Market impact and trading profile of large trading orders in stock ...

2data sets each transaction is labeled with codes identifyingthe members of the exchange who made the transaction.In most cases the member is acting as a broker,i.e. they are handling a trade for another institutionwho is not a member of the exchange. In other cases themember may trade for their own account. Thus a singlemembership code may lump together trades from manydifferent institutions, originated by many different tradingstrategies. As a consequence several hidden ordersfrom different trading accounts may be active with thesame broker at the same time, making it impossible tobe certain that all orders are correctly identified.The detection problem is aided by the fact that thesize of hidden orders has a heavy-tailed distribution, andlarge hidden orders can cause dramatic changes in therate at which a given membership code participates intrades to either buy or to sell. The method we use wasoriginally developed for detection of local stationary regionsin physiological time series [28]. As we use it here, itlooks for time intervals in the time series of orders comingthrough a given membership code when the firm acts asa net buyer or seller at an approximately constant rate.As in reference [27] we interpret these series of tradesas belonging to hidden orders. This method can neverbe perfectly accurate, but a variety of tests performedin reference [27] suggest that the reconstruction is goodenough to recover the most important statistical propertiesof hidden orders reasonably well. An importantadvantage of this approach is that we are able to studylarge samples of hidden orders coming from the wholemarket rather than only a subset belonging to a specificinstitution.The paper is organized as follows. In Section II wediscuss our data sets, the algorithm for detecting hiddenorders and the investigated variables. In Section IIIwe discuss the statistical properties of the variables characterizingthe hidden orders. In Section IV and V wepresent our empirical results on the impact of hidden ordersand on their trading profile, respectively. Section VIconcludes.II.DATA AND INVESTIGATED VARIABLESOur databases contain the on-book (SETS) markettransactions of the London Stock Exchange (LSE) fromJanuary 2002 to December 2004 and the electronic openbookmarket (SIBE) of the Spanish Stock Exchange(BME, Bolsas y Mercados Españoles) from January 2001to December 2004. Roughly 62% of the transactionsat the LSE are executed in the open book market androughly 90% of the transactions at the BME are executedin the electronic market.We have initially considered a subset consisting of themost heavily traded stocks in the two markets, 74 stockstraded in the LSE and 23 stocks traded in the BME. Forboth markets we have considered exchange members whomade at least one trade per day for at least 200 tradingdays per year and with a minimum of 1000 transactionsper year. This filter yielded approximately 60 exchangemember firms per stock. We then applied the algorithmfor detecting hidden orders described in Ref. [27], whichwe have already discussed, to identify hidden orders thatconsist of at least ten transactions. It is worth notingthat the detected patches are not necessarily composedof the same type of trades (buy or sell) but that at least75% of the transacted volume in the patch must have thesame sign. The algorithm detected 90,393 hidden ordersin the LSE and 55,309 in the BME.This study is based entirely on trades that take placethrough a continuous double auction. “Continuous”refers to the fact that trading takes places continuouslyand asynchronously, and “double” to the fact that bothbuyers and sellers are allowed to place and cancel ordersat any time. There are two fundamentally different waysto execute an order in such a market. One is to use alimit order, in which an order is placed inside the orderbook, which is essentially a list of unexecuted orders atdifferent prices. The other is to place a market order,which we define as any order that results in an immediatetransaction. Every transaction involves a marketorder transacting against a limit order. A given real ordermight act as both, e.g. part of it might result in animmediate transaction and part of it might be left in theorder book. We only consider transactions, so in the exampleabove we would treat the first part as a marketorder and treat the second part as a limit order, but thesecond part will enter our analysis only if it eventuallyresults in a transaction. The LSE database allows us toidentify whether the initiator of the transaction was thebuyer or the seller. For BME this information is not availableand we infer it with the Lee and Ready algorithm[29]A hidden order is characterized by several variables.These are• The execution time T (in seconds) of the hidden order,measured as the trading time interval betweenthe first and the last transaction of the hidden order.• The number N of transactions of the hidden order.We consider hidden orders of length N > 10.• The volume V of the hidden order defined asV =N∑v j , (1)j=1where v j is the signed volume of each transactionof the hidden order. For buy trades v i > 0 and forsell trades v i < 0. We consider the hidden order tobe a buy order if V > 0 and a sell order if V < 0.The buying/selling nature of a hidden order is thusencoded in its sign, ɛ = sign(V ). The volume is theproduct of the number of shares times the price andis measured in Pounds (LSE) or in Euro (BME).

