1. INTRODUCTION
This research aims to investigate, through simulation models, how
the interaction among agents in an artificial stock market can affect
the dynamics of asset prices. Thus, the study follows a different
methodology for the analysis of prices by exploring the simulation of
agents' behavior in an artificial stock market. Commonly used
approaches in finance involve induction and deduction. Thus, mechanisms
of analysis that relies on simulation of an artificial market are
relevant methodological alternatives that can bring new perspectives to
the study of asset prices. It is important to highlight that the
approach based on simulation is considered a recent methodology to
assess economic phenomena (Tesfatsion, 2006).
From the defining characteristics of agents, we set up an
artificial stock market in which individuals interact, by demanding and
supplying assets, driving the price of a stock to an equilibrium value.
In this artificial stock market, one can structure a controlled
environment to study specific phenomena, isolated from other external
effects. As suggested Ehrentreich (2007), the study of stock prices in
an artificial market is important because it allows the elimination of
various restrictive assumptions required for the analytic investigation
of asset prices. Additionally, empirical analysis of real markets may be
compromised due to the influence of several variables that are difficult
to control or measure. In the case of a real stock market, given that
each observation is influenced by specific situations that are unlikely
to occur again, mechanisms of artificial market simulation provide an
analysis of variables under controlled conditions that can be repeated.
In this study, we evaluate a particular process of diffusion of
dividends and its impact on the equilibrium of price of assets. More
particularly, using a dynamic in which the dividends follow a mean
reversion process, as suggested in the artificial market described by
Ehrentreich (2007), this study analyzes heterogeneous agents that
interact in an environment where the dividends of a period depend on the
realization of a dividend in the previous period. The results suggest
that, under the assumption of utility maximizers agents with different
expectations about future dividends, asset prices may under-react. Thus,
the gradual change of prices observed in the sub-reaction confronts the
efficient market hypothesis, in which all information is instantly
reflected in the price.
2. AGENT BASED MODELS
Agent Based Modeling (ABM) is a technique that is being
increasingly used in various applications in social sciences (Gilbert,
2007). Agent-based models have origins in the 1940s, with the study of
cellular automata. However, the ABM had a higher growth only beginning
in the 1990s, when advances in computer processing have made feasible
more realistic simulations of social phenomena (Buchanan, 2009).
Gilbert (2007) defines agent-based modeling as a computational
method that allows a researcher to create, analyze and experiment with
models in which agents interact in a controlled environment. Thus, the
ABM is an abstract representation of reality in which: (i) a variety of
objects interact with each other and the environment, (ii) the objects
are autonomous and therefore do not follow a central control and (iii)
the result of their interactions is numerically computed (Richiardi,
2011).
In this context, agent-based modeling is an important type of
simulation, characterized by the existence of several agents that
interact with little or no central direction, facilitating the emergence
of general properties through a bottom-up process (Axelrod, 2006). Thus,
from the definitions of the specific characteristics of each individual
or agent and of the ways agents interact, one can study general behavior
of the group as a whole.
Therefore, through ABM, one can evaluate situations in which
individuals have different behaviors from those established by
traditional financial models, enabling the identification of new
phenomena. Using computer simulations, ABM allows (i) to formalize
theories about complex processes, (ii) to conduct experiments and (iii)
to observe the emergence of some occurrence or event (Gilbert and Terna,
2000), representing an important methodology to investigate asset
prices.
3. MODELING THE BEHAVIOR OF STOCK PRICES
In this artificial stock market, we simulate the behavior of N
traders or agents that have (i) an amount of cash, which yields the risk
free interest rate, and (ii) a specified number of shares of a single
stock, with risky returns. The equilibrium model for dividends is based
on Ehrentreich's (2007) description of the Santa Fe Artificial
Stock Market Model, in which the stock pays dividends that are generated
by a stochastic Ornstein-Uhlenbeck autoregressive mean reversion process
given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are,
respectively, dividends in periods t + 1 and t' [[mu].sub.d] is the
average dividend, [rho] is the mean reversion parameter and [member of]
represents the random shocks in dividends. These shocks are normally
distributed with zero mean and variance [[sigma].sup.2] (Ehrentreich,
2007).
