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Multifractal Volatility - 1st Edition - ISBN: 9780121500139, 9780080559964

Multifractal Volatility

1st Edition

Theory, Forecasting, and Pricing

Authors: Laurent Calvet Adlai Fisher
eBook ISBN: 9780080559964
Hardcover ISBN: 9780121500139
Imprint: Academic Press
Published Date: 2nd September 2008
Page Count: 272
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Table of Contents

Chapter 1 Introduction
Chapter 2 Background
2.1 Autoregressive Volatility Modeling
2.2 Multifractals in the Natural Sciences
Chapter 3 The Multifractal Volatility Model: The MMAR
3.1 The Multifractal Model of Asset Returns
3.2 An Extension with Autocorrelated Returns
3.3 Empirical Evidence
3.4 Discussion
Chapter 4 The Marko-Switching Multifractal (MSM) in Discrete Time
4.1 MSM Construction in Discrete Time
4.2 Maximum Likelihood Estimation
4.3 Empirical Results
4.4 Comparison with Alternative Models
4.5 Discussion
Chapter 5. Multivariate MSM
5.1 Comovement of Univariate Volatility Components
5.2 A Bivariate Multifrequency Model
5.3 Inference
5.4 Empirical Results
5.5 Extension to Many Assets
5.6 Discussion
Chapter 6 The Marko-Switching Multifractal in Continuous Time
6.1 MSM in Continuous - Time
6.2 The Financial Model
6.3 Forecasting the Distribution of Returns
6.4 Discussion
Chapter 7 Multifrequency News and Stock Returns
7.1 An asset pricing model with regime-switching dividents
7.2 Volatility feedback with multifrequency shocks
7.3 Empirical results with fully informaed investros
7.4 Learning about volatility and endogenous skewness
7.5 Robustness checks, preference implications, and extension
7.6 Discussion
Chapter 8 Multifrequency Jump Diffusions
8.1 An Equilibrium Model with Endogenous Price Jumps
8.2 A Multifrequency Jump-Difussion for Equilibrium Stock Prices
8.3 Price Dynamics with an Infinity of Frequencies
8.4 Recursive Utility and Priced Jumps
8.5 Discussion
Chapter 9 Conclusion


Calvet and Fisher present a powerful, new technique for volatility forecasting that draws on insights from the use of multifractals in the natural sciences and mathematics and provides a unified treatment of the use of multifractal techniques in finance. A large existing literature (e.g., Engle, 1982; Rossi, 1995) models volatility as an average of past shocks, possibly with a noise component. This approach often has difficulty capturing sharp discontinuities and large changes in financial volatility. Their research has shown the advantages of modelling volatility as subject to abrupt regime changes of heterogeneous durations. Using the intuition that some economic phenomena are long-lasting while others are more transient, they permit regimes to have varying degrees of persistence. By drawing on insights from the use of multifractals in the natural sciences and mathematics, they show how to construct high-dimensional regime-switching models that are easy to estimate, and substantially outperform some of the best traditional forecasting models such as GARCH. The goal of Multifractal Volatility is to popularize the approach by presenting these exciting new developments to a wider audience. They emphasize both theoretical and empirical applications, beginning with a style that is easily accessible and intuitive in early chapters, and extending to the most rigorous continuous-time and equilibrium pricing formulations in final chapters.

Key Features

  • Presents a powerful new technique for forecasting volatility
  • Leads the reader intuitively from existing volatility techniques to the frontier of research in this field by top scholars at major universities
  • The first comprehensive book on multifractal techniques in finance, a cutting-edge field of research


Finance practitioners, academics, and students, and econometriciansSecondary readership: Mathematicians, statisticians, and natural scientists interested in fractals


No. of pages:
© Academic Press 2008
2nd September 2008
Academic Press
eBook ISBN:
Hardcover ISBN:


