Abstract
Background
For complex financial systems, the negative and positive return-volatility correlations, i.e., the so-called leverage and anti-leverage effects, are particularly important for the understanding of the price dynamics. However, the microscopic origination of the leverage and anti-leverage effects is still not understood, and how to produce these effects in agent-based modeling remains open. On the other hand, in constructing microscopic models, it is a promising conception to determine model parameters from empirical data rather than from statistical fitting of the results.
Methods
To study the microscopic origination of the return-volatility correlation in financial systems, we take into account the individual and collective behaviors of investors in real markets, and construct an agent-based model. The agents are linked with each other and trade in groups, and particularly, two novel microscopic mechanisms, i.e., investors’ asymmetric trading and herding in bull and bear markets, are introduced. Further, we propose effective methods to determine the key parameters in our model from historical market data.
Results
With the model parameters determined for six representative stock-market indices in the world, respectively, we obtain the corresponding leverage or anti-leverage effect from the simulation, and the effect is in agreement with the empirical one on amplitude and duration. At the same time, our model produces other features of the real markets, such as the fat-tail distribution of returns and the long-term correlation of volatilities.
Conclusions
We reveal that for the leverage and anti-leverage effects, both the investors’ asymmetric trading and herding are essential generation mechanisms. Among the six markets, however, the investors’ trading is approximately symmetric for the five markets which exhibit the leverage effect, thus contributing very little. These two microscopic mechanisms and the methods for the determination of the key parameters can be applied to other complex systems with similar asymmetries.
Introduction
In recent years, the understanding of complex systems has been undergoing rapid development. Financial markets are important examples of complex systems with many-body interactions. The possibility of accessing large amounts of historical financial data has spurred the interest of scientists in various fields, including physics. Plenty of results have been obtained with physical concepts, methods and models [1]–[13].
There are several stylized facts in financial markets. Besides the fat tail in the probability distribution of price returns, it is well-known that the volatilities are long-range correlated in time, which is the so-called volatility clustering [14]. However, our knowledge on the dynamics of the price itself is still limited. Since the auto-correlation of returns is extremely weak [2], [3], nonzero higher-order time correlations become important, especially the lowest-order one among them. In financial markets, this lowest-order nonzero correlation turns out to be the return-volatility correlation, on which we lay emphasis in this paper. In 1976, a negative return-volatility correlation is first discovered by Black [15]. This is the so-called leverage effect, which implies that past negative returns increase future volatilities. The leverage effect is actually observed in various financial systems, such as stock markets, futures markets, bank interest rates and foreign exchange rates [6], [15]–[21]. We have studied about thirty stock-market indices, and all of them exhibit the leverage effect. To the best of our knowledge, the leverage effect exists in almost all stock markets in the world. In Chinese stock markets, however, a positive return-volatility correlation is detected, which is called the anti-leverage effect [6], [19]. This effect is also observed in other economic systems, such as bank interest rates of early years and spot markets of non-ferrous metals.
The leverage and anti-leverage effects are crucial for the understanding of the price dynamics [6], [15], [19], [20], and important for risk management and optimal portfolio choice [22], [23]. However, the origination of the return-volatility correlation is still disputed, even at the macroscopic level [19], [20], [24]–[29]. According to Black, the leverage effect arises because a price drop increases the risk of a company to go bankrupt and leads the stock to fluctuate more. So far, various macroscopic models have been proposed to understand the return-volatility correlation [4], [19], [30]–[33]. The retarded volatility model is an enlightening one, which can produce both the leverage and anti-leverage effects [4]. However, it is a model with only one degree of freedom, and both the initial time series of returns and the function of the feedback return-volatility interaction, are actually input. Hence, the model is phenomenological in essence, and the generation mechanism of the leverage and anti-leverage effects is macroscopic. In very recent years, many researches have been devoted to the return-volatility correlation, but how to produce the return-volatility correlation with a microscopic model remains open.