Note that in the model, dividends follow a process of mean
reversion, suggesting therefore the existence of a negative correlation
between one-period lagged returns of the asset. It is important to
establish that when [rho] is zero, the dividend in the next period is
defined by the average dividend plus a random variable [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] with a normal distribution with
zero mean, and is therefore independent from the dividend of the
previous period. As [rho] increases in the range of positive numbers,
the dividend of the next period t + 1 is adjusted, taking into account
not only a stochastic component [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] but also the difference ([d.sub.t] -
[[mu].sub.g]) between the dividend of the previous period t and the
average dividend. In the model, all agents have the same expected
utility function that reflects a constant absolute risk aversion:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where [gamma] is the degree of risk aversion and [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] is the expected wealth by agent i
in the next period t + 1. Agents are therefore homogeneous with respect
to their risk aversion.
Considering equation 2, agents make decisions based only on
expectations of wealth for the subsequent period. According to
Ehrenthreich (2007), given that [x.sub.i,t] is the number of shares that
the agent i holds in period t, the budget constraint is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
In this context, the optimum quantity of shares [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] that each agent must have in the
portfolio, assuming normal distribution of returns, is given by (Le
Baron, Arthur and Palmer, 1999):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the
expectation of agent i about the sum of the stock price and the value of
the dividend in period [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] is the variance at time t of price changes, including gains or
losses associated with dividends, and [gamma] is a measure of risk
aversion, equal to all agents.
In this study, for simplicity of analysis, it is established that
the only source of randomness of the model involves dividends and thus
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where
[p.sup.*.sub.t] represents the realization of the asset price at time t.
It should be noted that, in the model described in equation 4, the
higher the risk aversion, the lower the quantity demanded, since the
agents would be less willing to acquire a risky asset. Similarly, the
higher the volatility of the asset, the lower the quantity demanded. In
contrast, when the expectation of an individual on the price plus the
dividend is high, the quantity demanded will increase. Eventually,
depending on the model parameters and expectations about future prices
and dividends, individuals should seek combination of risk free and
risky assets that maximize their expected utility.
Therefore, although individuals have the same utility function, the
portfolio depends on particular expectations about the stock price and
its dividend in the next period. If [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] then the individual will have incentives to
change the composition of its portfolio. If [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII], the agent will demand more stocks. If
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the agent will seek
to sell some of the shares. Thus, individual expectations of each agent
influence the supply and demand for assets and therefore the stock
price. The equilibrium price is obtained through an iterative process in
which (i) the forces of supply and demand change the stock price
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and (ii) the price
changes alleviate the imbalance between supply and demand, changing the
value of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In the model developed in this research, different expectations
about the position [[??].sub.i,t] in the risky asset at time t + 1
depend on a behavioral characteristic of the individual: the agents are
divided into groups of optimistic, pessimistic or neutral investors. So
while the model of the Santa Fe Institute depends on a mechanism based
on a genetic algorithm to set the expectations of each agent on future
dividends, in this study the heterogeneity of expectations is the result
of the existence of agents that are optimists and pessimists.
Agents establish optimistic expectations by giving a positive bias
on the behavior of future dividends of the asset, while agents that have
pessimistic expectations are negatively biased. Neutral agents set
expectations without bias. To address the optimistic and pessimistic
agents, the expected value of the total price and the dividend in next
period is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
represents an increase or decrease in the dividend estimated by agent;
for period t + 1, with uniform distribution between 0 and 1, [beta] is
an adjustment factor to maintain the forecasted dividend within an
appropriate range, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] if the agent is neutral, [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] if the agent is optimist and [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] if the agent is pessimist.
Therefore, if the agent is optimist, there is a positive correction in
the expected dividends of the next period and if the agent is pessimist,
there is a negative adjustment and the projected dividend is decreasing.
4. ANALYSIS OF RESULTS
Initially, we arbitrarily set the base values for model parameters,
as denoted in Table 1. From these initial values are carried out
comparative statics analysis to identify the sensitivity of model
results in relation to the main parameters.
Results are less sensitive to changes in the initial wealth of each
individual [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the
initial price of the asset [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] and risk free interest rate [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII]. Importantly, although the results are not
significantly sensitive to the risk free interest rate, calibration of
this parameter is critical. The risk-free interest rate should be set at
a value lower than the expected profitability of the risky asset,
reflecting the fact that the higher the risk, the higher the expected
return. Other parameters such as, for example, the initial cost of the
stock, the initial wealth of agents, the average dividend, were also
calibrated to be compatible with the other model parameters. Since each
simulation leads to different results, the following tables summarize
average values often simulations. To simplify notation, F denotes
results for a fair price or intrinsic value and M, results for market
prices or equilibrium value due to a balance between supply and demand.
Thus, the fair value is based solely on realizations of the dividends
and the market value represents equilibrium prices. The first analysis
shows that the parameter of risk aversion, common to all agents in the
model, does not influence the characteristics associated with the
returns, considering separately fair prices (F) and market prices (M).
