Complexity

Volume 2017 (2017), Article ID 6752086, 16 pages

https://doi.org/10.1155/2017/6752086

## Connectivity, Information Jumps, and Market Stability: An Agent-Based Approach

Financial Engineering, Stevens Institute of Technology, Hoboken, NJ, USA

Correspondence should be addressed to Khaldoun Khashanah

Received 2 April 2017; Accepted 18 July 2017; Published 22 August 2017

Academic Editor: Benjamin M. Tabak

Copyright © 2017 Khaldoun Khashanah and Talal Alsulaiman. 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

We propose a metamodel to assess simulated market stability by introducing information connectivity in an agent-based network. The market is occupied by heterogeneous agents with different behaviors, strategies, and information connectivity. A jump-diffusion process simulating events that may occur in the market is introduced. Agents information awareness varies along with agents propensity to respond to the information jump and jump size. A jump reshuffles market positions based on agents risk preferences determined by behavior and strategy. We examine the effect of information awareness on the volatility index of the simulated market in a scale-free market network. The analysis is performed by developing five experiments wherein the first one corresponds to systemic information ignorance state. Three experiments examine the role of hubs, normal agents, and hermits in the network when intermediate combinations of agent types have information awareness. The fifth experiment corresponds to the systemic information awareness with all agents being informed. The results show that the simulated market is driven to instability in a similar manner to patterns observed in a crisis where all agents become homogeneous in information awareness of events. Hubs contribute to increased connectivity and act as amplifiers of good, bad, or inaccurate information or sentiment.

#### 1. Introduction

How should a* financial economic environment simulator* (FEES) be designed and what should it inform? Our agent-based approach ultimately provides such a FEES just as a flight simulator offers the pilot a close-to-reality flying experience. However, the stage of reaching a truly complex financial environment simulator is not yet mature enough and this paper is one more step in that direction as it introduces information jumps, heterogeneous agent behavioral responses, information connectivity attributes, and a regulatory model in one financial network metamodel; the term metamodel in this paper is used in the sense that the ABM approach for a financial market network simulation constitutes a* model of models*. For the desired level of complexity, the financial economic dynamics cannot be, thus far, expressed analytically through a set of neatly justified equations. Thus, the advantage of agent-based metamodels lies in their ability to create sufficiently rich scenarios once the appropriate components of a financial system are included and endowed with their rules of interaction and transaction along with an initial state.

In this paper, we build on the previous work of Khashanah and Alsulaiman [1] where bounded rational heterogeneous market metamodel of network of networks was developed to explore the effect of the interaction of heterogeneous agents on the market volatility and capital. This paper develops the first metamodel for incorporating information jumps as shocks in an agent-based approach and to equip agents with heterogeneous response to information jump stimuli. Shocks are essential for meaningful financial systemic risk simulations. It is our philosophy to emphasize the quantitative-simulative-empirical approach, which aims to quantify the quantitative aspects of the problem, simulate the nonquantifiable aspects of the problem, and calibrate the metamodel parameters with empirical data. We do not agree with the excessive emphasis of some economic literature on analyzing systemic risk from the perspective of banking systems only. Markets, banks, intermediaries, media, and many other types of agents collectively interact as nonlinear dynamical systems in the financial ecosystem. Markets act as forward looking indicators of collective intelligence in all sectors including the banking system.

The novelty of our approach in this paper is to equip agents with new features representing their response to information flows including jumps. The idea that prices are information-driven was explained by many researchers starting with the work of Merton [2], yet those models assume that information has been factored into the price and may cause a jump in the price. Therefore, we obtain a jump-diffusion model depicting postulated price dynamics. However, market metamodels are interested in the mechanics by which the network supports the conversion of news into price impact. So far, market models in ABM have not included information sources as drivers of emerging patterns in market metamodels. As a first attempt to addressing this shortcoming, we observe that news are mappings of events into information from variable agent perspectives. Information packets flow in the network wherever there is a channel that supports that flow. Therefore, agents connectivity in our network refers to the ability to conduct information between two nodes. Thus we call it information connectivity. Since events map into information and events come in bursts of variable sizes, it makes sense to model information flow in a financial network as a stochastic jump-diffusion process that accounts for market events. Agents become receivers of information flows. To equip agents with this feature, we introduce the idea of agent* information awareness* in the network. Agents awareness variability of events and their variability of mapping events into actionable information both impact financial systemic patterns appearing in the financial system. In general, agents have heterogeneous information awareness; otherwise, agents are said to be homogeneous in information awareness.

