An Agent-Based Model to Explain the Emergence of Stylised Facts in Log Returns
AAn Agent-Based Model to Explain the Emergence of StylisedFacts in Log Returns
Elena Green ∗ Department of Theoretical Physics, National University of Ireland Maynooth,Maynooth, Co. Kildare, IrelandDaniel Heffernan † Department of Theoretical Physics, National University of Ireland Maynooth,Maynooth, Co. Kildare, IrelandSchool of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin 4, Ireland
Abstract
This paper outlines an agent-based model of asimple financial market in which a single assetis available for trade by three different typesof traders. The model was first introduced inthe PhD thesis of one of the authors, see ref-erence [1]. The simulated log returns are ex-amined for the presence of the stylised facts offinancial data. The features of leptokurtosis,volatility clustering and aggregational Gaus-sianity are especially highlighted and studied indetail. The following ingredients are found tobe essential for the production of these stylisedfacts: the memory of noise traders who makerandom trade decisions; the inclusion of techni-cal traders that trade in line with trends in theprice and the inclusion of fundamental traderswho know the “fundamental value” of the stockand trade accordingly. When these three ba-sic types of traders are included log returnsare produced with a leptokurtic distributionand volatility clustering as well as some furtherstatistical features of empirical data. This en-hances and broadens our understanding of thefundamental processes involved in the produc-tion of empirical data by the market. ∗ corresponding author, [email protected] † [email protected] This paper outlines the construction of anAgent-Based Model (ABM). The motivationfor this new model is to add to the understand-ing of the reasons for some of the stylised factsof financial data. It is built in the same vein asthe minimal model by Alfi et al [2]. The goal isto reproduce the key features of financial datawith an even simpler model. Although Alfi etal claim that their model is “minimal”, newmodels can continue to add to our understand-ing of the features of financial data.The stylised facts discussed in this paper arethose of leptokurtic log returns, volatility clus-tering and aggregational Gaussianity. We feelthat these are the most distinctive features oflog returns and they are the ones we are in-terested in explaining. We achieve further un-derstanding of the origins of these features bymeans of the ABM presented here.
Where some models may investigate the effectof market microstructure on the price or log re-turn process, the ABM presented in this paperis chiefly focused on trader behaviour and itseffect on the qualitative form of the log returns.It is concerned with the stylised facts of finan-cial data and the explanation of them from atrader-behaviour perspective. We do not at-1 a r X i v : . [ q -f i n . T R ] J a n empt to create a complete realistic market mi-crostructure. The agents in the model are builtin a way that attempts to capture essential fea-tures in an extremely simplified fashion.In this model, as in the minimal model byAlfi et al [2] and in the Lux and Marchesimodel [3], there is just one asset available fortrade. Each trader may only buy or sell oneunit of the asset at a time and trading takesplace at discrete points in time. At each timestep each trader might buy, sell, or stay inac-tive. The price is calculated from these tradedecisions and all trades are executed at thisprice. The agents do not learn or adapt theirstrategies during the simulation.The price update mechanism is multiplica-tive. After agents express their trade decisionthe excess demand D t is calculated. D t is de-fined as the number of buyers minus the num-ber of sellers at time t ; D t = N buy ,t − N sell ,t .Following other models [4, 5, 2, 6] the price S t at time t is then generated as a function ofthe excess demand D t as follows S t = (cid:18) m D t N (cid:19) S t − where m is a parameter limiting the largestproportional change in the price in one itera-tion of the model and N is the total number oftraders which is fixed for the entire simulation.The parameter m measures the impact oftrading on the price. D t / N is the proportionof traders with the majority opinion ( − ≤ D t / N ≤ m controls how muchinfluence this majority has on the price. When m < m = 0 the price is static. When m > D t / N = − m to 0 < m < N noise traderswith a knowledge of just the most recent pricechange, N with a memory of the last 5 pricechanges and N with a memory of the last21 price changes (think of these as day, weekand month traders). There are N T technicaltraders and N F fundamental traders. Thereare a total of N = N + N + N + N T + N F agents in the model.Whether it is realistic to classify all tradersas belonging to one of some set of predefinedtypes may seem unlikely due to our experienceof a very heterogeneous world. However, a pa-per by Tumminello et al. [7] goes some way tojustify this classification by their finding thattraders do tend to form discrete clusters whichperform trades synchronised in both directionand time. Noise traders base their trading decisions onlyon the current state of the market and not donot take into account any historical prices. Atevery iteration of the model each agent mustmake two decisions. Firstly, each agent decideswhether or not to get involved in trading. Al-lowing agents to be inactive has been foundto be crucial to the presence of the stylisedfacts [6, 8, 9, 10]. For example in the model ofAlfi et al [2], in an attempt to explain the self-organisation of markets into the intermittentstate which produces stylised facts, agents onlytrade if their personal price signal is greaterthan a minimum threshold.We incorporate this concept in our model.The number of agents who are actively trad-ing changes in response to the history of theproportional price change R t , where R t = S t − S t − S t − . n : R t,n = w ( n ) R t + (1 − w ( n )) R t − ,n . The weight w ( n ) = n +1 where n > n to take on a variety of values for dif-ferent agents.The value R t,n is then used by agents todecide if they will trade or not in the cur-rent trading period according to the functionΩ t = Ω( R t,n ) where Ω t is the proportion ofagents that will trade after observing R t,n ,Ω t = 1 + de − a ( | R t,n |− b ) e − a ( | R t,n |− b ) . (1) a , b and d are constant parameters which willbe discussed below. The number of agentswith memory length n that trade at time t is [ N n Ω( R t,n )] where N n is the total numberof traders with memory n and [ · ] denotes thenearest integer function. This new equation 1plays a fundamental role in our model. See thediscussion below and in particular Section 3.1.Thus in this model it is a collective deci-sion about the proportion of traders who traderather than a personal decision by each traderbased on a personal idiosyncratic signal. R t,n close to zero will lead to the minimum numberof permitted active traders submitting a traderequest while R t,n far from zero will lead toΩ t = 1 and all traders will attempt to trade.Figure 1 shows a graph of this function for twodifferent parameter selections.The number of noise agents active in themodel is therefore dynamic but is boundedabove by the total number of traders N n andbelow by (cid:20) de ab e ab N n (cid:21) ≈ [ dN n ] Ω ( R t ) a = 50b = 0.1d = 0.05a = 4000b = 0.02d = 0.05 R t Figure 1: Graph of Ω, equation 1, theproportion of active traders as a function ofthe latest proportional price change, shownfor various parameter values as indicated onthe graph.for the parameter values used in this paper.The parameter d , 0 < d <
1, controls the min-imum proportion of agents who are active atany time. Since the number of active noisetraders is rounded to the nearest whole num-ber it may be 0 if d is small.The steepness of the function is controlledby a . For high values of a the transition be-tween Ω t being at a minimum and a maximumis sharp in R t,n , so the number of active tradersis usually at one of these extremes. For lower a there is more scope for variations in the num-ber of active traders.The parameter b controls the width of theinterval of values of R t,n for which the mini-mum number of agents are active. The widthof this range grows with b .The second decision that the active agentsmust make is whether to buy or sell. Only oneshare can be traded by each agent at each timestep. The decision to buy or sell is made ran-domly according to a probability distribution P t which is based on the previous period’s pro-portional price change R t,n . It is the same foreach noise trader with memory n . P t = P [buy | R t,n ] = 11 + e − uR t,n (2)where u controls the steepness of the func-tion P t . P t is shown in Figure 2 for variousvalues of the parameter u .The parameter u can be interpreted as a“herding strength” parameter. When u issmall, P t is quite flat. This means that the3 = 2u = 5u = 1010.80.60.40.20-1 -0.8 -0.6 -0.4 -0.2 . . . . p r obab ili t y o f bu y i ng Figure 2: Graph of P t , equation 2, theprobability of buying.probability of buying will be at its extremes0 and 1 for only very extreme R t,n . This al-lows