3• The volume fraction of market orders f mo . A hiddenorder can be implemented with very differentliquidity strategies, i.e. with different compositionsof market and limit orders. In order to quantifythis we define the fraction (in volume) of marketorders within a hidden order as∑ Nj=1f mo =|v j,mo|∑ Nj=1 |v , (2)j|where v j,mo is the traded volume at each transactiondone through market orders. Values of f moclose to zero mean that the broker completed thehidden order by using mainly limit orders, whilevalues of f mo close to one imply the broker usedmainly market orders during the execution of thehidden order.• The participation rate α of a hidden order definedas∑ Ni=1α =|v i|, (3)V Mwhere V M is the unsigned volume of the stocktraded in the market concurrently with the hiddenorder. Values of α close to zero imply the hidden orderwas negligible compared to the activity in themarket, while values of α close to one mean thatmost of the activity in the market came from thetransactions of the hidden order.Summarizing, we expect that the market impact of ahidden order is a functionr = f(N, V, T, f mo ), (4)plus possibly other variables specific of the stock, suchas the participation rate, the capitalization, the volatility,or the spread. We will now try to simplify this byunderstanding some of the relationships between the dependentvariables and by conditioning on some of themin our analysis. Note that in all the analyses and figureswe compute error bars as standard errors. It should beborn in mind that this procedure underestimates the errorsdue to the heavy tails of the fluctuations and due topossible long-memory properties of the data.III.STATISTICAL PROPERTIES OF HIDDENORDERSWe investigate the statistical properties of the variablescharacterizing hidden orders. Ref. [27] considered a set of3 most capitalized stocks traded at the BME and studiedthe probability distribution of the variables characterizingthe hidden orders and the scaling relations betweenthese variables. In Ref. [27] no restriction on the lengthor on the fraction of market orders was set on the hiddenorders. The authors of Ref. [27] found that the distributionof hidden order size is fat tailed and consistent witha distribution with infinite variance. They also showedthat this broad distribution is due to an heterogeneity ofscales among different brokerage firms rather than to theheterogeneity of scales within the hidden orders of eachbrokerage firm. By using Principal Component Analysis(PCA) on the logarithm of the variables characterizingthe hidden orders, it was found that N, V and T arerelated through scaling relationshipsN ∼ V g1 , T ∼ V g2 , N ∼ T g3 . (5)where g 1 ≃ 1, g 2 ≃ 2 and g 3 ≃ 0.66 for 3 highly capitalizedstocks in the BME and including all hidden orders.We repeat the two dimensional PCA analysis of [27] onour much larger data set. Figure 1 shows the value ofthe three exponents for all the stocks as a function ofthe number of hidden orders per year. We observe thatfor stocks with a small number of hidden orders the heterogeneityin the value of the exponents is pretty large,while, as the number of hidden orders detected by the algorithmincreases, the exponent estimations become lessnoisy and tend to converge to similar values. Moreoverfor BME stocks there is a clear trend of the exponents asa function of the number of hidden orders. In order tomeasure market impact in a statistically reliable way, wepool together data from different stocks. We need thereforean homogeneous sample of stocks. To this end in thefollowing analysis we restrict our dataset to those stocksfor which our algorithm detects at least 250 orders peryear. These stocks are TEF, SAN, BBVA (as in [27]) butalso REP, ELE, IBE, POP and ALT for the BME marketand AZN, BSY, CCH, DVR, GUS, KEL, PO, PSON,SIG, TATE and TSCO for the LSE market. Moreover, inthis paper we will focus mainly on short hidden orders,considering the set of hidden orders of time duration Tsmaller than one trading day. The reason for this choice,detailed below, is to obtain stable statistical averages forthe market impact. Applying these two restrictions, weobtain a final dataset that contains 14,655 hidden ordersin the BME and 11,165 orders for the LSE (see Table I).We repeat the two-dimensional PCA analysis of [27] onthe pooled set of hidden orders from different stocks. Wefind for the BME market the following exponentsg 1 = 0.81 (0.79; 0.82), (6)g 2 = 1.57 (1.43; 1.72), (7)g 3 = 0.67 (0.65; 0.68), (8)where quantities in parenthesis are 95% confidence intervalsobtained through bootstrapping the data. Theserelations explains 83%, 61% and 80%, respectively, of thevariance observed in the data. For the LSE dataset wegetg 1 = 0.99 (0.98; 1.01), (9)g 2 = 2.41 (2.29; 2.52), (10)g 3 = 0.58 (0.57; 0.59), (11)

