Physica A 392 (2013) 4075–4082
Contents lists available at SciVerse ScienceDirect
Physica A
journal homepage: www.elsevier.com/locate/physa
Buying on margin, selling short in an agent-based
market model
Ting Zhang, Honggang Li ∗
Department of Systems Science, School of Management, Beijing Normal University, Beijing 100875, China
highlights
•
•
•
•
We study credit trading in financial market by an agent-based model.
We explore leverage’s effect on market indicators and individual wealth.
Simulation results confirm price discovery function of credit trading.
Leverage ratio has positive influence on price volatility and trading volume.
article
info
Article history:
Received 12 December 2012
Received in revised form 31 March 2013
Available online 9 May 2013
Keywords:
Buying on margin
Selling short
Leverage
Financial market
abstract
Credit trading, or leverage trading, which includes buying on margin and selling short,
plays an important role in financial markets, where agents tend to increase their leverages
for increased profits. This paper presents an agent-based asset market model to study the
effect of the permissive leverage level on traders’ wealth and overall market indicators. In
this model, heterogeneous agents can assume fundamental value-converging expectations
or trend-persistence expectations, and their effective demands of assets depend both
on demand willingness and wealth constraints, where leverage can relieve the wealth
constraints to some extent. The asset market price is determined by a market maker, who
watches the market excess demand, and is influenced by noise factors. By simulations, we
examine market results for different leverage ratios. At the individual level, we focus on
how the leverage ratio influences agents’ wealth accumulation. At the market level, we
focus on how the leverage ratio influences changes in the asset price, volatility, and trading
volume. Qualitatively, our model provides some meaningful results supported by empirical
facts. More importantly, we find a continuous phase transition as we increase the leverage
threshold, which may provide a further prospective of credit trading.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
Leverage trading plays an important role in financial systems and has effects on many aspects of financial markets [1].
Investors tend to increase their leverages by buying on margin or selling short for increased profits, enlarging their trading
amounts because of less wealth constraints. As a result, leverage trading contributes to the liquidity of the market. In
addition, leverage also influences the price level and volatility [2–4] and therefore magnifies risk, which connects leverage
with financial crises [5]. After the most recent global financial crisis, many countries took measures to adjust the regulations
on leverage trading. Therefore, how does leverage trading, specifically the leverage ratio of traders, influence a financial
market and traders’ wealth? This is not only a fundamental topic for theoretical research but also a practical issue for the
∗
Corresponding author. Tel.: +86 10 58802732; fax: +86 10 58210078.
E-mail address: [email protected] (H. Li).
0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.physa.2013.04.052
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T. Zhang, H. Li / Physica A 392 (2013) 4075–4082
creation of reasonable regulations. To answer this question, in this paper, we focus on leverage’s influence on the asset price
level and volatility, trading volume, and traders’ wealth.
There are numerous empirical studies related to how credit trading and leverage influence the financial market and
wealth of traders (see Section 2). This paper, however, explores these issues by a theoretical market model. We adopt the
agent-based modeling, which is an effective way to simulate trading activity in a financial market. There are two advantages
of a computationally oriented agent-based model compared to a theoretically oriented one. First, computer simulation
can link many behavioral aspects at the micro level with those at the macro level. Second, more realistic market features,
including budget or wealth constraints, can be readily incorporated into the market microstructure. Therefore, in this paper,
we incorporate traders’ balance sheets and market dynamics into an agent-based model to study the trading leverage ratio’s
effect on both market price behavior and individual wealth evolution. According to the simulation results, at the individual
level, we find that agents’ wealth accumulation is amplified as the leverage ratio increases. At the market level, the results
show that average price and final price converge more easily to the fundamental price as the leverage threshold value
increases, which can be regarded as the price discovery function of credit trading. In addition, we also observe that stock
price volatility and trading volume are magnified as the leverage ratio becomes larger. These results are consistent with
most of the empirical data, and we can reveal some stylized facts of financial markets in the time series of returns. More
importantly, by simulations we find a continuous phase transition as we increase the leverage threshold, which may provide
further insight of credit trading.