In ABM we calibrate the model to either theoretical finance or empirical observations and then explore alternative scenarios that describe the relationship between macrobehaviors and micromotivations via the change of market microstructure. This relationship may be explored to produce the systemic patterns of markets under various scenarios. In this study, we take the maximum volatility index as a proxy measure of the systemic stability of the simulated market.

For the paper to be manageable, the analysis is carried out by developing five scenarios for agents information awareness of market events. The first and last scenarios are extreme cases where it is assumed that none of the agents is informed and, in the other, all agents are uniformly informed of the events. The second scenario assumes that the hub agents in the network are aware of events and they pass this information to the agents that are directly connected to them. Similarly, the third and fourth scenarios assume that agents with normal connections and hermits are aware of information jumps in the market, respectively.

The outline of this paper is as follows: Section 2 offers a literature review. Section 3 presents definitions of the market metamodel, models, and rules of interactions. Section 4 extends the connectivity model and Section 5 shows the simulation results and analysis of the experiments. We conclude with Section 6.

#### 2. Literature Review

Researchers attempted to discover the causes of the anomalies in the stock market even before the development of the ABM. For over two decades researchers used ABM to explore complexities of a financial market. This includes the pioneering work of Kim and Markowitz [3] in 1989, the zero-intelligence model of Gode and Sunder [4] in 1993, the heterogeneous belief system of Brock and Hommes [5] in 1998, the Santa Fe artificial stock market [6] in 1999, and the Ising Model by Iori [7] in 2002.

Rules of interactions among market agents often find its roots outside the ABM literature. For example, Campbell and Kyle [8] derived an equilibrium model to examine the dynamics of stock prices. The model was constructed in a way that noise traders interact with smart money investors “fundamental traders.” They conclude that “overreaction” to the news about the fundamental value creates an important type of noise to stock movements; the term “noise” has a different connotation than the one used in signal processing.

Existing models for analyzing financial systems are based on top-down approaches, as explained in Bookstaber [9], and focus on a comprehensive view while endeavoring to break down the system into components, which may result in losing a portion of variability such as behavior and agent interactions. Additionally, Bookstaber categorizes the applications of ABM in economics and finance into banks and assets management; real-estate markets; mortgage payments; payment systems and credit risk; financial market microstructure; and macroeconomics. In [10] Bookstaber et al. suggest that the new version of the stress test should require more emphasis on institutional interactions and feedback effects. Therefore, Bookstaber et al. recommend using more adaptable models, such as agent-based model, that are capable of capturing more elements of systemic complexity and can evolve freely from the history in a characteristic design similar to the real systems.

Sornette [11] called attention to utilizing models of complexity theory, such as network analysis and agent-based models (ABM) to assess systemic risk. Helbing [12] expressed that the catastrophe engendered by human factors can not be clarified by analytical approaches only but rather requires a collective comprehension of the social dynamics.

Thurner et al. [13, 14], as an implementation of ABM in banking and asset management, studied the effect of leverage on market behavior by building an ABM that allows borrowing long assets with a margin. The results of such actions cause fluctuations in prices, fat tail returns, and volatility clustering. As a mechanism for contagion, a bank may liquidate some of its assets to raise sufficient capital in response to a margin call. Asset liquidation to cover margin calls becomes a systemic risk when fire sale liquidations cause a downward spiral price destruction and eventual crashes in extreme cases.

