Heavy tailed distributions in closing auctions
HHeavy tailed distributions in closing auctions
M. Derksen , , B. Kleijn and R. de Vilder , Deep Blue Capital N.V., Amsterdam Korteweg-de Vries Institute for Mathematics, University of Amsterdam
December 21, 2020
Abstract
We study the tails of closing auction return distributions for a sample of liquidEuropean stocks. We use the stochastic call auction model of Derksen et al. (2020a),to derive a relation between tail exponents of limit order placement distributions andtail exponents of the resulting closing auction return distribution and we verify thisrelation empirically. Counter-intuitively, large closing price fluctuations are typicallynot caused by large market orders, instead tails become heavier when market ordersare removed. The model explains this by the observation that limit orders aresubmitted so as to counter existing market order imbalance.
Key Words:
Closing auction; Closing prices; Stochastic models; Price formation;Heavy tails;
During the trading day, most securities change hands in continuous double auctions, inwhich buy and sell orders are immediately matched if possible. However, to determineopening and closing prices, call auctions are often conducted. In a call auction, ordersare aggregated for an interval of time, after which all possible transactions are conductedagainst a single clearing price that maximizes trading volume. In this paper we studythe tails of closing auction return distributions.Nowadays it is widely recognized that distributions of (stock) price changes exhibit heavytails: extreme price changes (of e.g. more than three standard deviations) are much morelikely than in a Gaussian model or other models with exponentially decaying tails. Thisissue was first adressed by Mandelbrot (1963) in his analysis of cotton prices, where heproposed L´evy stable distributions to model price fluctuations. It is generally assumedthat the tails follow a power law asymptotically. That is, the distribution of a return X a r X i v : . [ q -f i n . T R ] D ec ver some time interval satisfies , P ( X > x ) ∼ Cx − a , as x → ∞ , (1)where C > L ( x ))and a > tail exponent , determining how heavy the tail is . In early work (Fama,1965), the exponent a was believed to be below 2 for stock prices (in line with the stabledistributions of Mandelbrot (1963)). However, subsequent analyses have shown that theexponent is more likely to be around 3 on intraday time scales (see e.g. Gopikrishnan etal. (1998, 1999); Gu et al. (2008); Pagan (1996); Plerou and Stanley (2008), among manyothers). Although it is generally accepted to model the tails as power laws, the exactfunctional form is also subject of debate. For example, Malevergne et al. (2005) concludethat the tails decay slower than stretched exponential distributions, but somewhat fasterthan power laws. In this paper, we do not aim to answer this question, but use powerlaws because they describe the tails in enough detail for our analysis. Theoretically,the functional form in equation (1) is justified by extreme value theory, in the Fr´echet(heavy tailed) case (see e.g.
Embrechts et al. (2003)).Although most part of the relevant literature focuses on description of the tails of stockprice return distributions, some effort has gone towards explanations of this tail be-haviour. Gabaix et al. (2003, 2006) argue that large price fluctuations are due to largeorders submitted by large market participants. However, Farmer et al. (2004) and Weberand Rosenow (2006) study the issue on the microscopic level and find that large returnsare not due to large transactions, but instead are caused by big gaps in the order book, i.e. fluctuations in liquidity. Mike and Farmer (2008) propose a simulation based modelfor continuous trading, which suggests heavy tails in return distributions are caused bymarket microstructure effects, such as heavy tails in limit order placement and longmemory in order flow. More theoretically, Bak et al. (1997) and Cont and Bouchaud(2000) propose models linking heavy tails to herd behaviour.
In this paper, we use the model of Derksen et al. (2020a) to study the distribution ofreturns in the closing auction. In the model, limit orders are submitted to the auctionrandomly, with a limit price that is sampled from an order placement distribution F A (for sell orders) or F B (for buy orders). We study the closing auctions of liquid Euro-pean stocks listed on Euronext exchanges and find that both return distributions andorder placement distributions exhibit heavy tails, with different tail exponents. Zovkoand Farmer (2002) conclude ‘It seems that the power law for price fluctuations should berelated to that of relative limit prices, but the precise nature and the cause of this rela-tionship is not clear.’ Here, we solve this problem in the context of the closing auction: Here, ∼ denotes asymptotic equivalence , defined as f ∼ g ⇔ lim x →∞ f ( x ) g ( x ) = 1.