This paper is organized as follows. In Section 2, we review the previous literature about credit trading. In Section 3, we
introduce the agent-based market model with credit trading. In Section 4, we provide an analysis of the simulation results
of the model and compare them with some observed empirical facts. Finally, in Section 5, we present our conclusion.
2. Literature review
Much of the extant literature has given attention to the effect of margin requirements, which determine the threshold
of leverage, on market price volatility. One major reason why the Securities and Exchange Act (of the United States) created
Federal Reserve margin requirements in 1934 was to reduce stock price volatility. The ‘‘pyramiding and anti-pyramiding’’
mechanism described by Bogen and Krooss [6] demonstrated the fact that margin loans would increase stock price
variability. Luckett [7] explained that a rise in the margin requirement is a signal sent by the market regulator to indicate that
the market is overheating and has excessive risks. This signal induces investors to reduce credit trading. Hardouvelis [2,3]
examined the relationship between margin requirements and the volatility of stock prices in the cash market and concluded
that increasing the margin requirements reduces market volatility. Hardouvelis and Theodossiou [8] found a negative
association between margin requirements and stock volatility in the Japanese stock market and claimed that margin
requirements represent an effective tool for influencing stock prices and market volatility. In addition, Hardouvelis and
Peristiani [9] studied the effects of margin regulation in the Japanese stock market over the last 35 years and showed that
higher margin requirements are associated with lower margin borrowing and lower volatility of daily returns. In further
research, Hardouvelis and Theodossiou [10] also argued that the relationship between margin requirements and volatility
across bull, normal, and bear periods is asymmetric. In other words, the negative relation is stronger in bull periods, but it
disappears during bear periods. Moreover, Fortune [11] formulated a model of the distribution of stock prices to examine
the relationship between the level of margin debt and stock returns during the period from 1975 to 2001, when no changes
in Fed margins occurred, and also found that leverage trading can aggravate market volatility. Clearly, all these scholars
supported the assertion that leverage has a positive effect on price volatility. However, in contrast, Moore [1] believed that
high maintenance margins would bring about more margin calls; thus, a low maintenance margin would be a good way
to eliminate much of the danger of forced sales. Afterwards, Officer [12] found that the decline of volatility in the stock
market was not a result of the rise of margin requirements because margin requirements are changed after the variability
in the market factor has already started to change. At the same time, Largay and West [13] found that the effects of margin
requirement increases and decreases are asymmetric and inconspicuous. In a similar manner, the studies of Grube et al. [14],
Schwert [15], Kupiec [16], Salinger [17], Hsieh [18], Lee and Yoo [19], and Kim and Oppenheimer [20] also revealed that
public policies about margin requirements cannot control stock volatility. These findings imply that empirical works have
not yet reached a consensus on leverage’s effect on market price volatility.
The existing literature reveals that the effects of changing margin requirements for stock volatility can be illustrated
in two ways. First, an increase in the margin requirement is a signal sent by the regulator of the market to indicate that
the market is overheating and has excessive risks; this signal induces investors to reduce credit trading, which decreases
volatility [7,21]. Kumar et al. called this the speculative effect. Second, some speculators may be discouraged and withdraw
from the market due to the increase of margin requirements. Therefore, liquidity is decreased, and in turn, stock market
volatility becomes larger, which is called the liquidity effect [21,22]. Because the two effects are opposites, markets under
various conditions perform differently. The inconsistent results of empirical studies also reflect this. In our model, we mainly
demonstrate the speculative effect.
As for price level, we focus on the discrepancy between the stock price and fundamental value. Galam [23] and Biondi
et al. [24] found that the discrepancy existed and it even leads to the formation of bubbles and crashes, while some empirical
works show that symmetrical credit trading can avoid this kind of overvaluation of stock price [4,25]. In terms of leverage’s
effect on trading volume, Hardouvelis [3] showed that higher margin requirements are associated with lower overall trading
T. Zhang, H. Li / Physica A 392 (2013) 4075–4082
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volume in the US stock market. In a further study, Hardouvelis and Peristiani [9] also found the same relationship between
margin requirements and trading volume in the Japanese stock market. The empirical evidence appears to present a clear
consensus: trading volume is inversely related to margin requirements [11]. With regard to leverage’s effect on traders’
wealth, Adrian and Shin [26] found a strongly positive relationship between changes in total assets and changes in leverage
using data on US security dealers and brokers. Thurner et al. [27] showed that investors’ wealth grows faster with larger
leverage ratios, and as their wealth grows, investors have a greater impact on the stock price. Consequently, leverage makes
price fluctuations become fat tails and display clustered volatility as leverage increases.
