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**Copula methods in finance.**
*(English)*
Zbl 1163.62081

Wiley Finance Series. Chichester: John Wiley & Sons (ISBN 0-470-86344-7). xvi, 293 p. (2004).

From the preface: This book is an introduction to the use of copula functions from the viewpoint of mathematical finance applications. The target audience for our work consists of academics and practitioners who are eager to master and construct copula applications to financial problems.

Chapter 1 reviews the state of the art in asset pricing and risk management, going over the major frontier issues and providing justifications for introducing copula functions. Chapter 2 introduces the reader to the bivariate copula case. It presents the mathematical and probabilistic background on which theplications are built and gives some first examples in finance. Chapter 3 discusses the flaws of linear correlations and highlights how copula functions, along with nonparametric association measures, may provide a much more flexible way to represent market comovements. Chapter 4 extends the technical tools to a multivariate setting. Readers who are not already familiar with copulas are advised to skip this chapter at first reading (or to read it at their own risk!).

Chapter 5 explains the statistical inference for copulas. It covers both methodological aspects and applications from market data, such as calibration of actual risk factors comovements and VaR measurements. Here the readers can find details on the classical estimation methods as well as on the most recent approaches, such as conditional copulas. Chapter 6 is devoted to an exhaustive account of simulation algorithms for a large class of multivariate copulas. It is enhanced by financial examples.

Chapter 7 presents credit risk applications, besides giving a brief introduction to credit derivative markets and instruments. It applies copulas to the pricing of complex credit structures such as basket default swaps and CDOs. It is shown how to calibrate the pricing model to market data. Its sensitivity with respect to the copula choice is accounted for in concrete examples. Chapter 8 covers option pricing applications. Starting from the bivariate pricing kernel, copulas are used to evaluate counterparty risks in derivative transactions and bivariate rainbow options, such as options to exchange. We also show how the barrier option pricing problem can be cast in a bivariate setting and can be represented in terms of copulas. Finally, the estimation and simulation techniques presented in Chapters 5 and 6 are put at work to solve the evaluation problem of a multivariate basket option.

Chapter 1 reviews the state of the art in asset pricing and risk management, going over the major frontier issues and providing justifications for introducing copula functions. Chapter 2 introduces the reader to the bivariate copula case. It presents the mathematical and probabilistic background on which theplications are built and gives some first examples in finance. Chapter 3 discusses the flaws of linear correlations and highlights how copula functions, along with nonparametric association measures, may provide a much more flexible way to represent market comovements. Chapter 4 extends the technical tools to a multivariate setting. Readers who are not already familiar with copulas are advised to skip this chapter at first reading (or to read it at their own risk!).

Chapter 5 explains the statistical inference for copulas. It covers both methodological aspects and applications from market data, such as calibration of actual risk factors comovements and VaR measurements. Here the readers can find details on the classical estimation methods as well as on the most recent approaches, such as conditional copulas. Chapter 6 is devoted to an exhaustive account of simulation algorithms for a large class of multivariate copulas. It is enhanced by financial examples.

Chapter 7 presents credit risk applications, besides giving a brief introduction to credit derivative markets and instruments. It applies copulas to the pricing of complex credit structures such as basket default swaps and CDOs. It is shown how to calibrate the pricing model to market data. Its sensitivity with respect to the copula choice is accounted for in concrete examples. Chapter 8 covers option pricing applications. Starting from the bivariate pricing kernel, copulas are used to evaluate counterparty risks in derivative transactions and bivariate rainbow options, such as options to exchange. We also show how the barrier option pricing problem can be cast in a bivariate setting and can be represented in terms of copulas. Finally, the estimation and simulation techniques presented in Chapters 5 and 6 are put at work to solve the evaluation problem of a multivariate basket option.

### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

60H20 | Stochastic integral equations |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

91-02 | Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance |

91G70 | Statistical methods; risk measures |