Linking agent-based models and stochasticmodels of financial marketsLing Fenga,b,c,1, Baowen Lia,b,1, Boris Podobnikc,d,e,f, Tobias Preisc,g,h, and H. Eugene Stanleyc,1

aGraduate School for Integrative Sciences and Engineering, National University of Singapore, Singapore 117456, Republic of Singapore; bDepartment ofPhysics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Republic of Singapore; cCenter forPolymer Studies and Department of Physics, Boston University, Boston, MA 02215; dFaculty of Civil Engineering, University of Rijeka, 51000 Rijeka,Croatia; eFaculty of Economics, University of Ljubljana, 1000 Ljubljana, Slovenia; fZagreb School of Economics and Management, 10000 Zagreb, Croatia;gChair of Sociology, in particular of Modeling and Simulation, Eidgenössische Technische Hochschule Zurich, 8092 Zurich, Switzerland; and hArtemisCapital Asset Management GmbH, 65558 Holzheim, Germany

Contributed by H. Eugene Stanley, March 29, 2012 (sent for review December 6, 2011)

It is well-known that financial asset returns exhibit fat-taileddistributions and long-term memory. These empirical featuresare the main objectives of modeling efforts using (i) stochasticprocesses to quantitatively reproduce these features and (ii)agent-based simulations to understand the underlying microscopicinteractions. After reviewing selected empirical and theoreticalevidence documenting the behavior of traders, we construct anagent-based model to quantitatively demonstrate that “fat” tailsin return distributions arise when traders share similar technicaltrading strategies and decisions. Extending our behavioral modelto a stochastic model, we derive and explain a set of quantitativescaling relations of long-term memory from the empirical behaviorof individual market participants. Our analysis provides a behavior-al interpretation of the long-termmemory of absolute and squaredprice returns: They are directly linked to theway investors evaluatetheir investments by applying technical strategies at differentinvestment horizons, and this quantitative relationship is in agree-ment with empirical findings. Our approach provides a possiblebehavioral explanation for stochastic models for financial systemsin general and provides a method to parameterize such modelsfrom market data rather than from statistical fitting.

complex systems ∣ power law ∣ scaling laws

Modeling price returns has become a central topic in thestudy of financial markets due to its key role in financial

theory and its practical utility. Following models by Engle andBollerslev (1, 2), many stochastic models have been proposedbased on statistical studies of financial data to accurately repro-duce price dynamics. In contrast to this stochastic approach,economists and physicists using the tools of statistical mechanicshave adopted a bottom-up approach to simulate the same macro-scopic regularity of price changes, with a focus on the behaviorof individual market participants (3–10). Although the second so-called agent-based approach has provided a qualitative under-standing of price mechanisms, it has not yet achieved sufficientquantitative accuracy to be widely accepted by practitioners.

Here, we combine the agent-based approach with the stochas-tic process approach and propose a model based on the empiri-cally proven behavior of individual market participants thatquantitatively reproduces fat-tailed return distributions andlong-term memory properties (11–14).Empirical and Theoretical Market BehaviorsWe start by arguing that technical traders (usually agents seekingarbitrage opportunities and make their trading decisions based onprice patterns) contribute much more to the dynamics of dailystock prices St (or log price st ≡ ln!St") than fundamentalists(who attempt to determine the fundamental values of stocks).Although fundamentalists hold a majority of the stocks, theytrade infrequently (see SI Appendix, Fig. S6). In contrast, techni-cal traders contribute most of the trading activities (15) by tradingtheir minority holdings more frequently than fundamentalists.

Market surveys (16–18) also provide clear evidence of the preva-lence of technical analysis. We consider here only technical tra-ders, assuming that fundamentalists contribute only to marketnoise. Our study is of the empirical data recorded prior to2006 and ignores the effect of high frequency trading (HFT) thathas become significant only in the past 5 y. We propose a beha-vioral agent-based model that is in agreement with the followingempirical evidence:

i. Random trading decisions made by agents on a daily basis. n0

technical traders use different trading strategies, hence theirdecisions to buy, sell, or hold a position appear to be random.A trading decision is made daily because empirical studiesreport the lack of intraday trading persistence in empiricaltrading data (19). Market survey (16) also shows that fundmanagers put very little emphasis on intraday tradings. We es-timate the probability p of having daily trade empirically fromtrading volumes.

ii. Price returns. The price return rt ≡ st − st−1 is controlled bythe imbalance dt between the demand and the supply ofstocks—the difference in the number of buy and sell tradeseach day. The excess in total demand or supply moves theprice up or down, where the largest rt occurs when all tradersact in unison, when they all either buy or sell their stocks. Weassume this relationship between price change rt and dt to belinear each day, as supported by empirical findings (20, 21).

iii. Centralized interaction mechanism of returns on technicalstrategies. For technical traders, an important input para-meter in their strategies is past price movement (22, 23).Consequently, prices and orders reflect a main interactionmechanism between agents. In many agent-based models,the interaction strength between agents need to be adjustedwith agent population size (5, 24, 25) or interaction structure(26) to sustain “fat” tails in return distributions. Here we pro-pose a centralized interaction mechanism (price change)among agents so that the strength of interaction grows withagent population and is unaffected by interaction structure.