Poledna et al. [15] developed an ABM for simulating asset credit regulation policy. The assessments were performed under three different scenarios where in the first scenario the regulator imposes limits on maximum leverage only. In the second and third scenarios, Poledna et al. assessed the Basel II policy and alternative hypothetical ones by which the banks utilize options to hedge the risk completely. Kuzubaş et al. [16] simulate a financial network to examine the leverage effect in the banking system. The results indicate that the difference in leverage significantly affects the measure of systemic risk. The paper studied the impact of Basel III regulations for charging higher capital requirements for banks with higher leverage. The results confirmed that such a rule would increase the resilience of the system. In real-estate market application, Gilbert et al. [17] built up an ABM to research the housing market of the UK. The model accounts for interactions between the buyers, seller, and realtors given a set of endogenous and exogenous parameters representing agents attributes such as wealth and regulator control factors like interest rate. The results of the model show that house prices are sensitive to interactions of interest rate and loan-to-value (LTV) ratio. In addition, agents tend to cluster with high/low priced houses according to wealth. Also, altering the tax rate has an insignificant effect on house prices. Furthermore, a shock in the market causes fluctuations in house prices. Erlingsson et al. [18] generated an ABM of credit network and performed a set of simulation experiments to inspect the requirements for bank mortgages. The results show that lowering the mortgage standards increases house prices in the short term but leads to an unstable economic system in the long run and may lead to a recession. However, limiting home loans prerequisites helps market stability despite the possibility of slowing down economic growth. Other studies of systemic risk due to real-estate markets include the work of Carstensen [19], Bjarnason [20], and Ge [21].

Also, researchers addressed the role of network theory and agents direct interactions of model such as the models of Panchenko et al. [22], Yeh and Yang [23], Delellis et al. [24, 25], and Khashanah and Alsulaiman [1, 26]. For more details of the ABM in market microstructure, readers may refer to [27–30].

In postmodern finance with superconnectivity, agents in markets trade information dynamics. This paper is the first to introduce an information-based connectivity to model a financial network of market agent interactions, including information jumps, with heterogeneous agent response system to information flows using the ABM methodology. This paper investigates the impact of information connectivity on systemic volatility, which is proposed as an elementary proxy for the simulated market stability.

#### 3. The Metamodel and Models

In this section, based on the metamodel of Khashanah and Alsulaiman [1, 26], new definitions are introduced and the new features are explained. In the previous work the authors showed how the market volatility changes under heterogeneous agents behavior. In this paper we simulate financial market realizations to assess what may cause market instability as a function of information flow with jumps with heterogeneous agent information awareness and variable connectivity. For this purpose, we introduce the following definitions

*Definition 1. *The set denotes agent behavioral types; the set denotes the set of strategies; and denotes a possible degree of an agent with the number of agents in the metamodel.

*Definition 2. *An agent is defined in terms of the attribute vector , , , , and .

*Definition 3. *The agent lattice is the set .

*Definition 4. *A market consists of an agent lattice , a set of assets , a set of regulatory frameworks , and a set of connectivity characteristics summarized as the 4-tuple .

*Definition 5. *A market is said to be homogeneous in* behavior* if agents exhibit one behavior so that is a singleton; otherwise the market is said to be heterogeneous in behavior. A market is said to be homogeneous in* strategy* if agents have one strategy so that is a singleton; otherwise, it is heterogeneous in strategy. A market is said to be homogeneous in connectivity if agents are equally likely to be connected forming a random network; otherwise, is said to be heterogeneous in connectivity.

*Definition 6. *A metamodel is a model of models. A market is a metamodel consisting of its component models and their interactions expressed symbolically as the 4-tuple .

Current practices of scientific methodology assume linear interactions of models to produce a metamodel whose accuracy is as good as the minimum accuracy of its component models; this topic deserves attention as it has impact on advancing both complexity science and science complexity. A new definition of approximation and intermodel propagation of error is being developed in the context of quantitative-simulative-empirical methodology.

The metamodel is designed so that agents may interact directly in the trading environment where the network topology follows a scale-free network type that is structured based on the preferential attachment algorithm [31] with the initial number of hubs . In the experiments of this paper, with Definitions 1 through 6, agents , , with attribute vector with , , and referring to an agent degree. The behavioral types set is identified with where stand for the risk averse, risk averse with overconfidence and risk averse with conservative types. In addition, stand for the loss averse, loss averse with overconfidence and loss averse with conservative types.

