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