2e provide analytical relations between the tails of order placement distributions andthe tails of the closing price return distribution. In a version of the model without mar-ket orders, the tails of the closing price distribution behave as the product of the tailsof the order placement distributions F A and F B . When we incorporate market orders,this relation changes, depending on a proportionality relation between market order andlimit order imbalances. We empirically verify the relations between tail exponents oforder placement and auction return distributions predicted by the model.In theory, large market orders are a possible cause of large price fluctuations. We showhowever that this is typically not the case in closing auctions, which is our second im-portant result. Somewhat counter-intuitively, the empirical study shows that closingauction return distributions would have heavier tails if market orders are removed, sug-gesting that market orders have a stabilizing effect on price formation in closing auctions.Theoretically, we show (for the right tail) that this (initially perhaps somewhat puzzling)empirical fact can only arise whenever0 < M B − M A N A − N B ≤ a A a B , (2)under the assumption that F B and F A have heavy right tails with tail exponents a B and a A satisfying a B > a A >
0. Here, N A is the sell limit order volume, N B the buylimit order volume and M A and M B denote the sell and buy market order volume. Thisequation poses two conditions that should be fulfilled to make it theoretically possiblethat tails of closing auction return distributions are heavier without market orders. First,limit order imbalance and market order imbalance should be of opposite signs (when M B > M A , it should hold that N A > N B and vice versa) and limit order imbalance N A − N B should be larger in absolute value than market order imbalance M B − M A ,meaning that limit orders overcompensate for market order imbalance. Second, a B should not be too large, i.e. the right tail of the buy limit order placement distributionneeds to be sufficiently heavy. We show that equation (2) is indeed satisfied on averageempirically, which is explained by the chronology of the closing auction: most of themarket orders are submitted in the first seconds, revealing early in the auction themarket order imbalance. This leads to strategic behaviour in which limit orders areplaced against the direction of the market order imbalance: when there are more buythan sell market orders, one can submit a (possibly large) sell order without adverselyimpacting the price. Our results suggest that large closing price fluctuations are notcaused by large market orders (at least, not directly), but by placement of limit orders,in accordance with the intraday results of Farmer et al. (2004) and Weber and Rosenow(2006). Also, our results suggest that heavy tails are market microstructure effects andthat the tail exponents vary between different stocks and different market mechanisms,in line with the view of Mike and Farmer (2008).The remainder of this paper is structured as follows. In section 2 the model is described3nd theoretical results are derived. Then in section 3 the empirical results are presentedand the relations that are predicted by the model are verified. Concluding remarks aremade in section 4 and proofs of the mathematical theory are collected in the appendix. In this section we recall the auction model of Derksen et al. (2020a) and derive analyticalexpressions for the tail behaviour of the return distribution, given the tails of orderplacement distributions.
In the standard call auction, orders are aggregated over an interval of time and thenmatched to transact at a clearing price that maximizes the total transacted volume.Suppose N A sell limit orders and N B buy limit orders are submitted to the auction(all orders have unit size). We assume market participants on both sides of the marketformulate their orders independently, according to certain order placement distributions F A and F B . Here, F A denotes the distribution of sell orders and F B the distribution ofbuy orders. That is, we model the sell order prices ( A , . . . , A N A ) as an i.i.d. samplefrom F A and the buy order prices ( B , . . . , B N B ) as an i.i.d. sample from F B .For convenience we consider the log return axis instead of the real price axis. We assumethere is some reference price x (for example the last traded price before the auctionstarts or a volume weighted averaged version thereof) and all prices are expressed aslog returns relative to this reference price. So F A and F B are distributions on ( −∞ , ∞ )and F A ( x ) or F B ( x ) denotes the probability that a sell or buy order price is below x e x . Given ( N A , N B ), we denote by F A and F B the empirical distribution functionscorresponding to the samples ( A , . . . A N A ) and ( B , . . . , B N B ), meaning F A ( x ) = 1 N A N A (cid:88) i =1 { A i ≤ x } , F B ( x ) = 1 N B N B (cid:88) i =1 { B i ≤ x } Furthermore, we define the (monotone increasing) supply curve , D A ( x ) = N A F A ( x )and the (monotone decreasing) demand curve , D B ( x ) = N B (1 − F B ( x )) . Of course these assumptions are not all realistic. In reality, orders have different sizes and marketparticipants may react to each other’s orders. Despite these simplifying assumptions, the model providesa reliable stochastic description of auction price formation (see Derksen et al. (2020a)). x ∈ R the number of sell orders below x e x , thedemand curve gives for every x ∈ R the number of buy orders above x e x . Given all buyand sell orders, the clearing price is the price that maximizes the transactable volumein the auction, which is the price where supply and demand curves cross. That is, the clearing price X is defined as the solution to the market clearing equation, D A ( X ) = D B ( X ) . (3)This definition of X may give rise to problems with uniqueness and existence of solutionsto equation (3), as illustrated in figure 1. To solve these issues, consider the followingdefinition. Definition
For given supply curve D A and demand curve D B , the lower clearingprice is defined by X = inf { x ∈ R : D A ( x ) ≥ D B ( x )) } (4) and the upper clearing price is defined by X = sup { x ∈ R : D A ( x ) ≤ D B ( x ) } = inf { x ∈ R : D A ( x ) > D B ( x )) } . (5) The interval [ X, X ) is the interval of all possible clearing prices. D A ( · ) D B ( · ) X X D A ( · ) D B ( · ) X = X Figure 1: Two examples of the supply curve D A ( · ) (the increasing (red) step function)and the demand curve D B ( · ) (the decreasing (blue) step function). Left panel: a situationin which there is no unique point of intersection, but an interval [ X, X ) of possibleclearing prices. Right panel: a situation in which there is a unique intersection point X = X . Remark X . So in order to study the right tail of the closing pricereturn distribution, we should study X . The same reasoning implies that for the left tailwe should consider X . Note that the model is symmetric when the roles of X and X and the sides of the market are interchanged. That is, the left tail of the distribution of X behaves the same as the right tail of the distribution of X , when F A and F B and N A and N B are interchanged. So without loss of generality we focus on the right tail of X .The distribution of the lower clearing price, conditional on ( N A , N B ), has an analyticallytractable distribution function, given in the following theorem (see Derksen et al. (2020a),theorem 2.3). Theorem
