Filip Świtała

Stochastic simulations of time series within Weierstrass-Mandelbrot walks

Abstract In the present work we extend Lévy walks to allow the velocity of the walker to vary. We call these extended Lévy walks Weierstrass-Mandelbrot walks. This is a generalized model of the Lévy walk type which is still able to describe both stationary and non-stationary stochastic time series by treating the initial step of the walker differently. The model was partly motivated by the properties of financial time series and tested on empirical data extracted from the Warsaw stock exchange since it offers an opportunity to study in an unbiased way several features of the stock exchange in its early stages. We extended the continuous-time random walk formalism but the (generalized) waiting-time distribution (WTD) and sojourn probability density still play a fundamental role. We considered a one-dimensional, non-Brownian random walk where the walker moves, in general, with a velocity that assumes a different constant value between the successive turning points, i.e. the velocity is a piecewise constant function. So far the models which have been developed take only one chosen value of this velocity into account and therefore are unable to consider more realistic stochastic time series. Moreover, our model is a kind of Lévy walk where we assume a hierarchical, self-similar in the stochastic sense, spatio-temporal representation of WTD and sojourn probability density. The Weierstrass-Mandelbrot walk model makes it possible to analyse both the structure of the Hurst exponent and the power-law behaviour of kurtosis. This structure results from the hierarchical, spatio-temporal coupling between the walker displacement and the corresponding time of the walks. The analysis makes use of both the fractional diffusion and the super-Burnett coefficients. We constructed the diffusion phase diagram which distinguishes regions occupied by classes of different universality. We study only such classes which are characteristic of stationary situations. We proved that even after taking a moving averaging of the stochastic time series which makes results stationary in the sense that they are independent of the beginning moment of the random walk, it is still possible to see the non-Gaussian features of the basic stochastic process. We thus have a model ready for describing data presented, e.g., in the form of moving averages. This operation is often used for stochastic time series, especially financial ones. Based on the hierarchical representation of WTD we introduce an efficient Monte Carlo algorithm which makes a numerical simulation of individual runs of stochastic time series possible; this facilitates the study of empirical stochastic time series.

Stochastic simulation of time series by using the spatial-temporal Weierstrass function

We extend the recently considered toy model of Weierstrass (or Levy) walks with varying velocity of the walker [1,2] by introducing a more realistic possibility that the walk can be occasionally intermitted by its momentary localization; the localizations themselves are again described by the Weierstrass (or Levy) process. The direct empirical motivation for developing this combined model is, for example, the dynamics of financial high-frequency time series or meteorological ones. This approach makes it possible to study by efficient stochastic simulations the whole spatial-temporal range. To describe empirical data, which are collected at discrete time-steps, we used in the continuous-time series produced by the model a discretization procedure. We observed that such a procedure constitutes a basis for long-time autocorrelations (of the variation of the walker single-step displacements) which appear to be similar to those observed, e.g., in financial time series [3,4,5,6,7,8], although single steps of the walker within the continuous time are uncorrelated.

Competition in Commercial Banks in Poland – Analysis of Panzar-Rosse H-Statistics

This paper aims to find out how intense is the competition in Polish commercial banks loan market. Using Panzar – Rosse H-statistics and applying several estimation techniques (GLS, one-step GMM and two-step GMM) we find that this intensity is sensitive to the estimator applied. Upon analysis of results, one can conclude that competition evolves differently across years in Poland. In some years, competition was relatively high, as the H-statistics reached the level of 0.75, which is relatively close to perfect competition. In other years it gradually decreased reaching its bottom line in 2010, and took upward trend in 2011 and 2012. Generally, the values of our competitive environment measure indicate at monopolistic competition in Poland.

Micro- and Macroprudential Liquidity Standards and Their Effects

This article focuses on liquidity standards whose significance has risen since the recent crisis of 2007/2008. The analysis concentrates on two areas. The first one is identification of common characteristics of liquidity standards recommended in Basel III and provisions included in directives and regulations in the European Union, as well as common traits of Polish supervisory liquidity rules and Basel III standards in this field. The other is identification of effects of liquidity standards. Analysis of Polish supervisory liquidity standards leads to conclusion that they have their counterparts in liquidity standards recommended by the Basel Committee. Simulations conducted by the Bank for International Settlements show that implementation of liquidity standards will bring more benefits than losses, for both banking (and financial) sector and the real economy. Unfortunately, contemporary empirical research only explains what are the effects of application of reserve requirements used in monetary policy. So little is known about the effects of other instruments whose track record is relatively short. However, even this scant evidence seems to support the prediction that reserve requirements could potentially be applied as a tool stabilizing financial sector and diminishing systemic risk.

Does Banking Sector Structure Affect Bank Lending and Its Sensitivity to Capital Ratio? A Cross-country Study

Does the banking sector’s structure affect bank lending and its sensitivity to the capital ratio? Looking at banks operating in over 60 countries in the years 2000–2011, this chapter aims to resolve this puzzle. Its goal is also to find out whether country development and capital account openness, and inclusion in the Central and Eastern Europe (CEE) region explain the potential diversity of the banking sector structure and lending nexus. To resolve this problem we apply a two-step generalized method of moments (GMM) robust estimator. We find that an increase in concentration of the banking sector results in reduced bank lending and that concentration seems to affect the link between lending and the capital ratio. This effect is particularly strong in CEE.

Competition between commercial banks in Poland - an analysis of Panzar-Rosse H-statistics

This paper aims to find out how intense the competition between Polish commercial banks is in the loan market. Using Panzar–Rosse H-statistics and employing several estimation techniques (GLS, one-step GMM and two-step GMM), we find that this intensity is sensitive to the estimator applied. Upon the analysis of results, it can be concluded that competition evolved differently over the years in Poland. In some years, competition was rather high as the H-statistic reached the level of 0.75, which is relatively close to perfect competition. In other years, it gradually decreased reaching its lowest value in 2010, and showed an upward trend in 2011 and 2012. Generally, the values of our competitive environment measure indicate monopolistic competition in Poland.

Discretized Continuous-Time Hierarchical Walks and Flights as possible bases of the non-linear long-term autocorrelations observed in high-frequency financial time-series

By using regular time-steps we define discrete-time random walks and flights on subordinate (directed) Continuous-Time Hierarchical (or Weierstrass) Walks and Flights, respectively. The obtained results can be considered as a kind of warning that indicates some persistent non-linear long-term autocorrelations (artifacts) accompanying the recording of empirical high-frequency financial time-series by regular time-steps, indeed.