Complexity

Volume 2018, Article ID 9072948, 9 pages

https://doi.org/10.1155/2018/9072948

## Traders’ Networks of Interactions and Structural Properties of Financial Markets: An Agent-Based Approach

Correspondence should be addressed to Linda Ponta; ti.eginu@atnop.adnil

Received 27 October 2017; Accepted 21 December 2017; Published 29 January 2018

Academic Editor: Ilaria Giannoccaro

Copyright © 2018 Linda Ponta and Silvano Cincotti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An information-based multiasset artificial stock market characterized by different types of stocks and populated by heterogeneous agents is presented and studied so as to determine the influences of agents’ networks on the market’s structure. Agents are organized in networks that are responsible for the formation of the sentiments of the agents. In the market, agents trade risky assets in exchange for cash and share their sentiments by means of interactions that are determined by sparsely connected graphs. A central market maker (clearing house mechanism) determines the price process for each stock at the intersection of the demand and the supply curves. A set of market’s structure indicators based on the main single-assets and multiassets stylized facts have been defined, in order to study the effects of the agents’ networks. Results point out an intrinsic structural resilience of the stock market. In fact, the network is necessary in order to archive the ability to reproduce the main stylized facts, but also the market has some characteristics that are independent from the network and depend on the finiteness of traders’ wealth.

#### 1. Introduction

The large availability of financial data has allowed the study of financial markets by means of the cooperation of different fields such as engineering, physics, mathematics, and economics [1–5]. This new multidisciplinary approach overcomes the limits of the classical approach and improves the knowledge about the price processes, discovering the so called stylized facts, that is, the main statistical properties of financial markets. In particular, focusing on the distribution of intertrade time between different financial transactions, previous works have demonstrated the presence of Weibull distribution [6]. Moreover, empirical study has demonstrated that the dynamics of price and volume of transactions, including the volatility over different time horizons, are influenced by the correlations and temporal patterns of the intertrade times. Furthermore, the rules that regulate the interactions among agents strongly depend on the regulatory mechanisms of each individual market [7]. In order to evaluate the correlations and to identify and quantify integrations among dynamical entities, such as agents on the stock market, special methods have been developed [8, 9]. Generally speaking, according to the classical approach, simple analytically tractable models with a representative, perfectly rational agent have been the main corner stones and mathematics has been the main tool of analysis. Conversely, the complexity science approach considers financial markets as complex systems where a large number of heterogeneous agents interact. In particular, the markets are populated by boundedly rational, heterogeneous agents using rule of thumb strategies. This approach fits much better with agent-based simulation models and computational and numerical methods have become an important tool of analysis [10]. Thus, a number of computer-simulated, artificial financial markets have been born with the aim of becoming a framework to perform computational experiments. Following the pioneering work done at the Santa Fe Institute [11–13], a large number of researchers have proposed model for artificial markets populated by heterogeneous agents endowed with learning and optimization capabilities [14, 15]. Moreover, the artificial financial markets are a useful framework to study the role of fraudulent agents and the corruption in financial markets, that is, how the fraudulent agents impact on the markets [16]. In particular, empirical analysis shows that corruption influences the economic growth rate and foreign investment [17]. For a detailed review on microscopic (“agent-based”) models of financial markets see [18, 19].

In this paper, using the Genoa Artificial Stock Market (GASM) developed in Genoa, the impact of the structural properties of traders’ networks of interaction on the emergent outcome in financial markets has been studied. In particular, starting from the information-based single-assets artificial market, a multiassets artificial stock market version of the GASM has been used [20–23]. In order to investigate this relationship, the market is populated by heterogeneous agents that are seen as nodes of sparsely connected graphs. The market is characterized by different types of stocks and agents trade risky assets in exchange for cash. Agents share their information by means of interactions that are determined by the graphs. Besides the amount of cash and assets owned, each agent is characterized by sentiments that summarize the agent’s information about the market and agent world. The sentiments include in one element the influence of the market trend, the influence of the neighbours agents, and the propensity for the market. Agents are subject to a portfolio choice on number and type of risky securities. The allocation strategy is based on sentiments and wealth. A central market maker (clearing house mechanism) determines the price process for each stock at the intersection of the demand and the supply curves.