The distribution of the lower clearingprice X , conditional on ( N A , N B ) , is given by its survival function, P ( X > M | N A , N B )= N A (cid:88) k =0 N B (cid:88) l = k +1 (cid:18) N A k (cid:19)(cid:18) N B l (cid:19) (1 − F A ( M )) N A − k F A ( M ) k (1 − F B ( M )) l F B ( M ) N B − l . (6)In the situation described above, only limit orders are submitted to the auction. However,market participants also have the possibility to submit market orders. We define the(possibly stochastic) market order imbalance by ∆ = M B − M A , where M B is the numberof buy market orders and M A is the number of sell market orders. Note that marketorders only play a role through ∆, as matching market orders are executed against eachother without affecting the price formation process. When market order imbalance ∆ istaken into account, the market clearing equation (3) becomes D A ( X ) = D B ( X ) + ∆and the definitions of X and X change accordingly. A positive (negative) value of ∆means there is more buy (sell) market order volume than sell (buy) market order volume,possibly pushing the price up (down). The market order imbalance alters the clearingprice distribution as in the following proposition (a special case of proposition 2.8 inDerksen et al. (2020a)). Proposition
When market order imbalance ∆ plays a role, the lower clearing price distribution ascomputed in theorem 2.3 modifies into P ( X > M | N A , N B , ∆)= N A (cid:88) k =0 N B (cid:88) l =max( k − ∆+1 , (cid:18) N A k (cid:19)(cid:18) N B l (cid:19) (1 − F A ( M )) N A − k F A ( M ) k (1 − F B ( M )) l F B ( M ) N B − l . .2 Limit order auctions Next we concentrate on the right tail of the lower clearing price return distribution, asa function of the tails of the order placement distributions F A and F B , initially withoutmarket orders. We make the following assumption on the tails of F A and F B . Assumption F A has a heavier right tail than F B . That is, there existsfunctions T A , T B such that1 − F A ( M ) ∼ T A ( M ) , − F B ( M ) ∼ T B ( M ) , as M → ∞ and lim M →∞ T B ( M ) T A ( M ) = 0 . This assumption is intuitively reasonable and empirically verified in section 3.1. Fur-thermore, we will assume that ( N A , N B ) follows a distribution P N A ,N B on N = { , . . . , N } × { , . . . , N } , for some N ∈ N , with probability mass function p N A ,N B assigning positive probabilityto any point in N (we exclude the possibilities that N A = 0 or N B = 0, which describefailing auctions in which clearing prices do not exist).In the following proposition we first derive an expression for the right tail of the lowerclearing price distribution, conditional on ( N A , N B ). Finding an expression for the tailof the clearing price distribution amounts to finding the slowest decaying term in thedouble sum of theorem 2.3. This is made formal in the following proposition, the proofof which is found in the appendix. Proposition
Under assumption 1, we have P ( X > M | N A , N B ) ∼ N B T B ( M ) T A ( M ) N A , as M → ∞ . (7) Remark l = 1 , k = 0, correspondingto the event that D A ( M ) = 0 , D B ( M ) = 1, meaning all sell orders, but only one buyorder, are above M . This is interpreted as an auction in which there is little consensusbetween both sides of the market (buy and sell orders do not overlap), but there is avery aggressive buyer willing to pay a high price.When the conditional result of proposition 2.5 is summed with respect to the distributionof ( N A , N B ), the unconditional tail of X is discovered again by selecting the slowestdecaying term. This leads to the main result of this subsection, a relation between7he tail of the closing price return distribution and the tail of the order placementdistributions in a setting without market orders (its proof is again postponed to theappendix). Theorem
Under assumption 1we have P ( X > M ) ∼ CT A ( M ) T B ( M ) , as M → ∞ , where C = (cid:80) Nn =1 np N A ,N B (1 , n ) = E [ N B { N A =1 } ] > . The constant C indicates that the slowest decaying term in the sum corresponds to theevent that N A = 1: large positive returns are possible if there are only few sell orders. In this subsection we incorporate market orders in the derivation of subsection 2.2. Firstconsider the following assumption for the market order imbalance ∆.
Assumption ∈ ( − N B , N A ) with probability one.This assumption is necessary, because otherwise the clearing prices attain the values ±∞ with non-zero probability. Under this assumption, the right tail of the conditional lowerclearing price distribution is given by the next proposition (the proof is again postponedto the appendix and x + = max( x,
0) and x − = max( − x,
0) denote the positive andnegative part of x ∈ R ). Proposition
Under assumptions 1 and 2, we have P ( X > M | N A , N B , ∆) ∼ K ( N A , N B , ∆ − T B ( M ) (∆ − − T A ( M ) N A − (∆ − + , (8) as M → ∞ , where K ( N A , N B , ∆) = (cid:0) N A ∆ (cid:1) if ∆ > (cid:0) N B − ∆ (cid:1) if ∆ ≤ . This proposition shows that market orders potentially influence the tails heavily: if ∆is positive and large (close to N A ) the influence of the faster decaying term T B ( M ) iserased and only the slower decaying term T A ( M ) is left, possibly leading to very heavytails. On the other hand, if ∆ is negative, the influence of the faster decaying term T B grows, leading to less heavy tails. However, which combinations are possible depends onthe joint distribution of ( N A , N B , ∆). Until now, the tails T A and T B were unspecifiedand few assumptions were made on the distribution of ( N A , N B ). To work towards anempirically testable theory, we will make the following assumptions on the distributionof ( N A , N B , ∆) and the tails of F A , F B . Empirically, these assumptions are verified insection 3. 8 ssumption N A , N B , ∆) follows a distribution P on { , . . . , N }×{ , . . . , N }×{− N, . . . , N } , with probability mass function denoted by p , for some N ∈ N . Further-more, assume that market order imbalance M B − M A is proportional to limit orderimbalance N A − N B (in the opposed direction), that is,∆ = M B − M A = c ( N A − N B ) , (9)almost surely for some c ∈ (0 ,
1) and P (∆ = 0) = 0 (as the case ∆ = 0 is already con-sidered in subsection 2.2). Finally, assume that all possible combinations have positiveprobability, i.e. p ( n, m, d ) > , for all n, m ∈ { , . . . , N } , d ∈ ±{ , . . . , N } such that d = c ( n − m ) . Equation (9) states that limit order imbalance points in the opposed direction of marketorder imbalance, which resembles that limit order submitters adjust their orders to themarket order imbalance. This relation ensures assumption 2 holds and is empiricallyverified in section 3.3.
