Large Events

Quantitative Finance, Vol. 6, No. 1, February 2006, 714

Large stock price changes: volume or liquidity?

PHILIPP WEBER and BERND ROSENOW*

Institut fur Theoretische Physik, Universitat zu Koln, Koln D-50923, Germany

(Received 22 January 2004; in final form 14 April 2005)

We analyse large stock price changes of more than five standard deviations for (i) TAQ datafor the year 1997 and (ii) order book data from the Island ECN for the year 2002. Weargue that a large trading volume alone is not a sufficient explanation for large pricechanges. Instead, we find that a low density of limit orders in the order book, i.e. asmall liquidity, is a necessary prerequisite for the occurrence of extreme price fluctuations.Taking into account both order flow and liquidity, large stock price fluctuations can beexplained quantitatively.

Keywords: Large stock price changes; Trading volume; Liquidity

1. Introduction

The existence of fat tails in the distribution of commodity(Mandelbrot 1963) and stock price changes (Lux 1996,Campbell et al. 1997, Gopikrishnan et al. 1998), togetherwith the relevance of these fat tails for the practicalproblem of risk management, has spurred a large amountof interest in the price process in financial markets. Thereis evidence that the cumulative distribution functionof returns for both individual stocks and stock indicesdecays as a power law with exponent around three(Lux 1996, Gopikrishnan et al. 1998).The idea of power law distributed returns is very

appealing as the appearance of a power law is reminiscentof universality and critical phenomena, thus suggestingthat there might be a basic and universal mechanismbehind the distribution of price changes. Many phenom-enological as well as microscopic models have beendeveloped that are able to explain the main stylizedfacts concerning financial time series (Campbell et al.1997, Takayasu 2002). However, so far there is no generalconsensus concerning the mechanism behind extremeprice fluctuations in the tail of the distribution.In order to understand the mechanism underlying

the empirically observed return distribution in detail,one needs to study the price impact of trades. Besidesthe influence of breaking news, stock prices changeif there is an imbalance between supply and demand.If more people want to buy than to sell, stock priceswill move up, and if more people want to sell than to

buy, they will move down. This relation is quantified bythe price impact function (Hasbrouck 1991, Hausmannet al. 1992, Kempf and Korn 1999, Evans and Lyons2002, Hopman 2002, Plerou et al. 2002, Rosenow 2002,Gabaix et al. 2003, Lillo et al. 2003, Potters andBouchaud 2003, Bouchaud et al. 2004), which describesstock price changes as a conditional expectation value ofthe order flow. The order flow is the difference betweenthe number of shares bought and the number of sharessold in a given time interval.Recently, Gabaix et al. (2003) suggested a quantitative

explanation for the power law distribution of returns.They describe the cumulative distribution of orderflows within time intervals of a fixed length !t by apower law with exponent !V 1:5. They model theprice impact function by a time-independent squareroot function and use it as a predictor for stock pricechanges. According to this argument, the exponent !Gof the cumulative return distribution is twice the exponent!V of the order flow distribution. In this model, largereturns are exclusively caused by large order flows.This approach was criticized by Farmer and Lillo

(2004) because the test for the square root relationbetween order flow and returns presented by Gabaixet al. (2003) lacks power in the presence of correlationsin the order flow and because the functional form usedto describe the price impact of large orders seems to varyfor different stock markets. Instead, Farmer et al. (2004)conclude from a tick by tick analysis that large pricechanges are due to the granularity of the order book,which gives rise to a time varying liquidity.The present study is a contribution to the discussion

concerning the origin of extreme returns. We present an*Corresponding author. Email: [email protected]

Quantitative FinanceISSN 14697688 print/ISSN 14697696 online # 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/14697680500168008

empirical study of extreme stock price changes withintime intervals of length !t 5min. We analyse data forthe 44 most frequently traded NASDAQ stocks containedin the Trades and Quotes (TAQ) data base for the year1997, which is published by the New York StockExchange. In addition, we analyse the year 2002 oforder book data from the Island ECN for the ten mostfrequently traded stocksy. For both data bases, we findlittle evidence that price changes larger than five standarddeviations can be explained by an extreme order flowalone. For the order book data, we are able to reconstructthe price impact function for time intervals with largeprice changes and find that large returns occur only intimes where the liquidity is below average. Large returnsare explained quantitatively when taking into accountboth the order flow and the time-dependent slope of theprice impact function. Our analysis suggests that an unu-sually small slope of the price impact function is a neces-sary ingredient for the explanation of extreme stock pricechanges.The concept of market liquidity encompasses various

