We study asset pricing dynamics in artificial financial markets model. The financial market is populated with agents following two heterogeneous trading beliefs, the technical and the fundamental prediction rules. Agents switch between trading rules with respect to their past performance. The agents are loss averse over asset price fluctuations. Loss aversion behaviour depends on the past performance of the trading strategies in terms of an evolutionary fitness measure. We propose a novel application of the prospect theory to agentbased modelling, and by simulation, the effect of evolutionary fitness measure on adaptive belief system is investigated. For comparison, we study pricing dynamics of a financial market populated with chartists perceive losses and gains symmetrically. One of our contributions is validating the agentbased models using real financial data of the Egyptian Stock Exchange. We find that our framework can explain important stylized facts in financial time series, such as random walk price behaviour, bubbles and crashes, fattailed return distributions, powerlaw tails in the distribution of returns, excess volatility, volatility clustering, the absence of autocorrelation in raw returns, and the powerlaw autocorrelations in absolute returns. In addition to this, we find that loss aversion improves market quality and market stability.
In 1987, the Wall Street Stock Market faced a severe financial crisis. That crisis provoked economists to realize that traditional economic theories such as the
Therefore, the high trading activities in financial markets present evidence for the presence of heterogeneous predictions for asset prices. Heterogeneous agent models aim to relax the classical hypothesis of a representative agent and the rational expectations towards heterogeneous bounded rational agents [
Frankel and Froot [
Financial markets are comprised of many traders with heterogeneous beliefs, attributes, and level of rationality [
Artificial financial markets are models developed using the agentbased modelling approach. The main aim of artificial financial markets is to understand the endogenous variables that cause aggregate behaviours and patterns to emerge at the macro level [
Meanwhile, behavioural finance is a relatively new paradigm seeking to link behavioural and cognitive psychological theories with finance to understand the bounded rational decisions of financial traders. Since 1979, Kahneman and Tversky provoked the idea of the choice under uncertainty. They spent many years to study this concept by conducting surveys and collecting data about the traders’ behaviour under uncertainty [
Although the prospect theory has been developed since 1979; yet there is no clear definition of gains and losses and how to measure them. Also, there is no clear identification of the reference point. Accordingly, its application into financial markets framework is very challenging. The model proposed in this paper provides a novel application of the prospect theory, where agents recognize their gains and losses in terms of an evolutionary fitness measure.
Many studies have been developed to model the switching dynamics between fundamental analysis and technical analysis, such as [
In this paper, we explore the agentbased modelling as a tool for studying loss aversion behavioural bias introduced by the prospect theory. Our model contributes to behavioural finance research by linking the macro and the micro behaviours. This link is ignored in the classical models studied behavioural finance. To our knowledge, no research has been conducted to study the impact of loss aversion behavioural bias on the adaptive belief system and asset pricing dynamics, which is considered as our main contribution in the current work.
The rest of this paper is organized as follows. In Section
In this section we introduce an agentbased financial model populated with heterogeneous agents with loss aversion behavioural bias. At the beginning we discuss the model definition and assumptions. In Section
The main assumptions of the proposed artificial financial market can be summarized as follows.
There is only one risky asset to be traded.
There are two types of agents, the market maker and the traders.
In each time step
If a trader chooses to submit an order, she/he can either follow technical or fundamental trading rule. It is assumed that, at time
Beliefs adaptation rule: the agents are bounded rational as they tend to choose the strategy performed well in the recent past and therefore display some kind of learning behaviour. It is assumed that the fitness of each trading strategy is available and publicly known by all agents.
The chartist agents are loss aversion so that they recognize losses more than twice recognizing their gains. Consequently, they consider a value function proposed by the prospect theory to evaluate the fitness of each trading strategy.
The fraction of traders following each strategy is determined via a discrete choice model.
The market maker correlates the orders and adjusts the asset price according to the net submitted orders. It is assumed that the market maker is a risk neutral and settles the asset prices without intervention.
Agents in our market interact indirectly through their impact on price adjustment which affects the performance of the trading rules which in turn affects the agent decision to select trading strategy and so on.
The behaviour of the market maker is described as in Farmer and Joshi [
The goal of the technical analysis used by the chartists is to exploit the price changes [
Fundamental analysis assumes that prices will revert to their fundamental values in the short run [
The evolutionary part of the model, inspired by Brock and Hommes [
While, in Westerhoff [
Following Manski and McFadden [
The higher attractive strategy will be chosen by the agents. The parameter
Model parameter settings are determined following Tversky and Kahneman [
The main idea behind choosing specific values of the parameters can be summarized as follows. The reaction parameters of technical and fundamental trading rules (multiplied by the price settlement parameter) are between zero and 0.1 for daily data.
To keep the autocorrelation
The value of
Parameters for the simulation of the financial markets under loss aversion behavioural bias.
Parameter  Value  Description of parameter 


