Fractality and Multifractality in a Stock Market's Nonstationary Financial Time Series

A financial time series, such as a stock market index, foreign exchange rate, or a commodity price, fluctuates heavily and shows scaling behaviors. Scaling and multi-scaling behaviors are measured for a nonstationary time series, such as stock market indices, high-frequency stock prices of individual stocks, or the volatility time series of a stock index. We review the fractality, multi-scaling, and multifractality of the financial time series of a stock market. We introduce a detrended fluctuation analysis of the financial time series to extract fluctuation patterns. Multifractality is measured using various methods, such as generalized Hurst exponents, the generalized partition function method, a detrended fluctuation analysis, the detrended moving average method, and a wavelet transformation.