Multifractality of Deutschemark / Us Dollar Exchange Rates

40 Pages Posted: 21 Apr 1998

See all articles by Adlai J. Fisher

Adlai J. Fisher

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

Laurent E. Calvet

EDHEC Business School - Department of Economics & Finance; CEPR

Benoit B. Mandelbrot

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

Date Written: September 15, 1997

Abstract

This paper presents the first empirical investigation of the Multifractal Model of Asset Returns ("MMAR"). The MMAR, developed in Mandelbrot, Fisher, and Calvet (1997) (See Mandelbrot, Fisher, and Calvet, 1997 at the following URL: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=78588 ), is an alternative to ARCH-type representations for modelling temporal heterogeneity in financial returns. Typically, researchers introduce temporal heterogeneity through time-varying conditional second moments in a discrete time framework or time-varying volatility in a continuous time framework. Multifractality introduces a new source of heterogeneity through time-varying local regularity in the price path. The concept of local Holder exponent describes local regularity. Multifractal processes bridge the gap between locally Gaussian (Ito) diffusions and jump-diffusions by allowing a multiplicity of Holder exponents. This paper investigates multifractality in Deutschemark / US Dollar currency exchange rates. After finding evidence of multifractal scaling, we show how to estimate the multifractal spectrum via the Legendre transform. The scaling laws found in the data are replicated in simulations. Further simulation experiments test whether alternative representations, such as FIGARCH, are likely to replicate the multifractal signature of the Deutschemark / US Dollar data. On the basis of this evidence, the MMAR hypothesis appears more likely. Overall, the MMAR is quite successful in uncovering a previously unseen empirical regularity. Additionally, the model generates realistic sample paths and opens the door to new theoretical and applied approaches to asset pricing and risk valuation. We conclude by advocating further empirical study of multifractality in financial data, along with more intensive study of estimation techniques and inference procedures.

JEL Classification: G12, F31, C22, G15

Suggested Citation

Fisher, Adlai J. and Calvet, Laurent E. and Mandelbrot, Benoit B., Multifractality of Deutschemark / Us Dollar Exchange Rates (September 15, 1997). Cowles Foundation Discussion Paper No. 1166; Sauder School of Business Working Paper. Available at SSRN: https://ssrn.com/abstract=78628

Adlai J. Fisher (Contact Author)

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

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

Laurent E. Calvet

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

France

CEPR ( email )

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London, EC1V 0DX
United Kingdom

Benoit B. Mandelbrot

Yale University - International Center for Finance ( email )

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

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United States
914-945 1712 (Phone)
914-945 4149 (Fax)

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