1  Price settlement parameter 

0.04  Extrapolating parameter 

0.04  Reverting parameter 

0.975  Memory parameter 

300  Intensity of choice parameter 

0.01  Standard deviation of the random factors affect the price settlement process 

0.05  Standard deviation of the additional random orders of technical trading 

0.01  Standard deviation of the additional random orders of fundamental trading 

2.25  Loss aversion parameter 
In this section we discuss the dynamics of our model by simulation. In Section
To implement the proposed artificial financial market, we develop an agentbased simulation model using Netlogo platform. NetLogo provides an environment for simulating natural and social phenomena [
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Many researchers study the empirical behaviour of financial returns, and they find that most financial markets share the same statistical properties [
In the following, we explore the extent to which our model is capable to replicate these stylized facts. Also, to investigate the effect of loss aversion on the macro and micro dynamics of the artificial market, a benchmark case, where
In the following, our exposition mainly rests on the behaviour of the stock index of Egypt (EGX 30). Its time series covers the period from January 1, 1998, to November 16, 2014, and consists of 4123 daily observations. The data was downloaded from the Egyptian Stock Exchange website
Before we start a comprehensive Monte Carlo analysis, we will first explore a representative simulation run. The simulation run contains 4120 daily observations, mirroring a period of about 16 years. Figure
The evolution of index prices (logprices) of (a) EGX 30, (b) the MOD1, and (c) the MOD2, respectively. The dashed lines give a benchmark of the
EGX 30
MOD1 (
MOD2 (
Figure
From Figure
Figure
The returns of (a) the EGX 30, (b) the MOD1, and (c) MOD2. The panels give evidence for excessive prices volatility and clustered volatility properties. The grey rectangles exemplify periods of high volatility. The clustered volatility property can be seen from the presence of sustained periods of high or low volatility.
EGX 30
MOD1 (
MOD2 (
Figures
Another stylized fact concerns the distribution of returns and its fat tail feature. Figure
The probability density functions of returns. The panels show the empirical distribution of (a) the EGX 30, (b) the MOD1, and (c) the MOD2 returns. For comparison, the dashed lines give the probability density function of normally distributed returns with the same means and standard deviations. The three returns time series at hand exhibit fattailed return distributions, that is, a higher concentration around the mean, thinner shoulders, and more probability mass in their tails.
EGX 30
MOD1 (
MOD2 (
A power law may give a useful approximation to the tail behaviour of empirical data, but there is no reason to anticipate that it will appear in every market. One way to investigate the powerlaw behaviour is suggested by Clauset et al. [
Loglog plot of the complement of the cumulative distribution of normalized (a) EGX 30, (b) MOD1, and (c) MOD2 returns, respectively. The three time series are scaled by their sample standard deviation and absolute returns have been taken to merge the positive and negative tails. For comparison, the dashed lines give the complements of the cumulative distributions of the standard normal distributions. The solid line in each panel presents a performed loglog regression on the largest 30 percent of the observations. The estimated exponent
EGX 30
MOD1 (
MOD2 (
Now, the exponent
To reduce computational efforts, we use the socalled Hill tail index estimator
The Hill plot of the smallest 10 percent observations (on the lefthand sides) and the largest 10 percent observations (on the righthand sides) for (a) the EGX 30, (b) the MOD1, and (c) the MOD2. Here we plot Hill tailindex estimator
EGX 30
MOD1 (
MOD2 (
Figure
The autocorrelation functions of raw and absolute returns, respectively, for (a) the EGX 30, (b) the MOD1, and (c) the MOD2. From Panel (a), the EGX 30 raw returns show autocorrelation coefficients that are not significant for almost all lags except for the first lag. However, the absolute return time series reveal autocorrelation coefficients that are significant for up to 100 lags. In Panels (a) and (b), furthermore, the raw returns display autocorrelation coefficients that are not significant for up to 100 lags while the absolute returns show significant long memory effects for more than 60.
EGX 30
MOD1 (
MOD2 (
Note that Figures
Another astonishing fact of the stock markets is the selfsimilarity in the sense of Mandelbrot [
Following Peng et al. [
Estimation of selfsimilarity parameter
EGX 30
MOD1 (
MOD2 (
The
Finally, Figure
Dynamics of the adaptive belief system: chartists (black region), fundamentalists (grey region), and inactive traders (white region) for (a) the MOD1 and (b) the MOD2, respectively.
MOD1 (
MOD2 (
Note the price appreciations between periods 3650 and 3800 in Figures
To check the robustness of the MOD2 dynamics to different parameter settings, Figure
Four repetitions of the simulation using different parameter values. Each set of the four panels shows from top to bottom the evolution of stock prices, the asset returns, and the market shares of chartists (black region), fundamentalists (grey region), and notrade (white region), respectively.
Summing up, features of the simulation run displayed in Figures
Table
Descriptive statistics. The table reports the mean, maximum, minimum, standard deviation, skewness, kurtosis, and JarqueBera (JB) of the EGX 30 and estimates of the mean and the 5 percent, 25 percent, 50 percent, 75 percent, and 95 percent quantiles of these statistics for the MOD1 and the MOD2, respectively. Computations are based on 5000 time series, each containing 4120 observations.
Series  Mean/quantile  Mean  Max.  Min.  Std. dev.  Skew.  Kurt.  JB 

