MULTIFRACTAL FLUCTUATIONS IN FINANCE

Francois Schmitt (), Daniel Schertzer and Shaun Lovejoy
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Francois Schmitt: Dept. of Fluid Mechanics VUB, 2 Pleinlaan, B-1050 Brussels, Belgium
Daniel Schertzer: LMM, University of Paris VI, 4, Place Jussieu, F-75005 Paris, France
Shaun Lovejoy: McGILL University, Physics Department, 3600 University Street, Montreal H3A2T8, Canada

International Journal of Theoretical and Applied Finance (IJTAF), 2000, vol. 03, issue 03, 361-364

Abstract: We consider the structure functionsS(q)(τ), i.e. the moments of orderqof the incrementsX(t + τ)-X(t)of the Foreign Exchange rateX(t)which give clear evidence of scaling(S(q)(τ)∝τζ(q)). We demonstrate that the nonlinearity of the observed scaling exponentζ(q)is incompatible with monofractal additive stochastic models usually introduced in finance: Brownian motion, Lévy processes and their truncated versions. This nonlinearity correspond to multifractal intermittency yielded by multiplicative processes. The non-analyticity ofζ(q)corresponds to universal multifractals, which are furthermore able to produce "hyperbolic" pdf tails with an exponentqD> 2. We argue that it is necessary to introduce stochastic evolution equations which are compatible with this multifractal behaviour.

Date: 2000
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DOI: 10.1142/S0219024900000206

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