3. The model
Our model is based on the analytical framework of Brock and Hommes [28]. In this model, if an individual expects that
a stock’s price will increase, it will buy a certain number of stocks according to its risk tolerance. If it does not have enough
money to afford the stocks it wants to buy, it can borrow money from a broker. However, the limitation is that the leverage
ratio, which equals the ratio of assets to equity, cannot be larger than a threshold value. Whereas if an individual expects
that a stock’s price will decrease, it will sell a certain number of stocks according to its risk tolerance. It can also borrow
stocks from a broker to relieve position restrictions. Likewise, the leverage ratio cannot be larger than the threshold value.
Similar to the real stock market, there will be a compulsory deleverage if an agent’s leverage exceeds the threshold value.
Namely, the longs must sell a certain number of stocks and the shorts must buy a certain number of stocks to decrease the
leverage until their leverage ratios are not larger than the threshold.
3.1. Heterogeneous agents
In our model, there are two types of boundedly rational speculative agents: fundamentalists, who trade based on a
fundamental value and predict that the stock’s prices will converge to the fundamental value in the long run, and trendchasing agents (chartist), who expect the latest trend will persist into the next period. Trend-chasing agents are denoted by
α , and fundamentalists are denoted by β . The total number of agents is N. Assume that θ is the proportion of chartists [29];
then, the number of chartists is θ N, and the number of fundamentalists is (1 − θ )N. The fundamentalists expect that the
stock prices will converge to a fundamental value pβ,t , and the trend-chasing agents believe that the latest trend of stock
prices will persist. Specifically, if a stock price is above the reference moving average value pα,t , the trend-chasers expect
that the price will rise in the next period. Otherwise, the trend-chasers expect that the stock will fall. Let Eiα,t (pt +1 ) and
β
Ei,t (pt +1 ) be the forecasts of price in the next trading period pt +1 of the two types of investors. The prediction rules are
given by
Eiα,t (pt +1 ) = pt + η(pt − pα,t )
(1)
β
Ei,t (pt +1 )
(2)
= pt + η(pβ,t − pt )
where i represents a certain heterogeneous agent, pα,t is the chartists’ reference value (i.e. moving average) of stock prices,
pβ,t follows a random walk with a standard deviation σf , and η ∈ [0, 1] is a response coefficient.
3.2. Leverage and deleverage
Every agent initially has the same cash and securities. Let wi,t be the wealth of each investor in trading period t, and let
ci,t and si,t be the net assets in cash and stocks:
wi,t = ci,t + pt si,t .
aci,t
asi,t
Assume that
and
are the assets in cash and securities and
of each investor in trading period t can be written as
(3)
lci,t
and
lsi,t
are the liabilities in cash and securities. The asset
ai,t = aci,t + pt asi,t = (ci,t + lci,t ) + pt (si,t + lsi,t ) = wi,t + li,t .
(4)
Therefore, the leverage ratio of each investor in trading period t is defined as
Li,t = ai,t /wi,t = (wi,t + li,t )/wi,t = 1 + li,t /wi,t .
(5)
In addition, there is a permitted maximum leverage ratio, or leverage threshold value, which is assumed to be L∗ .
Therefore, 1/L∗ corresponds to the minimal percentage margin requirement.
The leverage ratio of each investor will be checked in the beginning of each trading period. If an investor’s leverage is
larger than the leverage threshold, it will meet a compulsory securities trading. Let gic,t denote the transaction amount by
mandatory securities sale because of excessive cash liabilities, and let gis,t denote the transaction amount by mandatory
securities purchase because of excessive stock liabilities. Under the restriction of leverage threshold value L∗ , gic,t =
min{max{li,t − (L∗ − 1)wi,t , 0}/pt , asi,t }, and gis,t = min{max{li,t − (L∗ − 1)wi,t , 0}, aci,t }/pt .