transactional properties of markets (Kyle 1985). Marketdepth denotes the amount of order flow innovation whichis required to change prices a given amount. Resiliencydescribes the speed with which prices recover from arandom uninformative shock, and tightness is the costfor turning around a certain amount of shares withina short period of time. Within a dynamic model ofinsider trading and sequential auctions, Kyle (1985)finds that both depth and volatility are constant intime. Glosten (1994) discusses the equilibrium priceschedule in an open limit order book and its robustnessagainst destabilizing strategies. Intraday patterns inprice discovery and transaction cost are discussed in anempirical study (Madhavan et al. 1997). Using a linearparameterization for the price impact of individual trades,a sharp drop of price impact after the first half tradinghour and a slight increase at the end of the trading dayare observed. A weak seasonality of the bid ask spreadand the average quote depth at the bid and ask price wasfound by Chordia et al. (2001) when these liquiditymeasures were averaged over a daily window. In addition,liquidity is found to be influenced by contemporaneousmarket returns, market trends, and market volatility. Ananalysis of the limit order book of the Stockholm StockExchange (Sandas 2001) shows that the depth calculatedfrom the order book is significantly smaller than what isexpected from a regression model. Similarly, a differencebetween hypothetical and actual price impact (Coppejanset al. 2002) is considered as evidence for discretionarytrading, i.e. large trades are more likely to be executedwhen the order book has sufficient depth. Returns arecorrelated with depth in the sense that rising pricesimply a liquidity increase on the offer side of the bookand vice versa for falling prices. Very recently, therelation between volatility and liquidity was studied

for the Euronext trading platform (Beltran et al. 2004).While in the framework of a two-state Markov switchingprocess the virtual price impact was found to increasein the high volatility state, a dynamical analysis basedon a VAR model was not conclusive with respect to theinfluence of volatility on liquidity.In a study of the average price impact for the same data

set studied here, Weber and Rosenow (2003) found thatthe virtual price impact calculated from the order bookdepth is four times stronger than the actual one. Thedifference between the two is accounted for by resiliency.Due to the important contribution of resiliency, pricechanges within time intervals of a finite length!t 5min are more difficult to interpret than a studyon a tick by tick basis. On the other hand, the answerto the question how do stock prices change in a certaintime interval in response to a given order flow is relevantfor the understanding of stock markets, which, after all,operate in real time. Since the distribution of order sizesfollows a power law (Gopikrishnan et al. 2000), largeorder flows are not the result of many small orders addingup, but are rather due to a single large transaction. Forthis reason, an analysis of intervals with fixed lengthshould be asymptotically correct when we are interestedin large orders.In order to have a theoretical framework in which the

origin of price changes can be discussed, we set up a priceequation,

Si Si"1 ci "iQi ui, 1

in the spirit of Glosten and Harris (1988). Here, theindex i labels successive transactions at times ti, Si is thetransaction price, ci the transitory spread component, "tthe slope of the virtual price impact at time ti, and ui is awhite noise which describes the fact that prices change notonly due to trading, but also due to the arrival of newpublic information. As we will mostly be concerned withthe analysis of midquote price changes in the empiricalsection, we let ci & 0 in the following. For the pricechange in an interval with a fixed length !t, one finds(Foster and Viswanathan 1993)

St!t " St X

ti2t, t!t("tiQti

Xti2t, t!t(

uti : 2

We would like to mention that the price impact of anorder volume Qti according to equations (1) and (2)is permanent and hence not easy to reconcile with thefact that long-range correlations in the order flow areobserved empirically (Lillo and Farmer 2004, Bouchaudet al. 2004).In light of equations (1) and (2) there are three

possible causes for large price changes: (i) large orderflows Qti; (ii) large price impacts (small liquidities) "ti ;and (iii) public information uti . We will present empiricalevidence that, in contrast to the theory (Gabaix et al.

yWe analysed the following companies (ticker symbols): AMAT, BRCD, BRCM, CSCO, INTC, KLAC, MSFT, ORCL,QLGC, SEBL.