Agent-based modeling is a powerful simulation technique, which is widely applied in various fields [34]–[41]. More recently, an agent-based model is proposed for reproducing the cumulative distribution of empirical returns and trades in stock markets [41]. It is a outstanding model with key parameters determined from empirical findings rather than from being set artificially. In this paper, we construct an agent-based model with asymmetric trading and herding to explore the microscopic origination of the leverage and anti-leverage effects. In the past decades, although the asymmetric trading and herding behaviors may have been touched macroscopically, they have not been taken into account in the microscopic modeling yet. Especially, we propose effective methods to determine the key parameters in our model from historical market data.
Methods
To study the microscopic origination of the return-volatility correlation in stock markets, we take into account the individual and collective behaviors of investors, and construct a microscopic model with multi-agent interactions. Further, we determine the key parameters in our model from historical market data rather than from statistical fitting of the results.
Our model is basically built on agents’ daily trading, i.e., buying, selling and holding stocks. Empirical studies indicate that investors make decisions according to the previous stock performance of different time windows [42], which suggests that their horizons of investment vary. This investment horizon is introduced to our model for a better description of agents’ market behavior. Most crucially, two important behaviors of investors are taken into account for understanding the return-volatility correlation.
1. Two Important Behaviors of Investors
Investors’ asymmetric trading in bull and bear markets. There are various definitions of bull and bear markets [43], [44]. The usual definition is that in stock markets, bull and bear markets correspond to the periods of generally increasing and decreasing stock prices respectively [43]. In this paper we adopt this definition, and simply define a market to be bullish on one day if the price return is positive, and bearish if the price return is negative. The asymmetric trading in bull and bear markets is an individual behavior, which is induced by investors’ different trading desire when the price drops and rises. To be more specific, an investor’s willingness to trade is affected by the previous price returns, leading the trading probability to be distinct in bull and bear markets.
Investors’ asymmetric herding in bull and bear markets. Herding, as one of the collective behaviors, is that investors cluster in groups when making decisions, and these groups can be large in financial markets [45]–[51]. Actually, the herding behavior in bull markets is not the same as that in bear ones [47], [52], [53]. For instance, previous study has shown that in the recent US market, the herding behavior in bear markets appears much more significant than that in bull ones [47]. Generally, investors may cluster more intensively in either bull or bear markets, leading the herding to be asymmetric.
2. Microscopic Model with Multi-agent Interactions
The stock price on day 
is denoted as 
, and the logarithmic price return is 
. In stock markets, the information for investors is highly incomplete, therefore an agent’s decision of buy, sell or hold is assumed to be random. Since intraday trading is not persistent in empirical trading data [54], we consider that only one trading decision is made by each agent in a single day. In our model, there are 
agents, and each operates one share every day. On day 
, each agent 
makes a trading decision 
,
 |
(1) |
and the probabilities of buy, sell and hold decisions are denoted as 
, 
and 
, respectively. The price return 
in our model is defined by the difference of the demand and supply of the stock, i.e., the difference between the number of buy agents and sell ones,
 |
(2) |
The volatility is defined as the absolute return 
.
The investment horizon is introduced since agents’ decision makings are based on the previous stock performance of different time horizons. It has been found that the relative portion 
of agents with 
days investment horizon follows a power-law decay, 
with 
[18]. The maximum investment horizon is denoted as 
, thus 
. With the condition of 
, we normalize 
to be 
. Agents’ trading decisions are made according to the previous price returns. For an agent having investment horizon of 
days, 
represents a simplified investment basis for decision making on day 
. We introduce a weighted average return 
to describe the integrated investment basis of all agents. Taking into account that 
is the weight of 
, 
is defined as
 |
(3) |
where 
is a proportional coefficient. We set 
, such that 
to ensure that the fluctuation scale of 
remains consistent with the one of 
(see Appendix S1). If 
, 
is just identical to 
. Actually, 
varies from market to market, and from time period to time period for a market. According to Ref. [42], the investment horizons of investors range from a few days to several months. We estimate the maximum investment horizon 
to be 150 in our model. For 
between 50 and 500, the simulated results remain qualitatively robust.
