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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: 27 March 2025 | Viewed by 6535

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Special Issue Editors

Dr. Shupei Huang

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Guest Editor

School of Economics and Management, China University of Geosciences, Beijing 100083, China
Interests: financial complexity; mining finance; resource and environmental economics and management; managing scientific complexity

Prof. Dr. Brian Lucey

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Guest Editor

Trinity Business School, Trinity College Dublin, Dublin, Ireland
Interests: applied finance; behavioral finance; corporate finance derivatives; finance theory; financial management; integrative cases in finance; introduction to finance; introduction to organizations and management investments port

Dr. Xueyong Liu

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Guest Editor

School of Management and Engineering, Capital University of Economics and Business, Beijing 100070, China
Interests: energy economics; time series analysis; Chinese economic studies; energy investment; energy policy; networks; econophysics; complex systems

Dr. Xinya Wang

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Guest Editor

University of International Business and Economics, Beijing 100029, China
Interests: financial market; financial network; financial complexity; financial risk

Special Issue Information

Dear Colleagues,

A financial system is a type of nonlinear dynamic system, but it is much more complex since human behavior is involved. One significant and effective way to explore and understand nonlinear dynamic systems is through complex network theory. By considering the financial system as a multilayer network with heterogeneous entities, including various financial markets, institutions, and stakeholders from different countries and regions, all these entities interact with each other through financial activities determined mainly by the available information. This network understanding of the financial system sheds light on the complexity of the financial system following the 2008 global financial crisis.

Information theory is a branch of applied mathematics that involves the quantification of information, while financial complexity is closely related to information. Specifically, key concepts from information theory involving entropy and mutual information offer effective tools for measuring uncertainties associated with a random variable and how knowledge about one random variable reduces uncertainty about another. All these contents are well aligned with the uncertainty and complex nature of the financial network.

Currently, the entire financial system is accelerating to face the occurrence of various uncertainties, such as geopolitical tensions (Israel and Iran in 2023), the Russian–Ukrainian crisis (since February 2022), and long COVID-19 (since 2020), leading to the highest degrees of complexity and uncertainties observed so far in the financial system. Therefore, it is necessary to further develop tools based on complexity, network, and information theory that could work well to complement existing economic modeling approaches.

Here, we collect various theoretical, modeling, and empirical contributions from the field of financial network. In particular, as we aim to align with the scope of the Entropy journal, manuscripts that integrate information theory are welcome.

Dr. Shupei Huang
Prof. Dr. Brian Lucey
Dr. Xueyong Liu
Dr. Xinya Wang
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

financial network analysis
information spillover in financial systems
multilayer networks in finance
uncertainties in financial network

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Published Papers (6 papers)

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Research

15 pages, 744 KiB

Open AccessArticle

Causal Hierarchy in the Financial Market Network—Uncovered by the Helmholtz–Hodge–Kodaira Decomposition

by Tobias Wand, Oliver Kamps and Hiroshi Iyetomi

Entropy 2024, 26(10), 858; https://doi.org/10.3390/e26100858 - 11 Oct 2024

Viewed by 350

Abstract

Granger causality can uncover the cause-and-effect relationships in financial networks. However, such networks can be convoluted and difficult to interpret, but the Helmholtz–Hodge–Kodaira decomposition can split them into rotational and gradient components which reveal the hierarchy of the Granger causality flow. Using Kenneth French’s business sector return time series, it is revealed that during the COVID crisis, precious metals and pharmaceutical products were causal drivers of the financial network. Moreover, the estimated Granger causality network shows a high connectivity during the crisis, which means that the research presented here can be especially useful for understanding crises in the market better by revealing the dominant drivers of crisis dynamics. Full article

(This article belongs to the Special Issue Complexity in Financial Networks)

