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Dynamic Copula Methods in Finance


Over the course of the past decade financial markets have witnessed a marked increase in the use of correlation dynamics models – new terms such as correlation trading and correlation products have now become mainstream, and, increasingly, trading and investment activities have involved more and more exposure to credit risks that are non-Gaussian by definition. By addressing the restrictions which must be imposed on copula functions to yield dynamically consistent results this book sets out the latest research into the application of copula functions to the solution of financial problems.

Beginning with a review of the issues surrounding dependence and correlation in finance and the basic concepts of copulas as they have been applied to financial problems up until now, the book goes on to introduce the theory of convolution-based copulas, and the concept of C-convolution within the mainstream of the Darsow, Nguyen and Olsen (DNO) application of copulas to Markov processes. The authors explain how the c-convolution approach can be exploited to address both spatial and temporal dependence – a twofold perspective which is entirely new to these applications – and demonstrate how it can be applied to the problems of evaluating multivariate equity derivatives, analyzing the credit risk exposure of a portfolio, and aggregating Value-at-Risk measures across risk-factors and business units.

It shows the reader how to build original and consistent copula-based solutions to problems such as:

  • The evaluation of multivariate and path dependent equity linked derivatives consistently with the no-arbitrage requirement imposed by financial theory and the “fair value” principle
  • The evaluation of multivariate credit derivatives with a focus on the price consistency of contracts of different maturities
  • A consistent strategy for aggregation and allocation of risk capital across different risk factors and business units
  • A new copula-based approach to the performance analysis of mutual funds and hedge funds

The culmination of five years original research at the University of Bologna on the use of copulas in finance, this book is essential reading for practitioners involved in pricing and risk management.

UMBERTO CHERUBINI is Associate Professor of Financial Mathematics at the University of Bologna, where he heads the Graduate Degree in Quantitative Finance. He is a fellow of the Financial Econometrics Research Center (FERC), a member of the Scientific Committees of Abiformazione – the professional education arm of the Italian banking association, and AIFIRM – the Italian Association of Financial Risk Managers. He has been consulting and teaching in the field of finance and risk management for more than ten years. Before joining academia he worked as an economist at the Economic Research Department of BCI Milan. He has published papers in finance and economics in international journals, and is co-author of six books on topics of risk management and financial mathematics, including Fourier Transform Methods in Finance, John Wiley & Sons, Ltd, 2009; and Copula Methods in Finance, John Wiley & Sons, Ltd, 2004.

FABIO GOBBI is a post-doctoral researcher at the University of Bologna. He has a PhD in Statistics from the University of Florence and his areas of research focus on probability and financial econometrics. This is his first book.

SABRINA MULINACCI is Associate Professor of Mathematical Methods for Economics and Finance at the University of Bologna, Italy. Prior to this Sabrina was Associate Professor of Mathematical Methods for Economics and Actuarial Sciences at the Catholic University of Milan. She has a PhD in Mathematics from the University of Pisa and has published a number of research papers in international journals on probability and mathematical finance. She is co-author of Fourier Transform Methods in Finance, John Wiley & Sons, Ltd, 2009.

SILVIA ROMAGNOLI is Assistant Professor of Mathematical Models for Economics and Actuarial and Financial Sciences at the University of Bologna. Her scientific research is mainly addressed to the applications of stochastic models to finance and insurance. She has published several research papers in international journals on mathematical finance.