iv. Opinion convergence due to price changes. This is the uniquemechanism that distinguishes our model from other models.It specifies the collective behavior of technical traders. Duffyet al. (27) found that agents learn from each other and tend toadopt the strategy that gives the most payoff. Given the pricepatterns at any point in time, a few most profitable technicalstrategies dominate the market because every technical trader

Author contributions: L.F., B.L., B.P., T.P., and H.E.S. designed research; L.F., B.L., B.P., T.P.,and H.E.S. performed research; L.F., B.L., B.P., T.P., and H.E.S. contributed new reagents/analytic tools; L.F., B.L., B.P., T.P., and H.E.S. analyzed data; and L.F., B.L., B.P., T.P., andH.E.S. wrote the paper.

The authors declare no conflict of interest.1To whom correspondence may be addressed. E-mail: g0801896@nus.edu.sg, phylibw@nus.edu.sg, or hes@bu.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1205013109/-/DCSupplemental.

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wants to maximize his/her profit by using the most profitablestrategy copied from each other (The most profitable strategywould become less profitable when most agents adopt it, and anew profitable strategy emerges from the new price trends;soon agents will flock to the new profitable strategy until itis no longer profitable. This is similar to the regime switchingphenomenon in various agent-based models.). On the otherhand, the individual strategies used by different technical tra-ders differ in their parameterizations of the buy/sell time,amount of risk tolerated, or portfolio composition (15). Sowhen the input signal—the previous price change rt−1—issmall, every agent acts independently. When the input signalis large, the agents act more in concert, irrespective of theirdifferences in trading strategies. During the spreading panicof market crashes, for example, most agents sell their stocks(and market makers are likely to make losses in such circum-stances). This supports the empirical finding that large priceswings occur when the preponderance of trades have thesame buy/sell decision indicated in the findings by GabaixX. et al. (28).

Behavioral Model and ResultsBased on the items of evidence (i)–(iv) listed above we constructa three-step behavioral model:

• Step 1. Based on evidence (i), we assume n0 agents, each ofequal size 1. Each day, a trading decision ! i!t" is made by eachagent i,

! i!t" ≡

8><

>:

1 with probability p ! buy;

−1 with probability p ! sell;

0 with probability 1 − 2p ! hold:

• Step 2. Based on evidence (ii), we define price change rt to beproportional to the aggregate demand dt, i.e., the difference inthe number of agents willing to buy and sell,

rt ≡ kdt # k!

n0

i#1

! i!t": [1]

Trading volume is equal to the total number of trades in thiscase because every agent has the same trading size of 1. Hencedaily trading volume Nt is defined as

Nt ≡!

n0

i#1

j! i!t"j [2]

and k is the sensitivity of price change with respect to dt. We setk to 1 because a choice for k does not affect the statistical prop-erties. Note that, according to Step 1, maximum (minimum) dtmeans that all agents are in collective mode when they behavethe same, all of them either buy or sell stocks.

• Step 3. Based on evidence (iii) and (iv), at day t$ 1 eachagent’s opinion is randomly distributed into each of the ct$1

opinion groups where

ct$1 # !n0∕jrtj"!; [3]

in which all agents comprising the same opinion group executethe same action (buy, sell, or hold) with the same probability pas in Step 1. Because 1 ! ct$1 ! n0, Eq. 3 implies (a) when theprevious return is maximum, jrtjmax # n0 (everyone buys/sells),there is only one trading opinion among the agents; (b) whenthe previous return is minimum jrtjmin # n!!−1"∕!

0 (see SIAppendix, Section 5), there are n0 opinions and every agent actsindependently of one another. We use ! # 1 in most of our

simulations because it produces results that are numericallyclose to empirical findings. The case of ! # 1 correspondsto jrtjmin # 1, and there is only one more buy/sell order thansell/buy order. The result with different ! is presented inthe end of this section. When market noise is considered,ct$1 ∼N!n0∕jrtj; "2

c ", whereN denotes the normal (Gaussian)distribution and "2

c # b · n0∕jrtj quantifies market noise due toexternal news events.

To obtain an empirical value for daily trading probability p,we choose 309 companies from the Standard and Poor’s 500index traded over the 10-y period 1997–2006. Only 309 of the500 total stocks were consistently listed during the entire 10-yperiod, and they are the ones we choose. We define the tradingvelocity V of an agent as the total number of shares he/she tradesin a year, divided by the number of shares, he/she owns on aver-age over a year

V # Number of tradesNumber of shares

: [4]

From the Compustat database, we divide the total number ofshares traded on the market by the number of outstanding sharesof each stock and obtain the average yearly trading velocity, V ,for these stocks through the 10-y period. Thus we obtain,

V # 1.64; [5]

where the value of V varies from 1.3 to 1.9 throughout the 10 y.Note that V > 1 means that, on average, each stock changes itsowner more than once during a year. From the data documentedby Yahoo! Finance, we assume that institutional owners are fun-damentalists and that the rest of the traders are predominantlytechnical traders. The percentage of outstanding shares held byinstitutional owners is usually higher than 60%, and the averagevalue is 83% (SI Appendix, Fig. S6A).

