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Contents

Cover

Series Page

Title Page

Copyright

Preface

Acknowledgments

About the Author

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

1.1 What are Financial Correlations?

1.2 What is Financial Correlation Risk?

1.3 Motivation: Correlations and Correlation Risk are Everywhere in Finance

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

1.5 A Word on Terminology

1.6 Summary

Appendix 1A: Dependence and Correlation

Appendix 1B: On Percentage and Logarithmic Changes

Practice Questions and Problems

References and Suggested Readings

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

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

2.2 Do Equity Correlations Exhibit Mean Reversion?

2.3 Do Equity Correlations Exhibit Autocorrelation?

2.4 How are Equity Correlations Distributed?

2.5 Is Equity Correlation Volatility an Indicator for Future Recessions?

2.6 Properties of Bond Correlations and Default Probability Correlations

2.7 Summary

Practice Questions and Problems

References and Suggested Readings

Chapter 3: Statistical Correlation Models—Can We Apply Them to Finance?

3.1 A Word on Financial Models

3.2 Statistical Correlation Measures

3.3 Should We Apply Spearman's Rank Correlation and Kendall's τ in Finance?

3.4 Summary

Practice Questions and Problems

References and Suggested Readings

Chapter 4: Financial Correlation Modeling—Bottom-Up Approaches

4.1 Correlating Brownian Motions (Heston 1993)

4.2 The Binomial Correlation Measure

4.3 Copula Correlations

4.4 Contagion Correlation Models

4.5 Summary

Appendix 4A: Cholesky Decomposition

Appendix 4B: A Short Proof of the Gaussian Default Time Copula

Practice Questions and Problems

References and Suggested Readings

Chapter 5: Valuing CDOs with the Gaussian Copula—What Went Wrong?

5.1 CDO Basics—What is a CDO? Why CDOs? Types of CDOs

5.2 Valuing CDOs

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

5.4 Summary

Practice Questions and Problems

References and Suggested Readings

Chapter 6: The One-Factor Gaussian Copula (OFGC) Model—Too Simplistic?