Preface ix

1 Correlation Risk in Finance 1

1.1 Correlation Risk in Pricing and Risk Management 1

1.2 Implied vs Realized Correlation 3

1.3 Bottom-up vs Top-down Models 4

1.4 Copula Functions 4

1.5 Spatial and Temporal Dependence 5

1.6 Long-range Dependence 5

1.7 Multivariate GARCH Models 7

1.8 Copulas and Convolution 8

2 Copula Functions: The State of the Art 11

2.1 Copula Functions: The Basic Recipe 11

2.2 Market Co-movements 14

2.3 Delta Hedging Multivariate Digital Products 16

2.4 Linear Correlation 19

2.5 Rank Correlation 20

2.6 Multivariate Spearman’s Rho 22

2.7 Survival Copulas and Radial Symmetry 23

2.8 Copula Volume and Survival Copulas 24

2.9 Tail Dependence 27

2.10 Long/Short Correlation 27

2.11 Families of Copulas 29

2.11.1 Elliptical Copulas 29

2.11.2 Archimedean Copulas 31

2.12 Kendall Function 33

2.13 Exchangeability 34

2.14 Hierarchical Copulas 35

2.15 Conditional Probability and Factor Copulas 39

2.16 Copula Density and Vine Copulas 42

2.17 Dynamic Copulas 45

2.17.1 Conditional Copulas 45

2.17.2 Pseudo-copulas 46

3 Copula Functions and Asset Price Dynamics 49

3.1 The Dynamics of Speculative Prices 49

3.2 Copulas and Markov Processes: The DNO approach 51

3.2.1 The * and Product Operators 52

3.2.2 Product Operators and Markov Processes 55

3.2.3 Self-similar Copulas 58

3.2.4 Simulating Markov Chains with Copulas 62

3.3 Time-changed Brownian Copulas 63

3.3.1 CEV Clock Brownian Copulas 64

3.3.2 VG Clock Brownian Copulas 65

3.4 Copulas and Martingale Processes 66

3.4.1 C-Convolution 67

3.4.2 Markov Processes with Independent Increments 75

3.4.3 Markov Processes with Dependent Increments 78

3.4.4 Extracting Dependent Increments in Markov Processes 81

3.4.5 Martingale Processes 83

3.5 Multivariate Processes 86

3.5.1 Multivariate Markov Processes 86

3.5.2 Granger Causality and the Martingale Condition 88

4 Copula-based Econometrics of Dynamic Processes 91

4.1 Dynamic Copula Quantile Regressions 91

4.2 Copula-based Markov Processes: Non-linear Quantile Autoregression 93

4.3 Copula-based Markov Processes: Semi-parametric Estimation 99

4.4 Copula-based Markov Processes: Non-parametric Estimation 108

4.5 Copula-based Markov Processes: Mixing Properties 110

4.6 Persistence and Long Memory 113

4.7 C-convolution-based Markov Processes: The Likelihood Function 116

5 Multivariate Equity Products 121

5.1 Multivariate Equity Products 121

5.1.1 European Multivariate Equity Derivatives 122

5.1.2 Path-dependent Equity Derivatives 125

5.2 Recursions of Running Maxima and Minima 126

5.3 The Memory Feature 130

5.4 Risk-neutral Pricing Restrictions 132

5.5 Time-changed Brownian Copulas 133

5.6 Variance Swaps 135

5.7 Semi-parametric Pricing of Path-dependent Derivatives 136

5.8 The Multivariate Pricing Setting 137

5.9 H-Condition and Granger Causality 137

5.10 Multivariate Pricing Recursion 138

5.11 Hedging Multivariate Equity Derivatives 141

5.12 Correlation Swaps 144

5.13 The Term Structure of Multivariate Equity Derivatives 147

5.13.1 Altiplanos 148

5.13.2 Everest 150

5.13.3 Spread Options 150

6 Multivariate Credit Products 153

6.1 Credit Transfer Finance 153

6.1.1 Univariate Credit Transfer Products 154

6.1.2 Multivariate Credit Transfer Products 155

6.2 Credit Information: Equity vs CDS 158

6.3 Structural Models 160

6.3.1 Univariate Model: Credit Risk as a Put Option 160

6.3.2 Multivariate Model: Gaussian Copula 161

6.3.3 Large Portfolio Model: Vasicek Formula 163

6.4 Intensity-based Models 164

6.4.1 Univariate Model: Poisson and Cox Processes 165

6.4.2 Multivariate Model: Marshall–Olkin Copula 165

6.4.3 Homogeneous Model: Cuadras Aug´e Copula 167

6.5 Frailty Models 170

6.5.1 Multivariate Model: Archimedean Copulas 170

6.5.2 Large Portfolio Model: Sch¨onbucher Formula 171

6.6 Granularity Adjustment 171

6.7 Credit Portfolio Analysis 172

6.7.1 Semi-unsupervised Cluster Analysis: K-means 172

6.7.2 Unsupervised Cluster Analysis: Kohonen Self-organizing Maps 174

6.7.3 (Semi-)unsupervised Cluster Analysis: Hierarchical Correlation Model 175

6.8 Dynamic Analysis of Credit Risk Portfolios 176

7 Risk Capital Management 181

7.1 A Review of Value-at-Risk and Other Measures 181

7.2 Capital Aggregation and Allocation 185

7.2.1 Aggregation: C-Convolution 187

7.2.2 Allocation: Level Curves 189

7.2.3 Allocation with Constraints 191

7.3 Risk Measurement of Managed Portfolios 193

7.3.1 Henriksson–Merton Model 195

7.3.2 Semi-parametric Analysis of Managed Funds 200

7.3.3 Market-neutral Investments 201

7.4 Temporal Aggregation of Risk Measures 202

7.4.1 The Square-root Formula 203

7.4.2 Temporal Aggregation by C-convolution 203

8 Frontier Issues 207

8.1 L´evy Copulas 207

8.2 Pareto Copulas 210

8.3 Semi-martingale Copulas 212

A Elements of Probability 215

A.1 Elements of Measure Theory 215

A.2 Integration 216

A.2.1 Expected Values and Moments 217

A.3 The Moment-generating Function or Laplace Transform 218

A.4 The Characteristic Function 219

A.5 Relevant Probability Distributions 219

A.6 Random Vectors and Multivariate Distributions 224

A.6.1 The Multivariate Normal Distribution 225

A.7 Infinite Divisibility 226

A.8 Convergence of Sequences of Random Variables 228

A.8.1 The Strong Law of Large Numbers 229

A.9 The Radon–Nikodym Derivative 229

A.10 Conditional Expectation 229

B Elements of Stochastic Processes Theory 231

B.1 Stochastic Processes 231

B.1.1 Filtrations 231

B.1.2 Stopping Times 232

B.2 Martingales 233

B.3 Markov Processes 234

B.4 Lévy Processes 237

B.4.1 Subordinators 240

B.5 Semi-martingales 240

References 245

Extra Reading 251

Index 259