The paper presents a study on how the traders’ networks and, thus, the sentiments’ components influence the market structure. In particular, this paper investigates the effects of changes in traders’ networks of interaction in the financial market. In order to perform this investigation, five different “market’s structure indicators” have been defined. The indicators have been defined considering the main univariate and multivariate stylized facts. Concerning univariate processes, the three main stylized facts taken as reference are the unitary root of price processes, the fat tails distribution of returns, and the volatility clustering. Concerning the multiassets environment the set of stylized facts consists in the statistical properties of the cross-correlation matrices of returns [24–26] and of the variance-covariance matrices of prices [27] that make reference to static and dynamic factors, respectively.

Thus, the indicators defined are the number of I processes, the number of heteroscedastic processes, the number of processes with fat tails, the number of sector presented in the market, and the number of common trends. The computational experiments show an intrinsic structural resilience of the stock market. In fact, some characteristics of the market are “structural” and depend on the agents’ budget constraint, whereas others are an emerging properties of the traders’ network of interactions.

The paper is organized as follows: Section 2 presents the model and Section 3 the “market’s structure indicators” and Section 4 shows the computational experiments and Section 5 the discussion of results. Finally, Section 6 provides the conclusion of the study.

#### 2. The GASM Model

##### 2.1. Overview of the GASM Model

The model presented in this paper is an enrichment of the Genoa Artificial Stock Market (GASM) developed at the University of Genoa [20, 28]. The GASM is an agent-based artificial stock market whose baseline originally includes heterogeneous agents that trade risky assets in exchange for cash [29]. They are modeled as liquidity traders; that is, decision making process is constrained by the finite amount of financial resources (cash + stocks) they own. At the beginning of the simulation, cash and stocks are distributed randomly among agents.

##### 2.2. Agents’ Networks

In order to investigate the effects of agents’ networks in financial markets, for each stock presented in the market, the heterogeneous agents have been organized in graphs, and in particular, according to a directed random graph, where the agents are the nodes and the branches represent the interactions among agents. The graphs are responsible for the changes in agent’s sentiments. The graphs are directed; that is, the interactions are assumed unidirectional (i.e., if agent -th influences agent -th not necessarily agent -th influences agent -th) and characterized by a strength , assuming a positive real number. Generally speaking, due to the presence of a directed graph, both an output node degree, related to the output branches of a given node, and an input node degree, related to the input branches, should be defined.

The agents are organized according to a Zipf law. For each stock an agent is randomly connected to a set of other agents whose number and strength (of the connection) are inversely proportional to his/her rank, that is, richer agents influence a larger number of agents with a higher strength. Consequently, the output degree distributions over the nodes are set to power laws and the input degree distributions result in power laws too. Each agent has a different belief about the assets depending on his/her rank. Agent is characterized by a sentiment (i.e., real number in the interval ) that represents a propensity to invest in asset . A positive average sentiment denotes a propensity to buy, whereas a negative average sentiment corresponds to a propensity to sell. The graphs are responsible for the changes in agent’s sentiments. At each time step , information is propagated through the market and sentiments of agent are updated.

Let be the set of agents that influences the behavior of trader* i*-th for the asset and the market price of the risky asset . The new sentiments of agent* i*-th for each asset are functions of the previous sentiments, of the log return (market feedback), of the influence of interacting agents and of average sentiment of the agent about the market behavior. The expression is whereis a smooth function that constrains agent sentiments in the range .

Furthermore, represents the market feedback, represents the influence of interacting agents, and models the global vision of agent -th for the market trend. The coefficients in (1) are inversely proportional to agent’s rank; that is, richer agents have stronger beliefs. Moreover a constraint on graph intersection is consideredthat is, self-interaction is a counterpart of graph interactions, with random (i.e., uniform distribution) changes in sign at each time step. Eq. (6) models a specific behavior of agents, that is, the fact that sometimes an agent changes idea about the sentiments of neighbour, and so he changes his reaction. In fact, (6) points out that agent that are strongly influenced by their previous sentiment (big traders, bank, mutual funds, etc.) and are poorly influenced by the neighbouring agents’ sentiment (e.g., small single investors) and represents the self-neighbouring sentiment balance coefficient [20].

