In this work, the air pollution index in three cities (Seoul, Busan, and Daegu) in South Korea was studied using multifractal detrended fluctuation analysis (MF-DFA). Hurst, Renyi, and Holder exponents were used to analyze the characteristics of the concentration time series of PM2.5 and NO2. The results showed that multifractality exists in each season interval and the multifractal degree of PM2.5 is stronger than that of NO2. To investigate the effects of the implementation of the “haze special law” on February 15, 2019, we analyzed the time series of PM2.5 during the time periods from February 15, 2018, to December 16, 2018, and February 15, 2019, to December 16, 2019. We found that the multifractal spectrum width after the implementation of the law was narrower than that before the law for all the cities, which shows that the enactment of the law has played a role in improving the efficiency of air pollution control in South Korea. We also conclude that the major effects of the law will be particularly visible in larger cities. To study the main causes of multifractality, the shuffled and phase-randomized series were analyzed using MF-DFA, and the results demonstrated that the fat-tailed distribution resulted in the multifractality of the time series before and after the implementation of the “haze special law” in Seoul and Daegu, whereas long-range correlation resulted in multifractality of the series before and after the implementation of the law in Busan.

In this paper, we use the finite difference methods to explore step-down Equity Linked Securities (ELS) options under the fractional Black-Scholes model. We establish Crank-Nicolson scheme under one asset and study the impact of Hurst exponent (H) on return of repayment under fixed stock price. We also explore the impact of stock price on return of repayment under different H. Through numerical experiments, it is found that the return of repayment of options is related to H, and the result of difference scheme will increase with the increase of H. In the case of two assets, we establish implicit scheme, and in the case of three assets, we use operator splitting method (OSM) method to establish semi-implicit scheme. We get the result that the H also influences the return of repayment in two and three assets. We also conduct Greeks analysis. Through Greeks analysis, we find that the long-term correlation of stocks has a huge impact on investment gains or losses. Therefore, we take historical volatility (fractal exponents) into account which can significantly reduce risk and increase revenue for investors.