Assumption F A , F B both have power law tails, that is,1 − F A ( M ) ∼ T A = M − a A , − F B ( M ) ∼ T B ( M ) = M − a B , as M → ∞ , for tail exponents a B > a A > c (controlling the relation between market and limit order imbalance) and a A and a B (controlling the heaviness of the tails of the buy and sell limit order placement distribu-tion). Theorem
Under assumptions 3 and 4, there exists a constant
C > , such that P ( X > M ) ∼ CM − a , as M → ∞ , where a = min (cid:18) ( c + 1) a A c , a A + 2 a B (cid:19) . (10)Note that without market order imbalance ∆ we have by theorem 2.7 a = a A + a B . Thistheorem makes testable predictions about the relation between the tails of the closingprice return distribution, the tails of the limit order placement distributions and thelimit and market order imbalance. In the next section we will investigate this relationempirically. 9 Empirical results
In this section we investigate empirically the relation between the tails of the closingauction return distributions and the tails of the limit order placement distributions.In order to do so, we obtain detailed order-by-order data over 2018 and 2019, for 100liquid European stocks (with market capitalization above EUR 1 bn) listed on Euronextexchanges in Amsterdam, Paris, Brussels or Lisbon.Estimating the tails of a distribution comes with a couple of problems. First, the powerlaw of equation (1) is not assumed to hold for all values of x , but only for the tail.This necessarily involves a starting point x min such that the power law holds for all x > x min (see Newman (2005) for a discussion). Unfortunately, the eventual estimatefor the tail exponent will depend on this cut-off point: if x min is taken too small, the bulkinstead of the tail will determine the estimates. The cut-off is often made through visualinspection of a double logarithmic plot. Then the second problem arises, because thecut-off eliminates most of the available data, leaving only a small fraction of the dataavailable for estimation. Finally, models are often designed to describe only ‘generic’situations well and are not intended to explain extreme events. It is a noteworthyadvantage of the call auction model of section 2 that it is suitable to model both thebulk of the data (as in Derksen et al. (2020a)) and extreme events, as in the currentpaper.Concerning the amount of data relevant for the tails, in every closing auction a largeamount of orders is submitted, so the tails of order placement distributions can be studiedper stock. Unfortunately, this is not possible for the closing auction return distribution:per stock, we have only around 500 trading days (two years of around 250 trading daysper stock) and thus only that many closing auction returns, which is far insufficient toexamine the tails. For example, if we take the 0.05-quantile for the cut-off point x min ,only about 25 data points reside in the tail, which is too few for meaningful statisticalanalysis. So to investigate the tails of the closing auction return distribution, we mergetogether the closing auction returns of all stocks in the sample.In the entire section, the reference price x will be the volume weighted average price overthe last five minutes of continuous trading. Closing auction returns will be measured inlog returns with respect to x . Following Bouchaud et al. (2002) and Zovko and Farmer(2002), limit order prices are measured in the number of ticks a limit order is placedaway from the reference price x . The mechanism of the call auction makes it possible to study both tails of both orderplacement distributions. In figure 2, both tails of the sell limit order distribution F A F B are shown in log-log plots, for four stocks thatare representative for the sample. x C D F ASML right tail F A SAINT GOBAINSIGNIFYUBISOFT (a) Right tail of F A . x C D F ASML right tail F B SAINT GOBAINSIGNIFYUBISOFT (b) Right tail of F B . x C D F ASML left tail F A SAINT GOBAINSIGNIFYUBISOFT (c) Left tail of F A . x C D F ASML left tail F B SAINT GOBAINSIGNIFYUBISOFT (d) Left tail of F B . Figure 2: Log-log plots of the tails of the order placement distributions for 4 selectedstocks (ASML Holding NV, Compagnie de Saint Gobain SA, Signify NV, Ubisoft Enter-tainment SA ). The x -axes show the number of ticks above (for the right tail) or below(for the left tail) the reference price x .Let us first focus on the right tails, i.e. the upper panels (a) and (b) of figure 2. The plotsof the right tails of F A show apparent power law behaviour in the range between 10 and1000 ticks above the reference price. After circa 1000 ticks the tails decay faster for awhile, but starting around 5000 ticks a new part of the distribution seems to start. Theplot is cut-off at 10 000 ticks, but some even reach until 100 000 ticks. These extremesdo not contribute to price formation in the auction at all. We focus on the interval ofthe price axis where price formation occurs: the intersection of the supports of F A and F B . For the right tail that means F B provides the effective upper bound (note that theclosing price can never take a value above the highest buy order). The support of F B ranges until around 1000-2000 ticks above the reference price so that is the region we usein our