Regularity in Stock Market Indices within Turbulence Periods: The Sample Entropy Approach

Expecting the Unexpected: Entropy and Multifractal Systems in Finance

Entropy serves as a measure of chaos in systems by representing the average rate of information loss about a phase point's position on the attractor. When dealing with a multifractal system, a single exponent cannot fully describe its dynamics, necessitating a continuous spectrum of exponents, known as the singularity spectrum. From an investor's point of view, a rise in entropy is a signal of abnormal and possibly negative returns. This means he has to expect the unexpected and prepare for it. To explore this, we analyse the New York Stock Exchange (NYSE) U.S. Index as well as its constituents. Through this examination, we assess their multifractal characteristics and identify market conditions (bearish/bullish markets) using entropy, an effective method for recognizing fluctuating fractal markets. Our findings challenge conventional beliefs by demonstrating that price declines lead to increased entropy, contrary to some studies in the literature that suggest that reduced entropy in market crises implies more determinism. Instead, we propose that bear markets are likely to exhibit higher entropy, indicating a greater chance of unexpected extreme events. Moreover, our study reveals a power-law behaviour and indicates the absence of variance.

Symbolic Encoding Methods with Entropy-Based Applications to Financial Time Series Analyses

Symbolic encoding of information is the foundation of Shannon's mathematical theory of communication. The concept of the informational efficiency of capital markets is closely related to the issue of information processing by equity market participants. Therefore, the aim of this comprehensive research is to examine and compare a battery of methods based on symbolic coding with thresholds and the modified Shannon entropy in the context of stock market efficiency. As these methods are especially useful in assessing the market efficiency in terms of sequential regularity in financial time series during extreme events, two turbulent periods are analyzed: (1) the COVID-19 pandemic outbreak and (2) the period of war in Ukraine. Selected European equity markets are investigated. The findings of empirical experiments document that the encoding method with two 5% and 95% quantile thresholds seems to be the most effective and precise procedure in recognizing the dynamic patterns in time series of stock market indices. Moreover, the Shannon entropy results obtained with the use of this symbolic encoding method are homogenous for all investigated markets and unambiguously confirm that the market informational efficiency measured by the entropy of index returns decreases during extreme event periods. Therefore, we can recommend the use of this STSA method for financial time series analyses.

Investigating Dynamical Complexity and Fractal Characteristics of Bitcoin/US Dollar and Euro/US Dollar Exchange Rates around the COVID-19 Outbreak

This article investigates the dynamical complexity and fractal characteristics changes of the Bitcoin/US dollar (BTC/USD) and Euro/US dollar (EUR/USD) returns in the period before and after the outbreak of the COVID-19 pandemic. More specifically, we applied the asymmetric multifractal detrended fluctuation analysis (A-MF-DFA) method to investigate the temporal evolution of the asymmetric multifractal spectrum parameters. In addition, we examined the temporal evolution of Fuzzy entropy, non-extensive Tsallis entropy, Shannon entropy, and Fisher information. Our research was motivated to contribute to the comprehension of the pandemic's impact and the possible changes it caused in two currencies that play a key role in the modern financial system. Our results revealed that for the overall trend both before and after the outbreak of the pandemic, the BTC/USD returns exhibited persistent behavior while the EUR/USD returns exhibited anti-persistent behavior. Additionally, after the outbreak of COVID-19, there was an increase in the degree of multifractality, a dominance of large fluctuations, as well as a sharp decrease of the complexity (i.e., increase of the order and information content and decrease of randomness) of both BTC/USD and EUR/USD returns. The World Health Organization (WHO) announcement, in which COVID-19 was declared a global pandemic, appears to have had a significant impact on the sudden change in complexity. Our findings can help both investors and risk managers, as well as policymakers, to formulate a comprehensive response to the occurrence of such external events.

The Complexity Behavior of Big and Small Trading Orders in the Chinese Stock Market

The Chinese stock market exhibits many characteristics that deviate from the efficient market hypothesis and the trading volume contains a great deal of complexity information that the price cannot reflect. Do small or big orders drive trading volume? We studied the complex behavior of different orders from a microstructure perspective. We used ETF data of the CSI300, SSE50, and CSI500 indices and divided transactions into big and small orders. A multifractal detrended fluctuation analysis (MFDFA) method was used to study persistence. It was found that the persistence of small orders was stronger than that of big orders, which was caused by correlation with time. A multiscale composite complexity synchronization (MCCS) method was used to study the synchronization of orders and total volume. It was found that small orders drove selling-out transactions in the CSI300 market and that big orders drove selling-out transactions in the CSI500 market. Our findings are useful for understanding the microstructure of the trading volume in the Chinese market.

CEGH: A Hybrid Model Using CEEMD, Entropy, GRU, and History Attention for Intraday Stock Market Forecasting

Intraday stock time series are noisier and more complex than other financial time series with longer time horizons, which makes it challenging to predict. We propose a hybrid CEGH model for intraday stock market forecasting. The CEGH model contains four stages. First, we use complete ensemble empirical mode decomposition (CEEMD) to decompose the original intraday stock market data into different intrinsic mode functions (IMFs). Then, we calculate the approximate entropy (ApEn) values and sample entropy (SampEn) values of each IMF to eliminate noise. After that, we group the retained IMFs into four groups and predict the comprehensive signals of those groups using a feedforward neural network (FNN) or gate recurrent unit with history attention (GRU-HA). Finally, we obtain the final prediction results by integrating the prediction results of each group. The experiments were conducted on the U.S. and China stock markets to evaluate the proposed model. The results demonstrate that the CEGH model improved forecasting performance considerably. The creation of a collaboration between CEEMD, entropy-based denoising, and GRU-HA is our major contribution. This hybrid model could improve the signal-to-noise ratio of stock data and extract global dependence more comprehensively in intraday stock market forecasting.

Entropy-Based Applications in Economics, Finance, and Management

The concept of entropy originated from physics (precisely, from thermodynamics), but it has been utilized in many research fields to characterize the complexity of a system and to investigate the information content of a probability distribution [...]