(i) Asymmetric trading
In Ref. [41], investors’ probabilities of buy and sell are assumed to be equal, i.e., 
, and 
is a constant. In our model, we adopt the value of 
estimated in Ref. [41], 
. We assume 
as well, but now 
and 
evolve with time since the agents’ trading is asymmetric in bull and bear markets. As the trading probability 
, we set its average over time 
. From the investors’ behavior (a) described in Subsec. 1 in Sect. Methods, we define the market performance of the previous 
days to be bullish if 
, and bearish if 
. The investors’ asymmetric trading in bull and bear markets gives rise to the distinction between 
and 
. Thus, 
should take the form
 |
(4) |
Here 
and 
are constants, and 
requires 
, i.e., 
and 
are not independent.
(ii) Asymmetric herding
The herding behavior implies that investors can be divided into groups. Here a herding degree 
is introduced to quantify the clustering degree of the herding behavior,
 |
(5) |
where 
is the average number of agents in each group on day 
. Herding should be related to previous volatilities [46], [55], and we set 
. Hence the herding degree on day 
is
 |
(6) |
This herding degree is symmetric for 
and 
. According to the investors’ behavior (b) described in Subsec. 1 in Sect. Methods, however, investors’ herding behaviors in bull and bear markets are asymmetric, i.e., herding is stronger in either bull markets or bear ones. More specifically, 
is not symmetric for 
and 
, and should be redefined to be
 |
(7) |
Here 
is the degree of asymmetry, and as 
grows, herding becomes more asymmetric. According to Eq. (5), 
. Therefore 
is the average number of agents in a same group. Thus we randomly divide 
agents into 
groups on day 
. Everyday, the agents in a group make a same trading decision (buy, sell or hold) with the same probability (
, 
or 
).
3. Determination of 
and 
This is the key step in the construction of our model. We emphasize that 
and 
are determined from the historical market data rather than from statistical fitting of the simulated results. Six representative stock-market indices are studied with our model, including the S&P 500, Shanghai, Nikkei 225, FTSE 100, Hangseng and DAX indices. We collect the daily data of closing price and trading volume, both of which are from 1950 to 2012 with 15775 data points for the S&P 500 Index, from 1991 to 2006 with 3928 data points for the Shanghai Index, from 2003 to 2012 with 2367 data points for the Nikkei 225 Index, from 2004 to 2012 with 1801 data points for the FTSE 100 Index, from 2001 to 2012 with 2787 data points for the Hangseng Index and from 2008 to 2012 with 1016 data points for the DAX Index. These data are obtained from “Yahoo! Finance” (http://finance.yahoo.com). For comparison of different time series of returns, the normalized return 
is introduced,
 |
(8) |
where 
represents the average over time 
, and 
is the standard deviation of 
.
The stock market is assumed to be bullish if 
, and bearish if 
. To determine 
, we first define an average trading volume 
for the bull markets, and 
for the bear ones,
 |
(9) |
Here 
and 
represent the number of positive and negative returns respectively, and 
is the trading volume on day 
. As displayed in Table 1, the ratio 
is 1.03 for the S&P 500 Index and 1.21 for the Shanghai Index. In our model, since the average trading volumes for bull markets (
) and bear markets (
) are 
and 
, the ratio of these two average trading volumes is
 |
(10) |
Table 1. The values of 
, 
, 
, 
, 
and 
for the six indices.