analysis, roughly in line with the intraday results from Zovko and Farmer (2002) . The sell orders (far) above this region can be thought of as coming from another distribution de-scribing patient sellers not relevant to the auction result. To sketch how irrelevant those orders are: thetick size of a stock is normally between 1 and 5 basis points. Assuming a tick size of 2.5 basis points,2000 ticks correspond to a return of 50%, while a closing auction return in the order of 1% is alreadyhigh. F B , but in the range of 100 until 1000 ticks powerlaw behaviour can be recognized for the liquid stocks ASML and Saint Gobain. For theless liquid stocks Signify and Ubisoft it stops earlier around 500 ticks, but this can alsobe due to smaller volumes of available data. The lower panels (c) and (d) of figure 2show the left tails of the order placement distributions. These are very similar to theright tails, when the roles of F A and F B are switched. Also, on the left side there isa real cut-off point, corresponding to price 0, which is found somewhere between 2000and 10 000 ticks. In figure 3 we zoom in on the right tails of F B and F A until around x C D F ASML right tail F A Linear fit, slope =-1.07ASML right tail F B Linear fit, slope =-2.37 x C D F GOBAIN right tail F A Linear fit, slope =-0.40GOBAIN right tail F B Linear fit, slope =-2.01 x C D F UBISOFT right tail F A Linear fit, slope =-0.58UBISOFT right tail F B Linear fit, slope =-3.63 x C D F SIGNIFY right tail F A Linear fit, slope =-0.89SIGNIFY right tail F B Linear fit, slope =-3.69
Figure 3: Log-log plots of the right tails of the order placement distributions for 4 stocks(ASML Holding NV, Compagnie de Saint Gobain SA, Signify NV, Ubisoft EntertainmentSA). The x -axes show the number of tick sizes above the reference price x . Linear fitsare also plotted, fitted on the 0.05-quantile of F B until the 0.001-quantile of F B , toestimate a B and a A a A and a B (the tail exponents of F A and F B as in assumption 4). We performlinear least square fits on the log-log plots of the tails of F B , starting at its 0.05-quantile.Visual inspection shows that in the extreme tails, available data points are too sparseto form a coherent picture. So we stop the fit at the 0.001-quantile of F B , which seemsreasonable when inspecting the plots and we make fits for F A on the same interval . Forexample for ASML, we obtain a A ≈ . , a B ≈ .
37, fitted on the interval of 168 until862 tick sizes. For all four stocks, F A shows a straight, slowly decaying line, resembling These choices are somewhat arbitrary, but cut-off choices need to be made in any practical tailanalysis (see Newman (2005)) and moreover, results do not change substantially when we extend the fitto e.g. the 0.0001-quantile, or e.g. start the fit at the 0.01-quantile.
12 power law with exponents around or even below 1. Furthermore, the tails of F B decayfaster than the tails of F A , with exponents between 2 and 4 (more results are discussedin section 3.4). For every stock i and day 1 ≤ d ≤ n we have a closing auction return X i,d , defined as X i,d = log( C i,d ) − log( x i,d ) , where C i,d is the closing price of stock i on day d and x i,d is the reference price of stock i on day d . Following e.g. Gopikrishnan et al. (1998) we standardize the returns perstock. That is, we divide for every stock i the return sample { X i,d : 1 ≤ d ≤ n } by itsstandard deviation and obtain a sample of standardized returns of size n ≈ a = 5 .
28 for the left tail and a = 4 .
74 for theright tail.This suggests closing auction returns are less heavy tailed than intraday returns overshort time intervals, for which a tail exponent a ≈ e.g. Gopikrishnan et al. (1998)). This difference might be explained in qualitativeterms by the large transacted volumes in the closing auctions. It is known that tails ofreturn distributions become thinner when longer time intervals are considered, an effectthat is known as aggregational Gaussianity (the empirical fact that return distributionsconverge to normal distributions when the interval length increases, see e.g.
Cont (2001)).This is theoretically supported by the call auction model: the clearing price distributionapproaches a normal distribution, when the number of orders tends to infinity (seeDerksen et al. (2020a), theorem 3.1). Moreover, the empirical effect is known to bestronger if time intervals are measured in trade time (Chakraborti et al., 2011). InEurope nowadays around 30% of the daily volume is transacted in the closing auction,which makes the duration of the closing auction in trade time similar to approximatelyhalf a day of continuous trading . The fraction of daily transacted volume that is transacted in closing auctions has increased greatlyover the past years, especially since the introduction of MiFID II, see Derksen et al. (2020b). x C D F Right tailLinear fit, slope= − . − . Figure 4: Log-log plot of the tails of the closing auction return distribution for all 100stocks in our sample. Blue dots show the right tail, that is P ( X > x ), red squares theleft tail, that is P ( X < − x ), the x -axis is in standardized returns. Linear fits are alsoplotted, giving a tail exponent of a = 4 .
74 for the right tail and a = 5 .
28 for the left tail.