8 P. Weber and B. Rosenow

2003), the order flow alone cannot explain extreme stockprice changes and that a low liquidity is a necessary con-dition for the occurrence of such events. To this end, weperform an event study of returns larger than five stan-dard deviations. Specifically, we focus on the question ofwhether the order flow is a good predictor for thesereturns under the assumption of a time-independentaverage price impact. We obtain a negative answer tothis question and argue instead that extreme returns areobserved only in times where the market liquidity is belowaverage. The magnitude of large returns can be explainedquantitatively when taking into account both order flowand market liquidity.The TAQ data base contains information about

transaction data such as the number of shares tradedand transaction price as well as information about quotes,i.e. the lowest sell offer (ask price Saskt) and the highestbuy offer (bid price Sbidt). The stock price change orreturn in a time interval !t is defined as

G!tt lnSt!t " lnSt: 3

For the analysis of TAQ data, S(t) is the price atwhich the last transaction before time t took place. Forthe analysis of order book data, S(t) is chosen as themidquote price SMt 12 Sbidt Saskt as we wantto make comparisons with hypothetical price impactscalculated from the order book. The market order flowQ in a time interval is the sum of all signed market ordervolumes executed between t and t!t. For the TAQdata, the sign of a transaction is determined by the Leeand Ready algorithm (Lee and Ready 1991), whichcompares the transaction price with the midquote price.The sign is positive for buy orders (transaction pricelarger than the midquote price) and negative for sellorders (transaction price smaller than the midquoteprice). For the order book data, the data base containsinformation about the direction of a trade. With thisdata set we were able to test the Lee and Ready algorithmby first computing the results using the algorithm andthen performing the same analysis with respect to thebuy and sell information contained in the order bookdata base. On the level of single events, the transactiondirections from the Lee and Ready algorithm deviatefrom the exact ones, but, upon averaging, both methodsyield a nearly identical price impact function.Returns G are normalized by their standard deviation

#G, which is well defined because the cumulativedistribution function of returns follows a power lawwith exponent around three (Lux 1996, Gopikrishnanet al. 1998). Since trading volume is described by acumulative distribution with power law exponent!V 1:5 (Gopikrishnan et al. 2000), its standarddeviation is not well defined. Hence, the order flow Q isnormalized by its first centred moment #Q hjQ" hQiji.

2. Average price impact and large events

The relation between price changes and market order flowis described by the price impact function,

ImarketQ hG!ttiQ: 4

It describes the average price change G caused by anorder flow Qy in the same time interval. We ask whetherthe average price impact function ImarketQ is able todescribe extremely strong price changes G > 5#G. Wedetermined all time intervals with price changes largerthan five standard deviations and checked carefully thatthese large price changes are not due to errors in the database but correspond to real events. While the order bookdata seem to be free of errors, some errors are containedin the TAQ data. We filtered the raw TAQ data againstrecording errors and apparent price changes due to thecombination of data from different ECNs (electroniccommunications networks). We used the algorithm ofChordia et al. (2001), which discards all trades forwhich the difference between trade price and midquoteprice is larger than four times the spread, which is definedas Sask " Sbid. In addition, we checked visually the returnand trading volume time series surrounding the largestprice changes on a tick by tick basis and did not findevidence for data errors after applying the filtering algo-rithm. The data filtering removes about 1% of all transac-tions and has a significant effect on the cumulativedistribution function PG > x. It is a common assump-tion that returns follow asymptotically a power law withPG > x ) x"$. The tail exponent $ was initially consid-ered to be smaller than two. As a consequence, the stan-dard deviation would not be well defined, and the timeaggregation of independent returns would be described bya Levy stable distribution (Mandelbrot 1963). However,subsequent works favour $ around three (Lux 1996,Gopikrishnan et al. 1998). As there is still no final con-sensus about whether a power law is the best descriptionfor the return distribution function, the present paperdoes not want to make a new contribution to this discus-sion. From a methodological point of view, we use powerlaw fits as a simple descriptive method to discuss differ-ences between data sets and to characterize nonlinearities.For the raw data without any filtering, we find $ 2:1,and after applying the filter we find $ 3:9 by fitting astraight line in a double logarithmic diagram. We notethat the filtering algorithm (Chordia et al. 2001) is veryrestrictive in the sense that it discards quite a few eventswhere the TAQ data set reports erratic and strong oscilla-tions (of several #G) of the price which are probably dueto the combination of data from different ECNs. Whilethe price has already reached its new true value in theleading ECN, there may still be limit orders at the oldprice in some smaller ECNs which are exploited byarbitrage traders. While these oscillations are true price

yWe do not include market orders executing hidden limit orders in the definition of Q(t) as we want to make a comparison with theorder book containing visible orders only.