Assuming institutional owners are fundamentalists with invest-ment (trading) horizons longer than 1 y (they trade less frequentlythan once a year), we calculate the value of p as follows. We letthe average yearly trading velocity of fundamentalists be Vf , thenthe average yearly trading velocity of technical traders Vc is

Vc # !V − 0.83Vf "∕!1 − 0.83": [6]

Because there are about 250 trading days in a year, we calculatethe daily trading probability p

p # Vc∕!250 · 2": [7]

Because fundamentalists have an investment horizon longerthan 1 y, Vf < 1 (see Eq. 4). We arbitrarily set Vf to be 0.2,0.4, 0.6, 0.8, and estimate the corresponding values of p as0.0174, 0.0154, 0.0134, 0.0115, respectively. For !Vf ; p" #!0.4; 0.0154", we find that approximately 80% of the tradingvolume is contributed by technical traders.

We compare the simulation results from our behavioral model(Steps 1, 2, and 3) with those obtained for the empirical set ofstocks comprising the S&P 500 index. For the cumulative distri-bution (CDF) of both absolute returns and number of trades.Fig. 1 shows that the model reproduces the empirical data in boththe central region and the tails. In particular, both model andempirical distributions have power-law tails (P!jrtj > x"∼x−"r ; P!nt > x" ∼ x−"n) (13, 29–31) where the power-law expo-nents "r and "n that we obtain for the empirical data by applyingthe Hill’s estimator are in agreement with the ones obtained forempirical data (32).

It is worth noting that with different values of ! in Eq. 3, thetail exponent "r for the absolute return distribution varies as

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shown in Table 1. We find that ! # 1 gives the tail exponent "rthat is closest to empirical finding (see SI Appendix, Section 5 forthe implication of different values in !).

Overall, we are able to generate power-law distributions inboth returns and number of trades in agreement with empiricaldata. The fact that we are able to capture the trends in the twohighly correlated quantities (33) implies a very plausible mechan-ism underlying our model. Furthermore, the tail exponent "r isinvariant with respect to different values of agent population sizen0 and fundamentalists’ investment horizon Vf (Table 2 and SIAppendix, Section 1). In particular, the insensitivity of the resultto the number of agents n0 distinguishes this model to most otheragent-based models (34, 35).

Extension to Stochastic ModelNext we study the underlying stochastic process of our behavioralmodel (Step 1 to Step 3). The mechanism of opinion convergenceunder price changes is incorporated into a mathematical form foran analytical understanding. Step 3 in our behavioral model in-dicates that the daily price change is not deterministic, where itsvariance "2

t ≡ E!rt − E!rt""2 is related to total number of opiniongroups ct, which is determined directly by previous return rt−1. Onaverage, each opinion group has jrt−1j agents, and we have

"2t ≡ E!r2t jrt−1" # ct · %p · r2t−1 $ p · !−rt−1"2 $ !1 − 2p" · 0&

"2t # 2pn0jrt−1j:

[8]

Because "t presents the standard deviation of price change rt,we can model price change as rt # "t#t, where #t$1 is a randomvariable with zero mean and unit variance, and its distribution isdetermined by Step 3. For simplicity, we use a normal distributionfor #t, although we find that other distributions, such as t-distri-bution, give similar results in our analysis (SI Appendix,Section 4).

Again we note that the variance "2t in Eq. 8 is not constant but

is time dependent, and time-dependent variances are commonlyfound in a variety of empirical outputs where phenomena are ran-ging from finance to physiology (2, 36). They are widely modeledwith the Autoregressive Conditional Heteroskedasticity (ARCH)process (1). Here we provide a possible explanation for theARCH effect found in financial data in terms of the behaviorof technical traders—larger previous price change rt−1 brings tra-ders’ opinions closer to each other, resulting in large subsequentprice fluctuations. Theoretical analysis on Eq. 8 leads to the fat-tailed return distributions (SI Appendix, Section 2) ubiquitouslyobserved in real data, and in our case a power-law tail. Previousworks have generated ARCH effect from other mechanisms thatis different from our model (37).

The time-dependent variance of Eq. 8 defined by the most re-cent price change rt−1 is therefore based on short memory in theprevious rt. To this end, we further extend our behavioral modelwith two more items of empirical evidence:

v. Technical strategies are applied at different return intervals. Incontrast to the simplistic realization in our behavioral model(Step 1 to Step 3) in which technical traders make their de-cisions only upon the most recent daily returns (see Eq. 8and Step 3), market survey (16) indicates that technical stra-tegies in practice are applied at different investment horizonsranging from 1 d to more than 1 y. This finding implies thatagents calibrate their technical strategies based on returns ofdifferent time horizons, i.e., how stock preformed during thelast day, week, month, up to year, or even longer. Most tech-nical strategies are focused on short-term returns of a fewdays, and fewer are focused at yearly returns according tothe survey (16).

vi. Increasing trading activity in volatile market conditions.Agents tend to trade more after large price movements. Dif-ferent technical strategies set different thresholds on prices totrigger trading decisions (38), so large price fluctuations arelikely to trigger more trades. Hence the probability of dailytrade p is directly related to past returns. Because a large pro-portion of technical strategies is applied with short investmenthorizons, price changes from previous days have larger impacton p than price changes from past year.