-810 100N10 1004g 1g 2g 31,510,50432100,80,60,40,20LSE50 100 200 5001,510,50432100,90,80,70,60,50,4#hidden orders (per year)BME50 100 200 500Figure 1: Exponents g i (i = 1, 2, 3) of the allometric relationsof Eq. 5 for each of the stocks considered in our LSE and BMEdatabases and for hidden orders with N ≥ 10 and T < 1 day,as a function of the number of detected hidden orders peryear. Error bars are 95% confidence intervals obtained bybootstrapping the data. In the analysis of market impact weconsider only stocks with at least 250 hidden orders per year(those in the white area of the figure).Table I: Statistics of the hidden order ensembles used in thepaper. Only hidden orders with T < 1 day and N > 10transactions are used.Market # orders 〈N〉 〈f mo〉 〈α〉 〈R〉 〈R〉 fmo>0.8BME 14,655 95.58 0.52 0.17 1.127 3.983LSE 11,165 97.53 0.53 0.34 0.587 2.156P(!)P(f mo)65432100 0,2 0,4 0,6 0,8 132,521,510,500 0,2 0,4 0,6 0,8 1BMELSE00 0,2 0,4 0,6 0,8 1f mo0,60,50,40,30,20,1〈α|fmo〉Figure 2: Ensemble statistics of the fraction of market ordersf mo and participation rate α of hidden orders in both theBME and LSE. Left panels show the probability distributionfunction of both parameters, while the right panel shows theconditional average of the participation rate conditioned on agiven value of f mo.lot in terms of the fraction of market orders used to completethem. In the investigation of the market impact ofhidden orders we will consider hidden orders characterizedby a restricted set of values of f mo to better characterizetheir profile with respect to the fraction of marketorders used to complete the hidden order. Specifically wewill use f mo > 0.8 (large fraction of market orders used)and f mo < 0.2 (large fraction of limit orders used). Thereasons why we expect this distinction to be criticallyimportant will be described in the next section.IV.MARKET IMPACTand these relations explain 88%, 75% and 86%, respectively,of the variance. These allometric relations areroughly consistent with those obtained in Ref. [27].The left panels of Fig. 2 show the probability densityfunction of f mo and of the participation rate α. Weobserve that the distribution of the fraction of marketorders is rather broad and is roughly centered aroundf mo = 0.5. In addition two peaks are observed forf mo ≃ 0 and f mo ≃ 1. For the BME the participationrate has a peak around α = 5%, while for the LSE thedistribution of α is broader, and peaks at a value closerto 20%. This value is pretty large and we do not havean explanation for the difference in the participation ratebetween the two markets. Finally, the two parameters αand f mo are not independent. Figure 2 shows the expectedvalue of α conditioned on f mo . For the BME theexpected value of α is almost constant except for verysmall values of f mo . In contrast, for the LSE the dependenceis much stronger. The participation rate is higherwhen f mo is at either of its extremes.The bottom left panel of Fig. 2 shows that f mo has abroad distribution. Hidden orders can therefore differ aA. DefinitionThe main focus of this paper is the empirical measurementof the market impact of hidden orders. Givena hidden order traded on stock i between times t andt + T , we measure the market impact by considering thechange in the log price of the stock between time t andtime t + T , i.e.r i (t, T ) = log p i,t+T − log p i,t , (12)where p i,t is the price of stock i at time t. We have usedfor p i the midprice, but our results do not depend onthis. Our objective is to study how r i (t, T ) changes asa function of the main properties of the hidden order.Different stocks have different scales of their price fluctuations.In order to be able to take the average of marketimpact across different stocks, we rescale it by dividingby the mean value of the spread s i of the stock duringthe year, where the spread is the difference between thelowest selling price (ask) and the highest buying price(bid). Specifically, we define the rescaled market impact

