Multifractality of Nonlinear Transformations with Application in Finances

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.