Journal of the Korean Physical Society, Vol. 77, No. 3, August 2020, pp. 186∼196 Review Articles Fractality and Multifractality in a Stock Market’s Nonstationary Financial Time Series Nam Jung, Quang Anh Le, Biseko J. Mafwele, Hyun Min Lee and Seo Yoon Chae Department of Physics, Inha University, Incheon 22212, Korea Jae Woo Lee ∗ Department of Physics, Inha University, Incheon 22212, Korea Institue of Natural Basic Sciences, Inha University, Incheon 22212, Korea and Institue of Advanced Computational Sciences, Inha University, Incheon 22212, Korea (Received 2 March 2020; revised 30 March 2020; accepted 12 May 2020) 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. Keywords: Multifractality, Financial market, Stock market, Econophysics, Detrended ﬂuctuation analysis DOI: 10.3938/jkps.77.186 I. INTRODUCTION Financial markets have been studied for the last three decades based on the concepts of complex systems [1–10]. In particular, time series in stock markets are recorded accurately all over the world. Stock market indices or stock prices of an individual stock are analyzed with nonlinear dynamics and complex systems methods. In the 1960s, Mandelbrot performed a pioneering empirical analysis for the distribution of income [1, 2], the price variation distribution of assets [3], and long-term cor- relation in an economic time series by using rescaled range analysis [4]. In the early 1990s, pioneering work on econophysics was conducted by statistical physicists [5– 8]. They observed the L´evy walks on the returns of the stock indices. The probability distribution of a return showed a scaling behavior and a fat-tailed distribution. Laherr`ere and Sornette reported stretched exponential distributions for a ﬁnancial time series [9]. Gopikrishnan et al. observed an inverse cubic law for the probability distribution of the return in stock market indices [10,11]. The ﬂuctuation of a ﬁnancial time series showed scaling and multi-scaling properties. The ﬁnancial time series showed stylized facts, includ- ing the power law distributions of the return and the ∗ E-mail: jaewlee@inha.ac.kr volatility, the long-range correlation of volatility, gain- loss asymmetry, volatility clustering, and scaling prop- erties [12–15]. Scaling and multi-scaling behaviors were studied in the early 1980s. Multifractality was observed in the turbulence of ﬂuid dynamics [16–19]. The con- cept of multifractality has been extended to the strange attractor in nonlinear dynamics, to the fractal growth processes (such as diﬀusion-limited aggregation), and to dendritic growth processes [20–24]. Fractal and multi- fractal objects were described by using the generalized fractal dimension, information dimension, and general- ized Renyi entropy [20–22]. Halsey et al. applied multi- fractal concepts to fractal growth processes [23,24]. They introduced multi-scaling exponents, τ (q), characterizing the generalized partition function, the exponent α of sin- gularity strength, and the exponent f (α) of the singular- ity spectrum. The exponent τ (q) is related to exponents α − f (α) via a Legendre transformation [24]. The origins of the scaling and the multiscaling prop- erties in the ﬁnancial time series are still not understood well. One is the power law distribution of the logarith- mic return. The other is the long-range autocorrelation of the absolute returns or the volatility. However, other reasons inﬂuencing multiscaling can be suggested. We believe that herding behaviors exist among the hetero- geneous multi-agents in the stock market. For example, the behaviors of the individual investors are very diﬀer- pISSN:0374-4884/eISSN:1976-8524 -186- c 2020 The Korean Physical Society