Fig. 1. Comparison of the distributional properties of simulations with empirical results for the parameter set Vf # 0.4, n0 # 210, b # 1.0. Vf characterizesthe how fast one agent’s shares is traded. Empirical data are from the 309 stocks out of S&P 500 index components that have been consistently listed during the10-y period 1997–2006. There are ≈800;000 data points in empirical data, and 1,000,000 sample points from simulation. (A) Cumulative distributions of dailyreturns, defined as rt ≡ log%xt;close& − log%xt;open&, the difference between the daily opening and closing price of a stock on day t. Thus we ignore overnightreturns arising, e.g., due to news events. The price for each stock is normalized to zero mean and unit variance before aggregation into a single distribution.The simulation results agree with the shape of the empirical distribution. The Hill estimator on 1% of tail region gives "r;S&P500 # 3.69' 0.07,"r;simulation # 4.08' 0.08. (B) The cumulative distribution of the same set of data but analyzed on the daily number of trades. Because the number of tradesincreases each year, we normalize the data by each stock mean number of trades on a yearly basis before aggregating them into the distributions. The simula-tion again reproduces the empirical results. The Hill estimator applied to 2.5% of the tail region gives "n;S&P500 # 4.02' 0.07, "n;simulation # 4.02' 0.08.

Table 1 Tail exponents of absolute return distribution withdifferent !

! 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.4 1.5 2.0"r 2.3 2.5 2.7 3.0 3.3 3.8 4.3 4.4 5.9 7.8 9.4

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The survey on USmarket (16) allows us to estimate the relativeproportion of technical analysis applied at different investmenthorizons. Precisely, agents having investment horizon i daysare affected by price change in the past i days, and the relativeportion of such traders is characterized by #i. Fig. 2 illustrates thisplot, and we find that the curve follows a power-law decay withexponent d # 1.12, i.e., #i ∝ i−d. As different agents look at re-turns over time scales of differing lengths, their trading opinionsare affected by the past returns of history longer than one day, sois the convergence of their collective opinions. Agents having in-vestment horizon i days are affected by price difference in thepast i days, and the relative portion of such traders are charac-terized by #i. Hence the convergence of opinions is not only af-fected by the previous day’s return but also the price difference oflonger durations. Therefore Eq. 3 is better written as

ct$1 #n0

!

Mi#1

#ijst − st−ij; [9]

where M is the maximum investment horizon for which technicalanalysis is applied, and #i is the proportion of agents focusing oninvestment horizon i and it decays with i as #i ∝ i−d as in Fig. 2.WhenM # 1, Eq. 9 is identical to Eq. 3, which is the scenario forhomogeneous investment horizon of 1 d. jst − st−ij is used as asimplified information content for technical traders with invest-ment horizon i. Because there can be at least one opinion group,this boundary condition requires∑M

i#1 #i # 1. From Eq. 9, Eq. 8 istransformed into

"2t$1 # 2pn0

!

M

i#1

#ijst − st−ij # 2pn0#!t; [10]

where # ≡ %∑Mi#1 i

−d&−1 and !t ≡ ∑Mi#1 i

−djst − st−ij.Note that Eq. 10 can be also intuitively derived. Because by #i

we denote the proportion of agents observing the price change inthe past i days, the price change jst − st−ij contributes to the over-

all opinion convergence with a weight #i. Even without knowingthe exact contribution of jst − st−ij in the overall opinion conver-gence, by taking its first order effect with weight #i, we wouldhave the functional form of Eq. 10.

To take into account the effect of increasing trading activitywhen market is more volatile, we assume that traders place tech-nical thresholds based on returns of different horizons, and thatthe proportion of traders with different horizons is determined bythe same survey results reported in ref. (16). Similar to the con-struction of cluster formation in Eq. 9, we define the time-depen-dent probability of trading as pt$1 ≈ p0 $ # 0!t. By p0 we denotethe base trading activity (largely due to fundamentalists) and # 0!tis the additional trade due to threshold crossing by technical tra-ders. Therefore we transform Eq. 10 to

"2t$1 ≈ 2pt$1n0#!t # 2#n0!p0 $ # 0!t"!t

# −#n0p20

2# 0 $ 2## 0n0

!!t $

p02# 0

"2

:

The first term is very small compared to the second term becausetechnical traders dominate trading activities. Hence, we canignore the first constant term and focus on the second quadraticterm. Therefore, the previous equation transforms to

"t$1 ≈###########2## 0

p !!t $

p02# 0

"# A$ B!t: [11]

Constant A is a scaling parameter defining the average size ofreturns, and B characterizes the relative portion of trades accom-plished by technical traders (due to crossing the thresholds) vs.background trading activities (mostly by fundamentalists). Eq. 11has similar functional form as "t$1 in the fractional integrated

Table 2 Tail exponents of absolute return distribution with different values of Vf and n0