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T. Zhang, H. Li / Physica A 392 (2013) 4075–4082
3.3. Excess demand of agents and market price dynamics
The agents are assumed to be myopic mean–variance maximizers. For simplicity, we suppose the interest rate is zero, and
risk aversion coefficient and the expected standard deviation of stock prices are assumed to be constant. Thus, the desired
optimum amount of stocks held by an agent is determined as

si,t = λ[Ei,t (pt +1 ) − pt ]
(6)
where λ is a constant value. The desired transaction amount is calculated as the desired optimum amount of stock holding
minus the actual stock holding (including the transaction amount by mandatory liquidation) x̂i,t = ŝi,t − si,t + gic,t − gis,t .
Considering the financial restriction with permission of credit trading, the effective transaction demand can be written as
follows
xi,t = min{x̂i,t , [aci,t + max{(L∗ − 1)wi,t − lct , 0}]/pt − gis,t } if x̂i,t ≥ 0
xi,t = − min{−x̂i,t , asi,t + max{(L∗ − 1)wi,t /pt − lsi,t , 0} − gic,t } if x̂i,t < 0.
After investors determine their effective transaction
trading, we
 + demand
 s and the amount of compulsory securities

 can
−
−
c
aggregate the demand of buying side as Xt+ =
i xi,t +
i gi,t and the demand of selling side as Xt = −
i xi,t +
i gi,t ,
where x+
with a positive value and x−
i,t is xi,t with a negative value. Therefore, the market excess demand of stock is
 c
i,t is xi,t
s
.
−
g
Xt =
x
+
g
i
,
t
i i ,t
i
i i,t
This model adopts a ‘‘quote-driven’’ price formation mechanism, and the market maker plays an important role in price
formation as Zhu et al. discussed [30]. On the one hand, as a liquidity provider, the market maker provides the market with
liquidity to accommodate transitory order imbalances in each period. When investors submit excess market orders at pt ,
the market maker clears the market by taking an offsetting position at the same price pt . Thus, the stock position of the
market maker is updated per trading period by Mt +1 = Mt − Xt . Therefore, the market volume is Vt = max{Xt+ , Xt− }. On the
other hand, as a price setter, the market maker wants to maintain a target position, which is assumed to be M d in the long
term. Here, the inventory of market maker is constant. The market maker will convey the intention of how much position
it wishes to transfer to the stock market and thus adjusts the stock price. Therefore, the stock price in the next period, pt +1 ,
is adjusted by the market maker according to
pt +1 = pt + γ [Xt + δ(M d − Mt )] + εt
(7)
where γ > 0 is a coefficient of price adjustment, and δ ∈ [0, 1] measures the marker maker’s willingness to adjust inventory,
which may have a relation to the degree of risk aversion. If the market maker tends to reach its target position quickly, it will
choose a large δ . When δ = 1, the market maker’s position change has maximal influence on price adjustment. εt is assumed
to be an i.i.d. normally distributed random variable with εt ∼ N (0, σ 2 ), implying some noise factors in price adjustment.
3.4. Update of agents’ balance sheets
Finally, based upon the above trading information, we can update the agents’ balance sheets. To maximize the benefits
of credit trading, the cash financing of agent i can be calculated as bci,t = max{pt (xi,t + gis,t ) − aci,t , 0}, and the stock
financing of agent i can be calculated as bsi,t = max{−xi,t + gic,t − asi,t , 0}. In this model, we assume that the agent will
pay back its liabilities as soon as it has surplus cash or securities. Therefore, the cash repayment is assumed to be hci,t =
min[aci,t + bci,t − pt xi,t − pt gis,t + pt gic,t , lci,t ], and the stock repayment is assumed to be hsi,t = min[asi,t + bsi,t + xi,t − gic,t + gis,t , lsi,t ].
Thus, the balance sheet of each agent will be updated as follows. lci,t and lsi,t , cash and securities liabilities, change to
lci,t +1 = lci,t + bci,t − hci,t and lsi,t +1 = lsi,t + bsi,t − hsi,t . In addition, the cash assets of each particular investor are adjusted as
aci,t +1 = aci,t − pt xi,t + pt gic,t − pt gis,t + bci,t − hci,t and the securities assets are adjusted as asi,t +1 = asi,t + xi,t − gic,t + gis,t + bsi,t − hsi,t .