Moreover, the amplitude of market feedback depends on rank, so that the coefficients are inversely proportional to agent ranks; that is, agents with higher ranks are less sensitive to the single asset trends. Finally, the term is a stabilizing element for the sentiment, so that the coefficient in (1) is always negative.

Agent’s trading decision is based on cash and stocks owned and on sentiment. In particular, the stock price process depends on the propagation of information among the interacting agents, on budget constraints and on market feedback. In this respect, also the coefficients in (1) are proportional to agent’s rank; that is, richer agents have stronger beliefs.

##### 2.3. Allocation Strategy

At each time step , a subset of agents is randomly chosen from a uniform distribution to operate as traders on the market. Let be the sentiment, the amount of cash, and the amount of asset owned by the -th trader at time .

If is the market price of the risky asset at time step , the risky wealth owned by trader at time step is whereas represents the total wealth of agent -th.

At each simulation step, trader -th tries to allocate in risky assets a fraction of his total wealth related to his vision of the market trend; that is, where .

is the average sentiments of all assets described by (5). The symbol denotes that is the amount that agent* i*-th is willing to allocate in the risky investment, whereas the real amount effectively allocated in stocks will depend on the trading process with the other agents. For each asset , a positive sentiment denotes a propensity to allocate, while a negative sentiment denotes a propensity to sell all the assets in the portfolio. In this model only long positions are allowed. Thus, if , the quantity desired by agent of risky asset is given by where is given by is the set of assets with positive sentiment. The symbol in (9) denotes the integer part. Conversely, if , asset is characterized by a desired quantity .

The amount of the order issued by trader -th at time step relative to stock is is the difference between the desired amount of stock at time step and the real amount held in the portfolio by agent -th. If the order is a buy order. Conversely, if the agent issues a sell order. Every order is associated with a limit price. Each limit price is determined according to (12) where is a random draw from a Gaussian distribution with average According to previous models [20, 30], we assume that buy (sell) orders cannot be executed at prices above (below) their limit price . It is worth noting that for a buy order (i.e., ) in average . Conversely, for a sell order (i.e., ) in average . Furthermore, the standard deviation is proportional to the historical volatility of the price of stock through the equation . Linking limit orders to volatility takes into account a realistic aspect of trading psychology: when volatility is high, uncertainty on the “true” price of a stock grows and traders place orders with a broader distribution of limit prices. In our model, is a constant for all agents, whereas is the standard deviation of log-price returns of asset , computed in a time window proper for agent -th randomly associated with the agent [20, 28].

All buy and sell orders issued at time step are collected and the demand and supply curves are consequently computed. The intersection of the two curves determines the new price (clearing price) of stock (see [20, 28] for more details on market clearing).

Buy and sell orders with limit prices compatible with are executed. After any transactions, traders’ cash, portfolio, and sentiments are updated. Orders that do not match the clearing price are discarded.

#### 3. Market’s Structure Indicators

As discussed in the previous Sections, we aim to investigate how the structural properties of traders’ networks of interactions affect the emergent outcome of financial markets. In order to measure the influence of the traders’ networks on financial markets, five different indicators have been defined, that is,(a)The number of prices processes that are integrated I processes(b)The number of returns processes that exhibit volatility clustering (heteroscedastic processes)(c)The number of returns processes whose distribution shows fat tails (power law distributions)(d)The number of static factors(e)The number of dynamic factors

These indicators make reference to the main stylized facts, empirically derived by the international literature on stock markets. In fact, the large availability of financial data has allowed both qualitative and quantitative investigations of financial markets by means of stylized facts. In particular, the indicators (a), (b), and (c) referred to the single asset statistical properties [31–35], whereas indicators (d) and (e) referred to the multiassets statistical properties [24–27].

The first indicator chosen is the number of prices processes that are integrated I processes and is indicated with [31]. In order to verify if a time series is integrated I process, the Augmented Dickey-Fuller and the Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) tests at the significance level of 5% are employed. It is worth remembering that the null hypothesis of the ADF test is that a univariate time series presents a unit root, whereas the null hypothesis of the KPSS test is that the time series is stationary [36, 37]. Figure 1(a) shows an I price process for a typical asset generated by the artificial stock market (GASM) presented in this paper.