Parameter value

n0 # 210, b # 1.0 Vf # 0.4, b # 1.0

Vf # 0.2 Vf # 0.4 Vf # 0.6 Vf # 0.8 n0 # 28 n0 # 210 n0 # 212 n0 # 214

Tail exponents "r 3.69 ± 0.07 3.69 ± 0.07 3.74 ± 0.07 3.89 ± 0.07 3.86 ± 0.07 3.70 ± 0.07 3.66 ± 0.07 3.69 ± 0.07

Fig. 2. Plot of survey result (16) on percent importance placed on technicalanalysis at different time horizons by U.S. fund managers. The plot shows apower-law function with exponent −1.12. We have combined the percentvalues for both flow and technical analysis at each time horizon, as flow ana-lysis in a broad sense is one type of technical analysis. Technical analysis isheavily used at investment horizons of days to weeks and decays to closeto zero at 500 d.

Fig. 3. Confirmation of the scaling relations Eq. 12. Dependence of $1against $2 for the 30 stocks in Dow Jones Indices components. Daily closingprices from 1971–2010 are used for each stock. The two exponents fulfill$1 ! $2 ! 2$1, so the scaling relation predicted by our stochastic model is con-sistent with these empirical data. The range of lag, 11 ! ! ! 250, is used be-cause theoretical power-law decay of the ACF is valid for large !, and for! > 250 the ACF values are close to the noise level.

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processes (39). However, there are some differences in a way howpast returns are used in "t$1—whereas in fractional integratedprocess "t$1 depends on past daily returns, in Eq. 11 we useabsolute values of past aggregate returns. Precisely, the absolutevalues of past daily, weekly, and monthly returns, to mention afew. Parke (40) has demonstrated how heterogeneous error dura-tions among traders could result in fractional integration, andhere we explicitly and quantitatively provide a behavioral inter-pretation of the long memory in absolute returns. The decayingdependence of standard deviation "t$1 on past aggregate returnsof different durations in Eq. 11 is a direct outcome of agentsapplying technical strategies at different investment horizons.We find this dependence to decay as a power law, and the power-law exponent d is calculated from empirical observations (16).

Analytical and Simulated ResultsIn general, the long memory in returns can be demonstratedby analyzing auto-correlation functions (ACF). We find thatthe ACFs of both absolute and squared returns (%! and % 0

!) decayas power laws with !, i.e., %!!jrtj; jrt−! j" ∝ !−$1 and % 0

!!r2t ; r2t−!" ∝!−$2 , where we obtain the scaling relations (SI Appendix,Section 2) similar to phase transitions in statistical mechanics,

8><

>:

$1 # 2d − 2;

2d − 2 ! $2 ! 4d − 4;

$1 ! $2 ! 2$1:

[12]

Whereas Ding et al. have shown that correlations in jrtj decaymore slowly than correlations in r2t (41), we, instead, show quan-titative scaling relations that have been derived and explained bythe behavior of individual market participants (characterized byd). Empirical verification with the stock components of DowJones Indices in Fig. 3 confirms the validity of the scaling relationof Eq. 12.

The behavioral understanding of our stochastic process Eq. 11allows us to perform simulations in which every parameter isbased on empirical calibration rather than on conventionalstatistical estimation. In Eq. 10 we replace the upper limit ofM in the summation by 500 trading days, which correspondsto ≈2 years of investment horizon as implied by the survey

(16). We set d # 1.2, in agreement with empirical finding (16).From the empirical trading volume fluctuations and uncondi-tional variance of price changes, we estimate A # 0.002 and B #0.05 (SI Appendix, Section 3). Therefore we obtain "t$1 ≈ 0.002$0.05∑500

i#1 i−1.2jst − st−ij. In Fig. 4 we compare the simulation re-

sults with the results for S&P 500 index. We demonstrate thatthe ACF of absolute and squared returns—for both simulationsand the S&P 500 index—fulfill our scaling relations of Eq. 12with $1;simulation # 0.40, $2;simulation # 0.53; $1;S&P500 # 0.44, and$2;S&P500 # 0.7.

Eq. 11 reflects the long-term memory of empirical data accu-rately while still being a stationary process (SI Appendix,Section 3) in contrast to many other power-law decaying pro-cesses (2, 39). In addition, the behavioral picture of Eq. 11 ex-plains the origin of long-term memory through heterogeneousinvestment horizons of different technical traders based on em-pirical evidence.

SummaryStarting from the empirical behavior of agents, we construct anagent-based model that quantitatively explains the fat tails andlong-term memory phenomena of financial time series withoutsuffering from finite-size effects (35). The agent-based modeland the derived stochastic model differ in construction but sharethe same mechanism of opinion convergence among technicaltraders. Whereas the agent-based model singles out the dominantmarket mechanism, the stochastic model allows a detailed ana-lytical study. Both approaches allow their parameter values tobe retrieved from market data with clear behavioral interpreta-tions, thus allowing an in-depth study of this highly complex sys-tem of financial market.

