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Multifractality In Asset Returns: Theory And Evidence

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Author Info
Laurent Calvet
Adlai Fisher

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Abstract

This paper investigates the multifractal model of asset returns (MMAR), a class of continuous-time processes that incorporate the thick tails and volatility persistence exhibited by many financial time series. The simplest version of the MMAR compounds a Brownian motion with a multifractal time-deformation. Prices follow a semi-martingale, which precludes arbitrage in a standard two-asset economy. Volatility has long memory, and the highest finite moments of returns can take any value greater than 2. The local variability of a sample path is highly heterogeneous and is usefully characterized by the local Hölder exponent at every instant. In contrast with earlier processes, this exponent takes a continuum of values in any time interval. The MMAR predicts that the moments of returns vary as a power law of the time horizon. We confirm this property for Deutsche mark/U.S. dollar exchange rates and several equity series. We develop an estimation procedure and infer a parsimonious generating mechanism for the exchange rate. In Monte Carlo simulations, the estimated multifractal process replicates the scaling properties of the data and compares favorably with some alternative specifications. © 2001 by the President and Fellows of Harvard College and the Massachusetts Institute of Technolog

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Publisher Info
Article provided by MIT Press in its journal The Review of Economics and Statistics.

Volume (Year): 84 (2002)
Issue (Month): 3 (August)
Pages: 381-406
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Handle: RePEc:tpr:restat:v:84:y:2002:i:3:p:381-406

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  1. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold, 2005. "Roughing it Up: Including Jump Components in the Measurement, Modeling and Forecasting of Return Volatility," NBER Working Papers 11775, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
    Other versions:
  2. Marina Resta & Davide Sciutti, . "A characterization of self-affine processes in finance through the scaling function," Modeling, Computing, and Mastering Complexity 2003 13, Society for Computational Economics. [Downloadable!]
  3. Laurent E. Calvet & Adlai J. Fisher, 2006. "Multifrequency Jump-Diffusions: An Equilibrium Approach," NBER Working Papers 12797, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  4. Ole E. Barndorff-Nielsen & Neil Shephard, 2005. "Variation, jumps, market frictions and high frequency data in financial econometrics," OFRC Working Papers Series 2005fe08, Oxford Financial Research Centre. [Downloadable!]
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  5. Laurent E. Calvet & Adlai J. Fisher, 2005. "Multifrequency News and Stock Returns," NBER Working Papers 11441, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  6. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold,, 2003. "Some Like it Smooth, and Some Like it Rough: Untangling Continuous and Jump Components in Measuring, Modeling, and Forecasting Asset Return Volatility," CFS Working Paper Series 2003/35, Center for Financial Studies. [Downloadable!]
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  7. jérôme Fillol & Fabien Tripier, 2003. "The scaling function-based estimator of the long memory parameter: a comparative study," Economics Bulletin, Economics Bulletin, vol. 3(23), pages 1-7. [Downloadable!]
  8. Cornelis A. Los, 2005. "The Degree of Stability of Price Diffusion," Finance 0508006, EconWPA. [Downloadable!]
  9. Liuren Wu, 2004. "Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns," Finance 0401001, EconWPA. [Downloadable!]
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  10. Ole E. Barndorff-Nielsen & Sven Erik Graversen & Jean Jacod & Neil Shephard, 2005. "Limit theorems for bipower variation in financial econometrics," OFRC Working Papers Series 2005fe09, Oxford Financial Research Centre. [Downloadable!]
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  11. Lux, Thomas, 2006. "The Markov-Switching Multifractal Model of asset returns : GMM estimation and linear forecasting of volatility," Economics Working Papers 2006,17, Christian-Albrechts-University of Kiel, Department of Economics. [Downloadable!]
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  12. Laurent E. Calvet & Adlai J. Fisher & Samuel B. Thompson, 2004. "Volatility Comovement: A Multifrequency Approach," NBER Technical Working Papers 0300, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
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  13. Alain Chaboud & Benjamin Chiquoine & Erik Hjalmarsson & Mico Loretan, 2007. "Frequency of observation and the estimation of integrated volatility in deep and liquid financial markets," International Finance Discussion Papers 905, Board of Governors of the Federal Reserve System (U.S.). [Downloadable!]
  14. Jérôme Fillol, 2003. "Multifractality: Theory and Evidence an Application to the French Stock Market," Economics Bulletin, Economics Bulletin, vol. 3(31), pages 1-12. [Downloadable!]
  15. Alain Chaboud & Benjamin Chiquoine & Erik Hjalmarsson & Mico Loretan, 2008. "Frequency of observation and the estimation of integrated volatility in deep and liquid financial markets," BIS Working Papers 249, Bank for International Settlements. [Downloadable!]
  16. Lisa Borland & Jean-Philippe Bouchaud & Jean-Francois Muzy & Gilles Zumbach, 2005. "The Dynamics of Financial Markets -- Mandelbrot's multifractal cascades, and beyond," Science & Finance (CFM) working paper archive 500061, Science & Finance, Capital Fund Management. [Downloadable!]
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