Consequently, the net assets in cash and net assets in securities can be calculated as ci,t +1 = ci,t − pt xi,t + pt gic,t − pt gis,t and
si,t +1 = si,t + xi,t − gic,t + gis,t . The market then moves to the next trading period, t + 1, and so on.
4. Simulation results
The following results are an average of 500 simulations with the same leverage threshold value. There are 1000 periods
in every simulation. The parameters are N = 1000, θ = 0.5, η = 0.55, γ = 0.00002, λ = 140, σ = 0.72, σf = 0.12,
δ = 0.1, M d = 1000. The moving average window of pα,t is set to 20 periods, and initial values are p1 = 101, ci,1 = 1000,
and si,1 = 10.
To test robustness of the simulation results, we have done further model simulations in a given parameter space. We find
that λ, γ and θ are important to our simulation. With λ ∈ [50, 200] and γ ∈ [0.000001, 0.0002], the model can reproduce
similar qualitative results. If λ and γ exceed the upper bounds, the stock price series will be divergent; if λ and γ are smaller
than their lower bounds, the qualitative results will disappear. If there are too many chartists in the market, the stock price
series may also be divergent. So θ should be kept in the range of [0, 0.6]. Moreover, M d , ci,1 and si,1 do not have a significant
effect on the qualitative results, and we just select them in a range as M d ∈ [0, 5000], ci,1 ∈ [0, 10000], and si,1 ∈ [0, 100].
T. Zhang, H. Li / Physica A 392 (2013) 4075–4082
a
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b
Fig. 1. The time series of prices (a) and returns (b), where L∗ = 4.
Fig. 2. The changes of the average leverage ratio under different leverage threshold values L∗ .
The random variable can be relaxed to σ ∈ [0.5, 1.2]. If σ exceeds the bounds, the prices will be much more volatile, and
to keep the stylized facts as volatility clustering, we set σf ∈ [0.01, 0.2]. Additionally, when the number of agents increases
from 1000 to 5000, the qualitative invariance of the results remains unchanged.
4.1. Prices and returns
In Fig. 1, we gain insight into the time series of prices and returns. Fig. 1(a) is a typical time series of prices. Fig. 1(b) shows
volatility clustering in a time series of returns. By further investigation, we confirmed that the return series accord with
stylized facts as ‘‘absence of autocorrelation in raw returns’’ and ‘‘slow decay of autocorrelation in absolute returns’’ [31].
4.2. Actual leverage ratio
In Fig. 2, we check the average actual leverage ratio of agents, which is the mean of the largest leverage ratios of all
investors in each trading period at different leverage thresholds. Fig. 2 shows that the actual leverage ratio increases as the
leverage threshold increases, which indicates that the use of the leverage by the agents is always pushed to the limit of the
threshold level. This implies that investors tend to enlarge their leverage ratios when their financial restrictions are relieved.
The increment step of the leverage threshold is 0.2 on the abscissas axis and is the same in the other figures below.
4.3. Agent’s average wealth
At the individual level, we focus on how the leverage threshold affects wealth evolution for both fundamentalists and
chartists, which is measured by the wealth index, a ratio of investors’ gross wealth in the final step to the wealth in the first
step. Fig. 3(a) shows that each agent’s average wealth index also increases with leverage. The larger the leverage threshold
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T. Zhang, H. Li / Physica A 392 (2013) 4075–4082
a
b
Fig. 3. The changes of traders’ wealth index (a) and wealth changes of market maker (b) under different leverage threshold values L∗ .
value is, the more likely it is for investors to make money. Fig. 3(b) shows that market maker’s wealth decreases with L∗ .
Through further analysis, we find that the increasing aggregate wealth growth of all traders derives from the capital loss
of the market maker. In the real market, the market maker can obtain revenues by setting the ask-bid spread and/or by
charging commission. Thus, if the market maker sets a large enough ask-bid spread or charges a large enough commission
rate of each transaction in our model, the market maker can get positive profits (see Zhu et al. [30]).