The universality of various empirical features implies a domi-nant mechanism underlying market dynamics. Here we proposethat this mechanism is driven by the use of technical analysisby market participants. In particular, past price fluctuationscan directly induce convergence or divergence of agents’ tradingdecisions, which in turn give rise to the “ARCH effect” in empiri-cal findings. Additionally, the heterogeneity in agents’ investmenthorizons gives rise to long-term memory in volatility.

Fig. 4. Comparison between empirical data and simulation. The empirical data (about 10,000 data points) are from the S&P 500 index daily closing prices inthe 40-y period 1971–2010. The simulation are carried out for 50,000 data points to obtain good convergence. Black Monday, October 19, 1989, is removedfrom the analysis of the ACF. (A) Autocorrelation of simulation vs. S&P 500 index. The simulation results show behavior similar to the S&P 500 for both absolutereturns and squared returns. The rate of decay is also similar numerically. (B) The CDF of absolute returns for simulations and the S&P 500. The simulationreproduces the return distribution well.

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ACKNOWLEDGMENTS We thank F. Pammolli, J. Cherian, and Y. Chen for dis-cussions and suggestions. Thework has been partially supported by the GrantR-144-000-285-646 from the National University of Singapore, the GermanResearch Foundation through Grant PR 1305/1-1, the Ministry of Science

of Croatia, the Keck Foundation Future Initiatives Program, Defense ThreatReduction Agency, and the National Science Foundation Grants PHY-0855453, CMMI-1125290, CHE-0911389, and CHE-0908218.

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APP

LIED

PHYS

ICAL

SCIENCE

SEC

ONOMIC

SCIENCE

S

SUPPLEMENTARY INFORMATION

This is the supplementary information for ‘Linking Agent-Based Models and Sto-

chastic Models of Financial Markets’.

1. Robustness checking of behavioral model

In order to see how the parameters a!ect our behavioral model, we vary the values

of probability of trading p, agent population size n0 and market noise level b to study

the robustness of the model. Value of p is determined indirectly by Vf as shown in

the paper, by varying the value of Vf between 0.2 to 0.8, typically smaller than 1.0

due to the nature of fundamentalists’ investment horizons longer than one year. n0

is chosen to vary from 256 to 16,384 covering a wide range. Market noise b is defined

in step 3 of the paper as: ct+1 ! Normal(n0/|rt|,!2c ), where !2

c = b · n0/|rt|. Value

of b is taken to be from 0 to 3.0 – no noise to extremely high noise. Figure 1 shows

the simulation results of our behavioral model with di!erent values of Vf and n0

varying individually. Each curve is shifted vertically for clarity. In 1a and 1b, it is

Figure 1. CDF plot for di!erent values of each parameter. Thecurves are shifted vertically for clarity. a) di!erent values of Vf b)di!erent n0. Di!erent values of fundamentalists’ trading frequency Vf

and agent population size n0 give almost identical results, showing therobustness of our model in producing the result matching empiricaldistributions.

1

2 SUPPLEMENTARY INFORMATION

Figure 2. CDF plot for di!erent values of noise b. The curves areshifted upwards for clarity. When market noise is high with a lot ofnews impacting the traders’ decisions, b is big, and the tail of CDFbecomes larger.

evident that the curves are almost identical in shape, demonstrating a robustness

of the result against the variation in these two parameters. Hill estimator gives

tail exponents close to empirical values, and the result is shown in the manuscript.

Figure 2 shows the influence of market noise b on the return distributions. Bigger

noise gives fatter tail, resulting in huge price swings as expected.

2. Mathematical analysis on the stochastic process

Excess Kurtosis of Behavioral Model

In our behavioral model, the conditional kurtosis of rt is Kurt(rt|rt!1). Since

there are n0/|rt!1| number of independent clusters with |rt!1| agents each, the excesskurtosis is equal to the kurtosis contributed by each cluster divided by the number

of clusters. The kurtosis of each cluster is 1/(2p), therefore we have the conditional

excess kurtosis Kurt(rt|rt!1) = ( 12p " 3) |rt!1|

n0, which varies between ( 1

2p " 3)/n0 to

( 12p " 3). In the limit n0 # $, we have 0 % Kurt(rt|rt!1) % ( 1

2p " 3) & 30 when

SUPPLEMENTARY INFORMATION 3

p = 0.015. Hence the conditional kurtosis of the distribution depends directly on

previous returns.

For simplicity, we assume rt = !t"t, where "t ! N(0, 1) is a standard normal

variable. It is shown later that t-distributions give similar simulation result. A

simple calculation would show that this construction gives an unconditional kurtosis

of

Kurt(r) ='rt4('rt2(2

='rt+1

4('rt+1

2(2

='!4

t "4t (

'2pn0|rt|(2

='(2pn0|rt|)2"4t ('2pn0|rt|(2

='!2

t "6t (

'|rt|2(2

> 3'!t|rt|

(2 =3#

2

> 3.

This demonstrates the excess kurtosis of our time series model is in line with empirical

observations.

It is worth noting that the real conditional distribution of "t in our model is not

Gaussian - it has larger kurtosis than Gaussian, and the conditional kurtosis value is

varying according to our earlier analysis. But this should not a!ect our conclusions

that the unconditional kurtosis is greater than 3, yet we could not conclude if the

value is finite.

