Lyapunov Exponents as Indicators of the Stock Market Crashes

Abstract

The frequent financial critical states that occur in our world, during many centuries have attracted scientists from different areas. The impact of similar fluctuations continues to have a huge impact on the world economy, causing instability in it concerning normal and natural disturbances [1]. The an- ticipation, prediction, and identification of such phenomena remain a huge chal- lenge. To be able to prevent such critical events, we focus our research on the chaotic properties of the stock market indices. During the discussion of the re- cent papers that have been devoted to the chaotic behavior and complexity in the financial system, we find that the Largest Lyapunov exponent and the spec- trum of Lyapunov exponents can be evaluated to determine whether the system is completely deterministic, or chaotic. Accordingly, we give a theoretical background on the method for Lyapunov exponents estimation, specifically, we followed the methods proposed by J. P. Eckmann and Sano-Sawada to compute the spectrum of Lyapunov exponents. With Rosenstein’s algorithm, we com- pute only the Largest (Maximal) Lyapunov exponents from an experimental time series, and we consider one of the measures from recurrence quantification analysis that in a similar way as the Largest Lyapunov exponent detects highly non-monotonic behavior. Along with the theoretical material, we present the empirical results which evidence that chaos theory and theory of complexity have a powerful toolkit for construction of indicators-precursors of crisis events in financial markets.

Machine learning for prediction of emergent economy dynamics

“…xvii This article highlights further research by the authors, begun in [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. The focus of the article "Irreversibility of financial time series: a case of crisis" [34] by Andrii O. Bielinskyi, Serhii V. Hushko (figure 5), Andriy V. Matviychuk, Oleksandr A. Serdyuk, Serhiy O. Semerikov and Vladimir N. Soloviev to measure the varying irreversibility of stock markets.…”

Econophysics of cryptocurrency crashes: a systematic review

“…Kristoufek [75], Li and Wang [76] consider that measures of financial and macroeconomic activity can be drivers of Bitcoin returns. Reviewing papers of the researches above, the experience of others and our own [77,78,79,80,81,82,83,84,85,86], we have revised our classification of such leaps and falls, relying on Bitcoin time series during the entire period (01.01.2011-21.01.2021) of verifiable fixed daily values of the Bitcoin price (BTC) (https://finance.yahoo.com/cryptocurrencies). We emphasize that • crashes are short and time-localized drops that last approximately two weeks, with the weighty losing of price each day.…”

“…One of the most common and popular algorithms have been applied by Wolf et al [222], Sano and Sawada [223], and later improved by Eckmann et al [224], Rosenstein et al [225], Parlitz [226], Balcerzak et al [227]. Here, we followed the methods proposed by Gao et al [228,229], Soloviev et al [230,82], Soloviev and Stratiychuk [231] to compute the spectrum of Lyapunov exponents. With Rosenstein's algorithm, we compute only the LLE from an experimental time series.…”

Correlational and Non-extensive Nature of Carbon Dioxide Pricing Market

“…Previously, some of such quantitative measures of complexity for cryptocurrencies, stock, and sustainability indices [2,[24][25][26][27][28][29] were studied. In this paper, in order to have the possibility to study trading opportunities, the prospects for investing in a market, particularly, to study various components that define the nature of carbon prices and the collective behavior of the whole carbon market which is of particular value for politicians of specific countries, such informative measures of complexity as Tsallis statistics and Random matrix theory are presented.…”