Výsledky projektu Fraktalita a multi-fraktalita finančních trhů: metody a aplikace

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The article introduces a new measure of capital market efficiency based on long-term memory, entropy and fractal dimension.

In this paper, we present the results of Monte Carlo simulations for two popular techniques of long-range correlation detection - classical and modified rescaled range analyses. A focus is put on an effect of different distributional properties on an ability of the methods to efficiently distinguish between short-term memory and long-term memory. To do so, we analyze the behavior of the estimators for independent, short-range dependent, and long- range dependent processes with innovations from eight different distributions. We find that apart from a combination of very high levels of kurtosis and skewness, both estimators are quite robust to distributional properties. Importantly, we show that R/S is biased upwards (yet not strongly) for short-range dependent processes, while M-R/S is strongly biased downwards for long-range dependent processes regardless of the distribution of innovations.

We investigate whether the fractal markets hypothesis and its focus on liquidity and investment horizons give reasonable predictions about the dynamics of the financial markets during turbulences such as the Global Financial Crisis of late 2000s. Compared to the mainstream efficient markets hypothesis, the fractal markets hypothesis considers the financial markets as complex systems consisting of many heterogenous agents, which are distinguishable mainly with respect to their investment horizon. In the paper, several novel measures of trading activity at different investment horizons are introduced through the scaling of variance of the underlying processes. On the three most liquid US indices - DJI, NASDAQ and S&P500 - we show that the predictions of the fractal markets hypothesis actually fit the observed behavior adequately.

We introduce a new method for the detection of long-range cross-correlations and multifractality —multifractal height cross-correlation analysis (MF-HXA)— based on scaling of q-th order covariances. MF-HXA is a bivariate generalization of the height-height correlation analysis of Baraba ?si and Vicsek (Baraba ?si A. L. and Vicsek T., Phys. Rev. A, 44 (1991) 2730). The method can be used to analyze long-range cross-correlations and multifractality between two simultaneously recorded series. We illustrate the utility of the method on both simulated and real-world time series.

We apply a method of time-dependent Hurst exponent, proposed in the series of papers by Grech and Mazur [Physica A 336, 335 (2004)], Grech and Pamula [Physica A 387, 4299 (2008)] and Czarnecki, Grech and Pamula [Physica A 387, 6801 (2008)], on the stock market of the Czech Republic for a period between 1997 and 2009. Our results support the findings of the authors so that the time-dependent Hurst exponent can give some crucial information before a critical event happens on a market. We also discuss some potentially weak points of the method.

In the paper, we research on the presence of long-range dependence in returns and volatility of Hungarian (BUX), Czech (PX) and Polish (WIG) stock indices between years 1997 and 2009 with a use of classical and modified rescaled range and rescaled variance analyses. Moving block bootstrap with pre-whitening and post-blackening is used for a construction of confidence intervals for the hypothesis testing. We show that there is no significant long-range dependence in returns of all examined indices. However, significant long-range dependence is detected in volatility of all three indices. The results for returns are contradictory with several studies which claim that developing markets are persistent. However, majority of these studies either do not use the confidence intervals at all or only the ones based on standard normal distribution. Therefore, the results of such studies should be reexamined and reinterpreted.

In this paper, we show how the sampling properties of the Hurst exponent methods of esti- mation change with the presence of heavy tails. We run extensive Monte Carlo simulations to find out how rescaled range analysis (R/S), multifractal detrended fluctuation analysis (MF-DFA), detrending moving average (DMA) and generalized Hurst exponent approach (GHE) estimate Hurst exponent on independent series with different heavy tails. For this purpose, we generate independent random series from stable distribution with stability exponent α changing from 1.1 (heaviest tails) to 2 (Gaussian normal distribution) and we estimate the Hurst exponent using the different methods. R/S and GHE prove to be robust to heavy tails in the underlying process. GHE provides the lowest variance and bias in com- parison to the other methods regardless the presence of heavy tails in data and sample size. Utilizing this result, we apply a novel approach of the intraday time-dependent Hurst ex- ponent and we estimate the Hurst exponent on high frequency data for each trading day separately. We obtain Hurst exponents for S&P500 index for the period beginning with year 1983 and ending by November 2009 and we discuss the surprising result which uncovers how the market’s behavior changed over this long period.

We reexamine the results of Serletis and Rosenberg [Serletis A, Rosenberg A. Mean reversion in the US stock market. Chaos, Solitons and Fractals 2009;40:2007–2015.] who claim that the returns of the most important US stock indices (DJI, NASDAQ, NYSE and S&P500) are strongly anti-persistent and thus mean reverting. We apply various methods to detect long-range dependence – detrending moving average, detrended fluctuation analysis, gen- eralized Hurst exponent approach, classical rescaled range analysis and modified rescaled range analysis. We show that there are no signs of anti-persistence in any of the indices. Moreover, we discuss that the authors did not find any anti-persistence but rather showed returns of the said assets do not follow the scaling power law around their moving average with varying window length. Anti-persistence is thus spurious and due to wrong application of detrending moving average method.

Long-term memory processes have been extensively examined in recent literature as they provide simple way to test for predictabilty in the underlying process. However, most of the literature interprets the results of estimated Hurst exponent simply by its comparison to its asymptotic limit of 0.5. Therefore, we use moving block bootstrap method for rescaled range and periodogram method. In our analysis of evolution of Hurst exponent between 1997 and 2009, we show that PX experien- ced persistent behavior which weakened in time. Nevertheless, the returns of PX remain close to confidence interval separating independent and persistent behavior.

We focus on finite sample properties of two mostly used methods of Hurst exponent H estimation—rescaled range analysis (R/S) and detrended fluctuation analysis (DFA). Even though both methods have been widely applied on different types of financial assets, only seve- ral papers have dealt with the finite sample properties which are crucial as the properties differ significantly from the asymptotic ones. Recently, R/S analysis has been shown to overestimate H when compared to DFA. However, we show that even though the estimates of R/S are truly significantly higher than an asymptotic limit of 0.5, for random time series with lengths from 2^9 to 2^17, they remain very close to the estimates proposed by Anis & Lloyd and the estimated standard deviations are lower than the ones of DFA. On the other hand, DFA estimates are very close to 0.5. The results propose that R/S still remains useful and robust method even when compared to newer method of DFA which is usually preferred in recent literature.