László Kullmann

Identification of clusters of companies in stock indices via Potts super-paramagnetic transitions

The clustering of companies within a specific stock market index is studied by means of super-paramagnetic transitions of an appropriate q-state Potts model where the spins correspond to companies and the interactions are functions of the correlation coefficients determined from the time dependence of the companies’ individual stock prices. The method is a generalization of the clustering algorithm by Domany et al. to the case of anti-ferromagnetic interactions corresponding to anti-correlations. For the Dow Jones industrial average where no anti-correlations were observed in the investigated time period, the previous results obtained by different tools were well reproduced. For the Standard & Poors 500, where anti-correlations occur, repulsion between stocks modify the cluster structure.

Characteristic times in stock market indices

In this study we analyze the Standard and Poors 500 index data of the New York Stock Exchange for more than 32 years. Using a simple random walk model we demonstrate that the proper variable to look at is the logarithmic return. In the statistical analysis we have done fittings to the Levy distribution using either the index data as such or pre-processing it with ARCH, GARCH or IGARCH methods, which tend to remove the time-dependent variance. For short times the truncated Levy distribution is found to fit the data quite well. Since this is not a stable distribution, the scaling behavior observed for short times should brake down for longer times. We demonstrate that the characteristic time where this cross-over starts is of the order of one day.

Preferential growth: solution and application to modeling stock market

We consider a preferential growth model where particles are added one by one to the system consisting of clusters of particles. A new particle can either form a new cluster (with probability q) or join an already existing cluster with a probability proportional to the size thereof. We calculate exactly the probability Pi(k,t) that the size of the ith cluster at time t is k. We applied our model as a background for a microscopic economic model.

Break-down of Scaling and Convergence to Gaussian Distribution in Stock Market Data

We analyze the Standard and Poors 500 index data of the New York Stock Exchange for more than 32 years. It was suggested earlier that the high frequency data are well described by a truncated Levy distribution and scaling with respect to the sampling time differences was found. The truncated character of the distribution implies that scaling must break down and that the distribution ultimately converges to a Gaussian. We show by comparing Levy and Gaussian fits that the characteristic time of the break-down of scaling is of the order of few days. The analysis of the dependence of the kurtosis on the time differences shows that this is much shorter than the time needed for the convergence to the Gaussian being of the order of months.

Crossover To Gaussian Behavior In Herding Market Models

We have analyzed possible mechanisms of the crossover to the Gaussian distribution of the logarithmic returns in the Cont–Bouchaud herding model of the stock market. Either the underlying cluster distribution is not in the Levy attraction regime, or a cut-off effect is responsible for the crossover. The cut-off can be due to the finite size of the system, where clusters are created. If such finite size effects are responsible for the crossover, a delicate interplay between the size dependence of the deviation from the Gaussian and of the number of values to be summed up in one step may result in a size-independent crossover value of the activity. It is shown that this is the case for percolation clusters in spatial dimensions from 2 to 6. A further origin of the cut-off can be the limited number of clusters taken into account.

Correlations and response: absence of detailed balance on the stock market

In and near thermal equilibrium microscopic reversibility resulting in detailed balance has specific consequences like the symmetry of time dependent cross correlation (TDCC) functions and fluctuation dissipation theorem. In this paper we study some consequences of the absence of microscopic reversibility on financial processes. By analyzing high resolution data we find asymmetric TDCC functions indicating dominance of some companies in the price formation procedure. We show that in the Lux–Marchesi multi agent market model spontaneous fluctuations decay differently from perturbations caused by external effects.

Measurement on the cavitating vortex shedding behind rectangular obstacles

Measurement results on the cavitating vortex shedding behind sharp-edged rectangular bodies are presented, intended to provide benchmark cases for the validation of unsteady cavitation models of CFD codes. Rectangular bodies of increasing aspect ratio (1, 2, 3 and 4) were used with a constant 25mm height (12.5% blockage ratio). The water velocity in the 0.2x0.05m test section of the channel was varied between 1 and 12 m/s resulting in a Reynolds number in the range of (0.4-3.5)x105. Pressure signals were measured at several locations, notably in the wake. Dominant frequencies and Strouhal numbers are reported from cavitation-free flow (classic von Karman vortex shedding) up to supercavitation as a function of the free-stream Reynolds number. The results are in good agreement with the literature in case of the square cylinder. We experienced a slight increase of the dominant Strouhal number with increasing aspect ratio. This result is somewhat inconsistent with the literature, in which a fall of the Strouhal number can be observed at side ratio 2. This may be the consequence of the different ranges of Reynolds numbers. It was also found that between the inception of cavitation and the formation of supercavitation the Strouhal number is not affected by cavitation.

Time dependent correlations and response in stock market data and models

We study some consequences of the absence of microscopic reversibility on financial processes. We analyze high resolution data and find asymmetric time dependent cross correlation functions indicating dominance of some companies in the price formation procedure. These effects can be summarised in a directed netowrk of influence. Furthermore, we show that in the Lux-Marchesi multi agent market model spontaneous fluctuations decay differently from perturbations caused by external effects. The latter are easily controlled in the model, however, in real data the separation of the internal and the external effects is a highly nontrivial task.