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Financial contagion refers to the spread of market turmoils, for example from one country or index to another country or another index. It is standardly assessed by modelling the evolution of the correlation matrix, for example of returns, usually after removing univariate dynamics with the GARCH model. However, significant events like crises visible in one financial market are usually reflected in other financial markets/countries simultaneously in several dimensions, i.e., in general, entire distributions of returns in other markets are affected. These distributions are determined/described by their expected value, variance, skewness, kurtosis and other statistics that determine the shape of the distribution function of returns, which can be based on higher (mixed) moments. These descriptive statistics are not constant over time, and, moreover, they can interreact within the given market and among the markets over time. In this article we propose, and use for the daily values of five indexes (CAC40, DAX30, DJIA, FTSE250 and WIG20) over the time period 2006–2017, a new, simple and computationally inexpensive methodology to automatically extend contagion evaluation from the evolution of the correlation matrix to the evolution of multiple higher mixed moments as well. Specifically, the joint distribution of normalized variables for each pair of indexes is modeled as a polynomial with time evolving coefficients estimated using an exponential moving average. As we can obtain any arbitrary number of evolving mixed moments this way, its dimensionality reduction using PCA (principal component analysis) is also discussed, obtaining a lower number of dominating and relatively independent features, which can each be interpreted through a polynomial that describes the corresponding perturbation of joint distribution. We obtain features that describe the interrelations among stock markets in several dimensions and that provide information about the current stage of crisis and the strength of the contagion process.

The properties of the time series of durations between consecutive trades of a particular stock have been studied by many contributors in the literature of financial econometrics. Among them are highly prominent scientists like Engle (2000) and Gourieroux and Jasiak (2001). The importance of this topic, accompanied by the growing availability of (ultra-)high-frequency data, has prompted an increase of contributions in recent years. Intensive research based on high-frequency data has several financial motivations. First of all, it is linked with microstructure theory. Secondly, it contributes to the literature on stochastic time deformation. But the most important need for research on the dynamics of trade durations is the necessity to manage liquidity risk. The reason is that durations between the following trades are a widely accepted measures of market liquidity. In addition, their volatility reflects the liquidity risk.

The logarithmic ACD model: The microstructure of the German and Polish stock markets1

This paper is concerned with a dependence analysis of returns, return volatility and trading volume for five companies listed on the Vienna Stock Exchange and five from the Warsaw Stock Exchange. Taking into account high frequency data for these companies, tests based on a comparison of Bernstein copula densities using the Hellinger distance were conducted. The paper presents some patterns of causal and other relationships between stock returns, realized volatility and expected and unexpected trading volume. There is a linear causality running from realized volatility to expected trading volume, and a lack of nonlinear dependence in the opposite direction. The authors detected strong linear and nonlinear causality from stock returns to expected trading volume. They did not find causality running in the opposite direction. In addition, the existence of fractional cointegration was examined. Despite the equality of the long memory parameters of realized volatility and trading volumes, they do not move together in the long term horizon. Članek obravnava analizo odvisnosti med donosnostjo, nihajnostjo in obsegom trgovanja z delnicami petih podjetij, ki kotirajo na dunajski borzi, in petih podjetij, ki kotirajo na varšavski borzi. Na podlagi hitrih podatkov za ta podjetja so bili narejeni preizkusi, ki temeljijo na primerjavi Bernsteinovih kopul gostot z uporabo Hellingerjeve distance. Članek predstavlja nekaj vzorcev vzročnih in ostalih povezav med donosnostjo delnic, realizirano nihajnostjo ter pričakovanim in nepričakovanim obsegom trgovanja. Obstaja linearna povezava med realizirano nihajnostjo in pričakovanim obsegom trgovanja, ni pa nelinearne povezave v nasprotni smeri. Avtorja sta ugotovila močno linearno in nelinearno povezavo med donosnostjo obveznic in pričakovanim obsegom trgovanja. Nista pa opazila povezave v nasprotni smeri. Preverila sta tudi, ali obstaja frakcijska kointegracija. Kljub enakovrednosti parametrov realizirane nihajnosti in obsega trgovanja v daljšem obdobju se ti parametri v daljših časovnih obdobjih ne gibljejo enako.