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Fig. 4. Power-law correlations for worldwide financial indices in (Upper) absolute values of price changes (|R|) and (Lower) absolute values of trading- volume changes (|R|). We use the method described by Eqs. 7 and 8. (Upper) The average return interval t vs. threshold q (in units of standard deviation) for absolute values of trading-volume changes. For each of 28 worldwide financial indices, we calculate the corresponding t, values. Then we collect all the t values obtained from different indices, and show Tt, versus q. Up to eight standard deviations, we find a power law with exponent & = 2.41+0.06. (Lower) The average return interval tg vs. threshold q for absolute values of price changes (see Eq. 3) for different levels of aggregation. For each of five different types of aggregation reported, we find that t versus q exhibits a power law with an exponent very close to a = 3. Finally, in addition to trading-volume changes, we employ for stock price changes our procedure for identifying power-law behavior in the pdf tails described in Methods (case ii). The pdf of

Figure 4 Power-law correlations for worldwide financial indices in (Upper) absolute values of price changes (|R|) and (Lower) absolute values of trading- volume changes (|R|). We use the method described by Eqs. 7 and 8. (Upper) The average return interval t vs. threshold q (in units of standard deviation) for absolute values of trading-volume changes. For each of 28 worldwide financial indices, we calculate the corresponding t, values. Then we collect all the t values obtained from different indices, and show Tt, versus q. Up to eight standard deviations, we find a power law with exponent & = 2.41+0.06. (Lower) The average return interval tg vs. threshold q for absolute values of price changes (see Eq. 3) for different levels of aggregation. For each of five different types of aggregation reported, we find that t versus q exhibits a power law with an exponent very close to a = 3. Finally, in addition to trading-volume changes, we employ for stock price changes our procedure for identifying power-law behavior in the pdf tails described in Methods (case ii). The pdf of

Related Figures (4)

Fig. 1. Autocorrelations and cross-correlations in absolute values of price changes |R;| of Eq. 3. and trading-volume changes |Rz| of Eq. 4 for daily returns of the S & P 500 Index. (A) The cross-correlation function C(R, R) between R and R, and the cross-correlation function C(|R|, |R|) between |R| and |RI. (B) For R(t), and R(t), we show the rms of the detrended variance Fpra(n) for |R| and |R| and also the rms of the detrended covariance (37), Fpcca(n). The two DFA exponents Aj\q, and 24, imply that power-law autocorrelations exist in both |R| and |R|. The DCCA exponent implies the presence of power- law cross-correlations. Power-law cross-correlations between |R| and |R| imply that current price changes depend upon previous changes but also upon previous volume changes and vice versa.
Fig. 1. Autocorrelations and cross-correlations in absolute values of price changes |R;| of Eq. 3. and trading-volume changes |Rz| of Eq. 4 for daily returns of the S & P 500 Index. (A) The cross-correlation function C(R, R) between R and R, and the cross-correlation function C(|R|, |R|) between |R| and |RI. (B) For R(t), and R(t), we show the rms of the detrended variance Fpra(n) for |R| and |R| and also the rms of the detrended covariance (37), Fpcca(n). The two DFA exponents Aj\q, and 24, imply that power-law autocorrelations exist in both |R| and |R|. The DCCA exponent implies the presence of power- law cross-correlations. Power-law cross-correlations between |R| and |R| imply that current price changes depend upon previous changes but also upon previous volume changes and vice versa.
Fig. 3. Pdf of absolute value of differences in logarithm of trading volume, R, of Eq. 4 for the members of the NYSE Composite Index. We use the method described in the Methods section—case ii—for normalized volatilities of Eq. 6 (A) From 1 o to 15 0 we show the linear-log plot of the pdf P(R). The straight line is exponential fit. The far tail of pdf deviates from the fit in the central region of pdf. (B) Log-log plot of pdf from 15 0 to 25 o. The tail part of the pdt can be explained by a power law R'*@ with & = 4.2+0.26. (C) For the absolute values of changes in trading volume (see Eq. 4), we show the average return interval tg versus threshold q (in units of a standard deviation). Up to 8 o, we show a power law with exponent & = 3.11 + 0.12 which leads to the inverse cubic law.
Fig. 3. Pdf of absolute value of differences in logarithm of trading volume, R, of Eq. 4 for the members of the NYSE Composite Index. We use the method described in the Methods section—case ii—for normalized volatilities of Eq. 6 (A) From 1 o to 15 0 we show the linear-log plot of the pdf P(R). The straight line is exponential fit. The far tail of pdf deviates from the fit in the central region of pdf. (B) Log-log plot of pdf from 15 0 to 25 o. The tail part of the pdt can be explained by a power law R'*@ with & = 4.2+0.26. (C) For the absolute values of changes in trading volume (see Eq. 4), we show the average return interval tg versus threshold q (in units of a standard deviation). Up to 8 o, we show a power law with exponent & = 3.11 + 0.12 which leads to the inverse cubic law.
Fig. 5. Cross-correlations between two time series generated from the sto- chastic process of Eqs. 14-17, with 8 = 6 = 0.65, a = &@ = 0.14, y = 7 = 0.2, and @ = @ = 0.01. In A, we show the time series « and € of Eqs. 14-17, where the latter time series is shifted for clarity. These two time series follow each other due to the terms y 4 0 and y 4 O. In B, we show the autocorrelation function A(n) for |e¢| and the cross-correlation function C(|é|, ||). The 95% confidence intervals for no cross-correlations are shown (solid lines) along with the best exponential fit of A(n) (dotted curve).
Fig. 5. Cross-correlations between two time series generated from the sto- chastic process of Eqs. 14-17, with 8 = 6 = 0.65, a = &@ = 0.14, y = 7 = 0.2, and @ = @ = 0.01. In A, we show the time series « and € of Eqs. 14-17, where the latter time series is shifted for clarity. These two time series follow each other due to the terms y 4 0 and y 4 O. In B, we show the autocorrelation function A(n) for |e¢| and the cross-correlation function C(|é|, ||). The 95% confidence intervals for no cross-correlations are shown (solid lines) along with the best exponential fit of A(n) (dotted curve).
Related topics:
FinanceStochastic ProcessTrade VolumeNew York Stock Exchange
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