| Index |
|
|
|
|
|
p-value |
|
| S&P 500 (1950–2012) | 1.03 | 0.993 | 1.127 | 1.01±0.01 | 0.067±0.007 | 6.7×10−4 | 3 |
| Shanghai (1991–2006) | 1.21 | 0.533 | 0.447 | 1.09±0.01 | −0.043±0.005 | 1.0×10−3 | −2 |
| Nikkei 225 (2003–2012) | 1.01 | 0.729 | 0.807 | 1.01±0.01 | 0.039±0.005 | 1.5×10−3 | 2 |
| FTSE 100 (2004–2012) | 0.98 | 0.673 | 0.729 | 0.99±0.01 | 0.028±0.003 | 7.3×10−4 | 2 |
| Hangseng (2001–2012) | 1.04 | 0.966 | 1.029 | 1.02±0.2 | 0.032±0.003 | 4.4×10−4 | 2 |
| DAX (2008–2012) | 0.96 | 0.797 | 0.822 | 0.98±0.02 | 0.013±0.002 | 2.9×10−3 | 1 |
, 
and 
are determined from the historical data for each index. We calculate 
from 
and 
, and 
from 
. Student’s t-test is performed to analyze the statistical significance of 
. A p-value less than 0.05 is considered statistically significant. We compute 
from the linear relation between 
and 
for all these indices. As 
for the Shanghai Index is negative, it is rounded down to the nearest integer, while 
for other indices are positive, and each of them is rounded up to the nearest integer.
Together with the condition 
, we determine 
from 
for the S&P 500 Index and 
for the Shanghai Index. Table 1 also shows the values of 
and 
for the Nikkei 225, FTSE 100, Hangseng and DAX indices. Several data series of different time periods are sampled from the historical market data, and the error is given for 
in this table. Student’s t-test is performed to analyze the statistical significance for 
deviating from 
, and a p-value less than 0.05 is considered statistically significant. The analysis shows that only the value 
of the Shanghai Index is significantly deviating from 1.0, with the 
. In our simulation, for simplicity, we approximate 
to be 1.0 for the S&P 500, Nikkei 225, FTSE 100, Hangseng and DAX indices, and 1.1 for the Shanghai Index.
Now we turn to 
. In real markets, herding is related to volatilities [46], [55]. Thus we introduce the average 
with the weight 
to describe the herding degree in a specific period. Thus the herding degrees of bull markets (
) and bear markets (
) are defined as
 |
(11) |
From empirical findings, the herding degrees of bull and bear stock markets are not equal, i.e., 
. In order to equalize 
and 
, we introduce a shifting to 
, denoted by 
, such that 
with 
. From this definition of 
, we derive (see Appendix S2)
 |
(12) |
Thus we obtain 
for the S&P 500 Index and 
for the Shanghai Index. In our model, we similarly compute the shifting to the time series 
, which equalize the herding degree 
in bull markets (
) and bear markets (
). Actually, one may prove that the shifting to 
is equivalent to the shifting to 
(see Appendix S3). If 
is replaced by 
, 
turns into 
, which is symmetric for bull and bear markets. Therefore, 
is the shifting to 
, and it is just the shifting to 
.
The time series of returns in different real markets and in our model fluctuate at different levels. For comparison, we normalize the returns with Eq. (8). Similarly, 
, the shifting to returns, should also be normalized to 
. However, in simulating the stock markets with our model, the parameter we need is 
. Therefore, we should first derive the relation of 
and 
. With the normalization of the time series 
, 
should be normalized to 
,
 |
(13) |
where 
represents the average over time 
, and 
is the standard deviation of 
. To determine the relation of 
and 
, 
is set to be −4, −3, −2, −1, 0, 1, 2, 3, 4, respectively, and 
is set to be 1.0 to produce time series 
. With 
simulated 100 times for each 
, we compute 
and average the results. As displayed in Fig. 1, the relation of 
and 
is linear, and 
. For 
between 0.9 and 1.1, the results remain robust. Thus, we determine 
for the simulation of the S&P 500 Index and 
for the simulation of the Shanghai Index. Table 1 shows the values of 
and 
, as well as the error of 
, for the Nikkei 225, FTSE 100, Hangseng and DAX indices. Due to the fluctuation of the empirical data, the error of 
is about 10 percent. Since the sign of 
determines that the simulation yields the leverage or anti-leverage effect, we perform Student’s t-test to analyze the statistical significance of 
, and the corresponding p-value is listed in Table 1. A p-value less than 0.05 is considered statistically significant.