EGX 30  5.39 
0.18  −0.18  0.018  −0.33  12.04  14101.9  


MOD1 ( 
Mean  7.36 
0.14  −0.14  0.024  0.01  3.69  — 
0.05  −6.73 
0.11  −0.18  0.021  −0.23  2.70  —  
0.25  −2.02 
0.13  −0.15  0.023  −0.08  3.21  —  
0.50  −7.27 
0.14  −0.14  0.024  0.01  3.64  —  
0.75  1.94 
0.15  −0.12  0.025  0.11  4.10  —  
0.95  6.82 
0.18  −0.11  0.027  0.25  4.89  —  


MOD2 ( 
Mean  −6.19 
0.14  −0.14  0.023  0.01  4.39  — 
0.05  −6.82 
0.11  −0.18  0.020  −0.25  3.28  —  
0.25  −1.86 
0.13  −0.15  0.021  −0.10  3.85  —  
0.50  −7.31 
0.14  −0.14  0.023  0.01  4.32  —  
0.75  1.57 
0.15  −0.13  0.024  0.01  4.82  —  
0.95  6.08 
0.18  −0.11  0.026  0.27  5.77  — 
Note the estimates of the standardized third moment
Table
The Hill tail index estimator
Series  Mean/quantile  Lefttail exponent  Righttail exponent  






 
EGX 30  3.39 (3.24, 3.55)  3.32 (3.24, 3.39)  3.47 (3.43, 3.51)  3.56 (3.43, 3.68)  3.72 (3.66, 3.78)  3.92 (3.88, 3.95)  


MOD1 ( 
Mean  3.29  3.54  3.45  3.31  3.56  3.47 
0.05  2.69  2.98  3.01  2.71  3.00  3.02  
0.25  3.02  3.29  3.24  3.05  3.31  3.27  
0.50  3.27  3.53  3.43  3.28  3.55  3.45  
0.75  3.53  3.78  3.63  3.55  3.80  3.66  
0.95  3.93  4.17  3.95  3.96  4.18  3.97  


MOD2 ( 
Mean  3.27  3.52  3.40  3.38  3.61  3.46 
0.05  2.68  2.97  2.96  2.68  2.97  2.96  
0.25  3.01  3.28  3.21  3.01  3.28  3.20  
0.50  3.25  3.50  3.39  3.25  3.50  3.39  
0.75  3.52  3.74  3.59  3.52  3.74  3.59  
0.95  3.90  4.13  3.90  3.90  4.13  3.90 
To continue investigating the robustness of our results, Table
The autocorrelation functions of raw and absolute returns. The table contains the autocorrelation function of raw returns
Series  Mean/quantile 








EGX 30  0.18  0.02  0.04  0.29  0.09  0.05  0.03  


MOD1 ( 
Mean  0.02  0.01  0.003  0.25  0.16  0.08  0.02 
0.05  −0.02  −0.03  −0.04  0.20  0.10  0.02  −0.03  
0.25  0.01  −0.01  −0.01  0.23  0.13  0.05  −0.003  
0.50  0.02  0.01  0.004  0.25  0.16  0.07  0.02  
0.75  0.04  0.02  0.02  0.27  0.18  0.10  0.04  
0.95  0.07  0.05  0.04  0.30  0.21  0.14  0.07  


MOD2 ( 
Mean  0.02  0.01  0.004  0.28  0.17  0.08  0.02 
0.05  −0.02  −0.04  −0.04  0.23  0.11  0.02  −0.03  
0.25  0.004  −0.01  −0.01  0.26  0.15  0.05  −0.001  
0.50  0.02  0.01  0.004  0.28  0.17  0.08  0.02  
0.75  0.04  0.02  0.02  0.30  0.20  0.11  0.04  
0.95  0.07  0.05  0.05  0.33  0.23  0.15  0.09 
Now, we check the robustness of the scaling power law. Table
The Hurst index for the raw and absolute returns, respectively. The table reports the scaling exponent of raw returns
Series  Mean/quantile 



EGX 30  0.57 (0.55, 0.59)  0.81 (0.76, 0.87)  


MOD1 ( 
Mean  0.48  0.86 
0.05  0.41  0.79  
0.25  0.44  0.83  
0.50  0.47  0.86  
0.75  0.51  0.89  
0.95  0.57  0.92  