6.1 The Original One-Factor Gaussian Copula (OFGC) Model

6.2 Valuing Tranches of A CDO with the OFGC

6.3 The Correlation Concept in the OFGC Model

6.4 The Relationship Between the OFGC and the Standard Copula

6.5 Extensions of the OFGC

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

6.7 Summary

Practice Questions and Problems

References and Suggested Readings

Chapter 7: Financial Correlation Models—Top-Down Approaches

7.1 Vasicek's 1987 One-Factor Gaussian Copula (OFGC) Model Revisited

7.2 Markov Chain Models

7.3 Contagion Default Modeling in Top-Down Models

7.4 Summary

Practice Questions and Problems

References and Suggested Readings

Chapter 8: Stochastic Correlation Models

8.1 What is a Stochastic Process?

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

8.3 Dynamic Conditional Correlations (DCCs) (Engle 2002)

8.4 Stochastic Correlation—Standard Models

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

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

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

8.8 Summary

Practice Questions and Problems

References and Suggested Readings

Chapter 9: Quantifying Market Correlation Risk

9.1 The Correlation Risk Parameters Cora and Gora

9.2 Examples of Cora in Financial Practice

9.3 Cora and Gora in Investments

9.4 Cora in Market Risk Management

9.5 Gora in Market Risk Management

9.6 Summary

Practice Questions and Problems

References and Suggested Readings

Chapter 10: Quantifying Credit Correlation Risk

10.1 Credit Correlation Risk in a CDS

10.2 Pricing CDSs, Including Reference Entity–Counterparty Credit Correlation

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

10.4 Correlation Risk in a Collateralized Debt Obligation (CDO)

10.5 Summary

Practice Questions and Problems

References and Suggested Readings

Chapter 11: Hedging Correlation Risk

11.1 What is Hedging?

11.2 Why is Hedging Financial Correlations Challenging?

11.3 Two Examples to Hedge Correlation Risk

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

11.5 Summary

Practice Questions and Problems

References and Suggested Readings

Chapter 12: Correlation and Basel II and III

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

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

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

12.4 Credit Value Adjustment (CVA) Approach Without Wrong-Way Risk (WWR) in the Basel Accord

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

12.6 How do the Basel Accords Treat Double Defaults?

12.7 Debt Value Adjustment (DVA): IF Something Sounds too Good to be True…

12.8 Funding Value Adjustment (FVA)

12.9 Summary

Practice Questions and Problems

References and Suggested Readings

Chapter 13: The Future of Correlation Modeling

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

13.2 New Developments in Artificial Intelligence and Financial Modeling

13.3 Summary

Practice Questions and Problems

References and Suggested Readings

Glossary

Index

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Preface

Correlation risk is the risk that the correlation between two or more financial variables changes unfavorably. Correlation risk was highlighted in the global financial crisis in 2007 to 2009, when correlations between many financial variables such as the default correlation between debtors or the default correlation between a debtor and an insurer increased dramatically. This led to huge unexpected losses for many financial institutions, which in part triggered the global financial crisis.

This book is the first to address financial correlation risk in detail. In Chapter 1, we introduce the basic properties of correlation risk, before we show in Chapter 2 how correlations behave in the real world. We then discuss whether correlation risk can be quantified using standard statistical correlation measures such as Pearson's ρ, Spearman's rank correlation coefficient, and Kendall's τ in Chapter 3. We address specific financial correlation measures in Chapter 4, and discuss whether the copula correlation model is appropriate to measure financial correlations in Chapter 5. Often, as in the Basel III framework, a shortcut to the Gaussian copula is applied, such as the one-factor Gaussian copula (OFGC) model. This approach, which is applied in the Basel framework to derive credit risk, is discussed in Chapter 6. In Chapter 7 we address a fairly new correlation family, the elegant but somewhat coarse top-down correlation models. Chapter 8 discusses stochastic correlation models, which are a new and promising way to model financial correlations. In Chapters 9 and 10, we introduce new concepts to quantify market and credit correlation risk. In Chapter 11 we address the challenging task of hedging correlation risk. Chapter 12 evaluates the proposed correlation concepts in the Basel III framework, which are designed to mitigate correlated credit and market risk. Chapter 13 deals with the future of correlation modeling, which may include neural networks, fuzzy logic, genetic algorithms, chaos theory, and combinations of these concepts.

Figure P.1 gives an overview of the main correlation models that will be addressed in this book. We will discuss the conceptual, mathematical, and computational properties of the models and evaluate their benefits and limitations for finance.

# 4.1 Target Audience

This book should be valuable to anyone who is exposed to financial correlations and financial correlation risk. So it should be of interest to upper management, risk managers, analysts, traders, compliance departments, model validation groups, controllers, reporting groups, brokers, and others. The book contains questions and problems at the end of each chapter, which should facilitate using the book in a classroom. The answers to the problems are available to instructors; please e-mail gunter@dersoft.com.

# Basel III

This book addresses new risk measures, especially the new correlation risk measures of the Basel III accord. We discuss the Basel-applied value at risk (VaR) concept, which includes correlated market risk, in the introductory Chapter 1, section 1.3.3. We address the one-factor Gaussian copula (OFGC) correlation model, which underlies the Basel credit correlation framework, in Chapter 6. We revisit the VaR concept for a multi-asset portfolio in Chapter 9, section 9.4. In Chapter 12, we discuss the Basel III correlation framework in detail, deriving credit value at risk (CVaR) and required capital (RC). In particular, we address credit value adjustment (CVA) with general and specific wrong-way risk (WWR), which includes the correlation between general market factors as well as the correlation between specific entities.

## Additional Materials

This book comes with 26 supporting spreadsheets, models, and documents. They can be downloaded at www.wiley.com/go/correlationriskmodeling; password: gunter123.

The supporting documents can also be downloaded from the author's website www.dersoft.com/correlationbook/downloads.

Below is a breakdown of the supporting documents by file.