for heterogeneity among the noise traderswhen making their trading choices, so the herd-ing strength is low. When u is large P t is steepand so it reaches its extremes of 0 and 1 formore modest R t,n . This means that there ismore agreement in the market, or that there isstrong herding among the noise traders. Thiscauses high volatility as everyone is trading inthe same direction which leads to large jumpsin the price.If u = 0, P t = / for all values of R t,n whichis equivalent to the noise traders choosing thedirection of their trades by simply tossing acoin. In order to produce dynamics which aredependent on the price moves we set u > u < Technical traders or chartists inform their trad-ing choices by indicators from past prices suchas moving averages. They use these indicatorsor signals to attempt to predict future pricemoves. For example on a price chart, if themoving average of the price crosses over theprice it shows that there has been a change inthe trend. Technical traders use signals likethis as a basis for their trading decisions.The chartists in the model by Lux andMarchesi [3] are divided into optimists and pes-simists. The optimists always buy and the pes-simists always sell. They do not analyse histor-ical prices at all. In the Minimal Model by Alfiet al [2] the chartists use a basic moving aver-age of historical prices compared to the currentprice to identify trends.The technical traders in this model use aslightly more involved technical analysis oftrends in the price in order to make their de-cisions as this was found to lead to more real-istic results. They calculate the Moving Av-erage Convergence Divergence (MACD). Al-though the aim of this model is to be as simpleas possible, the rationale for using this morecomplicated technique lies in its realism. Italso leads to richer dynamics in the results asthe traders have a fuller picture of price trendsthan that afforded by the basic moving aver-age. Unlike the original noise traders, the tech-nical traders’ trading decisions are completelydeterministic given the price history.The MACD technique first involves takingtwo EMAs of the price, A and B , of differentlengths l A and l B . Then find the difference M t between these two moving averages. This dif-ference is called the MACD. Next calculate anEMA of the MACD, s t , of length l >
0. These4teps are described by the following equations: A t = w ( l A ) S t + (1 − w ( l A )) A t − B t = w ( l B ) S t + (1 − w ( l B )) B t − M t = A t − B t s t = w ( l ) M t + (1 − w ( l )) s t − The weight w depends on the length; w ( l ) = l +1 . A comparison between the MACD M t and its EMA s t indicates trends in the price. M t > s t indicates that the price is on an up-ward trend and the technical traders will re-spond by buying the stock. M t < s t showsthat the price is on a downward trend and thetechnical traders will respond to this signal byselling. The technical traders therefore amplifythe price trends they detect.This leads to the excess demand by the tech-nical traders: D t = N buy − N sell = N T sgn( A t − B t − s t )where N T is the total number of technicaltraders and sgn( x ) is the sign function givenby sgn( x ) := , x > , x = 0 − , x < . There is a problem with having technicaltraders in the model. The amplification oftrends leads the price to either grow to infinityor drop toward zero very quickly. All tradersalso have unlimited buying power and so ex-tremely large prices are a common occurrence.The price can also drop to extremely small val-ues.Another type of trader is necessary to keepthe market reasonably stable. Fundamentaltraders will fill this role.
In order to have fundamental traders in themodel, there must first of all be a defined “fun-damental value” for the traded asset. In a realtrading environment, the fundamental value ofa stock can be estimated as the current valueof expected future dividend payments. This sort of calculation clearly cannot be performedwithin this model. Other models which havefundamental traders often set the fundamentalvalue to some constant level for the duration ofthe simulation [3, 2]. Allowing the fundamen-tal value to vary or giving fundamental tradersheterogeneous beliefs may allow for more inter-esting dynamics [16, 17].In this model all the fundamental tradersagree with each other on what the fundamentalvalue f is at any moment. f is set to follow arandom walk: f t = f t − (1 + µ f + σ f (cid:15) t ) (3) µ f and σ f are the mean and variance of f , and (cid:15) t is a random number taken from a standardnormal distribution.The fundamental traders know the funda-mental value of