Efficient method of finding scaling exponents from finite-size Monte-Carlo simulations

Monte-Carlo simulations are routinely used for estimating the scaling exponents of complex systems. However, due to finite-size effects, determining the exponent values is often difficult and not reliable. Here we present a novel technique of dealing with the problem of finite-size scaling. This new method allows not only to decrease the uncertainties of the scaling exponents, but makes it also possible to determine the exponents of the asymptotic corrections to the scaling laws. The efficiency of the technique is demonstrated by finding the scaling exponent of uncorrelated percolation cluster hulls.Comment: The "previous version" of this is arXiv:0804.1911. This version is published in EPJ

Leptokurtic Portfolio Theory

The question of optimal portfolio is addressed. The conventional Markowitz portfolio optimisation is discussed and the shortcomings due to non-Gaussian security returns are outlined. A method is proposed to minimise the likelihood of extreme non-Gaussian drawdowns of the portfolio value. The theory is called Leptokurtic, because it minimises the effects from "fat tails" of returns. The leptokurtic portfolio theory provides an optimal portfolio for investors, who define their risk-aversion as unwillingness to experience sharp drawdowns in asset prices. Two types of risks in asset returns are defined: a fluctuation risk, that has Gaussian distribution, and a drawdown risk, that deals with distribution tails. These risks are quantitatively measured by defining the "noise kernel" -- an ellipsoidal cloud of points in the space of asset returns. The size of the ellipse is controlled with the threshold parameter: the larger the threshold parameter, the larger return are accepted for investors as normal fluctuations. The return vectors falling into the kernel are used for calculation of fluctuation risk. Analogously, the data points falling outside the kernel are used for the calculation of drawdown risks. As a result the portfolio optimisation problem becomes three-dimensional: in addition to the return, there are two types of risks involved. Optimal portfolio for drawdown-averse investors is the portfolio minimising variance outside the noise kernel. The theory has been tested with MSCI North America, Europe and Pacific total return stock indices.Comment: 10 pages, 2 figures, To be presented in NEXT-SigmaPh

Intersections of moving fractal sets

Intersection of a random fractal or self-affine set with a linear manifold or another fractal set is studied, assuming that one of the sets is in a translational motion with respect to the other. It is shown that the mass of such an intersection is a self-affine function of the relative position of the two sets. The corresponding Hurst exponent h is a function of the scaling exponents of the intersecting sets. A generic expression for h is provided, and its proof is offered for two cases --- intersection of a self-affine curve with a line, and of two fractal sets. The analytical results are tested using Monte-Carlo simulations

Properties of low variability periods in financial time series

Properties of low-variability periods in the time series are analysed. The theoretical approach is used to show the relationship between the multi-scaling of low-variability periods and multi-affinity of the time series. It is shown that this technically simple method is capable of reveling more details about time-series than the traditional multi-affine analysis. We have applied this scaling analysis to financial time series: a number of daily currency and stock index time series. The results show a good scaling behaviour for different model parameters. The analysis of high-frequency USD-EUR exchange rate data confirmed the theoretical expectations.Comment: 14 pages, 5 figures, 3 tables, Submitted to Physica

Econophysics studies in Estonia

A short review of the econophysics research done in Estonia, devoted to the 15th anniversary of the term "econophysics".Comment: Submitted to the special issue on "Econophysics" of the journal Science & Culture (http://www.scienceandculture-isna.org/journal.htm), a publication of the Indian Science News Association, established in 193