Large stock price changes: volume or liquidity? 9

changes in the sense that they are not due to recordingerrors, they are an artifact of the trading system andwere not included in our analysis.Figure 1 shows both the price impact function and

those events with price changes larger than five standarddeviations #G. We find 1198 such events for the TAQdata base and 210 for the Island ECN data. The largeevents cluster at quite small values of Q where theprice impact function is significantly below G 5#G.For some of these events the signs of Q and G do notagree. We believe that this disagreement is (i) caused bythe inaccuracy of the Lee and Ready algorithm, as suchsituations are less frequent for the order book data, and(ii) due to the analysis of intervals with a fixed lengthrather than the analysis of individual transactions.We note that even for large order flows the average

price impact function is several standard deviations(measured by the statistical error of the mean) belowthe line G & 5#G for the TAQ data and at least onestandard deviation of the mean below the line G & 5#Gfor the order book data. We conclude that order flowalone cannot explain the occurrence of large returnsbut must be accompanied by another effect. An obviousexplanation is that price impact is stronger thanaverage during the occurrence of large returns. Inthe following, we will argue that this is indeed thecorrect explanation.

3. Time varying price impact

As the average price impact function does not providefor a satisfactory explanation of large returns, we studythe time dependence of price impact. In order to achievethis goal, it is insufficient to calculate the price impactfunction as a conditional average over many timeintervals. Instead, one needs to estimate the strength ofprice impact within short time intervals. This goal canonly be achieved when using additional informationabout limit orders. In an electronic market place, marketorders are matched with limit orders stored in theorder book. A buy limit order indicates that a trader iswilling to buy a specified number of shares at a given orlower price, while a sell limit order signals that a traderwants to sell a certain number of shares at a given orhigher price. The buy limit order with the highest pricedetermines the bid price, and the sell limit order withthe lowest price the ask price. The price change due toa given market order is determined by the limit ordersstored in the order book. If a trader places a buy marketorder with volume q, it executes as many limit ordersas necessary to fill that volume. In this way, the orderbook determines the price change due to a single marketorder. We describe limit orders by their density %book&i, tas a function of time t and the discrete coordinate

&i lnSlimit " lnSbid=!&(!& limit buy order,lnSlimit " lnSask=!&(!& limit sell order,

(5

which describes the position of orders in the orderbook. Here, the function x( denotes the smallest integerlarger than x. The total volume of limit orders placed inthe interval i" 1!&, i!&( is given by %booki!&, t!&with integer i. The use of a discrete grid improvescomputational efficiency, reduces the amount ofcomputer memory needed for data analysis, and speedsup calculations by more than one order of magnitude.In our analysis, we chose !& 0:3#G as a compromisebetween computational speed and accuracy.A market buy order with volume q executes limit

sell orders stored in the order book beginning at the askprice until the whole volume q is traded. The lowestremaining sell limit order forms the new ask price.Assuming a constant spread, the relation between returnG and order volume q is given by

qG X&i*G

%book&i, t!&, 6

which is just the market depth for a given return G. Thereturn G in equation (6) is denoted as the instantaneousor virtual price impact of the order q. In the frameworkof the model equation (1), the order book density isapproximated as constant and the depth would be justqG G="t. From equation (6) one sees that the sameorder volume q can be related to quite different returns Gdepending on the function %book&i, t. In the following,we will argue that it is this time dependence of the orderbook which is important for the occurrence of large price

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Volume Q

Retu

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Retu

rn G

(a)

(b)

Figure 1. (a) Average price impact function for the 44most frequently traded NASDAQ stocks in the year 1997 withstandard deviation of the mean. Price changes larger than fivestandard deviations cluster in the region of small volumeimbalance, and all of them are clearly outside the error bars.(b) Same as (a) but for 2002 data from the Island ECN orderbook for the ten most frequently traded stocks.