**For a general refresher on the basics of mathematical finance:**

Math refresher.docx

**Chapter 1**

2-asset VaR.xlsx

Matrix primer.xlsx

Exchange option.xls

Quanto option.xls

Dependence and Correlation.xlsm

Log returns.xlsx

**Chapter 2**

Correlation fitting.docx

**Chapter 3**

Lookback option.xls

**Chapter 4**

GBM path with jumps.xlsm

2-asset default time Copula.xlsm

**Chapter 5**

CDO Gauss educational.xlsm

**Chapter 6**

OFGC educational.xls

Base correlation generation.xlsm

**Chapter 7**

Base correlation generation.xlsm

**Chapter 8**

GBM path with jumps.xlsm

Stochastic correlation.xlsx

**Chapter 9**

VaR educational.xlsm

VaR n asset cora gora.xlsm

Exchange option cora.docx

Math refresher.docx

**Chapter 10**

CDS with default correlation.xlsm

CDS three correlated entities pricing code.docx

**Chapter 11**

CDS with default correlation.xlsm

Option on the better of two.xlsm

Correlation swap.xls

Interest rate swap pricing model.xls

**Chapter 12**

CVAR.xlsm

Basel double default.xlsm

I welcome feedback. If you have a suggestion or comment, or if you spot an error, please email me at gunter@dersoft.com. There is an errata page at www.dersoft.com/correlationbook/errata.docx.

Acknowledgments

Many of my students, colleagues, and friends have significantly contributed to the writing of this book. I would like to thank the master of financial engineering (MFE) classes of 2012 and 2013 at the Shidler College of Business at the University of Hawaii for detailed discussions and number crunching, especially for the empirical Chapter 2. In particular, Martin Chang, Zhen Chen, Clint Davis, Charles Demarest, Barrett Gady, Susan Globokar, Ziyan Jiang, Thuy Le, Stefan Mayr, Dongfang Nie, Amirarsalan Pakravan, Babak Saadat, Alex Schnurrer, Jun Sheng, Wenjing Tang, Manogaran Thanabalan, Ryuichi Umeda, Eugene Wong, Amir Yousefi, and Elke Zeller contributed strongly.

I would like to thank Ranjan Bhaduri and Edgar Lobachevskiy for discussions on mathematical issues. Seth Rooder programmed two of the models that are referenced in the book. King Burch, Sidy Danioko, Brendan Lane Larson, Stefan Mayr, Rudolf Meissner, Eric Mills, Jason Mills, and Pedro Villarreal did an excellent job proofreading the book, finding errors, and suggesting improvements. Pedro Villarreal also helped to solve small and big computer problems and derived complex graphics.

I would also like to thank the editors Gemma Diaz, Chris Gage, Emilie Herman, Nick Wallwork, and Jules Yap of John Wiley & Sons, for their encouragement, support, and competent work.

*Why don't you make the book more fun?*

—Jasmine Meissner, 7

Yeah, well, while I enjoyed writing this book, I can only hope that the reader also enjoys the book and learns from it. I am happy to receive feedback; you can e-mail me at gunter@dersoft.com.

About the Author

After a lectureship in mathematics and statistics at the Economic Academy Kiel, Gunter Meissner, PhD, joined Deutsche Bank in 1990, trading 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, which at the time were lookback options, multi-asset options, quanto options, average options, index amortizing swaps, and Bermuda swaptions. In 1995/1996 Gunter was Head of Options at Deutsche Bank Tokyo. From 1997 to 2007, he was Professor of Finance at Hawaii Pacific University and from 2008 to 2013 Director of the master in financial engineering program at the Shidler College of Business at the University of Hawaii. Currently, he is President of Derivatives Software (www.dersoft.com), founder and CEO of Cassandra Capital Management (www.cassandracm.com), and Adjunct Professor of Mathematical Finance at NYU-Courant.

Gunter Meissner has published numerous papers and books on derivatives and risk management, and is a frequent speaker at conferences and seminars. He can be reached at gunter@dersoft.com.