In this paper, high frequency (per 5 min) data of Shanghai Stock Exchange Composite index (SSEC) from January 1999 to July 2001 is analyzed by multifractal. We find that the correlation of the parameters of the multifractal spectra with the variation of daily return ZZ in SSEC is noticeably different from that in previous studies of Heng Seng index in Hong Kong stock market [Sun et al., Phys. A 291 (2001) 553–562; Sun et al., Phys. A 301 (2001) 473–482]. So, we suppose that there may not be a universal rule for the dependence of the parameters of the multifractal spectra with daily return of a stock index. Then, we construct a new measurement of market risk based on multifractal spectra, and test its ability of predicting index fluctuations with a more thorough method than that in Sun et al.
In Refs. [1] and [2], Hang Seng index in Hong Kong stock market is analyzed by multifractal, and two main parameters of multifractal spectrum of daily high frequency indices, ΔαΔα and ΔfΔf, are calculated. Some interesting empirical results are discovered. For example, it is found that the amount of the variation of daily return is correlated with the amount of ΔαΔα of that day, and the average of ΔfΔf increases with increasing daily return. Therefore, the predictability of today's gain probability (G%)(G%) and the day's index increase probability (N%)(N%) with ΔfΔf of the previous 3 days are studied.
However, in our multifractal analysis of Shanghai Stock Exchange Composite index (SSEC), the most important index in China stock market, we find that the average of ΔfΔf does not increase, but decreases with increasing daily return, ZZ. So, we suppose that there may not be a universal rule for the dependence of ΔfΔf with daily return of a stock index. Moreover, the statistical significance of the dependence of the average of ΔfΔf with ZZ of SSEC is noticeably poorer than that of a new market risk measurement RfRf, which is constructed by the two main parameters of multifractal spectra in Section 3, with daily return ZZ. We also find that today's market risk measurement RfRf can be used to predict the next day's gain probability (G%)(G%) and index increase probability (N%)(N%), but a more thorough method than that in Ref. [2] is used.
This paper is organized as follows. In Section 2, the basic multifractal analysis is carried out, using high-frequency data (a quotation per 5 min) of SSEC, from 19 January 1999 to 6 July 2001 (586 trading days), totally 28,128 quotes. In Section 3, a new market risk measurement RfRf, based on multifractal spectra, is introduced, and the dependence of RfRf with daily return of SSEC is also investigated. In Section 4, predictability of the next day's gain probability (G%)(G%) and index increase probability (N%)(N%) based on today's RfRf is tested, and some conclusions are made in Section 5.
It has been proved that multifractal is a powerful tool to describe the anomalous phenomena in financial markets by Mandelbrot and many other researchers [3], [4], [5], [6], [7], [8], [9] and [10]. However, many such researches concentrate only on the examining of multifractal characteristics of different kinds of financial data, such as, stock index and exchange rate. How to make full use of the empirical statistic results in the multifractal analysis to improve the practice of financial management is still lacking. Recently, Sun et al. [1] and [2] have found that ΔfΔf, one of the main parameters of multifractal spectra, is in direct correlation of the daily return ZZ of the Hang Seng index. Based on this finding, they have studied the predictability of the gain probability G%G% and increase probability N%N% of the index.
However, in our research, we find a different empirical result that ΔfΔf is in inverse correlation of the daily return ZZ of SSEC. So, we suppose that there may not be a universal rule for the dependence of ΔfΔf with daily return of a stock index. To make better use of the statistical information in multifractal spectra, we construct a new market risk measurement, RfRf, which contains the integrated information of the two parameters of multifractal spectra, ΔfΔf and ΔαΔα. We also study the predictability of the gain probability G%G% and increase probability N%N% of SSEC with a subtler method, and explain the reasons of such predictability using the theories of technical analysis.
In conclusion, multifractal is a powerful tool in the researches of price fluctuations of financial markets. Lots of statistical information may be obtained in the multifractal analysis to help us to understand the complexity of financial markets.
