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Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Physica A: Statistical Mechanics and its Applications, Volume 347, 1 March 2005, Pages 626–638
By applying the multifractal detrended fluctuation analysis to the high-frequency tick-by-tick data from Deutsche Börse both in the price and in the time domains, we investigate multifractal properties of the time series of logarithmic price increments and inter-trade intervals of time. We show that both quantities reveal multiscaling and that this result holds across different stocks. The origin of the multifractal character of the corresponding dynamics is, among others, the long-range correlations in price increments and in inter-trade time intervals as well as the non-Gaussian distributions of the fluctuations. Since the transaction-to-transaction price increments do not strongly depend on or are almost independent of the inter-trade waiting times, both can be sources of the observed multifractal behaviour of the fixed-delay returns and volatility. The results presented also allow one to evaluate the applicability of the Multifractal Model of Asset Returns in the case of tick-by-tick data
نتیجه گیری انگلیسی
We study the multifractal properties of the most liquid stocks from the German stock market. Our original data consisting of the recordings of time and price at which all the transactions took place, allowed us to separate the complete process of the stock trading into its pure price and pure time components. Since both components can contribute to the fixed-Dt volatility and returns, studying the properties of these components can help to identify the microscopic sources of the observed multifractality of the fixed-delay returns. We show that both the signals for the transaction-to-transaction price increments and for the inter-trade waiting times exhibit the characteristics that can be interpreted in terms of multifractality. Its degree expressed by, e.g., the widths of the singularity spectra f ðaÞ varies across different stocks but these properties are entirely company-specific and are not related to industry sectors, company size, average transaction frequency or any other characteristics of this type. The multifractal properties of Gs and Ts are of different nature; though on a qualitative level the corresponding spectra can be related. The relevant relation  between the f ðaÞ spectra for the returns and for the multifractal time yðtÞ does not however apply fully quantitatively. This is because the c.d.f. of the microscopic price increments and the c.d.f. of the inter-trade time intervals are not related through the Brownian motion as required in Ref. . If the price and the time components of the trading are independent or they at most weakly depend on each other, the compelling next step in this kind of analysis is to perform a fully 2-D multifractal approach in order to link the fractal nature of the price and the time increments into one unified frame.