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16 pages, 2616 KiB

Open AccessArticle

Wandering Drunkards Walk after Fibonacci Rabbits: How the Presence of Shared Market Opinions Modifies the Outcome of Uncertainty

by Nicolas Maloumian

Entropy 2024, 26(8), 686; https://doi.org/10.3390/e26080686 - 13 Aug 2024

Viewed by 623

Abstract

Shared market opinions and beliefs by market participants generate a set of constraints that mediate information through a not-so-unstable system of expected target prices. Price trajectories, within these sets of constraints, confirm or disprove the likelihood of participant expectations and cannot, de facto, be considered permutable, as literature has shown, since their inner structure is dynamically affected by their own progress, suggesting per se the presence of both heat and cycles. This study described and discussed how trajectories are built using different alphabets and suggests that prices follow an ergodic course within structurally similar tessellation classes. It is reported that the courses of price moves are self-similar due to their a priori structure, and they do not need to be complete in order to create the conditions, in resembling conditions, for the appearance of the well-known and commonly used Fibonacci ratios between price trajectories. To date, financial models and engineering are mostly based on the mathematics of randomness. If these theoretical findings need empirical validation, such a potential infrastructure of ratios would suggest the possibility for a superstructure to exist, in other words, the emergence of exploitable patterns. Full article

(This article belongs to the Special Issue Complexity in Financial Networks)

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Figure 1
Different possible combinations after six trades, ‘a’ marking any price change, ‘H’ marking no change. Full article ">Figure 2
Different possible combinations posting a six-letter chain. Full article ">Figure 3
Different ways to see the same rise in price from a bottom to a top as depicted in ‘a’, with ‘b’ presenting an l-composition (l standing for left) of ‘a’, and with ‘c’ presenting an r-composition (r standing for right) of ‘a’. On both ‘b’ and ‘c’, each horizontal line is an ‘H’. Full article ">Figure 4
Grouping different ‘tiles’ of an eight-letter chain composed with ‘1’s’ (one tick up or down) and ‘0’s’ (no change). (A) shows the set of tiles for moves going up, and (B) shows the set of tiles for moves going down. Full article ">Figure 5
Probability structure of Fibonacci n-letter chains, or classes, probability levels on the y-axis. Full article ">Figure 6
Showing how the tile sets T{9} T{8} T{7} are related. Black squares indicate no price changes (‘H’ or ‘0’), and all other squares indicate a price change of one unit of price (‘1’). Full article ">

17 pages, 2886 KiB

Open AccessArticle

Study on the Stability of Complex Networks in the Stock Markets of Key Industries in China

by Zinuoqi Wang, Guofeng Zhang, Xiaojing Ma and Ruixian Wang

Entropy 2024, 26(7), 569; https://doi.org/10.3390/e26070569 - 30 Jun 2024

Cited by 1 | Viewed by 966

Abstract

Investigating the significant “roles” within financial complex networks and their stability is of great importance for preventing financial risks. On one hand, this paper initially constructs a complex network model of the stock market based on mutual information theory and threshold methods, combined with the closing price returns of stocks. It then analyzes the basic topological characteristics of this network and examines its stability under random and targeted attacks by varying the threshold values. On the other hand, using systemic risk entropy as a metric to quantify the stability of the stock market, this paper validates the impact of the COVID-19 pandemic as a widespread, unexpected event on network stability. The research results indicate that this complex network exhibits small-world characteristics but cannot be strictly classified as a scale-free network. In this network, key roles are played by the industrial sector, media and information services, pharmaceuticals and healthcare, transportation, and utilities. Upon reducing the threshold, the network’s resilience to random attacks is correspondingly strengthened. Dynamically, from 2000 to 2022, systemic risk in significant industrial share markets significantly increased. From a static perspective, the period around 2019, affected by the COVID-19 pandemic, experienced the most drastic fluctuations. Compared to the year 2000, systemic risk entropy in 2022 increased nearly sixtyfold, further indicating an increasing instability within this complex network. Full article

(This article belongs to the Special Issue Complexity in Financial Networks)