Figure 1. The relation of 
and 
.
With 
set to be −4, −3, −2, −1, 0, 1, 2, 3 and 4 respectively, time series 
is simulated 100 times for 
. The corresponding 
is computed and averaged for each 
. This plot shows a linear relation of 
and 
, i.e., 
, and this result remains robust for 
between 0.9 and 1.1.
To further validate the methods for the determination of the key parameters and the simulations for the leverage and anti-leverage effects, eight more indices are studied (see Appendix S4). The simulation of each index correctly produces the leverage or anti-leverage effect.
4. Simulation
The number of agents in our simulations is 10000, i.e., 
. With 
and 
determined for each index, our model produces the time series of returns 
in the following procedure. Initially, the returns of the first 150 time steps are set to be 0. On day 
, we calculate 
according to Eq. (3), then 
and 
according to Eq. (4) and Eq. (7), respectively. Next, we randomly divide all agents into 
groups. The agents in a group make a same trading decision (buy, sell or hold) with the same probability (
, 
or 
). After all agents have made their decisions, we calculate the return 
with Eq. (1) and Eq. (2). Repeating this procedure, we obtain the return time series 
. 
data points of 
are produced in each simulation, but the first 10000 data points are abandoned for equilibration.
Results
To describe how past returns affect future volatilities, the return-volatility correlation function 
is defined,
 |
(14) |
with 
and 
[8]. Here 
represents the average over time 
.
As displayed in Fig. 2, 
calculated with the empirical data of the S&P 500 Index shows negative values up to at least 15 days, and this is the well-known leverage effect [4], [6], [15]. On the other hand, 
for the Shanghai Index remains positive for about 10 days. That is the so-called anti-leverage effect [6], [19]. Fitting 
to an exponential form 
, we obtains 
and 8 days for the leverage and anti-leverage effects, respectively. Compared with the short correlating time of the returns, the order of minutes [2], [3], both the leverage and anti-leverage effects are prominent. As the lowest-order nonzero correlations of returns, the leverage and anti-leverage effects are theoretically crucial for the understanding of the price dynamics [6], [15], [19], [20]. In practical application, these effects are important for risk management and optimal portfolio choice [22], [23]. After the time series 
produced in our model is normalized to 
, we compute the return-volatility correlation function, and the result is in agreement with that calculated from empirical data on amplitude and duration for both the S&P 500 and Shanghai indices, as shown in Fig. 2. This is the first time that the leverage and anti-leverage effects are produced with a microscopic model.
Figure 2. The return-volatility correlation functions for the S&P 500 and Shanghai indices, and for the corresponding simulations.
The S&P 500 and Shanghai indices are simulated with 
and 
, respectively. Dashed lines show an exponential fit 
with 
and 
for the S&P 500 Index and the Shanghai Index.
For the Nikkei, FTSE 100, Hangseng and DAX indices, the volume data of early years are not available to us. However, 
is computed from only price data. In order to reduce the fluctuation of 
, we collect the price data of a longer period, which are from 1984 to 2012 with 7092 data points for the Nikkei 225 Index, from 1984 to 2012 with 7227 data points for the FTSE 100 Index, from 1988 to 2012 with 6181 data points for the Hangseng Index and from 1990 to 2012 with 5514 data points for the DAX Index. As displayed in Fig. 3, 
for the simulations is in agreement with that for the corresponding indices. Table 2 shows the values of 
and 
of the exponential fit 
for the six indices and the corresponding simulations. Since 
is obviously non-zero, the p-value of Student’s t-test is only listed for 
.
