Detailed study of the financial empirical correlation matrix of the 30 companies comprised by DAX within the period of the last 11 years, using the time-window of 30 trading days, is presented. This allows to clearly identify a nontrivial time-dependence of the resulting correlations. In addition, as a rule, the draw downs are always accompanied by a sizable separation of one strong collective eigenstate of the correlation matrix which, at the same time, reduces the variance of the noise states. The opposite applies to draw ups. In this case the dynamics spreads more uniformly over the eigenstates which results in an increase of the total information entropy.PACS numbers: 01.75.+m Science and society -05.40.+j Fluctuation phenomena, random processes, and Brownian motion -89.90.+n Other areas of general interest to physicists

A hypothesis that the financial log-periodicity, cascading self-similarity through various time scales, carries signatures of a law is pursued. It is shown that the most significant historical financial events can be classified amazingly well using a single and unique value of the preferred scaling factor lambda=2, which indicates that its real value should be close to this number. This applies even to a declining decelerating log-periodic phase. Crucial in this connection is identification of a "super-bubble" (bubble on bubble) phenomenon. Identifying a potential "universal" preferred scaling factor, as undertaken here, may significantly improve the predictive power of the corresponding methodology. Several more specific related results include evidence that: (i) the real end of the high technology bubble on the stock market started (with a decelerating log-periodic draw down) in the begining of September 2000; (ii) a parallel 2000-2002 decline seen in the Standard & Poor's 500 from the log-periodic perspective is already of the same significance as the one of the early 1930s and of the late 1970s; (iii) all this points to a much more serious global crash in around 2025, of course from a level much higher (at least one order of magnitude) than in 2000.Comment: Talk given by S. Drozdz at International Econophysics Conference, Bali, August 28-31, 2002; typos correcte

Effects connected with the world globalization affect also the financial markets. On a way towards quantifying the related characteristics we study the financial empirical correlation matrix of the 60 companies which both the Deutsche Aktienindex (DAX) and the Dow Jones (DJ) industrial average comprised during the years 1990-1999. The time-dependence of the underlying cross-correlations is monitored using a time window of 60 trading days. Our study shows that if the time-zone delays are properly accounted for the two distant markets largely merge into one. This effect is particularly visible during the last few years. It is however the Dow Jones which dictates the trend.

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.

Detailed analysis of the log-periodic structures as precursors of the financial crashes is presented. The study is mainly based on the German Stock Index (DAX) variation over the 1998 period which includes both, a spectacular boom and a large decline, in magnitude only comparable to the so-called The fact that a healthy and normally functioning financial market may reveal certain properties common to complex systems is fascinating and, in fact, seems natural. Especially interesting in this context is the recently suggested analogy of the financial crashes to critical points in statistical mechanics [1][2][3][4][5][6]. Criticality implies a scale invariance which in mathematical terms, for a properly defined function F (x) characterizing the system, means that for small x F (λx) = γF (x).(1)A positive constant γ in this equation describes how the properties of the system change when it is rescaled by the factor λ. The simplest solution to this equation reads:where α = log(γ)/ log(λ). This is a standard power-law that is characteristic of continuous scale-invariance and α is the corresponding critical exponent.More interesting is the general solution [7] to Eq. (1):where P denotes a periodic function of period one. In this way the dominating scaling (2) acquires a correction which is periodic in log(x). This solution accounts for a possible discrete scale-invariance [8] and can be interpreted [9,10] in terms of a complex critical, which corresponds to the first term in a Fourier expansion of (3). Thus, if x represents a distance to the critical point, the resulting spacings between consecutive minima x n (maxima) of the log-periodic oscillations seen in the linear scale follow a geometric contraction according to the relation:Then, the critical point coincides with the accumulation of such oscillations. 3Existence of the log-periodic modulations correcting the structureless pure power-law behaviour has been identified in many different systems [8]. Examples include diffusionlimited-aggregation clusters [11], crack growth [12], earthquakes [9,10] and, as already mentioned, the financial market where x is to be interpreted as the time to crash. Especially in the last two cases this is an extremely interesting feature because it potentially offers a tool for predictions. Of course, the real financial market is exposed to many external factors which may distort its internal hierarchical structure on the organizational as well as on the dynamical level. Therefore, the searches for the long term, of the order of few years, precursors of crashes have to be taken with some reserve, as already pointed out in Ref. [13].A somewhat related example is shown in Fig. 1 which displays the S&P 500 versus DAX charts between 1991 and February 1999. While the global characteristics of the two charts are largely compatible there exist several significant differences on shorter time-scales. It is the purpose of the present paper to explore more in detail the emerging short-time behaviour of the stock market indices.On the more ...