4.4. Price level and fluctuation
At the market level, we focus on how the securities price and price volatility change with the leverage threshold. We
can see from Fig. 4(a) and (b) that the average price and final price converge to the fundamental value with the increase
of leverage threshold. This phenomenon implies that trading with a high leverage ratio contributes to price discovery. By
analyzing the trading data, we can see that the capital of chartists decreases over time when the leverage threshold value is
high. By contrast, fundamentalists accumulate more wealth, especially with high leverage ratios. Therefore, fundamentalists
can exert more influence on price through transactions of greater amounts and thus move the price back to fundamental
value. Furthermore, from Fig. 4(a) and (b), we can observe the existence of a continuous phase transition. Below the phase
transition point, the difference between average/final price and the fundamental value varies linearly with L∗ , going to zero
continuously as the leverage threshold approaches the transition. Whereas, when the leverage threshold value is above the
phase transition, the difference basically remains vanishingly small. In other words, the market has fully exerted the function
of price discovery with the leverage threshold values above the phase transition; thus, continuing to increase the leverage
threshold value almost has no influence on the converging of price.
As for the average rate of logarithmic return, Fig. 4(c) shows that the rate declines as the leverage threshold value
increases, which implies that the leverage ratio has a negative relationship with average rate of returns. In Fig. 4(c), the
phase transition also exists. When the leverage threshold value is below the phase transition, the average rate of logarithmic
returns has linear changes. Whereas, when the leverage threshold value is above the phase transition point, the average rate
of logarithmic returns becomes stable.
However, the standard deviation of the logarithmic returns, which reveals the volatility of the market, becomes larger
as the leverage threshold value increases, as shown in Fig. 4(d). This result is consistent with the empirical data indicating
that a large leverage will bring more volatility to the stock market.
4.5. Market volume
Increasing the leverage threshold relieves investors’ financial restrictions; thus, the effective transaction demand of each
investor is enlarged. As a result, the average trading volume in the market becomes larger, which can be seen in Fig. 5.
This phenomenon has also been observed by many scholars in the real stock market, which implies that leverage trading
amplifies market depth and increases liquidity.
5. Conclusions
In this paper, we apply the agent-based model and focus our discussion on the effects of leverage from the aspects
of individuals and the financial market. At the market level, the simulation results show that as the leverage threshold
value increases, market prices converge more easily to the fundamental value, price volatility increases, and the average
T. Zhang, H. Li / Physica A 392 (2013) 4075–4082
a
b
c
d
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Fig. 4. The results of how differences between the average price (a), final price (b) and fundamental value, average rate of logarithmic returns (c), and
standard deviation of logarithmic returns (d) in the market change as the leverage threshold increases.
Fig. 5. The changes of trading volume under different leverage threshold values L∗ .
trading volume in the market increases. These results are consistent with most of empirical studies. In addition, we find
a phase transition as we increase the leverage threshold in some simulation results. Namely, below the phase transition,
the differences between market prices and fundamental value and average rate of logarithmic returns have linear changes;
however, they remain stable when the leverage threshold value is above the phase transition. This discovery may provide
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T. Zhang, H. Li / Physica A 392 (2013) 4075–4082
further insight into the research and regulation of margin requirement. At the individual level, we observe that each agent’s
average leverage ratio increases as the leverage threshold increases, and wealth index has a positive relationship with the
leverage threshold value.
The regulation of financial markets in different countries or during different times is not the same; thus, the model does
not coincide with all real markets perfectly. Some countries have constraints on selling short, such as the uptick rule, which
only allow investors to sell short with a price higher than the most recent transaction price. Much of the existing literature
[4,25] has discussed the effects of such constraints on the market price, volatility and volume. If the uptick rule is
implemented, then the actual leverage will be reduced under the same leverage threshold; thus, the effect on the market
may be weakened to some extent.
Acknowledgments
We wish to thank Xuezhong He and Zengru Di for useful comments and discussions, and two anonymous referees for
helpful comments. None of the above is responsible for any of the errors in this paper. This work was supported by the
Fundamental Research Funds for the Central Universities under Grant No. 2012LZD01 and the National Science Foundation
of China under Grant No. 61174165.
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