5Return0,0020-0,002-0,004-0,006-0,008FTSE100IBEX352001 2002 2003 2004Year50001Days10000ing the years 2001-2004 stock markets were in a substantialdecline for more than two years, only recovering atthe end of 2003 and 2004 (see the inset of Figure 3).Figure 3 shows the conditional average 〈R|T 〉 of therescaled market impact of the hidden orders as a functionof their time duration T . We observe that for Tlarger than one day, rescaled impact is on average negative,irrespectively of the sign of the hidden order. Thereason for this phenomenon is that market-wide movementswere mostly negative for values of T larger thanone day. Only for hidden orders of duration close to orbelow one day do we observe negligible changes in marketindexes when compared to price changes during hiddenorder completion. This is the main motivation of ourchoice of restricting our study to hidden orders of durationT less or equal to one day 1 .Figure 3: Conditional average 〈R|T 〉 of the rescaled impact ofhidden orders (Eq. (13)) as a function of their time duration T(symbols) compared to the average return of the stock marketindex over random periods of the same time duration (solidlines). The inset shows the price of the FTSE100 and IBEX35indices over the period of study. In this figure we are usingall detected hidden orders without any conditioning on T orf mo values but with N > 10.asR i (t, T ) = ɛ i r i (t, T )/s i . (13)where, as before, ɛ i = +1 for a buy hidden order andɛ i = −1 for a sell hidden order. Although we observe asmall asymmetry between the market impact of buy vs.sell orders, similar to that observed elsewhere [31], forthe purpose of our study here we lump together buy andsell hidden orders in order to obtain better statistics.B. The noisy nature of market impactWhile a given hidden order is trading there are typicallymany other orders trading at the same time, as wellas news arrival, and thus there is a considerable amountof noise in the price change associated with any particularhidden order. The price change associated with a hiddenorder functionally depends on several factors, which canbe writtenr i (t, T ) = R[r M (t, T ), ρ i (t, T ), η i (t, T )]. (14)where r M corresponds to market-wide movements [25],ρ is the average market impact of the hidden order, andη i is the background uncorrelated noise coming from thetrading of the rest of the market [12]. While the backgroundnoise can be controlled by taking averages overdifferent orders with the same properties or restrictingour analysis to very small values of T , market-wide movementsremain large, especially for large values of T . Dur-C. Impact of limit orders vs. market ordersIt is important to stress that market impact comesabout through changes in supply and demand, and thatthis causes a strong a priori difference in the impact oneexpects to observe in the execution of a limit order vs. amarket order. For example consider buy orders. A buymarket order reflects an increase in demand at the currentprice. If sufficiently large it will cause a positive pricechange. Since in a continuous double auction market ordersalways execute against limit orders, this implies thatthe sell limit order that the buy market order executesagainst will generate a positive market impact. We thereforeexpect that executed limit orders have the oppositeimpact of market orders: Buying drives the price downand selling drives it up.The problem with this line of reasoning is that we areconsidering only executed limit orders, which creates astrong selection bias. To measure the impact of limitorders correctly we need to condition on all orders thatare placed, rather than only on those that are executed.When this is done the impacts for limit orders should beroughly the same as for market orders, as otherwise itwould be possible to make a profit by simply using limitorders instead of market orders. If a buy limit order isplaced below the current price it is executed only if theprice drops. The probability of execution of a limit orderdepends on future price movements: under an adverseprice movement the probability of execution is higherthan for a favorable price movement. This is caused inpart by the mechanical dynamics of a random walk, butalso by asymmetric information: Placing a limit ordergives others the option of executing at their will, when1 Other authors [25] have proposed to use industrial sector indexesas proxies for market-wide movements of a given stock and thusthe study can be extended to larger values of T . We do not followthis procedure due to lack of that information.