However, the dynamics of the proposed artificial capital market implies
interesting results. Regardless of the degree of risk aversion of
agents, the average return of assets calculated from the fair prices,
i.e., from the prices defined by the stochastic Ornstein-Uhlenbeck
autoregressive mean reversion process that generates dividends, is
positive and substantially greater than the average return of the asset
considering the equilibrium price, in which agents with different
profiles interact on the artificial stock market.
Considering equation 1, it is expected that, on average, the fair
value of assets shows an increase in price over time, since
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The positive
returns in Table 2 for the fair price (F) confirm the model dynamics.
However, when agents with different expectations about dividends
interact in the market, average returns of market prices become
negative. This result differs substantially from expected values if the
price was set only by dividends generated by equation 1.
In addition to reducing the average returns of the asset, the
presence of agents in the market decreases the level of volatility.
Although volatility is high relative to average returns, mainly due to
the calibration of the initial parameters, the results indicate that the
interaction among agents can lead to an under-reaction reflected in
lower variability of returns. Table 2 also shows that the
auto-correlation between lagged returns in one period also suffers
considerable influence of the presence of heterogeneous agents.
Although in this simulation the percentage of pessimistic and
optimistic agents are equal and therefore, not favoring any market
trend, the interaction among agents induces market returns with small
but positive auto-correlation. This result contrasts with the negative
auto-correlation that would be expected when considering only the fair
price. In fact, the characteristic of mean reversion of the dividend, as
reflected in equation 1, would imply a negative auto-correlation.
Importantly, the model also reinforces the under-reaction of the market,
given the reduction in volatility and the positive auto-correlation
between returns of subsequent periods. One can also observe that the
auto-correlation of a period is not significantly influenced by the
parameter of risk aversion.
By comparing the market value and the fair value, similar results
on the average return, on the autocorrelation and on the standard
deviation of returns are obtained when the coefficient 3 varies, as
outlined in Table 3. The parameter [beta] is representative of the
adjustment factor of optimistic and pessimistic agents established in
equation 5. The higher the [beta], the more optimistic or pessimistic
agents are, adding or subtracting a larger value in their estimates of
dividends.
Results regarding the composition of optimistic and pessimistic
agents in the market are summarized in Table 4 and also support the
reduction of the standard deviation of returns of the market price and
the exchange of signal in the auto-correlation of returns on lagged
periods. Average returns are substantially higher when one considers
fair prices in contrast to market prices that are obtained by balancing
supply and demand for stocks.
It should be noted, however, that the isolated analysis of the fair
market data does not reveal that the variation in the parameter [beta]
and the change in the composition of optimists and pessimists in the
market impact the characteristics of simple returns. For example, the
apparent lack of a positive relationship between [beta] and average
return, or between the percentage of optimists and the standard
deviation in Tables 3 and 4, suggests the need for further analysis.
In particular, future studies should investigate possible nonlinear
relationships, as intuition would suggest that the higher the percentage
of optimistic agents in the market, the more likely the average returns
to be positive. However, the dynamics of dividends, due to a mean
reversion feature, may, for example, imply estimates for subsequent
periods have characteristics that reduce the average return.
The smoothing of the variability of returns can also be seen in the
analysis of the influence of the parameter of mean reversion of
dividends, as shown in Table 5. The results suggest that the higher the
value of [rho], the greater the observed auto-correlation in the returns
of the asset based on market prices. Thus, the greater influence of the
difference between the dividend in the previous moment and the average
dividend in the next period are reflected in a higher auto-correlation.
Note that this trend of increasing correlation is observed also in fair
prices that take into account only the behavior of dividends, without
interaction among agents.
Finally, the model also allows the analysis of the impact of shocks
of dividends [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on
prices. Following Ehrenthreich (2007), the shocks to dividends have a
normal distribution with zero mean and variance [[sigma].sup.2]. The
higher the [sigma], the greater the degree of dispersion of the
dividends. Table 6 illustrates the results when [sigma] varies. As might
be expected, the standard deviation of the asset returns, for the fair
price and the market price, rises as the parameter related to
variability of dividends increases. Not surprisingly, the more volatile
the dividend, the more volatile the price of the stock. This increase in
the volatility of returns is accompanied by an increase in the returns
themselves. However, at least for a lag of just one period, it is not
possible to explore the auto-correlation, which is different from zero,
to obtain extraordinary gains.