Autocorrelations of Stochastic Model

4 SUPPLEMENTARY INFORMATION

As absolute values are di"cult to deal with, we approximate them to |st" st!i| &(!i

j=1 |ri!j|)/)i. Autocorrelation of absolute returns at lag l is given by

$l ! '(|rt|" '|rt|()(|rt+l|" '|rt+l|()(

! '(!t " '!t()(!t+l " '!t+l()(

! '(#t " '#t()(#t+l " '#t+l()(

! '""

i,j=1

i!dj!d(|st " st!i|" '|st " st!i|()(|st+l " st+l!j|" '|st+l " st+l!j |()(

!""

i,j=1

i!d!0.5j!d!0.5$ijl,

where $ijl = '!i

m=1 |rt!m|!j

n=1 |rt+l!n| " '!i

m=1 |rt!m|('!j

n=1 |rt+l!n|((. When

j < l,$ijl & 0. When l < j < i+ l, we have

$ijl = $i,k+l,l & '!k

m=1 |rt!m|2 " '|rt!m|(2( & k!2|r| & (j " l)!2

|r|.

When j > i+ l, we have

$ijl = $i,i+l+k,l & '!k

m=1 |rt!m|2 " '|rt!m|(2( & i!2|r|.

Therefore we have

$l !""

i,j=1

i!d!0.5j!d!0.5$ijl

=""

i>k

i!d!0.5(k + l)!d!0.5k!2|r| +

""

k>i

i!d!0.5(k + l)!d!0.5i!2|r|.

Replacing the summation with continuous integral, we obtain

$l !# "

0

# "

ki!d!0.5k(k + l)!d!0.5di dk +

# "

0

# k

0i!d+0.5(k + l)!d!0.5di dk

!1

d" 0.5

# "

0

d1.5!d

(k + l)d+0.5dk +

1

1.5" d

# "

0

k1.5!d

(k + l)d+0.5dk

!B(1.5" d, 2d" 1)

(d" 0.5)(1.5 " d)l2!2d

* l2!2d.

Therefore we obtain $l * l2!2d = l!1 , where %1 = 2 " 2d. For autocorrelation in

squared returns, we have

$#l ! '(r2t " 'r2t ()(r2t+l " 'r2t+l()(

! '(!2t " '!2

t ()(!2t+l " '!2

t+l()(.

There are two extreme situations here:

SUPPLEMENTARY INFORMATION 5

1) If p0 + 2&##t, where #t ,!M

i=1 i!d|st"st!i|, representing the fact that trading

tendency due to past price fluctuations is much higher than background value of p0,

we obtain !t+1 & B#t. Assuming that (st " st!i) and (st!i " st!j) are weakly

correlated, we obtain corr(|st " st!i|, |st " st!j |) & 0. Thus

!2t " '!2

t ( ! 2"

i$j

(ij)!d(|st " st!i||st " st!j|" '|st " st!i||st " st!j |()

& 2""

i=1

i!2d((st " st!i)2 " '(st " st!j)

2().

With similar treatment to $l, we obtain

$#l ! '""

i,j=1

i!2dj!2d((st " st!i)2 " '(st " st!i)(2)((st+l " st+l!j)

2"

'(st+l " st+l!j)(2)( ! l4!4d.

i.e. $#l ! '(r2t " 'r2t ()(r2t+l " 'r2t+l()( ! l!2 where %2 = 4d " 4. For d = 1.12, we have

%2 = 0.48.

2) In the case that p0 - 2&##t, which is approximately the case of our behavioral

model, we again have the decay in squared returns as $#l ! l2!2d.

Therefore, the value of %2 fluctuates between %1 and 2%1 depending on the relative

size of trades triggered by crossing thresholds placed by technical traders.

Overall, we obtained the scaling relations

%1 = 2d" 2

2d" 2 % %2 % 4d" 4

%1 % %2 % 2%1.

3. Determination of parameters in the stochastic process

The ratio of background trading volume p0 against trading volume due to price

impact &#t is approximated from empirical daily trading volume data of a stock.

Since the market landscape has been changing during the last half a century due to

computerization in late 90s and the adoption of high-frequency trading in the recent

decade, there have been structural changes in trading volumes. Hence we only focus

on the 10 year period 1997-2006 during which high frequency trading has not been

massively adopted and program trading has already become a norm.

6 SUPPLEMENTARY INFORMATION

For stock AA, daily trading volume vi is used from 1997 to 2006 with i . [1, 2516].

Since trading volume has increased through the ten-year period, we fit the data of viv.s. i with a linear function yi, and form a new detrended time series of v#i = vi/yi.

From v#i, the average value of the smallest 1% of the data is assumed to be repre-

senting p0, as we know that pt / p0. The average of the whole time series is assumed

to be 'pt(.From our calculation, 'pt( & 3p0, which means 'p0+&##t( & 3p0. Hence we obtain

the relation p0 & 0.5'&##t(. This indicates trading volume due to price impact is

twice of background trading volume p0. This implies the relation

A =1

4'B#t( =

1

4B

500"

i=1

i!d'|st " st!i|(.

