The latest tools and techniques for pricing and risk management This book introduces readers to the use of copula functions to represent the dynamics of financial assets and risk factors, integrated temporal and cross-section applications. The first part of the book will briefly introduce the standard the theory of copula functions, before examining the link between copulas and Markov processes. It will then introduce new techniques to design Markov processes that are suited to represent the dynamics of market risk factors and their co-movement, providing techniques to both estimate and simulate such dynamics. The second part of the book will show readers how to apply these methods to the evaluation of pricing of multivariate derivative contracts in the equity and credit markets. It will then move on to explore the applications of joint temporal and cross-section aggregation to the problem of risk integration.

About the authors

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 RiskManagers. 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.

Klappentext

Dynamic copula methods in finance

Umberto Cherubini, Fabio Gobbi, Sabrina Mulinacci and Silvia Romagnoli

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 consistent 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.

Zusammenfassung
The latest tools and techniques for pricing and risk management

This book introduces readers to the use of copula functions to represent the dynamics of financial assets and risk factors, integrated temporal and cross-section applications. The first part of the book will briefly introduce the standard the theory of copula functions, before examining the link between copulas and Markov processes. It will then introduce new techniques to design Markov processes that are suited to represent the dynamics of market risk factors and their co-movement, providing techniques to both estimate and simulate such dynamics. The second part of the book will show readers how to apply these methods to the evaluation of pricing of multivariate derivative contracts in the equity and credit markets. It will then move on to explore the applications of joint temporal and cross-section aggregation to the problem of risk integration.


Inhalt

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 _ Produ…