Before we study the influence of market orders on the tail behaviour of closing auctionreturn distributions, we first investigate the relation between the market order imbalanceand the limit order imbalance. In figure 5 the market order imbalance ∆ = M B − M A in every closing auction is plotted against the limit order imbalance N A − N B in thatclosing auction, for the four stocks that were also studied in section 3.1. The figure showsthat the proportionality relation between ∆ and N A − N B introduced in equation (9)holds approximately, with values of c in the range 0.2-0.4, estimated using linear leastsquare regression. This means that limit order imbalance is generally in the oppositedirection of market order imbalance. An explanation for this lies in the chronologyof the closing auction. We observe in auction data that the vast majority of marketorders is submitted in the first seconds of the closing auction, revealing the marketorder imbalance early in the auction (during the accumulation phase of the auction,information on the imbalance and an indicative price is released, so it is possible to acton this information). Subsequently, limit orders are placed against the direction of themarket order imbalance, reflecting strategic behaviour: when there is a large positivemarket order imbalance (more buy market orders than sell market orders), one can14ubmit a (possibly large) sell order without adversely affecting the price. ∆ N A − N B ASML, c =0.329 ∆ N A − N B GOBAIN, c =0.236 ∆ N A − N B UBISOFT, c =0.204 ∆ N A − N B SIGNIFY, c =0.252 Figure 5: The difference N A − N B plotted against the market order imbalance ∆,showing limit order imbalance goes against the direction of market order imbalance.The dashed red line is the result of linear least square regression, to estimate the valueof c in equation (9), which is the slope of the dashed red line (outliers of more than fourstandard deviations away from the mean are removed).Next, we will investigate the effect of market orders on the tail exponents. Consider figure6, where two auction results are shown. Supply and demand curves are represented bythe solid lines and the point of intersection is the closing price, indicated by the blackstar. When market orders are removed, translated supply and demand curves (plottedby the dashed lines) lead to an alternative closing price, represented by the black square.The upper panel shows a situation in which a large positive closing auction return iscaused by a high market order imbalance. When the market order imbalance would beremoved, the closing price would be much lower (black square). The lower panel shows avery different situation: a small positive closing auction return, but a strongly negativemarket order imbalance. If in this case the market order imbalance would be removed,the closing auction return would get much higher (black square).The two scenarios presented in figure 6 raise the question which is more common: arelarge closing auction returns caused by large market order imbalances or is this potentialeffect cancelled by limit order imbalance and are limit orders usually the driver of large15
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Price V o l u m e D A ( x ) + M A D B ( x ) + M B D A ( x ) D B ( x )Reference price x Closing priceAlternative closing price (a) Closing auction ASML, 2018-03-16.
Price V o l u m e D A ( x ) + M A D B ( x ) + M B D A ( x ) D B ( x )Reference price x Closing priceAlternative closing price (b) Closing auction KPN, 2018-02-07.
Figure 6: Two closing auction results. Solid lines are the supply (red) and demand(blue) curves of the particular closing auction, including market orders (for conveniencesell (buy) market orders are placed just below (above) the lowest sell (highest buy) limitorder). Dashed lines show the supply and demand curves without market orders. Theblack dot denotes the reference price x , the black star denotes the closing price and theblack square denotes the alternative price when only limit orders are considered.16eturns? To answer this question, we also investigate the tails of the return distribution ofthe alternative closing price , defined as the intersection point of the supply and demandcurves when the market orders are removed (black squares in figure 6). So, for everystock i and day d we have an alternative closing auction return ˜ X i,d , defined as˜ X i,d = log( ˜ C i,d ) − log( x i,d ) , where ˜ C i,d is the alternative closing price of stock i on day d . We again standardize thesereturns per stock, giving for every stock around 500 alternative closing auction returns,which are merged to study the tails. In figure 7 the tails of the alternative closing pricereturn distribution are shown, together with the tails of the real closing price returndistribution from figure 4. The figure shows that the tails become heavier when marketorders are removed. For the right tail we document a tail exponent a = 3 .
75 withoutmarket orders, compared to a = 4 .
74 with market orders. For the left tail, the tailexponent becomes a = 3 . a = 5 .
28 when market orders are included. x Without market ordersLinear fit, slope= − . − . (a) Right tail return distribution, P ( X > x ) x Without market ordersLinear fit, slope= − . − . (b) Left tail return distribution, P ( X < − x ) Figure 7: Log-log plot of the tails of the closing auction return distribution for all100 stocks in the sample. Blue dots show the tails for the real closing auction returndistribution, red dots the tails for the alternative closing auction returns that emergewhen market orders are removed.It is thus concluded that large closing price fluctuations are in general not caused by alarge market order imbalance (at least, not directly). The explanation for this counter-intuitive result lies in the chronology of the auction and the placement of limit orders:when the market order imbalance is positive (negative), there are more sell (buy) limitorders submitted ( cf. figure 5). Theorems 2.7 and 2.9 give the model’s view on the matter17nd state that without market orders the tail exponent is equal to a A + a B and withmarket orders it is equal to min( ( c +1) a A c , a A + 2 a B ). This means that tails get heavierwithout market orders, whenever c ≤ a A a B . (11)This equation in fact resembles two conditions that should be fulfilled to make it possi-ble that tails are heavier without market orders (see also equation (2)). First, c shouldbe small and positive, reflecting that the abovementioned strategic behaviour is strong:when there is a large market order imbalance, in general the limit order difference over-compensates for this. Second, a B should not be too large compared to a A . This is acondition on the right tail of the buy limit order distribution. Without market orders,the highest buy limit order serves as an upper bound for the closing price. So to obtainheavier tails without market orders, the right tail of F B should be sufficiently heavy(small a B ). It turns out that condition (11) is indeed satisfied for most of the stocks: forexample, for ASML we obtained estimators a B ≈ . , a A ≈ . c ≈ .
