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Abstract

We are looking for the agent-based treatment of the financial markets considering necessity to build bridges between microscopic, agent based, and macroscopic, phenomenological modeling. The acknowledgment that agent-based modeling framework, which may provide qualitative and quantitative understanding of the financial markets, is very ambiguous emphasizes the exceptional value of well defined analytically tractable agent systems. Herding as one of the behavior peculiarities considered in the behavioral finance is the main property of the agent interactions we deal with in this contribution. Looking for the consentaneous agent-based and macroscopic approach we combine two origins of the noise: exogenous one, related to the information flow, and endogenous one, arising form the complex stochastic dynamics of agents. As a result we propose a three state agent-based herding model of the financial markets. From this agent-based model we derive a set of stochastic differential equations, which describes underlying macroscopic dynamics of agent population and log price in the financial markets. The obtained solution is then subjected to the exogenous noise, which shapes instantaneous return fluctuations. We test both Gaussian and q-Gaussian noise as a source of the short term fluctuations. The resulting model of the return in the financial markets with the same set of parameters reproduces empirical probability and spectral densities of absolute return observed in New York, Warsaw and NASDAQ OMX Vilnius Stock Exchanges. Our result confirms the prevalent idea in behavioral finance that herding interactions may be dominant over agent rationality and contribute towards bubble formation.

Citation: Gontis V, Kononovicius A (2014) Consentaneous Agent-Based and Stochastic Model of the Financial Markets. PLoS ONE 9(7): e102201. doi:10.1371/journal.pone.0102201

Editor: Tobias Preis, University of Warwick, United Kingdom

Received: April 15, 2014; Accepted: June 14, 2014; Published: July 16, 2014

Copyright: © 2014 Gontis, Kononovicius. This is an open-access article distributed under the terms of the Creative Commons Attribution License. which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its Supporting Information files.

Funding: The authors performed this research as staff researchers of Vilnius University. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Statistical physics has got the edge over socio-economic sciences in the understanding of complex systems [1] –[7]. This happened due to the fact that physicists were able to start from the understanding of simple phenomena via simple models and later built the complexity up together with the increasing complexity of the considered phenomena. On the other hand socio-economic sciences had to face complexity right from the start as socio-economic systems are in no way simple systems — they are intrinsically complex at many different levels at the same time. Financial markets are one of the most interesting examples of such complex systems. Unlike in physics we have no direct way to gain insights into the nature of microscopic interactions in financial markets, thus the understanding of the financial market fluctuations may become rather limited and very ambiguous. Yet the understanding might be improved indirectly through the further development of the complex systems approach [8]. First of all, currently there are huge amounts of the available empirical data, which itself is attracting representatives of the experimental sciences [9]. Also there is an agent-based modeling framework, which may provide qualitative and quantitative understanding of the financial markets. The intense applications of these ideas is still ongoing [9] –[11] and the challenge is still open.

Agent-based modeling has become one of the key tools, which could improve the understanding of the financial markets as well as lead to the potential applications [12] –[17]. Currently there are many differing agent-based approaches in the modeling of the financial markets. Some of them aim to be as realistic as possible, yet they usually end up being too complex to posses analytical treatment. One of the most prominent examples of these kind of models is the Lux-Marchesi model [18]. The more recent approaches in the similar direction consider modeling order books [19] –[21]. Other approaches, on the other hand, aim to capture the most general properties of the many complex socio-economic systems (some of the examples include [22]. [23] ). Though there are also some interesting approaches which combine realism and analytical tractability, e.g. Feng et al. [24] have used both empirical data and trader survey data to construct agent-based and stochastic model for the financial market. Looking for the ideal agent-based approach we would consider as a primary necessity to build bridges between microscopic, agent based, and macroscopic, phenomenological, modeling [25]. Following this trace of thought it would be rational to combine two origins of the noise: exogenous one, related to the information flow, and endogenous one, arising form the complex stochastic dynamics of agents. Such integral view of the financial markets can be achieved only with very simple zero-intelligence agent-based models and macroscopic, phenomenological, approaches incorporating external information flow. This is the main idea of our present consideration of the financial markets.