Figure 3. The return-volatility correlation functions for the four indices and the corresponding simulations.
The Nikkei 225, FTSE 100, Hangseng and DAX indices are simulated with 
, 
, 
and 
, respectively. Dashed lines show an exponential fit 
with 
for the Nikkei 225 Index, 
for the FTSE 100 Index, 
for the Hangseng Index and 
for the DAX Index.
Table 2. The values of 
and 
of the exponential fit 
for the six indices and the corresponding simulations.
|
|
p-value | |
| S&P 500 | −0.36±0.02 | −0.053−0.005 | 4.5×10−4 |
| simulation | −0.30±0.01 | −0.032±0.001 | 5.7×10−6 |
| Shanghai | 0.61±0.12 | −0.133±0.014 | 6.9×10−4 |
| simulation | 0.30±0.02 | −0.066±0.004 | 7.9×10−5 |
| Nikkei 225 | −0.25±0.01 | −0.038±0.004 | 6.9×10−4 |
| simulation | −0.27±0.01 | −0.042±0.001 | 1.9×10−6 |
| FTSE 100 | −0.33±0.03 | −0.055±0.007 | 1.4×10−3 |
| simulation | −0.26±0.01 | −0.36±0.001 | 3.6×10−4 |
| Hangseng | −0.50±0.06 | −0.098±0.001 | 1.2×10−3 |
| simulation | −0.22±0.01 | −0.027±0.001 | 1.1×10−5 |
| DAX | −0.20±0.01 | −0.026±0.002 | 2.0×10−4 |
| Simulation | −0.22±0.01 | −0.31±0.001 | 6.5×10−4 |
Student’s t-test is performed to analyze the statistical significance of 
. A p-value less than 0.05 is considered statistically significant.
Our model also produces other features of the real markets, such as the long-term correlation of volatilities and the fat-tail distribution of the returns. Here we take the S&P 500 and Shanghai indices as examples. The auto-correlation function of volatilities is defined as
 |
(15) |
where 
[19], and 
represents the average over time 
. As shown in Fig. 4, 
for the simulations is consistent with that for the empirical data. The cumulative distributions 
of absolute returns are shown in Fig. 5, where the fat tail in the distribution of empirical returns can be observed in that of the simulated returns as well.
Figure 4. The auto-correlation functions of volatilities for the S&P 500 and Shanghai indices, and for the corresponding simulations.
For clarity, the curves for the S&P 500 Index have been shifted down by a factor of 10.
Figure 5. The cumulative distributions of absolute returns for the S&P 500 and Shanghai indices, and for the corresponding simulations.
For clarity, the curves for the S&P 500 Index have been shifted left by a factor of 8.5.
By the definitions, both 
and 
are not dependent on the number of agents (denoted by 
) in the model. However, the slope of the linear relation between 
and 
increases with 
. Therefore, the magnitude of 
becomes larger as 
grows. For the simulation results, the amplitude of 
increases with 
, but gradually converges for larger 
(see Appendix S5). For 
and 
, the cases are similar.
Discussion
In our model, the crucial generation mechanisms of the return-volatility correlation are the agents’ asymmetric trading and herding behaviors in bull and bear markets. Now we discuss how these two mechanisms contribute to the leverage and anti-leverage effects, and which one is more significant. According to Eq. (4) and 
, 
is symmetric about 
if 
, and asymmetric if 
. On the other hand, 
in Eq. (7) is asymmetric about 
if 
. In our model, the S&P 500 and Shanghai indices are simulated with 
and 
, respectively. Therefore, 
is symmetric in the simulation of the S&P 500 Index, but asymmetric in the simulation of the Shanghai Index. 
is asymmetric in the simulation of both the S&P 500 and Shanghai indices. With other parts of the model remain unchanged, we consider the following controls: (a) 
is replaced by a symmetric one in the simulation of the Shanghai Index; (b) 
is replaced by a symmetric one in the simulation of both the S&P 500 and Shanghai indices; (c) both 
and 
are replaced by the symmetric ones in the simulation of the Shanghai Index.
