Multifractality of Nonlinear Transformations with Application in Finances

• Authors: D. Grech , G. Pamuła
• Languages of publication: EN

Abstracts

EN

We study the multifractal effects of nonlinear transformations of monofractal, stationary time series and apply the found results to measure the "true" unbiased multifractality generated only by multiscaling properties of initial (primary) data before transformations. A difference is stressed between "naive" observed multifractal effects calculated directly within detrended multifractal analysis as the spread Δh of the generalized Hurst exponents h(q) and the more reliable unbiased multifractality received after subtraction of residual bias effects generated by nonlinear transformations of initial data and coupled with finite size effects in time series. This property is investigated for volatile series of the real main world financial indices. A difference between multifractal properties of intraday and interday quotes is also pointed out in this context for the Warsaw Stock Exchange WIG index. Finally, based on the observed feature of real nonstationary data, a new measure of unbiased multifractality in signals is introduced. This measure comes from an analysis of the whole generalized Hurst exponent profile instead of looking just at its edge behavior h^{±} ≡ h(q→ ±∞). Such an approach seems to be particularly useful when h(q) is not a monotonic function of the moment order q. Interesting examples with extreme events from finance are presented. They convince that an analysis directed only on investigation of the edges h^{±} in multifractal spectrum may be misleading.

Keywords

EN

05.45.Df; 05.45.Tp; 89.65.Gh; 89.75.Da; 89.75.-k; 89.20.-a; 05.40.-a

Discipline

• 05.45.Df: Fractals(see also 47.53.+n Fractals in fluid dynamics; 61.43.Hv Fractals; macroscopic aggregates in structure of solids)
• 89.75.-k: Complex systems(for complex chemical systems, see 82.40.Qt; for biological complexity, see 87.18.-h)
• 05.45.Tp: Time series analysis
• 89.20.-a: Interdisciplinary applications of physics
• 89.65.Gh: Economics; econophysics, financial markets, business and management(for economic issues regarding production and use of renewable energy, see 88.05.Lg)
• 89.75.Da: Systems obeying scaling laws
• 05.40.-a: Fluctuation phenomena, random processes, noise, and Brownian motion(for fluctuations in superconductivity, see 74.40.-n; for statistical theory and fluctuations in nuclear reactions, see 24.60.-k; for fluctuations in plasma, see 52.25.Gj; for nonlinear dynamics and chaos, see 05.45.-a)

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2013
• Volume: 123
• Issue: 3
• Pages: 529-537

Dates

published

2013-03

Contributors

author

D. Grech

• Institute of Theoretical Physics, University of Wrocław, pl. M. Borna 9, PL-50-204 Wrocław, Poland

author

G. Pamuła

• Institute of Theoretical Physics, University of Wrocław, pl. M. Borna 9, PL-50-204 Wrocław, Poland

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Identifiers

DOI

10.12693/APhysPolA.123.529