Complex Systems: From Nuclear Physics to Financial Markets

We compare correlations and coherent structures in nuclei and financial markets. In the nuclear physics part we review giant resonances which can be interpreted as a coherent structure embedded in chaos. With similar methods we investigate the financial empirical correlation matrix of the DAX and Dow Jones. We will show, that if the time-zone delay is properly accounted for, the two distinct markets largely merge into one. This is reflected by the largest eigenvalue that develops a gap relative to the remaining, chaotic eigenvalues. By extending investigations of the specific character of financial collectivity we also discuss the criticality-analog phenomenon of the financial log-periodicity and show specific examples.Comment: 6 pages, 10 figures, elsarticle clas

Decomposing the stock market intraday dynamics

The correlation matrix formalism is used to study temporal aspects of the stock market evolution. This formalism allows to decompose the financial dynamics into noise as well as into some coherent repeatable intraday structures. The present study is based on the high-frequency Deutsche Aktienindex (DAX) data over the time period between November 1997 and September 1999, and makes use of both, the corresponding returns as well as volatility variations. One principal conclusion is that a bulk of the stock market dynamics is governed by the uncorrelated noise-like processes. There exists however a small number of components of coherent short term repeatable structures in fluctuations that may generate some memory effects seen in the standard autocorrelation function analysis. Laws that govern fluctuations associated with those various components are different, which indicates an extremely complex character of the financial fluctuations.Comment: 15 pages, 13 PostScript figure

Alternation of different fluctuation regimes in the stock market dynamics

Based on the tick-by-tick stock prices from the German and American stock markets, we study the statistical properties of the distribution of the individual stocks and the index returns in highly collective and noisy intervals of trading, separately. We show that periods characterized by the strong inter-stock couplings can be associated with the distributions of index fluctuations which reveal more pronounced tails than in the case of weaker couplings in the market. During periods of strong correlations in the German market these distributions can even reveal an apparent L\'evy-stable component.Comment: 19 page

Nature of order from random two-body interactions

We investigate the origin of order in the low-lying spectra of many-body systems with random two-body interactions. Our study based both on analytical as well as on numerical arguments shows that except for the most $J$-stretched states, the ground states in the higher $J$-sectors are more orderly and develop larger energy gaps than the ones in the J=0-sector. Due to different characteristic energy scales in different $J$-sectors the J=0 ground states may predominate only when all the states are taken together.Comment: LaTeX2e-RevTeX, 11 pages, 5 Postscript figure

Time scales involved in market emergence

In addressing the question of the time scales characteristic for the market formation, we analyze high frequency tick-by-tick data from the NYSE and from the German market. By using returns on various time scales ranging from seconds or minutes up to two days, we compare magnitude of the largest eigenvalue of the correlation matrix for the same set of securities but for different time scales. For various sets of stocks of different capitalization (and the average trading frequency), we observe a significant elevation of the largest eigenvalue with increasing time scale. Our results from the correlation matrix study go in parallel with the so-called Epps effect. There is no unique explanation of this effect and it seems that many different factors play a role here. One of such factors is randomness in transaction moments for different stocks. Another interesting conclusion to be drawn from our results is that in the contemporary markets the emergence of significant correlations occurs on time scales much smaller than in the more distant history.Comment: 13 page

Identifying Complexity by Means of Matrices

Complexity is an interdisciplinary concept which, first of all, addresses the question of how order emerges out of randomness. For many reasons matrices provide a very practical and powerful tool in approaching and quantifying the related characteristics. Based on several natural complex dynamical systems, like the strongly interacting quantum many-body systems, the human brain and the financial markets, by relating empirical observations to the random matrix theory and quantifying deviations in term of a reduced dimensionality, we present arguments in favour of the statement that complexity is a pheomenon at the edge between collectivity and chaos.Comment: Talk given by S. Drozdz at "Horizons in Complex Systems", Messina, December 5-8, 200