The strategies set, , consists of four strategies: the first strategy accounts for arbitrary traders that make investment decisions randomly, called zero-intelligence agents, and denoted by . The second investment strategy accounts for fundamental traders, denoted by , who concentrate on the fundamental value of the asset, not on historical prices. The third type is referred to as momentum traders or technical traders, denoted by , and the fourth type is called adaptive traders, denoted by , with artificial intelligence capabilities using neural networks (NN); thus, . The last two types distinguish themselves from the first two types in their practice that historical prices contain relevant information to future decisions.

Using Definition 5, the market under consideration is heterogeneous in behavior, strategy and connectivity. The agent lattice in this experiment is the collection of agents possible states. Thus the market contains possible types of agents as a result of combinations of behaviors and strategies. Allowing for degrees to change, we obtain the number of points in the agent lattice to be , however, only 211 points in the agent lattice can be occupied at a given state of the market. Putting a practical upper limit on the agent degree reduces the agent lattice cardinality to reduce computational complexity.

The set of assets in this experiment contains two assets: the risk-free rate asset and one risky asset. Furthermore, the regulatory set with the risk-free rate , taxation , and buying and selling power and as control parameters. The regulatory environment can be designed to approximately map its real constructs in various financial economic systems.

The set of* information connectivity* characteristics depends on the network topology of the market. In this paper, our metamodel is heterogeneous in information connectivity in the sense of Definition 5 since the underlying network is scale-free. As part of connectivity characteristics, we introduce the new feature of heterogeneous ability to acquire information and be aware of information flows and call it* information awareness* in the market network. We consider this feature to be a function of agent connectivity and the coordinates of the agent in the agent lattice. More precisely, it is postulated that this property is a function of betweenness centrality of an agent. In our metamodel, heterogeneous agent awareness and the variability of mapping events into actionable information depending on agent awareness impact systemic patterns. The relationship between the interconnectivity and systemic risk depends on the definition of connectivity. In most studies on connectivity and systemic risk, connectivity is implied by the level of cash flow and risk flow, which are two important dimensions. In our context, information connectivity in terms of information flow is emphasized and, with third-party settlement, there is no direct transactional connectivity in this model. There is a reason for this type distinction between connectivity types: information dynamics, including market sentiments, lead to decisions on positions (such as buy, sell, or hold), which result in some transactions. In general, an information connectivity network is not identical to the corresponding transactional or cash flow connectivity network of the same market but rather the networks are dynamically linked with a variable lag. In an upcoming work, multidimensional connectivity types are considered including agent-bank, transactional, cash flow, and risk connectivity.

Agents belong to the market to achieve their objectives. Agents myopically aim to maximize their utility function given the wealth constraint [32]. The solution of the optimization problem is given bywhere denotes the behavior of the agent, which is identified with the set of behavioral types . With keeping the order, so that, for example, corresponds to risk averse type and so on. The expectation is the expected price and dividend for the next time step, which is crucial for the determination of optimal holding. The expectations of heterogeneous agents are by necessity diverse and they are determined based on investment strategies explained in the next section. Here is the conditional standard deviation of price and dividend at time . For simplicity, is assumed to be constant of unit value. The change in the sign in the above equation opposite the state of makes . By the change in the sign, we mean that the negative sign follows the positive state of and the positive sign follows the negative state of .

Agent behaviors are quantified by assigning values to coefficients. Hence, the risk aversion coefficient , loss aversion coefficient as , and . Consider overconfidence/conservative where as or , if or , and as . The intuition behind the agent behaviors quantification is that the overconfidence/conservative agents tend to hold, sell higher, or lower portion of stock over the neutral traders. Also, the loss averse traders tend to reduce their positions as confronted with a loss at a given reference point as it has been indicated by Kahneman and Tversky [33]—individuals are impacted by losses more than profits.

The expectation in (1) is a critical variable in the model. It is estimated individually given the agent trading strategy. The zero-intelligence agents quantitatively and randomly expect future prices and dividends as where and are fixed at and in this study to reduce computational complexity and is the uniform distribution on whose parameters can be calibrated.