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22 pages, 1744 KiB

Open AccessArticle

SF-Transformer: A Mutual Information-Enhanced Transformer Model with Spot-Forward Parity for Forecasting Long-Term Chinese Stock Index Futures Prices

by Weifang Mao, Pin Liu and Jixian Huang

Entropy 2024, 26(6), 478; https://doi.org/10.3390/e26060478 - 30 May 2024

Viewed by 707

Abstract

The complexity in stock index futures markets, influenced by the intricate interplay of human behavior, is characterized as nonlinearity and dynamism, contributing to significant uncertainty in long-term price forecasting. While machine learning models have demonstrated their efficacy in stock price forecasting, they rely solely on historical price data, which, given the inherent volatility and dynamic nature of financial markets, are insufficient to address the complexity and uncertainty in long-term forecasting due to the limited connection between historical and forecasting prices. This paper introduces a pioneering approach that integrates financial theory with advanced deep learning methods to enhance predictive accuracy and risk management in China’s stock index futures market. The SF-Transformer model, combining spot-forward parity and the Transformer model, is proposed to improve forecasting accuracy across short and long-term horizons. Formulated upon the arbitrage-free futures pricing model, the spot-forward parity model offers variables such as stock index price, risk-free rate, and stock index dividend yield for forecasting. Our insight is that the mutual information generated by these variables has the potential to significantly reduce uncertainty in long-term forecasting. A case study on predicting major stock index futures prices in China demonstrates the superiority of the SF-Transformer model over models based on LSTM, MLP, and the stock index futures arbitrage-free pricing model, covering both short and long-term forecasting up to 28 days. Unlike existing machine learning models, the Transformer processes entire time series concurrently, leveraging its attention mechanism to discern intricate dependencies and capture long-range relationships, thereby offering a holistic understanding of time series data. An enhancement of mutual information is observed after introducing spot-forward parity in the forecasting. The variation of mutual information and ablation study results highlights the significant contributions of spot-forward parity, particularly to the long-term forecasting. Overall, these findings highlight the SF-Transformer model’s efficacy in leveraging spot-forward parity for reducing uncertainty and advancing robust and comprehensive approaches in long-term stock index futures price forecasting. Full article

(This article belongs to the Special Issue Complexity in Financial Networks)

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Transformer model for stock index futures price forecasting. The time series data are input into the encoder, which employs multiple attention layers to extract features for forecasting. Simultaneously, the forecasting horizons, marked as zeros and accompanied by the previous historical time series data, are fed into the decoder. The decoder, integrating features from the encoder, predicts the values of forecasting horizons using multiple attention layers and a fully connected layer. Full article ">Figure 2
Architecture of SF-Transformer. (a) Sequential spot-forward (SF) parity values, stock index futures values, and global time constitute the input representation to the SF-Transformer. This input generates embeddings for SF via value/position/time embedding. (b) The SF-Transformer utilizes the embeddings of the encoder and decoder inputs to generate forecasts of stock index futures prices in a generative manner. Model training involves the use of mean squared error (MSE) to measure the difference between forecasted values and ground-truth values. Full article ">Figure 3
1-day- to 28-days-ahead out-of-sample forecasting errors of SF-Transformer, SF-LSTM, and SF-MLP for (a) IF, (b) IH, and (c) IC Stock Index Futures measured by MAPE. Note that arbitrage-free is not illustrated here due to its significantly higher and even inapplicable MAPEs. Full article ">Figure 4
1-day- to 28-days-ahead out-of-sample forecasting errors of Transformer, LSTM, and MLP for (a) IF, (b) IH, and (c) IC Stock Index Futures measured by MAPE. Full article ">Figure 5
Variation of mutual information (VarMI) with and without spot-forward parity across forecasting horizons from 1 to 28 days for (a) IF, (b) IH, and (c) IC Stock Index Futures. Full article ">

