) 159


8.3 Dynamic Conditional Correlations (DCCs) (Engle 2002) 160


8.4 Stochastic Correlation—Standard Models 162


8.5 Extending the Heston Model with Stochastic Correlation (Buraschi et al. 2010; Da Fonseca et al. 2008) 168


8.6 Stochastic Correlation, Stochastic Volatility, and Asset Modeling (Lu and Meissner 2012) 172


8.7 Conclusion: Should We Model Financial Correlations with a Stochastic Process? 176


8.8 Summary 177


Practice Questions and Problems 177


References and Suggested Readings 178


CHAPTER 9
Quantifying Market Correlation Risk 181


9.1 The Correlation Risk Parameters Cora and Gora 182


9.2 Examples of Cora in Financial Practice 184


9.3 Cora and Gora in Investments 187


9.4 Cora in Market Risk Management 189


9.5 Gora in Market Risk Management 197


9.6 Summary 198


Practice Questions and Problems 199


References and Suggested Readings 200


CHAPTER 10
Quantifying Credit Correlation Risk 201


10.1 Credit Correlation Risk in a CDS 203


10.2 Pricing CDSs, Including Reference Entity–Counterparty Credit Correlation 205


10.3 Pricing CDSs, Including the Credit Correlation of All Three Entities 215


10.4 Correlation Risk in a Collateralized Debt Obligation (CDO) 227


10.5 Summary 231


Practice Questions and Problems 232


References and Suggested Readings 233


CHAPTER 11
Hedging Correlation Risk 235


11.1 What Is Hedging? 235


11.2 Why Is Hedging Financial Correlations Challenging? 238


11.3 Two Examples to Hedge Correlation Risk 239


11.4 When to Use Options and When to Use Futures to Hedge 247


11.5 Summary 248


Practice Questions and Problems 249


References and Suggested Readings 249


CHAPTER 12
Correlation and Basel II and III 251


12.1 What Are the Basel I, II, and III Accords? Why Do Most Sovereigns Implement The Accords? 251


12.2 Basel II and III’s Credit Value at Risk (CVaR) Approach 252


12.3 Basel II’s Required Capital (RC) for Credit Risk 258


12.4 Credit Value Adjustment (CVA) Approach without Wrong–Way Risk (WWR) in The Basel Accord 261


12.5 Credit Value Adjustment (CVA) with Wrong–Way Risk in the Basel Accord 264


12.6 How Do the Basel Accords Treat Double Defaults? 269


12.7 Debt Value Adjustment (DVA): If Something Sounds Too Good to Be True . . . 274


12.8 Funding Value Adjustment (FVA) 276


12.9 Summary 278


Practice Questions and Problems 280


References and Suggested Readings 280


CHAPTER 13
The Future of Correlation Modeling 283


13.1 Numerical Finance: Solving Financial Problems Numerically with the Help of Graphical Processing Units (GPUs) 283


13.2 New Developments in Artificial Intelligence and Financial Modeling 287


13.3 Summary 298


Practice Questions and Problems 300


References and Suggested Readings 300


Glossary 303


Index 315