Large Deviations and the Distribution of Price Changes

30 Pages Posted: 22 Apr 1998

See all articles by Laurent E. Calvet

Laurent E. Calvet

EDHEC Business School - Department of Economics & Finance; CEPR

Adlai J. Fisher

University of British Columbia (UBC) - Sauder School of Business

Benoit B. Mandelbrot

Yale University - International Center for Finance; IBM Corporation - Thomas J. Watson Research Center

Date Written: September 15, 1997

Abstract

The Multifractal Model of Asset Returns (See Mandelbrot, Fisher, and Calvet, 1997 ) proposes a class of multifractal processes for the modelling of financial returns. In that paper, multifractal processes are defined by a scaling law for moments of the processes' increments over finite time intervals. In the present paper, we discuss the local behavior of multifractal processes. We employ local Holder exponents, a fundamental concept in real analysis that describes the local scaling properties of a realized path at any point in time. In contrast with the standard models of continuous time finance, multifractal processes contain a multiplicity of local Holder exponents within any finite time interval. We characterize the distribution of Holder exponents by the multifractal spectrum of the process. For a broad class of multifractal processes, this distribution can be obtained by an application of Cramer's Large Deviation Theory. In an alternative interpretation, the multifractal spectrum describes the fractal dimension of the set of points having a given local Holder exponent. Finally, we show how to obtain processes with varied spectra. This allows the applied researcher to relate an empirical estimate of the multifractal spectrum back to a particular construction of the stochastic process.

JEL Classification: C22, G12, C14

Suggested Citation

Calvet, Laurent E. and Fisher, Adlai J. and Mandelbrot, Benoit B., Large Deviations and the Distribution of Price Changes (September 15, 1997). Cowles Foundation Discussion Paper No. 1165; Sauder School of Business Working Paper. Available at SSRN: https://ssrn.com/abstract=78608

Laurent E. Calvet (Contact Author)

EDHEC Business School - Department of Economics & Finance ( email )

France

CEPR ( email )

33 Great Sutton Street
London, EC1V 0DX
United Kingdom

Adlai J. Fisher

University of British Columbia (UBC) - Sauder School of Business ( email )

2053 Main Mall
Vancouver, BC V6T 1Z2
Canada
604-822-8331 (Phone)
604-822-4695 (Fax)

HOME PAGE: http://finance.sauder.ubc.ca/~fisher

Benoit B. Mandelbrot

Yale University - International Center for Finance ( email )

Box 208200
10 Hillhouse Avenue
New Haven, CT 06520-8200
United States
914-945 1712 (Phone)
914-945 4149 (Fax)

IBM Corporation - Thomas J. Watson Research Center ( email )

Route 134
Kitchawan Road
Yorktown Heights, NY 10598
United States
914-945 1712 (Phone)
914-945 4149 (Fax)

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