The expected properties of such model lead us to the return fluctuations, characterized by the power law distributions and the power law autocorrelations of absolute return considered in [26] –[32]. We investigate an agent-based herding model of the financial markets, which proves to be rather realistic and also simple enough to be analytically tractable [33]. [34]. Namely we consider a three agent states’ model [34] and incorporate it into the standard model of the stock price described by the geometric Brownian motion or into process with statistical feedback [35]. exhibiting Tsallis statistics. We find that the improved three state agent-based herding model reproduces the power law statistics observed in the empirical data extracted from the NYSE Trades and Quotes database, Warsaw Stock Exchange and NASDAQ OMX Vilnius Stock Exchange.

We start by discussing the possible alternatives in macroscopic and phenomenological modeling providing some insight into the possible connection to the agent-based microscopic approach. Next we develop the microscopic approach by defining the herding interactions between three agent groups and incorporate it into a consistent model of the financial markets. Further we couple the endogenous fluctuations of the agent system with the exogenous information flow noise incorporated in macroscopic approach and provide detailed comparison with the empirical data. Finally we discuss the obtained results in the context of the proposed double stochastic model of the return in the financial markets.

Methods

Macroscopic and phenomenological versus microscopic and agent-based treatment of the financial markets

It is the natural peculiarity of the social systems to be treated first of all from the macroscopic and phenomenological point of view. In contrast to the natural sciences microscopic treatment of the social systems is ambiguous and hardly can be considered as a starting point for the consistent modeling. The complexity of human behavior leaves us without any opportunity to consider human agent in action as a determined dynamic trajectory. The financial markets as an example of the social behavior first of all are considered as a macroscopic system exhibiting stochastic movement of the variables such as asset price, trading volume or return [10]. Despite lack of knowledge regarding microscopic background of the financial systems there is considerable progress in stochastic modeling producing very practical applications [36]. [37]. The standard model of stock prices, . referred to as geometric Brownian process, is widely accepted in financial analysis (1)

In the above Wiener process can be considered as an external information flow noise while accounts for the stochastic volatility. Though one must consider the model of stock prices following geometric Brownian motion as a hypothesis which has to be checked critically, this serves as a background for many empirical studies and further econometric financial market model developments. Acknowledgment that analysis taking and constant have a finite horizon of application has become an important motivation for the study of the ARCH and GARCH processes [38] –[40] as well as for the stochastic modeling of volatility [37] .

We acknowledge this phenomenological approach as a good starting point for the macroscopic financial market description incorporating external information flow noise and we will go further by modeling volatility as an outcome of some agent-based herding model. The main purpose of this approach is to demonstrate how sophisticated statistical features of the financial markets can be reproduced by combining endogenous and exogenous stochasticity.

Here we consider only the most simple case, when fluctuations are slow in comparison with external noise . In such case the return, . in the time period can be written as a solution of Eq. (1) (2)

This equation defines instantaneous return fluctuations as a Gaussian random variable with mean and variance . Let us exclude here from the consideration long term price movements defined by the mean as we will define the dynamics of price from microscopic agent-based part of model. This assumption means that we take from phenomenological model only the general idea how to combine exogenous and endogenous noise. Then Eq. (2) simplifies to the instantaneous Gaussian fluctuations with zero mean and variance .

In [41]. [42]. while relying on the empirical analysis, we have assumed that the return, . fluctuates as instantaneous q-Gaussian noise with some power-law exponent . and driven by some stochastic process defining second parameter of fluctuations . was introduced as a linear function of absolute return moving average calculated from some nonlinear stochastic model [42] (3) where parameter serves as a time scale of exogenous noise and quantifies the relative input of exogenous noise in comparison with endogenous one described by .