changes. We note that, from the order book, one obtainsonly information about the price change as a function ofbuy or sell volume. Order book information can berelated to time aggregated signed order volumes onlyunder the assumption (i) that the order book is symmetricaround the midquote price and that (ii) nonlinearities canbe neglected. Both assumptions are generally not satis-fied. For this reason, we will consider either the buy orthe sell volume ~QQ in a given 5-min interval, depending onthe direction of the return in that interval. In this way, ~QQis equal to the volume of buy market orders if G!t > 0 inthe 5-min interval. For G!t < 0, on the other hand, ~QQ isequal to the volume of sell market orders and has anegative sign. We have recalculated Imarket as a functionof ~QQ by averaging with respect to either the sell orthe buy volume. This new ~IImarket is quite similar to theoriginal one.We try to find a quantitative explanation of extreme

price changes by taking into account not only orderflow but also market liquidity as described by marketdepth and market tightness in the beginning of a giventime interval. The depth D is the size of the market orderrequired to change the price by a given amount 5#G and isobtained from equation (6). The tightness T is the costof a round trip (buying and selling a volume of 2#Q overa short period of time). To determine the tightness for agiven time interval, we calculated the virtual price impactIbook ~QQ by inverting the relation in equation (6), anddefine the tightness as

T 1jIbook2#Qj jIbook"2#Qj : 7

In the framework of the model equation (1), the orderbook density is approximated as constant and thetightness would be just T 1=4#Q"t.We compare the ratio of the actual price change G!tt

and the predicted price change,

Gpredt ~IImarket ~QQt, 8

with the inverse liquidity as described by the inversedepth and the inverse tightness. We analyse 5-min returnslarger than five standard deviations for the ten mostliquid stocksy contained in the 2002 Island ECN dataset. The liquidity measures depth and tightness are nor-malized by the average depth D and the average tightnessT calculated from the average order book. In order tocalculate Gpred, we computed ~IImarket ~QQt up toj ~QQj 18#Q as the statistics is insufficient for j ~QQj > 18#Q.Therefore, we had to discard 11 events with j ~QQj > 18#Qfrom this analysis. In addition, for eight events the orderbook did not contain enough limit orders to trade avolume of 2#Q, so we were not able to compute the tight-ness T. In the analysis of the inverse tightness as liquiditymeasure, these events are excluded. For reasons of con-sistency, we also removed two events with D=D > 30 in

figure 2. A scatter plot for events with jG!tj > 5#G isshown in figures 2 and 3. Contrary to the expectationthat the ratio of actual and predicted price change isexplained by a small liquidity, there is only a moderatecorrelation between liquidity and returns for both depthand tightness. This visual impression is confirmed by cor-relation coefficients R2 0:14 and R2 0:11 for depthand tightness, respectively.When studying the price impact of the order flow in

a given time interval, it is not sufficient to invoke theorder book density %book&i, t at one instant of time.In addition, one has to consider changes in theorder book which occur within a given time interval.Weber and Rosenow (2003) showed that the virtualprice impact of a given order volume is roughly fourtimes stronger than the actual one. This difference isdue to additional limit orders placed in reaction to aprice change. Hence, the inclusion of dynamical effectsis crucial for calculating the correct price impact.

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D_/D

G/G

pred

Figure 2. Ratio of actual price change to predicted price changeplotted against the inverse market depth for large 5-min returnscontained in the 2002 Island data. A linear regression (full line)has a correlation coefficient R2 0:14.

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T_

/T

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pred

Figure 3. Ratio of actual price change to predicted price changeplotted against the inverse market tightness for large 5-minreturns contained in the 2002 Island data. A linear regression(full line) has a correlation coefficient R2 0:11.

yWe analysed the following companies (ticker symbols): AMAT, BRCD, BRCM, CSCO, INTC, KLAC, MSFT, ORCL,QLGC, SEBL.