5. FINAL COMMENTS
In this study, we investigated how different expectations about
future prices affect market dynamics, taking into account a mean
reverting process for the diffusion of dividends. In the simulation
model, agents have access to different information. The information
asymmetry is based on the fact that the projected probability
distributions for dividends differ from the intrinsic or real
probability distribution of dividends. Moreover, agents are
heterogeneous and are classified into different groups of individuals:
neutral, optimistic and pessimistic, as their average expectations about
future dividends may differ.
Besides information asymmetry and heterogeneity of agents, the
model incorporates the assumption that dividends follow an
Ornstein-Uhlenbeck autoregressive mean reversion. Thus, the dividend in
a given period is, by construction, dependent on the dividend of the
previous period. Dividends projected by agents also follow similar
process, but with a setting that reflects their optimism or pessimism.
The presence of heterogeneous agents in the market brings
interesting results, especially with regards to auto-correlation between
subsequent returns. Through the process of generating dividends, based
on mean reversion, the fair price of the asset should theoretically lead
to returns with negative one-period auto-correlation. However, the
equilibrium prices when agents interact in the market, make
autocorrelations of returns positive, although low.
These results suggest an under-reaction that eventually could be
exploited by agents. However, we should emphasize that the model does
not include momentum agents, who could influence the behavior of market
prices by trading stocks based solely on past returns. In this case, the
presence of an autocorrelation different from zero does not guarantee
the possibility of strategies that lead to extraordinary gains, since
the very implementation of momentum strategies can distort market
prices.
BIBLIOGRAPHY:
Axelrod, R. The evolution of cooperation: revised edition. Basic
Books, 2006.
Buchanan, M. "Meltdown modeling. Could agent-based computer
models prevent another financial crisis?" Nature, 460(6): 680-682,
2009.
Ehrentreich, N. Agent-based modeling. The Santa Fe Institute
Artificial Stock Market Model Revisited. Springer, 2007.
Gilbert, N. Agent-based Models. Quantitative Applications in the
Social Sciences. Sage, 2007.
Gilbert N.; Terna, P. "How to build and use agent-based models
in social science". Mind & Society, 1(1): 57-72, 2000.
Le Baron, B.; Arthur, W. B.; Palmer, R. "Time series
properties of an artificial stock market". Journal of Economic
Dynamics and Control, 23(9-10): 1487-1516, 1999.
Richiardi, M. "Agent-based computational economics: a short
introduction". Forthcoming The Knowledge Engineering Review,
Special Edition, 27(1), 2012. Available at
www.agsm.edu.au/bobm/KER/FinalDrafts/ker-matteo-final.pdf. Access in
05/10/2011.
Tesfatsion, L. Agent-based computational economics: a constructive
approach to economic theory. In Tesfatsion, L.; Judd, K. L. (Eds.).
Handbook of Computational Economics, vol 2, Agent-based computational
economics: 831-880, 2006.
Herbert Kimura, Universidade Presbiteriana Mackenzie, Sao Paulo,
Brazil
Fabiano Guasti Lima, Universidade de Sao Paulo, Ribeirao Preto,
Brazil
Luiz Carlos Jacob Perera, Universidade Presbiteriana Mackenzie, Sao
Paulo, Brazil
Roberto Borges Kerr, Universidade Presbiteriana Mackenzie, Sao
Paulo, Brazil
Dr. Herbert Kimura earned his Ph.D. at University of Sao Paulo,
Brazil, in 2002. Currently he is a professor of Finance of the
Pos-Graduation Program in Administration at Mackenzie Presbyterian
University in Sao Paulo, Brazil.
Dr. Fabiano Guasti Lima earned his Ph.D. at University of Sao
Paulo, Brazil, in 2004. Currently he is a professor of Risk Models in
Finance and Accountability in the Program in Controllership and
Accounting at Universidade de Sao Paulo, Ribeirao Preto, Brazil.
Dr. Luiz Carlos Jacob Perera earned his Ph.D. at Universidade de
Sao Paulo, Brazil in 1998 and did his postdoctoral work at Universite
Pierre Mendes France, in Grenoble, France. Currently he is a professor
of the Pos-Graduation Program in Accountability Science at Universidade
Presbiteriana Mackenzie in Sao Paulo, Brazil.
Dr. Roberto B. Kerr earned his Ph.D. at Mackenzie Presbyterian
University, Sao Paulo, Brazil, in 2008. Currently he is a professor of
Financial Markets and Corporate Finance at the same University.