The simulations are performed 100 times for average. We conclude that for the leverage and anti-leverage effects, both the investors’ asymmetric trading and herding are essential generation mechanisms. As displayed in Fig. 6, the anti-leverage effect is weakened significantly and the leverage effect disappears after we replace the asymmetric 
with the symmetric one. On the other hand, the anti-leverage effect recedes after the asymmetric 
is replaced by the symmetric one. It is worth mentioning that for the five stock markets exhibiting the leverage effect, the S&P 500, Nikkei 225, FTSE 100, Hangseng and DAX, 
is approximately symmetric, thus contributing very little to the leverage effect. The investors’ asymmetric trading in the Shanghai market may result from the fact that the Shanghai market is an emerging market. Investors are somewhat speculative, and rush for trading as the stock price increases [6].
Figure 6. The return-volatility correlation functions for the simulated results of the S&P 500 and Shanghai indices, and for those of the controls.
The S&P 500 and Shanghai indices exhibit the leverage and anti-leverage effects, respectively. For the leverage effect, we consider two cases: 
is asymmetric; 
is symmetric. The latter is the control. For the anti-leverage effect, we consider the following cases: both 
and 
are asymmetric; only 
is asymmetric; only 
is asymmetric; both 
and 
are symmetric. The last three cases are controls. For each case, the simulation is performed for 100 times, and the average 
is displayed.
Conclusion
Based on investors’ individual and collective behaviors, we construct an agent-based model to investigate how the return-volatility correlation arises in stock markets. In our model, agents are linked with each other and trade in groups. In particular, two novel mechanisms, investors’ asymmetric trading and herding behaviors in bull and bear markets, are introduced. There are four parameters in our model, i.e., 
, 
, 
and 
. We adopt 
estimated in Ref. [41], and estimate the only tunable parameter 
to be 
. 
and 
, the key parameters, are induced by the asymmetries in trading and herding, respectively. Specifically, we determine 
from the ratio of the average trading volume when stock price is rising and that when price is dropping, and 
from investors’ different herding degrees in bull and bear markets.
We collect the daily price and volume data of six representative stock-market indices in the world, including the S&P 500, Shanghai, Nikkei 225, FTSE 100, Hangseng and DAX indices. With 
and 
determined for these indices respectively, we obtain the corresponding leverage or anti-leverage effect from the simulation, and the effect is in agreement with the empirical one on amplitude and duration. Other features, such as the long-range auto-correlation of volatilities and the fat-tail distribution of returns, are produced at the same time. Further, it is quantitatively demonstrated in our model that both the investors’ asymmetric trading and herding are essential generation mechanisms for the leverage and anti-leverage effects at the microscopic level. However, the investors’ trading is approximately symmetric for the five stock markets exhibiting the leverage effect, thus contributing very little to the effect. These two microscopic mechanisms and the methods for the determination of 
and 
can also be applied to other complex economic systems with similar asymmetries in individual and collective behaviors, e.g., to futures markets, bank interest rates, foreign exchange rates and spot markets of non-ferrous metals.
Supporting Information
Derivation of
.
(PDF)
Derivation of
.
(PDF)
Equivalence of the shifting to
and that to
.
(PDF)
The values of
,
and
for eight more indices.
(PDF)
How
affects the model parameters and simulation results.
(PDF)
Funding Statement
This work was supported in part by NNSF of China under Grant No. 11075137 and Zhejiang Provincial Natural Science Foundation of China under Grant No. Z6090130. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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Associated Data
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Supplementary Materials
Derivation of
.
(PDF)
Derivation of
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(PDF)
Equivalence of the shifting to
and that to
.
(PDF)
The values of
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and
for eight more indices.
(PDF)
How
affects the model parameters and simulation results.
(PDF)