In order to calculate the density of limit orders arrivingin a given time interval, we fix a reference frame by thebid and ask price in the beginning of the interval. Sell limitorders arriving at a price lower than this ask price arecounted as if they were arriving at the ask price, andvice versa for buy limit orders. While %book&i, t describesthe density of limit order volume at a depth & i in thebeginning of the time interval t, t!t(, we defineanother density function %flow&i, t,!t describing thedensity of limit order volume placed at a depth &iminus the limit order volume removed during this timeinterval with

%flow&i hQadd!t &i "Qcanc!t &ii, 9where Qadd!t &i is the volume of limit orders added tothe book at a depth &i, and Q

canc!t &i is the volume of

orders canceled from the book. Thus, %flow&i, t!& isthe net limit order volume arriving in the time intervalt, t!t( and in the price interval i" 1!&, i!&(. Thetotal limit order density available for transactions is thengiven by

%&i, t %book&i, t %flow&i, t,!t: 10The relation between %&i, t and the order flow ~QQ is

~QQG X&i*G

%&i, t!&: 11

By inverting this relation we calculate a price impactfunction Iactual ~QQ. The sell order side of this functionfor ten events with price changes larger than 5#G isshown in figure 4. In figure 5, the average over all suchevents is compared with the average price impact function~IImarket ~QQ. One sees that the slope of Iactual ~QQ is muchlarger than the slope of ~IImarket ~QQ. As a consequence, intime intervals with large price changes there are less limitorders available than on average. Hence, we suggest theuse of the slope of the actual price impact function as ameasure of market liquidity.The curves displayed in figure 4 look quite linear, and

also the average of the Iactual for all large events (seefigure 5) is approximately lineary. Accordingly, we expecta linear fit to the actual price impact functions to be agood description for the strength of price impact. Foreach time interval with jG!tj > 5#G, we define a suscept-ibility '(t) by a linear fit through the origin to the actualprice impact function Iactual ~QQ up to a return G 5#G orG "5#G, depending on the sign of G!t. Although in thesimple model equation (1) the order book density isapproximated as constant and dynamical effects are notincluded, one can formally identify 't "t.Liquidity is measured by the inverse 1='t. In this way

a large slope of the price impact function corresponds to alow liquidity.

In figure 6 the ratio of Gpred and G!t is plotted againstthe susceptibility '=' for all events with jG!tj > 5#G. Wehave normalized ' by ', the slope of a linear fit to theaverage price impact function ~IImarket up to jG!tj 5#G.To make this analysis consistent with the analysis oftightness and depth we removed one event with extremelysmall liquidity ('=' > 60). The data points in figure 6cluster in the vicinity of a linear fit with R2 0:79.In comparison with the two liquidity measures usedabove, this result is a considerable improvement. Webelieve that the improved description of large returnswith the help of ' is due to the fact that the susceptibility' takes into account the dynamics of the order book,which is important for describing liquidity. From thisanalysis, we conclude that the time-dependent slope ofthe price impact function has a large explanatory powerfor the occurrence of extreme price changes.As additional evidence for the idea that the return in a

given time interval is caused by a combination of theorder flow and the time varying liquidity, we discuss

yWe tested Iactual ~QQ for each time interval with price change larger than 5#G for nonlinearities. As a simple descriptive methodwe fitted these curves with power laws. The exponents we found vary between 0.15 and 2.35 with a mean of 1.32 and they scatterwith a standard deviation of 0.41. On the other hand, a power law fit to the average of Iactual for all such events yields an exponentof 1.03, which is approximately linear.

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Figure 4. Price change as a function of buy or sell volume forten of the largest price changes in the Island ECN data.

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Figure 5. Price change as a function of buy or sell volumeaveraged over all time intervals with returns larger than 5#G(connected black circles). The price change averaged over alltransactions (connected grey circles) is much smaller than thatfor the extreme events.


returns as a function of both order flow ~QQ in the directionof the price change and the susceptibility '='. For thisanalysis, the susceptibility ' needs to be redefinedbecause the order book density at a depth &i > G!t doesnot affect the price dynamics. As this effect would weakenthe explanatory power of ' for time intervals with smallreturns G!t < 5#G, we define a new susceptibility 'G bya linear fit through the origin to the actual price impactfunction Iactual ~QQ up to the actual return G!t. In timeintervals with jG!tj < 1#G, the linear fit extends up to#GsgnG!t in order to include enough data points for areliable fit.In figure 7, the average return is plotted as a function

of both market order flow and liquidity measured by'='G. The magnitude of returns is coded in a grey scalefrom bright grey for small returns to black for the largestones. One observes quite sharp borders between regimesof different expected returns, again demonstrating that,for a given order flow, the magnitude of the returndepends on liquidity. In addition, one sees that largereturns occur only for liquidities below average, whilesmall returns can be found even for very large volumes.