MOD2 ( 
Mean  0.48  0.87 
0.05  0.40  0.80  
0.25  0.44  0.85  
0.50  0.47  0.88  
0.75  0.51  0.90  
0.95  0.57  0.93 
To summarize our results so far, the illustrated figures and the performed Monte Carlo analysis show that MARKET2 is able to generate return time series possessing detailed stylized facts of real financial data. These properties include fattailed return distributions, absence of autocorrelation in raw returns, persistent long memory of volatility, excess volatility, volatility clustering, and powerlaw tails.
We perform a Monte Carlo analysis to check the robustness of the MOD1 and the MOD2 evolutionary dynamics. Table
Statistical properties and evolutionary dynamics of the agentbased model. The table contains estimates of the mean and the 5 percent, 25 percent, 50 percent, 75 percent, and 95 percent quantiles of the volatility, the distortion, and strategy weights:
Series  Mean/quantile  Volatility  Distortion 




MOD1 
Mean  1.70  12.44  34  35  30 
0.05  1.51  8.80  29  33  27  
0.25  1.61  10.46  32  34  29  
0.50  1.70  12.02  34  36  31  
0.75  1.79  13.96  36  37  32  
0.95  1.92  17.80  40  38  34  


MOD2 
Mean  1.60  11.53  29  38  33 
0.05  1.40  8.02  23  35  29  
0.25  1.50  9.65  26  37  32  
0.50  1.59  11.07  29  38  33  
0.75  1.68  12.91  31  39  35  
0.95  1.82  16.44  35  41  37 
Now, the impact of loss aversion on the fractions of agents within each strategy block deserves greater attention. Is the technical analysis least appealing due to the loss aversion behavioural bias? What is the effect of loss aversion on the adaptive belief system and on the pricing dynamics? To answer these questions we consider the benchmark case,
In 1979, Kahneman and Tversky proposed their famous psychological theory, the prospect theory, in order to understand the psychological motivations for traders’ behaviours. The prospect theory considers loss aversion as one of the main behavioural biases that affect traders’ decisions under uncertainty. The theory states that traders recognize their losses more than twice their recognition of gains.
To increase our understanding of traders’ behaviour and their adaptive beliefs, we develop an agentbased financial market model. Agentbased modelling provides the link between macro and micro dynamics. In our model, agents can trade following either stochastic technical or stochastic fundamental trading rules. While technical analysis builds decisions upon past price trends, fundamental analysis advises betting on mean reversion. Since chartists are loss averse, any losses following technical analysis cause a rapid switching to other groups. Price is adjusted by the market maker according to the net submitted orders without any intervention from her/him.
Simulations reveal that our model is capable of explaining a number of important stylized facts of stock markets, such as random walk price behaviour, bubbles and crashes, fattailed return distributions, excess volatility, and volatility clustering. In addition to these, we investigate the presence of powerlaw tails. The observed estimates of the exponent
The dynamics of our model can be summarized as follows. The farther the asset prices deviate from their fundamental values, the more aggressive the chartists will become. The increase in market shares of the chartists will increase the volatility causing a bubble or a crash to emerge. However, the loss aversion behavioural bias improve the market by minimizing its volatility and distortion. As the distortion reaches its maximum value the fundamental analysis becomes more appealing for traders to follow. The increase in switching to the fundamental analysis will pull asset prices to their fundamentals and the volatility diminishes. Due to the market dynamics, no trading strategy dominates the others. This causes substantial long memory effects in returns volatility.
To sum up, loss aversion directly affects the adaptive belief system as recognized losses stimulate chartists to adopt fundamental trading or stay inactive. This adaptation works for the market stability and prices efficiency. The simulation results proposed the agentbased model to successfully replicate the macro behaviour of real financial markets and, consequently, to enhance our understanding of asset pricing dynamics. Therefore, our model serves as a good testbed for policy makers to explore the effect of different regulatory policies which improves the decision making process.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the respected reviewers for their comments and valuable suggestions.
This result was observed by Arthur [
The fundamental value is assumed to be constant, such that
A discrete choice model specifies probabilities
Autocorrelation function (ACF) studies the linear dependence between
The returns are defined as
This data was extracted from the EGX annual reports from 2006 to 2009.
Distortion and volatility are considered to be important determinants of market quality [
Such as a value of 0.86 percent for FTSE 100, 0.86 percent for S&P 500, and 1.09 percent for DAX; these statistics are our own calculations of data collected for the FTSE 100, the S&P 500, and the DAX from
Hill tailindex estimator
To put it simply, a nonstationary stochastic process is said to be
To estimate the selfsimilarity parameter we follow Peng et al. [
The JarqueBera (JB) test statistic for normality is defined as follows [
For fattail distributions the first and second moments are not enough to describe the data [