Therefore we have

'!t( = A+B500"

i=1

i!d'|st " st!i|(

=5

4B

500"

i=1

i!d'|st " st!i|(

&5

4B

500"

i=1

i0.5!d'|rt|(

=5

4B

500"

i=1

i0.5!d'!t($

2/#

=5

4

$

2/#500"

i=1

i0.5!dB'!t(.

When d = 1.2, we have B & 0.05. The empirical value of '!t( & 0.01 for S&P500,

hence we have A = 15'!t( = 0.002. Analysis on other major stocks gives similar re-

sults. The estimation of A and B implies that the time series has finite unconditional

variance.

4. Using t-distribution for "t instead of normal distribution

In our stochastic process, rt = !t"t, "t is assumed to follow standard normal dis-

tribution. This does not reflect the excess kurtosis of the conditional distribution

discussed above. Hence we use t-distribution of di!erent degrees of freedom to per-

form simulation. The degree of freedom is taken to be 6, 8, 10 so that the excess

SUPPLEMENTARY INFORMATION 7

kurtosis is 3, 1.5, 1. It can be seen from Figure 3 to 5 that the simulation results are

still in agreement with empirical finding with di!erent values of degree of freedom.

The real underlying dynamics has a time varying distribution with excess kurtosis

changing between 0 and 30 shown earlier. Since we demonstrate that fixed distribu-

tions with di!erent excess kurtosis values give the same outcome, we can safely say

the real dynamic process would exhibit the same scaling relations as well.

Figure 3. Plots of simulation results using "t following t-distribution with degree of freedom 6. Left) ACF plot of absoluteand squared returns for simulation v.s. S&P500 index. Simulationgives %1 = 0.38, %2 = 0.51, which are in agreement with empiricalfinding. Right) CDF of absolute returns.

8 SUPPLEMENTARY INFORMATION

Figure 4. Plots of simulation results using "t following t-distribution with degree of freedom 8. Left) ACF plot of absoluteand squared returns for simulation v.s. S&P500 index. Simulationgives %1 = 0.31, %2 = 0.49, which are in agreement with empiricalfinding. Right) CDF of absolute returns.

Figure 5. Plots of simulation results using "t following t-distribution with degree of freedom 10. Left) ACF plot of absoluteand squared returns for simulation v.s. S&P500 index. Simulationgives %1 = 0.38, %2 = 0.55, which are in agreement with empiricalfinding. Right) CDF of absolute returns.

5. Analysis on di!erent ' values in the equation of opinion cluster

formation ct+1 = (n0/|rt|)"

In Eq. (3) of the paper, we define the total number of opinion clusters at time

t+ 1 as ct+1, and it is directly determined by the previous price changes as ct+1 =

SUPPLEMENTARY INFORMATION 9

(n0/|rt|)". Since there is at least one opinion among the agents, and that happens

when every agent does the same buy/sell action, we have the boundary condition

that

ct+1,min = 1 when |rt|max = n0,

which fulfills ct+1 = (n0/|rt|)" for any value of '.

Since there are n0 agents in the market, there are at most n0 opinion groups in

the market. According to Eq. (3), this occurs when

ct+1,max = n0

(n0/|rt|min)" = n0

|rt|min = n("!1)/"0 .

This means whenever |rt| = |rt|min = n("!1)/"0 , the market would have the maxi-

mum of n0 opinions and every agent acts independently.

When ' = 1.0, we have |rt|min = 1, and it means when there is exactly one more

buy/sell trade than sell/buy trade in the market in the previous day, the market will

have totally diverse opinions.

When ' = 0.5, we have |rt|min = n!10 . We have ct+1 =

)n0 + ct+1,max = n0

(for large n0) when |rt| = 1, meaning the market has some convergence in opinions

when there is exactly one more buy/sell trade than sell/buy trade in the previous

day. Hence the agents are more likely to converge in their opinions than the case for

' = 1, and return distribution has a fatter tail.

When ' = 2.0, we have |rt|min =)n0. This means the market would have totally

diverse opinions whenever |rt| is smaller than)n0. Hence the agents are less likely

to converge in their opinions than the case for ' = 1, and return distribution has a

thinner tail.

According to our simulation result, ' = 1 gives the absolute return distribution

closest to empirical findings.

10 SUPPLEMENTARY INFORMATION

Figure 6. a. Percentage of outstanding shares held by institutionalinvestors of the 309 companies. Data is from Yahoo! Finance. Ma-jority of the shares are held by institutional owners as the percentageis mostly larger than 70%. b. Estimated results on the average per-centages of shares being held and traded by the two di!erent typesof traders – fundamentalists and technical traders. Data used arefrom Yahoo! Finance and COMPUSTAT. Left: Average percent-age of outstanding shares held by institutional owners and technicaltraders. The majority stake is held by institutional owners who investfor long-term profits. Right: approximately 80% of trading volume iscontributed by technical traders despite a minority holding by them.This number is calculated assuming fundamentalists have an invest-ment horizon of 2.5 years. It is evident that although fundamentalistsare holding the majority of the stocks, they only contribute a minorityof the trading activity.