329 ( cf. figures3 and 5), satisfying the condition in equation (11). Indeed, theorems 2.7 and 2.9 implythat the tail exponent for closing auction returns of ASML is a A + a B = 3 .
44 withoutmarket orders and ( c +1) a A c = 4 .
32 with market orders. In the next subsection we willverify the theoretical results on the whole sample consisting of 100 stocks.
In this subsection the relations predicted by the model are tested over the whole sampleof 100 stocks. For every stock we estimate the tail exponents of the order placementdistributions ( a A and a B ) and the value of the parameter c (as in equation (9)). Theresults are shown in table 2 (for 50 stocks with the lower market capitalizations) andtable 3 (for 50 stocks with the higher market capitalizations). To estimate the parameter c , we use linear least squares regression and to estimate the values of a A and a B we usethe method described in section 3.1: for every stock, we make linear least square fitson double logarithmic plots as in figure 3, on the interval between the 0.05- and 0.001-quantiles of F B . The absolute values of the resulting slopes are the estimators for a A and a B . For example, for ASML we obtain in this way estimators a B ≈ . , a A ≈ .
07 andfor Ubisoft we find a B ≈ . , a A ≈ . cf. figure 3. In tables 2 and 3 the results areshown for all stocks in the sample, the columns a B ( r ) and a A ( r ) give the estimated tailexponents for the right tails of F B and F A . For the left tails, the same method applieswhen the roles of F B and F A are interchanged. On the left side, F B has a heavier tailand F A provides the effective lower bound.In figure 8 the left tails of the order placement distributions are shown for ASML andUbisoft, as well as the linear least square fits, showing that for the left tails a A ≈ . , a B ≈ .
17 for ASML and a A ≈ . , a B ≈ .
87 for Ubisoft. In tables 2 and 3 the18 x C D F ASML left tail F A Linear fit, slope =-2.50ASML left tail F B Linear fit, slope =-1.17 x C D F UBISOFT left tail F A Linear fit, slope =-2.81UBISOFT left tail F B Linear fit, slope =-0.87
Figure 8: Log-log plots of the left tails of the order placement distributions for ASMLHolding NV and Ubisoft Entertainment SA. The x -axis shows the number of tick sizesabove the reference price x . Linear fits are also plotted, fitted on the 0.05-quantile of F A until the 0.001-quantile of F A , to estimate a B and a A columns a B ( l ) and a A ( l ) give the estimated tail exponents for the left tails of F B and F A .Estimates for a A , a B and c give rise to an estimate for the tail exponent a for the returndistribution of that particular stock. With market orders a = min( ( c +1) a A c , a A + 2 a B )( cf. theorem 2.9) and without market orders a = a A + a B ( cf. theorem 2.7) . Ideally, wewould test these predictions against the realized tail exponents of the return distribution for every stock . However, as noted in the beginning of this section, this is not possible,because we only have around 500 closing auction returns per stock. Instead, we canverify the predictions over groups of stocks, by comparing estimated tail coefficientswith the model’s average predicted values.First, consider the whole sample of 100 stocks. In figure 7 it was shown that the righttail of the closing price return distribution has an estimated tail exponent of a = 4 . a = 3 .
75 if market orders are removed. If we take the average of themodel’s predictions over all 100 stocks, we find an average predicted tail exponent of4.89 with market orders (column ‘ a ( r ) MO’ in tables 2 and 3) and 3.89 without marketorders (column ‘ a ( r ) no MO’ in tables 2 and 3). Furthermore, figure 7 shows that the lefttail of the closing price return distribution has an estimated tail exponent of a = 5 . a = 3 .
90 if the market orders are removed. For the left tail, the averagepredicted tail exponent over all 100 stocks equals 5.01 with market orders (column ‘ a ( l )MO’ in tables 2 and 3) and 3.72 without market orders (column ‘ a ( l ) no MO’ in tables 2and 3). The predicted tail exponents vary a lot between the different stocks, suggestingthat the heaviness of the tails depends on the stock. To additionally test if these perstock predictions give information about the real tail exponents, we split our sample into50 stocks with the lower market caps (those in table 2) and 50 stocks with the highermarket caps (table 3). In that way, the groups are kept large enough to examine thetails of the closing auction return distributions. Note that for the left tails the roles of a A and a B need to be interchanged (see also remark 2.2). x C D F Without market ordersLinear fit, slope= − . − . (a) Small caps, right tail return distribution. x C D F Without market ordersLinear fit, slope= − . − . (b) Small caps, left tail return distribution. x C D F Without market ordersLinear fit, slope= − . − . (c) Large caps, right tail return distribution. x C D F Without market ordersLinear fit, slope= − . − . (d) Large caps, left tail return distribution. Figure 9: Log-log plots of the tails of the closing auction return distributions for the50 small cap stocks of table 2 (upper panel) and 50 large cap stocks of table 3 (lowerpanel). Blue dots show the tails for the real closing auction return distribution, red dotsthe tails for the alternative closing auction returns that emerge when market orders areremoved. 20n figure 9 the tails of the closing auction return distribution for the 50 small capsand the 50 large caps are shown in double logarithmic plots, again with and withoutmarket orders (similar to figure 7). The linear fits to the double logarithmic plots are therealized tail exponents for the both groups, which can again be compared to the averagepredicted values in tables 2 and 3. The results are summarized in table 1, showing firstof all that the model’s predicted exponents are quite close to the realized exponents.Given that estimation of tails (and tail exponents in particular) is generally thought ofas a difficult statistical problem, the congruence is quite remarkable. Second, based onthe modelling assumption in equation (9), the model predicts correctly that the tails getheavier if market orders are removed, and by how much. The theoretical predictions areespecially accurate for the case without market orders, which is not surprising: theorem2.7 holds very generally and follows directly from the mechanics of the closing auction.For the case with market orders, more assumptions were made (see assumption 3). Mostimportantly, we assumed equation (9) holds true, which of course in reality holds onlyapproximately (see also figure 5). When looking at tables 2 and 3, the predictions forthe case with market orders vary strongly between the stocks. We do not claim that themost extreme values that are predicted are close to reality, but we have shown that, onaverage, model predicted and realized tail exponents match remarkably well.