26 pages, 887 KiB

Open AccessArticle

Underwriter Discourse, IPO Profit Distribution, and Audit Quality: An Entropy Study from the Perspective of an Underwriter–Auditor Network

by Songling Yang, Yafei Tai, Yu Cao, Yunzhu Chen and Qiuyue Zhang

Entropy 2024, 26(5), 393; https://doi.org/10.3390/e26050393 - 30 Apr 2024

Viewed by 1031

Abstract

Underwriters play a pivotal role in the IPO process. Information entropy, a tool for measuring the uncertainty and complexity of information, has been widely applied to various issues in complex networks. Information entropy can quantify the uncertainty and complexity of nodes in the network, providing a unique analytical perspective and methodological support for this study. This paper employs a bipartite network analysis method to construct the relationship network between underwriters and accounting firms, using the centrality of underwriters in the network as a measure of their influence to explore the impact of underwriters’ influence on the distribution of interests and audit outcomes. The findings indicate that a more pronounced influence of underwriters significantly increases the ratio of underwriting fees to audit fees. Higher influence often accompanies an increase in abnormal underwriting fees. Further research reveals that companies underwritten by more influential underwriters experience a decline in audit quality. Finally, the study reveals that a well-structured audit committee governance and the rationalization of market sentiments can mitigate the negative impacts of underwriters’ influence. The innovation of this paper is that it enriches the content related to underwriters by constructing the relationship network between underwriters and accounting firms for the first time using a bipartite network through the lens of information entropy. This conclusion provides new directions for thinking about the motives and possibilities behind financial institutions’ cooperation, offering insights for market regulation and policy formulation. Full article

(This article belongs to the Special Issue Complexity in Financial Networks)

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20 pages, 3707 KiB

Open AccessArticle

Systemic Importance and Risk Characteristics of Banks Based on a Multi-Layer Financial Network Analysis

by Qianqian Gao, Hong Fan and Chengyang Yu

Entropy 2024, 26(5), 378; https://doi.org/10.3390/e26050378 - 29 Apr 2024

Viewed by 1158

Abstract

Domestic and international risk shocks have greatly increased the demand for systemic risk management in China. This paper estimates China’s multi-layer financial network based on multiple financial relationships among banks, assets, and firms, using China’s banking system data in 2021. An improved PageRank algorithm is proposed to identify systemically important banks and other economic sectors, and a stress test is conducted. This study finds that China’s multi-layer financial network is sparse, and the distribution of transactions across financial markets is uneven. Regulatory authorities should support economic recovery and adjust the money supply, while banks should differentiate competition and manage risks better. Based on the PageRank index, this paper assesses the systemic importance of large commercial banks from the perspective of network structure, emphasizing the role of banks’ transaction behavior and market participation. Different industries and asset classes are also assessed, suggesting that increased attention should be paid to industry risks and regulatory oversight of bank investments. Finally, stress tests confirm that the improved PageRank algorithm is applicable within the multi-layer financial network, reinforcing the need for prudential supervision of the banking system and revealing that the degree of transaction concentration will affect the systemic importance of financial institutions. Full article

(This article belongs to the Special Issue Complexity in Financial Networks)

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Figure 1
Multi-layer financial network of the Chinese banking system. (Red: banks; green: financial assets; blue: firms). Full article ">Figure 2
Distribution of degree and total loan amount of banks in the bank–firm credit network. (BOC: Bank of China; CMB: China Merchants Bank; ICBC: Industrial and Commercial Bank of China; CCB: China Construction Bank; CIB: Industrial Bank; SPDB: Shanghai Pudong Development Bank; BCM: Bank of Communications; CMBC: China Minsheng Bank; ABC: Agricultural Bank of China; CITIC: China CITIC Bank; CDB: China Development Bank; ADBC: Agricultural Development Bank of China; PSBC: Postal Savings Bank of China). Full article ">Figure 3
Distribution of the degree of asset classes and the total amount of assets in the bank–asset portfolio network. Full article ">Figure 4
Degree of banks and total lending amount in the interbank lending network. Full article ">Figure 5
Degree, the amount of transactions and the PageRank index of the bank in each financial market. Full article ">Figure 6
The proportion of each asset in total assets and the average degree in various networks of different types of banks in various networks. Full article ">Figure 7
Counterparties of the six largest state-owned banks across trading markets. (ICBC: Industrial and Commercial Bank of China; ABC: Agricultural Bank of China; CCB: China Construction Bank; BOC: Bank of China; PSBC: Postal Savings Bank of China; BCM: Bank of Communications). Full article ">

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