the asset. At time t , theycompare the price S t to f t and decide if theasset represents good value. They will buy ifthe price is below the fundamental value andsell if it is above. Their trading strategy pullsthe price back towards the fundamental value.They have the opposite effect on prices to thetechnical traders and help to stabilise the mar-ket. Like the technical traders their decisionsare deterministic once f t is known and S t is re-vealed. All of the fundamental traders trade inthe same direction.The demand of the fundamental traders attime t is therefore given by D t = N F sgn( f t − S t )where N F is the total number of fundamentaltraders in the model. In this section the log returns generated by theABM will be tested for the stylised facts ofempirical log returns. The stylised facts whichhave been found to be present in the syntheticlog return time series include leptokurtosis,volatility clustering and aggregational Gaus-sianity. These features are discussed below.The number of each trader type present in themodel described as Trader Sets A and B are5odel set-up N N N N T N F Trader Set A 4 4 8 2 2Trader Set B 0 0 16 2 2Table 1: The number of the different types oftraders in the ABM for the results presentedbelow. N , N and N are noise traders withmemories of 1, 5, and 21 times stepsrespectively.given in Table 1. The parameters used in themodel are shown in Table 2. The log returns Z produced by the modelhave been found to be consistently leptokur-tically distributed independent to the presenceof technical and fundamental traders. See ex-amples in Figure 4. In each case the log re-turns produced had a leptokurtic shape andthe Shapiro-Francia test rejected normality ata significance level of 0 . simulated dataGaussian fit normalised log returns p r obab ili t y den s i t y (a) Trader Set A simulated dataGaussian fit -6 -4 -2 0 2 4 6 normalised log returns p r obab ili t y den s i t y (b) Trader Set B Figure 3: Examples of the distribution ofnormalised log returns produced by the ABMwhen different numbers of traders are present.A Gaussian fit is shown for comparison.This function adjusts the number of activenoise traders according to the previous pricechange. The value of d is critical. If d = 1,Ω = 1 and the number of noise traders active inthe ABM is constant. Keeping the number of This test is suitable for leptokurtic log returns. Itsorts the data into ascending order and finds the corre-lation between this ordered data and the expected orderstatistics for data from a normal distribution [18]. active noise traders at a constant level resultsin log returns which are well described by aGaussian distribution. This is the case evenwhen technical and fundamental traders, whoare independent of Ω, are also present in theABM.The function Ω, defined in Equation 1, mim-ics realistic trading patterns. In real trading ifthere is a large price move, perhaps as a resultof some news arriving to the market, traderswho are normally not very active may be mo-tivated to review their portfolio and make sometrades. This leads to more log returns close tozero when these more casual investors are nottrading and extreme log returns when every-body wants to trade because they have seen alarge price move.This finding is consistent with other studieswhich have related the leptokutric distributionof log returns to the varying rate of trading [19,20, 21, 22, 23, 24]. High volatility is related toperiods of high trading volume. Since withinthe ABM each trader can only trade one shareat a time, the number of active traders [Ω N N ]+ N T + N F is a proxy for volume. Volatility clustering occurs in the log returnsonly when technical and fundamental tradersare added to the market. Clusters can be iden-tified by eye in Figure 4. trading time steps no r m a li s ed l og r e t u r n s (a) Trader Set A trading time steps no r m a li s ed l og r e t u r n s (b) Trader Set B Figure 4: Examples of the log returnsproduced by the ABM when different numbersof traders are present. They are normalised tobe in units of standard deviation.Examples of the autocorrelation function(ACF) are shown in Figure 5. The magnitudesof log returns generated by Trader Sets A andB are long-term correlated. A slow decay inthe ACF of absolute values of log returns is a6 P Ω MACD fm u a b d l A l B l µ f σ f . .
02 0 .
05 12 26 9 3 · − . au t o c o rr e l a t i on time lag A(Z)A(|Z|) (a) Trader Set A
A(Z)A(|Z|) time lag au t o c o rr e l a t i on (b) Trader Set B Figure 5: Examples of the ACF of the logreturns and their absolute values producedwith different numbers of traders in the ABM.