4. Discussion

In the last section, we argued that a lack of liquidity isa necessary condition for the occurrence of large returnsand that a combination of low liquidity and orderflow allows for a quantitative explanation of returns. Sofar, we have not yet discussed the possible influence ofpublic information on large stock price changes. In anearlier empirical study using a structural model ofintraday price formation (Madhavan et al. 1997), publicinformation was found to account for 35 to 46% ofthe volatility of transaction price movements. As publicinformation seems to influence price volatility, it isconceivable that public information also influencesunusually large price changes. In the standard pricingmodel equations (1) and (2), the white noise term utdescribes price changes due to the arrival of new publicinformation. In a limit order market, this type of pricechange takes place via cancelation and placementof limit orders. To be specific, if there is good newsabout a given company in the beginning of a 5-min inter-val, e.g. a positive earnings report, prices might go up dueto the cancelation of sell limit orders in the vicinity of thecurrent ask price and the placement of new sell limitorders at a higher price, and vice versa for buy limitorders. In such a situation, we would observe a negative%flow in the vicinity of the ask price and a positive %flowsome distance away from it. Thus, the total densityof limit orders %book %flow is reduced in the vicinity ofthe ask price and the slope ' of a linear fit to the orderbook is reduced as well. In other words, our liquiditymeasure 1='t reflects price movements due to publicinformation. In this sense, our liquidity measure describesthe combined influence of order book depth, resiliency,and public information.In our analysis, we have explained large price

movements by a reduction in liquidity, i.e. we haveargued that there is a causal relation between lowliquidity and large price movements. In a recent studyof the turbulent October 1987 period (Goldstein andKavajecz 2004) the limit order book spread was shownto have increased significantly on October 28, the dayafter the market drop on October 27. Although the sizeof the spread does not affect midquote price changes,this result supports the possibility that causality maygo from large price movements to a reduced liquidity aswell. While we cannot make general statements aboutthe possibility that large price movements may lead toa liquidity reduction, it is clear from our analysis thatduring the 5-min interval with the large price drop areduced liquidity is causally responsible for the largeprice movement in the sense that an average order volumecauses an above average price change. We did notstudy the reason for the below average liquidity insuch a time interval, i.e. we do not address the possibilitythat a price movement at an earlier time may havecaused the reduction of liquidity.In summary, we have studied two alternative

approaches to explain large stock price changes: largefluctuations in trading volume and time varying liquidity.

Volume Q~

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Figure 7. Expected return hG!tti ~QQ,'=' as a function of orderflow ~QQ and liquidity '='. For every combination of ~QQ and '='we plotted (i) the average return if there is more than one match-ing time interval, (ii) the return if there is only one event or (iii)nothing if the combination never occurred. The magnitude ofthe return is coded from bright grey for small returns to blackfor the largest ones.

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/_

Figure 6. Ratio of actual price change to predicted price changeplotted against the slope of the actual price impact functionnormalized by the slope of the average price impact function.The data points cluster in the vicinity of a linear fit withR2 0:79.


We find little evidence that extreme stock price changesare caused exclusively by a large trading volume. Usingorder book data, we have reconstructed the price impactof trading volume for all time intervals with returns largerthan five standard deviations. We find that the priceimpact in these time intervals is much stronger than theaverage one and that such an anomalously large priceimpact is a necessary prerequisite for the occurrence ofextreme price changes. The combined effect of tradingvolume and time varying liquidity can account forextreme price changes quantitatively.

References

Beltran, H., Durre, A. and Giot, P., How does liquidity reactto stress periods on a limit order market? NBB WorkingPaper No. 49, 2004.

Bouchaud, J.-P., Gefen, Y., Potters, M. and Wyart, M.,Fluctuations and response in financial markets: thesubtle nature of random price changes. QuantitativeFinance, 2004, 4, 176190.

Campbell, J.Y., Lo, A.W. and MacKinlay, A.C., TheEconometrics of Financial Markets, 1997 (PrincetonUniversity Press: Princeton, NJ).

Chordia, T., Roll, R. and Subrahmanyam, A., Market liquidityand trading activity. Journal of Finance, 2001, 56, 501530.

