Autoregressive conditionally heteroscedastic (ARCH) models were introduced by Engle in an article published in Econometrica in the early 1980s (Engle, 1982). The proposed application in that article focused on macroeconomic data and one could not imagine, at that time, that the main field of application for these models would be finance. Since the mid-1980s and the introduction of generalized ARCH (or GARCH) models, these models have become extremely popular among both academics and practitioners. GARCH models led to a fundamental change to the approaches used in finance, through an efficient modeling of volatility (or variability) of the prices of financial assets. In 2003, the Nobel Prize for Economics was jointly awarded to Robert F. Engle and Clive W.J. Granger ‘for methods of analyzing economic time series with time™varying volatility (ARCH)’.
Since the late 1980s, numerous extensions of the initial ARCH models have been published (see Bollerslev, 2008, for a (tentatively) exhaustive list). The aim of the present volume is not to review all these models, but rather to provide a panorama, as wide as possible, of current research into the concepts and methods of this field. Along with their development in econometrics and finance journals, GARCH models and their extensions have given rise to new directions for research in probability and statistics. Numerous classes of nonlinear time series models have been suggested, but none of them has generated interest comparable to that in GARCH models. The interest of the academic world in these models is explained by the fact that they are simple enough to be usable in practice, but also rich in theoretical problems, many of them unsolved.
This book is intended primarily for master’s students and junior researchers, in the hope of attracting them to research in applied mathematics, statistics or econometrics. For experienced researchers, this book offers a set of results and references allowing them to move towards one of the many topics discussed. Finally, this book is aimed at practitioners and users who may be looking for new methods, or may want to learn the mathematical foundations of known methods.
Some parts of the text have been written for readers who are familiar with probability theory and with time series techniques. To make this book as self-contained as possible, we provide demonstrations of most theoretical results. On first reading, however, many demonstrations can be omitted. Those sections or chapters that are the most mathematically sophisticated and can be skipped without loss of continuity are marked with an asterisk. We have illustrated the main techniques with numerical examples, using real or simulated data. Program codes allowing the experiments to be reproduced are provided in the text and on the authors' web pages. In general, we have tried to maintain a balance between theory and applications.
Readers wishing to delve more deeply into the concepts introduced in this book will find a large collection of exercises along with their solutions. Some of these complement the proofs given in the text.
The book is organized as follows. Chapter 1 introduces the basics of stationary processes and ARM A modeling. The rest of the book is divided into three parts. Part I deals with the standard univariate GARCH model. The main probabilistic properties (existence of stationary solutions, representations, properties of autocorrelations) are presented in Chapter 2. Chapter 3 deals with complementary properties related to mixing, allowing us to characterize the decay of the time dependence. Chapter 4 is devoted to temporal aggregation: it studies the impact of the observation frequency on the properties of GARCH processes.
Part II is concerned with statistical inference. We begin in Chapter 5 by studying the problem of identifying an appropriate model a priori. Then we present different estimation methods, starting with the method of least squares in Chapter 6 which, limited to ARCH, offers the advantage of simplicity. The central part of the statistical study is Chapter 7, devoted to the quasi-maximum likelihood method. For these models, testing the nullity of coefficients is not standard and is the subject of Chapter 8. Optimality issues are discussed in Chapter 9, as well as alternative estimators allowing some of the drawbacks of standard methods to be overcome.
Part III is devoted to extensions and applications of the standard model. In Chapter 10, models allowing us to incorporate asymmetric effects in the volatility are discussed. There is no natural extension of GARCH models for vector series, and many multivariate formulations are presented in Chapter 11. Without carrying out an exhaustive statistical study, we consider the estimation of a particular class of models which appears to be of interest for applications. Chapter 12 presents applications to finance. We first study the link between GARCH and diffusion processes, when the time step between two observations converges to zero. Two applications to finance are then presented: risk measurement and the pricing of derivatives.
Appendix A includes the probabilistic properties which are of most importance for the study of GARCH models. Appendix B contains results on autocorrelations and partial autocorrelations. Appendix C provides solutions to the end-of-chapter exercises. Finally, a set of problems and (in most cases) their solutions are provided in Appendix D.
For more information, please visit the author’s website http://perso.univ-lille3.fr/~cfrancq/Christian-Francq/book-GARCH.html.