Another recognised feature of financial data isthat as the time lag is increased the distribu-tion of the log returns begins to more closelyresemble a Gaussian [25, 11, 26]. Specificallyat long time scales, such as annual log returns,the empirical distribution is reasonably fittedby a Gaussian.Let Z t, ∆ = log( S t +∆ ) − log( S t ) . So far the logreturns of successive prices (∆ = 1) generatedby the ABM have been examined. In order tolook for a scale-dependent distribution log re-turns at different time scales ∆ must be found.If ∆ is allowed to increase the shape of the dis-tribution does indeed change, as is shown inFigure 7. lag au t o c o rr e l a t i on |Z|power law, exponent β = -0.3 (a) Trader Set A lag au t o c o rr e l a t i on |Z|power law, exponent β = -0.3 (b) Trader Set B Figure 6: Graphs of the autocorrelation of theabsolute log returns generated by the ABMon doubly logarithmic scales. Both are shownwith a pure power law for comparison. Thepower law provides a good fit in both cases. Δ = 100 Δ = 500 Δ = 1,000 Δ = 10,000 Δ = 100,000Standard Normal -1 -2 -3 -4 p r obab ili t y den s i t y log returns -10 -8 -6 -2 0 4 6 8 10-4 2 Figure 7: Graph of the distribution ofnormalised log returns Z t, ∆ calculated overdifferent lags ∆ for a long simulation( T = 5 · ) with Trader Set B and varying f .The solid black line shows a standard normaldistribution. The vertical scale is logarithmic.7he leptokurtic distribution begins to breakdown for large ∆. At ∆ = 10 ,
000 there is areasonable fit to a Gaussian distribution within3 σ of the mean, but beyond this the tails aremuch too fat to be explained by a Gaussian.However at ∆ = 100 ,
000 all values of Z t, ∆ fallroughly on the Gaussian distribution. The re-sults are shown on a semi-log scale to allow forgreater visibility of the tails.The reason for the aggregational Gaussian-ity lies with the fundamental value f . The logreturns of f have a normal distribution due toits dependence on the Gaussian random num-ber (cid:15) t , see equation 3. At large ∆ large eventsbecome rare and the consistent Gaussian in-fluence of f on the fundamental traders domi-nates Z t, ∆ . At large lags any short term trendsinstigated by technical traders are not felt andthe shape of the distribution is influenced prin-cipally by the fundamental traders.To confirm that this is the reason for theaggregational Gaussianity, the same analysiswas carried out on log returns generated bythe ABM with f set to a constant value forthe entire simulation. Figure 8 shows the re-sult. Even at large ∆ there is no agreementwith a Gaussian distribution in this case. Thisis because there is no Gaussian influence onany traders and the log returns retain their fattails. These results are similar to those foundby Alfi et al in the analysis of their model [16]. Δ = 100 Δ = 500 Δ = 1,000 Δ = 10,000 Δ = 100,000Standard Normal p r obab ili t y den s i t y log returns -1 -2 -3 -4 -5 -20 -15 -10 -5 0 5 10 15 20 Figure 8: Graph of the distribution ofnormalised log returns Z t, ∆ calculated overdifferent lags ∆ for a long simulation( T = 5 · ) with Trader Set B and constant f . The solid black line shows a standardnormal distribution. The vertical scale islogarithmic. In this paper a new ABM has been developed.The motivation behind this new ABM was tofind a very simple model which can reproducesome of the key stylised facts of empirical fi-nancial time series. This enriches the under-standing of the origin of the stylised facts inempirical data. This ABM focuses on traderbehaviour rather than market microstructure.As is the case with many ABMs useful resultsare only obtained from this model in a limitedarea of the parameter space [14, 27].Leptokurtic log returns are generated by thenoise traders in the ABM. It has been shownthat the varying number of active traders is thesource of this feature in the results. This mim-ics the behaviour of real traders and offers anexplanation for the leptokurtosis of empiricallog returns.Volatility clusters come from having somememory in the noise traders along with techni-cal traders who analyse historical prices look-ing for patterns. Technical traders bring mem-ory to the system as they detect trends andamplify them. Neither the technical tradersalone nor the memory of noise traders aloneis enough to produce this feature. Both ofthese are necessary. The presence of technicaltraders necessitates the presence of fundamen-tal traders to keep the price reasonably sta-ble. It is also the fundamental traders whotrigger the bursts of high volatility. Three es-sential ingredients have thus been identified forthis model to produce this stylised fact of fi-nancial data. They are the memory of thenoise traders, the inclusion of technical traderswho trade in line with trends in the price andthe inclusion of fundamental traders who tradeaccording to the “fundamental value” of thestock.Transition of the distribution of the log re-turns to a Gaussian has also been identified asa statistical property of the log returns gener-ated by the ABM. This is caused by the fun-damental value and indicates that many realtraders may also be under the influence of aGBM process.We have shown that some of the most dis-8inctive stylised facts of financial data havebeen produced by this model with just a fewsimple elements. This contributes to our un-derstanding of the processes behind some of themain features of empirical data, in particularthe leptokurtic distribution, volatility cluster-ing and aggregational Gaussianity.
Acknowledgments
We thank WilliamHanan for useful discussions on the role ofNoise Traders during the early stages of thiswork. The research was supported in partby Science Foundation Ireland under grantnumber 08/SRC/FM1389.
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