6they have information that indicates it is favorable todo so. This phenomenon is called adverse information.When this is properly taken into account, limit ordershave impact in the direction one would expect, i.e. buyinghas positive impact and selling has negative impact[32, 33]. Furthermore the magnitude of the impact oflimit orders when the selection effects are properly takeninto account is comparable to that of market orders.For the BME we have a record of transactions but notof orders. Thus to measure market impact and avoid theselection bias associated with executed limit orders weare forced to use only those hidden orders that are predominantlybuilt out of market orders. For consistencywe analyze both the BME and the LSE data in the sameway.In Table I we show the mean value of the rescaledmarket impact R of Eq. (13) for hidden orders of durationless than one day. We also show the mean value〈R〉 fmo>0.8 of the rescaled market impact computed overthe set of hidden orders with a large fraction of marketorders (f mo > 0.8). 〈R〉 fmo>0.8 is significantly largerthan 〈R〉 indicating that hidden orders mainly composedby market orders have on average a larger market impactthan hidden orders composed of both limit and marketorders.D. Impact vs. NFigure 4 shows the average over all hidden orders ofthe rescaled market impact 〈R|N〉 as a function of theconditioning variable N. This grows slightly as a functionof N, but one must keep in mind that the meaning of thisis difficult to interpret in view of the discussion above,since we are averaging together a roughly equal numberof market orders and executed limit orders.To investigate the average market impact and minimizethe effect of the selection bias, we divide the data into twogroups: liquidity providing hidden orders, with f mo 0.8. As expected, for the former group the market impactis on average negative, while for the latter it is positive.Using ordinary least squares, we find that for both groupsthe dependence of 〈R|N〉 on N is well described by thepower law|〈R|N〉| = A N γ . (15)The estimated parameters are in Table II. In summary,we find that the market impact of hidden orders dominatedby market orders is consistent with〈r|N〉 ∝ ɛsN γ (16)where ɛ is the sign of the order and s is the spread. Forhidden orders dominated by limit orders the market impactis very similar to minus the impact of hidden ordersdominated by market orders.Table II: Parameters of the fitting of the market impact withEq. 15.Market A fmo>0.8 γ fmo>0.8 A fmo

8BMELSEVI.CONCLUSIONSv(t) / < v(t) >1,31,21,110,90,80 0,2 0,4 0,6 0,8 1t / T1,31,21,110,9HiddenTotal0,80 0,2 0,4 0,6 0,8 1t / TFigure 6: Trading profile inside the hidden order. Averagevolume of the transactions within the hidden order dividedby the average volume in the hidden order as a function ofthe normalized time t/T . Circles are the results for all hiddenorders, while squares are the volume traded in the market (inthe same stock) concurrently with the hidden order. Data isonly for hidden orders with f mo > 0.8.In this paper we have empirically studied the mainproperties of the impact and of the trading protocolof intradaily hidden orders using a large fraction of eithermarket orders or limit orders. We have found thatthe temporary impact of hidden orders is concave androughly described by a square root function of the hiddenorder size. Moreover the price reverts after the completionof the hidden order in such a way that the permanentimpact is equal to roughly 0.5 − 0.7 of the temporary impact.We have also studied how the order is completed intime and we have shown that more volume of the hiddenorder is traded at the beginning and at the end of thehidden order. When we take into account that hiddenorders are more likely to start at the beginning of a dayand are more likely to end near the end of the day, thisroughly matches the volume traded in the market.The fact that we observe similar behavior in both theLondon and Spanish stock exchanges, and that othershave also observed this in the New York Stock Exchange,suggests the possibility of a “law” for market impact. Itwill be very interesting to see whether this hypothesizedlaw continues to hold up under future studies.BMELSE56AcknowledgmentsP(t)432100 0,2 0,4 0,6 0,8 1t (days)54321t init fin00 0,2 0,4 0,6 0,8 1t ( days )Figure 7: Initial and final times of the hidden orders. Probabilitydistributions of the initial time t i and final time t f ofthe hidden orders, measured with respect of the time of theday. Data is only for hidden orders with f mo > 0.8FL and JDF acknowledge Bence Toth and HenriWaelbroeck for useful discussion. GV, FL, andRNM acknowledge financial support from the PRINproject 2007TKLTSR “Computational markets designand agent-based models of trading behavior”. EM,GV, FL, and RNM acknowledge Sociedad de Bolsasfor providing the data and the Integrated Action Italy-Spain “Mesoscopics of a stock market” for financial support.EM acknowledges partial support from MEC(Spain) through grants Ingenio-MATHEMATICA andMOSAICO and Comunidad de Madrid through grantSIMUMAT-CM. JDF, JV and AG would like to thankBarclays Bank, Bill Miller, and National Science Foundationgrant 0624351 for support. Any opinions, findings,and conclusions or recommendations expressed in thismaterial are those of the authors and do not necessarilyreflect the views of the National Science Foundation.[1] R. Kissell and M. Glantz. Optimal Trading Strategies:Quantitative Approaches for Managing Market Impactand Trading Risk. American Management Association,(2003).[2] J. B. Berk and R. C. Greene, Journal of Political Economy,112 1269 (2004)[3] Y. Schwarzkopf and J. D. Farmer, Time Evolutionof the Mutual Fund Size Distribution, preprint atarXiv:0807.3800 (2008).[4] J.P. Bouchaud, J. D. Farmer, and F. Lillo, How marketsslowly digest changes in supply and demand, in Handbookof Financial Markets: Dynamics and Evolution,T. Hens and K.R. Schenk-Hoppé (Editors), 57 (North-Holland, 2009).[5] F. Lillo, J. D. Farmer and R. N. Mantegna, Nature 421129-130 (2003).

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