TABLE 1. INITIAL PARAMETERS OF THE MODEL
Initial parameters of agents
Number of agents N 100
Initial number of shares by each agent X 1
Initial wealth of each agent [W.sub.O] 20000
Initial parameters of stock market
Initial price of stock [P.sub.O] 10
Risk free interest rate R 0.01
Parameters of dividend dynamics
Average dividend [[mu].sub.d] 0.5
Mean reversion parameter [rho] 0.3
Standadr deviation of shocks [sigma] 0.5
Characteristics of agents
Risk aversion parameter [lambda] 0.4
Percentage of optimists [p.sub.o] 0.2
Percentage of pessimists [p.sub.p] 0.2
Dividend adjustment parameter [beta] 0.1
TABLE 2. RESULTS FOR DIFFERENT VALUES
OF THE RISK AVERSION PARAMETER [lambda]
[lambda] Average return Standard Auto-
deviation correlation
F M F M F M
0.1 0.47% 0.11% 14.58% 11.43% -0.45 0.26
0.2 0.38% -0.10% 14.83% 10.99% -0.47 0.24
0.3 0.30% -0.03% 13.84% 10.99% -0.44 0.28
0.4 0.20% -0.18% 13.62% 10.01% -0.44 0.11
0.5 0.22% -0.22% 13.88% 10.04% -0.48 0.20
0.6 0.61% -0.05% 16.57% 11.35% -0.46 0.25
0.7 0.32% -0.22% 14.28% 10.38% -0.46 0.23
0.8 0.38% -0.11% 13.71% 9.71% -0.49 0.20
0.9 0.53% -0.09% 15.60% 10.60% -0.51 0.18
1.0 0.65% -0.01% 16.83% 12.03% -0.46 0.26
1.1 0.58% -0.03% 16.20% 12.11% -0.45 0.25
1.2 0.39% -0.10% 15.96% 11.58% -0.53 0.16
TABLE 3. RESULTS FOR DIFFERENT VALUES OF THE ADJUSTMENT
IN DIVIDENDS PROVIDED BY OPTIMISTS AND PESSIMISTS
[beta] Average return Standard Auto-
deviation correlation
F M F M F M
0.1 0.42% 0.00% 13.19% 10.43% -0.44 0.26
0.2 0.42% -0.05% 13.52% 9.82% -0.47 0.20
0.3 0.56% 0.17% 15.12% 12.27% -0.45 0.24
0.4 0.57% 0.08% 14.71% 10.78% -0.52 0.21
TABLE 4. RESULTS FOR DIFFERENT COMPOSITION OF OPTIMISTS
AND PESSIMISTS IN THE MARKET
O P Average return Standard Auto-
deviation correlation
F M F M F M
0.0 0.2 0.26% -0.12% 12.57% 9.37% -0.44 0.19
0.1 0.2 -0.60% 0.62% 25.36% 18.48% -0.39 0.23
0.2 0.2 0.31% -0.17% 13.30% 9.47% -0.53 0.18
0.3 0.2 0.60% 0.21% 15.61% 13.29% -0.47 0.18
0.4 0.2 0.26% -0.12% 12.57% 9.37% -0.44 0.19
0.2 0.0 0.26% -0.11% 11.74% 8.41% -0.46 0.25
0.2 0.1 1.08% 0.36% 18.00% 13.34% -0.46 0.18
0.2 0.3 0.33% -0.20% 14.48% 10.36% -0.52 0.21
0.2 0.4 0.48% -0.04% 14.88% 11.07% -0.47 0.19
TABLE 5. RESULTS FOR DIFFERENT VALUES
OF DIVIDEND MEAN REVERSION [rho]
[rho] Average return Standard Auto-
deviation correlation
F M F M F M
0.1 0.67% -0.11% 15.68% 9.64% -0.61 0.03
0.2 0.36% -0.15% 13.96% 9.43% -0.56 0.11
0.3 1.84% -0.27% 31.90% 17.00% -0.42 0.24
0.4 1.81% 0.06% 24.46% 15.92% -0.36 0.30
0.5 0.49% 0.15% 15.68% 12.94% -0.29 0.33
TABLE 6. RESULTS FOR DIFFERENT VALUES OF VOLATILITY OF DIVIDEND SHOCKS
[sigma] Average return Standard Auto-
deviation correlation
F M F M F M
0.1 -0.52% -0.56% 2.70% 2.27% -0.22 0.35
0.2 -0.40% -0.47% 4.93% 3.75% -0.44 0.27
0.3 -0.26% -0.41% 7.96% 6.24% -0.47 0.26
0.4 0.05% -0.27% 11.34% 8.34% -0.52 0.21
0.5 0.29% -0.22% 13.66% 10.23% -0.49 0.23