A more solid background for this kind of approach can be found in the work by L. Borland [43]. The idea to replace geometric Brownian process of market price by process with statistical feedback [35] leads to the equation of return as function of time interval given by (4) where evolves according to the statistical feedback process [35] (5) and satisfies the nonlinear Fokker-Planck equation (6)

The explicit solution for in the region of values can be written as one of the Tsallis distributions [44] (7) where and are as follows (8)

Assuming as slow stochastic process in comparison with from Eq. (4) one gets that . This sets PDF for the same as for Eq. (7), one just has to replace and, defined in Eqs. (8) by and accordingly. This gives a Tsallis distribution for as (9) where as new parameter related to previous one can be written as (10) (11)

Now we are prepared to combine two phenomenological approaches introduced by Eqs. (2) and (4) with agent-based endogenous three state herding model. serves as a measure of system volatility in both of the phenomenological approaches. It is reasonable to assume that financial market is in the lowest level of possible volatility when assets market value is equal to the it’s fundamental value . lets define it as constant . Volatility of financial system increases when market value of the asset deviates from the fundamental value. These deviations can be accounted as . Further in this contribution we will assume that volatility is defined by through the linear relation (12) where parameter serves as a scale of exogenous noise and quantifies the relative input of endogenous noise. Both parameters and have to be defined from empirical data. To complete the model we have to propose agent-based consideration of log price . In the following section we present the three state herding model giving stochastic equations for the log price .

The three state herding model as a source for the endogenous stochastic dynamics

Having discussed a macroscopic view of the financial fluctuations we now switch to the microscopic consideration of the endogenous fluctuations. Let us derive the system of stochastic differential equations defining the endogenous log-price fluctuations from a setup of appropriate agent groups composition. We consider a system of N heterogenous agents — market traders continually changing their trading strategies between three possible choices: fundamentalists, chartists optimists and chartists pessimists. We further develop this commonly used agent group setup [24] by considering all transitions between three agent states to be a result of binary herding interactions between agents during their market transactions.

Fundamentalists are traders with fundamental understanding of the true stock value, which is commonly quantified as the stock’s fundamental price, . We exclude from our consideration any movements of the fundamental price. The assumption of constant fundamental price means that we further will consider price fluctuations around its fundamental value. Taking into account a long-term rational expectations of the fundamentalists their excess demand, . might be assumed to be given by [45] (13) where is a number of the fundamentalists inside the market and is a current market price. The rationality of fundamental traders keeps asset price around its fundamental value as they sell when . and buy when .

Short term speculations cause unpredictable price movements. There is a huge variation of speculative trading strategies, so it is rational, from the statistical physics point of view, to consider these variations as statistically irrelevant. We make only two distinctions between chartists: optimists suggest to buy and pessimist suggest to sell at a given moment. Thus the excess demand of the chartist traders, . can be written as: (14) where is a relative impact factor of the chartist trader and further will be integrated into a certain empirical parameter, and are the total numbers of optimists and pessimists respectively. So, as you can see, we replace a big variety of “rational” chartist trading strategies by herding kinetics between just two options: buy or sell.

The proposed system of heterogenous agents defines asset price movement by applying the Walrasian scenario. A fair price reflects the current supply and demand and the Walrasian scenario in its contemporary form may be written as (15) here is a speed of the price adjustment, a total number of traders in the market, and . By assuming that the number of traders in the market is large, . one obtains: (16)

Stochastic dynamics of the proposed agent-based system is defined by the occupations of the three agent states: (17)

One can model the evolution of occupations as a Markov chain with some reasonable assumptions for the sake of simplicity. There are six one agent transition possibilities in three group setup, see Fig. 1 and [34]. Few assumptions are natural for the financial markets as there is some symmetry. With the notation of agent transitions from state to as subscripts to any parameter . where i and j take values from the set . we will use following assumptions, for the rates of spontaneous transitions: . . . and for the herding transitions: . Finally it is reasonable to assume that transitions between chartist states are much faster than between chartists and fundamentalists . . . . Taking into account the restraint and having in mind that transitions are equivalent to . one can can write one step herding transition rates between and groups for given and as [33] (18)