Coppejans, M.T., Domowitz, I.H. and Madhavan, A., Liquidityin an automated auction (December 21, 2001). AFA 2002Atlanta Meetings.

Evans, M.D. and Lyons, R.K., Order flow and exchangerate dynamics. Journal of Political Economy, 2002, 110,170180.

Farmer, J.D., Gillemot, L., Lillo, F., Mike, S. and Sen, A., Whatreally causes large price changes. Quantitative Finance, 2004,4, 383397.

Farmer, J.D. and Lillo, F., On the origin of power law tails inprice fluctuations. Quantitative Finance, 2004, 4, C7C11.

Foster, F. and Viswanathan, S., Variations in trading volume,return volatility, and trading costs: evidence on recent priceformation models. Journal of Finance, 1993, 48, 187211.

Gabaix, X., Gopikrishnan, P., Plerou, V. and Stanley, H.E.,A theory of power-law distributions in financial marketfluctuations. Nature, 2003, 423, 267270.

Glosten, L.R., Is the electronic open limit order book inevitable?Journal of Finance, 1994, 49, 11271161.

Glosten, L. and Harris, L., Estimating the components of thebidask spread. Journal of Financial Economics, 1988, 21,123142.

Goldstein, M. and Kavajecz, K., Trading strategies duringcircuit breakers and extreme market movements. Journal ofFinancial Markets, 2004, 7, 301333.

Gopikrishnan, P., Meyer, M., Amaral, L.A.N. andStanley, H.E., Inverse cubic law for the probabilitydistribution of stock price variations. European PhysicsJournal B, 1998, 3, 139.

Gopikrishnan, P., Plerou, V., Gabaix, X. and Stanley, H.E.,Statistical properties of share volume traded in financialmarkets. Physical Review E, 2000, 62, 44934496.

Hasbrouck, J., Measuring the information content of stocktrades. Journal of Finance, 1991, 46, 179207.

Hausmann, J., Lo, A. and MacKinlay, C., An ordered probitanalysis of transaction stock prices. Journal of Finance andEconomics, 1992, 31, 319379.

Hopman, C., Are supply and demand driving stock prices?MIT Working Paper, 2002.

Kempf, A. and Korn, O., Market depth and order size. Journalof Financial Markets, 1999, 2, 2948.

Kyle, A.S., Continuous auctions and insider trading.Econometrica, 1985, 53, 13151335.

Lee, C.M. and Ready, M.J., Inferring trade direction fromintraday data. Journal of Finance, 1991, 46, 733746.

Lillo, F. and Farmer, J.D., The long memory of the efficientmarket. Studies in Nonlinear Dynamics & Econometrics,2004, 8(3), Article 1.

Lillo, F., Farmer, J.D. and Mantegna, R.N., Master curve forprice-impact function. Nature, 2003, 421, 129.

Loretan, M. and Phillips, P.C.B., Testing the covariancestationarity of heavy-tailed time series: an overview of thetheory with applications to several financial datasets.Journal of Empirical Finance, 1994, 1, 211.

Lux, T., The stable Paretian hypothesis and the frequency oflarge returns: an examination of major German stocks.Applied Financial Economics, 1996, 6, 463475.

Madhavan, A., Richardson, M. and Roomans, M., Why dosecurity prices change? A transaction-level analysis ofNYSE stocks. Review of Financial Studies, 1997, 10,10351064.

Mandelbrot, B.B., The variation of certain speculative prices.Journal of Business, 1963, 36, 394419.

Plerou, V., Gopikrishnan, P., Gabaix, X. and Stanley, H.E.,Quantifying stock price response to demand fluctuations.Physical Review E, 2002, 66, 027104[1]027104[4].

Potters, M. and Bouchaud, J.-P., More statistical propertiesof order books and price impact. Physica A, 2003, 324,133140.

Rosenow, B., Fluctuations and market friction in financialtrading. International Journal of Modern Physics C, 2002,13, 419425.

Sandas, P., Adverse selection and competitiv market making:empirical evidence from a limit order market. Review ofFinancial Studies, 2001, 14, 705734.

Takayasu, H., editor, Empirical Analysis of Financial Fluctua-tions, 2002 (Springer: Berlin).

Weber, P. and Rosenow, B., Order book approach to priceimpact. Eprint cond-mat/0311457, 2003.


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