Quantifying dynamics of the financial correlations

A novel application of the correlation matrix formalism to study dynamics of the financial evolution is presented. This formalism allows to quantify the memory effects as well as some potential repeatable intradaily structures in the financial time-series. The present study is based on the high-frequency Deutsche Aktienindex (DAX) data over the time-period between November 1997 and December 1999 and demonstrates a power of the method. In this way two significant new aspects of the DAX evolution are identified: (i) the memory effects turn out to be sizably shorter than what the standard autocorrelation function analysis seems to indicate and (ii) there exist short term repeatable structures in fluctuations that are governed by a distinct dynamics. The former of these results may provide an argument in favour of the market efficiency while the later one may indicate origin of the difficulty in reaching a Gaussian limit, expected from the central limit theorem, in the distribution of returns on longer time-horizons.Comment: 10 pages, 7 PostScript figures, talk presented by the first Author at the NATO ARW on Econophysics, Prague, February 8-10, 2001; to be published in proceedings (Physica A

Wavelet versus Detrended Fluctuation Analysis of multifractal structures

We perform a comparative study of applicability of the Multifractal Detrended Fluctuation Analysis (MFDFA) and the Wavelet Transform Modulus Maxima (WTMM) method in proper detecting of mono- and multifractal character of data. We quantify the performance of both methods by using different sorts of artificial signals generated according to a few well-known exactly soluble mathematical models: monofractal fractional Brownian motion, bifractal Levy flights, and different sorts of multifractal binomial cascades. Our results show that in majority of situations in which one does not know a priori the fractal properties of a process, choosing MFDFA should be recommended. In particular, WTMM gives biased outcomes for the fractional Brownian motion with different values of Hurst exponent, indicating spurious multifractality. In some cases WTMM can also give different results if one applies different wavelets. We do not exclude using WTMM in real data analysis, but it occurs that while one may apply MFDFA in a more automatic fashion, WTMM has to be applied with care. In the second part of our work, we perform an analogous analysis on empirical data coming from the American and from the German stock market. For this data both methods detect rich multifractality in terms of broad f(alpha), but MFDFA suggests that this multifractality is poorer than in the case of WTMM.Comment: substantially extended version, to appear in Phys.Rev.

The foreign exchange market: return distributions, multifractality, anomalous multifractality and Epps effect

We present a systematic study of various statistical characteristics of high-frequency returns from the foreign exchange market. This study is based on six exchange rates forming two triangles: EUR-GBP-USD and GBP-CHF-JPY. It is shown that the exchange rate return fluctuations for all the pairs considered are well described by the nonextensive statistics in terms of q-Gaussians. There exist some small quantitative variations in the nonextensivity q-parameter values for different exchange rates and this can be related to the importance of a given exchange rate in the world's currency trade. Temporal correlations organize the series of returns such that they develop the multifractal characteristics for all the exchange rates with a varying degree of symmetry of the singularity spectrum f(alpha) however. The most symmetric spectrum is identified for the GBP/USD. We also form time series of triangular residual returns and find that the distributions of their fluctuations develop disproportionately heavier tails as compared to small fluctuations which excludes description in terms of q-Gaussians. The multifractal characteristics for these residual returns reveal such anomalous properties like negative singularity exponents and even negative singularity spectra. Such anomalous multifractal measures have so far been considered in the literature in connection with the diffusion limited aggregation and with turbulence. We find that market inefficiency on short time scales leads to the occurrence of the Epps effect on much longer time scales. Although the currency market is much more liquid than the stock markets and it has much larger transaction frequency, the building-up of correlations takes up to several hours - time that does not differ much from what is observed in the stock markets. This may suggest that non-synchronicity of transactions is not the unique source of the observed effect

Spectral Decorrelation of Nuclear Levels in the Presence of Continuum Decay

The fluctuation properties of nuclear giant resonance spectra are studied in the presence of continuum decay. The subspace of quasi-bound states is specified by one-particle one-hole and two-particle two-hole excitations and the continuum coupling is generated by a scattering ensemble. It is found that, with increasing number of open channels, the real parts of the complex eigenvalues quickly decorrelate. This appears to be related to the transition from power-law to exponential time behavior of the survival probability of an initially non-stationary state.Comment: 10 Pages, REVTEX, 4 PostScript figure