The fundamental agents keep an eye on the fundamental process of the asset price where it is assumed that the dividends follow the process:where is the growth rate of the dividend, is the dividend volatility, and is a Wiener process with normal distributions ~. The variance may be estimated using GARCH (1,1): where , , and . The fundamental prices of the asset then follow Williams [34]: where is the risk-free rate. This fundamental value method works well as long as .

The expected dividends and prices for agent areThe momentum agents verify the saying “the trend is your friend,” and they buy/sell if the previous returns are positive/negative:where .

The last trading strategy in in the strategy model is an adaptive trading strategy where the neural network is utilized for prediction execution. The designed neural network is a feed-forward network where the input to nodes represents past returns of the stock that are uniformly distributed among the agents with the minimum of one previous return and maximum of ten past returns. Also, the neural network consists of one hidden layer that may be composed of one to ten nodes with equal probability for agent . The output of the neural network is the expected return at time that will be mapped to and comparative approach applies to . Agents will learn by updating the weights inside the neural network from the training data where the training follows uniform distribution from 10 to 100. The neural network is invoked customarily according to parameter . The stopping criteria are subject to maximum iteration or base error .

Agents may change their initial decisions of stock holding as a consequence of the interaction with other agents. Whenever agents have a direct interaction with each other with a chance to share their market sentiment, agents may be influenced to change their outlook on the market as a result of the interaction. Notice that market sentiment sharing is a form of information connectivity. The final holding decision is then constructed as the weighted average of the agent initial decision and initial decisions of the connected agents: where is the final decision for agent and is a given weight for the initial decision of holding shares of stock for agent , is the total number of agents, and if is connected to and zero otherwise.

The price formation will follow the price adjustment method [35–38]. The price adjustment process achieves local price equilibrium in the market based on the aggregate bids and offers where is the market price at time , is the price adjustment speed relative to the spread, that is, a simplified form of market efficiency. Further, represents the total number of bids among all agents and is the total number of offers.

#### 4. Exogenous Factors and Market Reactions

In this study, information flow including jumps is introduced in the ABM metamodel. The market “consumes” information and converts it through agent responses into new realizations of prices and positions. The market may be influenced by nonsystematic events such as political, economic, and natural disasters, wars, or, in this day and age, tweets and fake news. For asset price dynamics, Merton’s jump-diffusion model superposes jumps on a diffusion process [2] with jumps following a compound Poisson process. Merton’s model is a special case of Lévy processes. Most literature in finance on jumps has been in conjunction with option pricing as the underlying asset undergoes a jump. A good reference on the subject of jumps is the book by Tankov [39]. For the purpose of this paper, and as a first approximation, we postulate that the information flow process follows a jump-diffusion process. The process of jumps follows the notation and construct in [40], with the usual definition of a probability space as a probability space with information filtration and the information process defined on the probability space with dynamics expressed by the stochastic differential equation (SDE)where is the drift coefficients and is the volatility—both are assumed to be constant. is a continuous Brownian motion process and is Poisson process adapted to the filtration with constant rate with . As in [40], the term is a jump amplitude process. The three processes, , , and , are assumed to be independent and the jump process, as in Merton model, is taken to be a compound Poisson process in the sense that we can sum jump amplitudes of Poisson jumps occurring over a given time interval. Using Itô calculus the solution of (10) takes the form

The price process is assumed to follow the same dynamics. Our implementation calibrates to empirical jumps of price time series with a predefined jump threshold. For example, for the S&P500 price process, for the year of 2008, with a threshold of , we collect return jumps in the set , which is a time subseries of the return time series.

The jump size distribution, , can be assumed to be normally distributed which sufficiently achieves our implementation for this paperwith constant being the jump mean and being the jump standard deviation. Kou and Wang assume double exponential distribution as in “option pricing under a double exponential jump-diffusion model,” Management Science, 50 (2004), pp. 1178–1192. For example, in the 2008 sample we find for negative jumps and for positive jumps. The is the jump standard deviation and .

Figure 1 shows the empirical jumps of the S&P500 in 2008 (a) and the simulated process for the corresponding period with . The empirical jumps exhibit the phenomenon of jump clustering, which invites, in an upcoming work, the use of more sophisticated jump processes such as Hawkes self-exciting processes.