Advance Praise for Multifractal Volitility

“I thoroughly enjoyed reading the book and highly recommend it. The authors masterfully present their work on the Markov-Switching Multifractal model and its implications for Asset Pricing. This is a wonderful contribution to the field of Financial Economics.” --Ravi Bansal, J.B. Fuqua Professor of Finance, Duke University Durham, NC

“I have always been intrigued by the multi-fractal approach to volatility modeling, forecasting and pricing pioneered by Calvet and Fisher. This book does a wonderful job in gathering together all of the fundamental ideas and results in a coherent framework, and I highly recommend it to anybody interested in learning more about these novel techniques and how they compare to the more traditional GARCH and stochastic volatility based modeling procedures.” --Tim Bollerslev, Juanita and Clifton Kreps Professor of Economics, Duke University, NC

“This starkly original work defines a key part of the research frontier, developing a ‘multifractal’ perspective on volatility that unifies regime-switching and long memory, in discrete and continuous time, univariate and multivariate. Simultaneously and astonishingly, it is of immediate practical relevance for asset management, asset pricing and risk management. This book is required reading, for academics and practitioners alike.” --Francis X. Diebold, J.M. Cohen Professor of Economics, University of Pennsylvania

"Calvet and Fisher have fashioned the definitive treatment of multi-fractal models of return volatility. Since Mandelbrot first challenged the standard paradigm, evidence supporting the parsiomony and flexibility of the multifractal approach has accumulated. Calvet and Fisher are uniquely positioned to finally unify this progress, much of which is based on their own research. The result is masterful and convincing, particularly for capturing return risk over multiple time horizons. I highly recommend their book." --Darrell Duffie, Dean Witter Distinguished Professor of Finance, Stanford University, CA

“This book offers a unified treatment of multifractal volatility, a remarkable approach developed by the authors in a series of earlier papers. The idea is to capture in a single, coherent framework the set of observed features of financial data, whether seen as “jumps” in continuous-time models, “fat tails” in data discretely sampled over short intervals, different characterizations of volatility persistence over intermediate and long horizons, and nonlinearities and skewness in the conditional distributions. The underlying framework is a Markov-switching model with a very large number of different regimes, with the nature of different regimes summarized by a much smaller set of parameters. The book provides an excellent illustration of just how successful this flexible yet parsimonious approach can be in terms of describing a wide variety of the characteristics of financial time series.” --James Hamilton, Professor of Economics, University of California, San Diego

“Calvet and Fisher provide a valuable and thorough development of a novel class of models of financial market volatility. The methods and models exposited so nicely in their book should be part of the toolkit of researchers interested in understanding and characterizing the stochastic nature of volatility fluctuations. Their book is simultaneously accessible and complete. It shows how to use these models in practice, and it provides a rigorous foundation for their application.” --Lars Hansen, Livingston Distinguished Service Professor, University of Chicago, IL

“Volatility is a central concern of modern financial econometrics, challenging econometricians to build plausible models and practical methods of inference. Calvet and Fisher draw together the ingredients of a promising new research agenda, integrating a decade of work on multifractal modeling into a masterful overview of the field of volatility, demonstrating the advantages of Markov switching multifractals in aggregating components of differing persistence and showing us how rare events need not be studied in isolation as curiosa. A compelling read for financial theorists and practitioners.” --Peter C. B. Philips, Sterling Professor of Economics & Statistics, Yale University, CT

“To accommodate the high persistence and variability of volatility in financial time series, Calvet and Fisher developed the class of Markov-Switching Multifractal models. This book, which summarizes ten years of their research, is of great interest to researchers in asset pricing and essential reading for practitioners working on risk management or volatility forecasting.” --Jose Scheinkman, Theodore Wells '29 Professor of Economics, Princeton University, NJ

Ratings and Reviews

About the Authors

Laurent Calvet

Affiliations and Expertise

Professor, Chair in Finance - Tanaka Business School, Imperial College London, UK

Adlai Fisher

Affiliations and Expertise

Faculty of Commerce, University of British Columbia, Vancouver, Canada