Left tails Right tailsMO No MO MO No MOPredicted Realized Predicted Realized Predicted Realized Predicted RealizedAll stocks 5.01 5.28 3.72 3.90 4.89 4.74 3.89 3.75Small caps 5.76 5.26 4.14 4.11 5.19 4.95 4.17 4.10Large caps 4.25 5.39 3.29 3.75 4.59 4.60 3.61 3.48
Table 1: Average predicted tail exponents compared to realized tail exponents. Pre-dicted exponents are averages over tables 2 and 3, realized exponents are the results ofthe linear fits in figures 7 (all stocks) and 9 (small and large caps), for the cases with(MO) and without (No MO) market orders.21 tock Exch. Mcap a A (l) a B (l) a A (r) a B (r) c a (l)no MO a (l)MO a (r)no MO a (r)MOASM INTL AMS 6.7 3.312 1.154 0.717 3.377 0.127 4.466 7.778 4.094 6.357AALBERTS AMS 3.8 3.178 0.763 1.207 3.301 0.174 3.941 5.135 4.508 7.808WDP REIT BRU 5.2 3.142 3.279 1.739 3.032 0.113 6.421 9.562 4.771 7.804REXEL PAR 3.2 2.105 1.782 1.465 2.626 0.184 3.887 5.992 4.091 6.718EURONEXT PAR 6.8 3.319 1.943 1.918 3.229 0.164 5.262 8.582 5.146 8.375IMCD GROUP AMS 6.0 3.328 0.972 1.859 3.016 0.158 4.300 7.118 4.875 7.892SIGNIFY AMS 4.6 3.485 1.181 0.888 3.690 0.252 4.666 5.866 4.579 4.413ALTEN PAR 2.8 3.443 1.476 1.017 2.420 0.137 4.919 8.362 3.437 5.857BIC PAR 1.9 3.025 1.644 0.916 3.820 0.324 4.669 6.720 4.736 3.746EUTELSAT COM PAR 1.9 2.969 0.993 0.677 3.752 0.201 3.962 5.972 4.429 4.068INGENICO GROUP PAR 8.5 1.534 0.970 0.740 2.996 0.129 2.504 4.038 3.735 6.461EURAZEO PAR 3.3 4.512 1.953 1.536 3.928 0.134 6.464 10.976 5.464 9.393AEGON AMS 5.0 1.552 0.801 0.421 2.561 0.15 2.353 3.905 2.982 3.220KPN KON AMS 10.0 2.363 1.011 0.584 3.942 0.207 3.373 5.736 4.526 3.413RANDSTAD AMS 8.4 3.071 1.130 0.939 3.603 0.291 4.201 5.016 4.541 4.166KLEPIERRE REIT PAR 3.5 3.242 1.016 0.503 3.317 0.371 4.259 3.754 3.820 1.858SUEZ PAR 9.9 1.416 0.866 0.724 2.588 0.250 2.281 3.697 3.312 3.613GALP ENERGIA LIS 6.8 4.467 2.173 1.438 3.977 0.484 6.640 6.663 5.415 4.410ARKEMA PAR 7.2 4.292 1.231 1.354 4.441 0.278 5.522 5.659 5.795 6.225COVIVIO PAR 5.2 3.400 3.485 2.230 3.279 0.304 6.884 10.284 5.509 8.788ICADE REIT PAR 3.4 4.179 1.649 1.371 3.499 0.231 5.828 8.798 4.870 7.314IPSEN PAR 6.5 2.515 1.463 0.844 2.594 0.200 3.978 6.493 3.439 5.072ORPEA PAR 5.9 1.605 0.786 0.987 1.683 0.126 2.391 3.997 2.671 4.354SCOR PAR 4.5 3.363 1.475 0.941 3.944 0.378 4.837 5.372 4.885 3.427GETLINK PAR 6.4 1.910 1.341 1.166 3.294 0.148 3.251 5.161 4.460 7.753J.MARTINS SGPS LIS 9.2 4.043 1.022 1.027 3.621 0.276 5.065 4.730 4.648 4.752DASSAULT AVIAT PAR 6.3 4.021 1.155 0.764 4.088 0.282 5.176 5.247 4.852 3.471EDENRED PAR 10.1 3.832 2.070 2.352 3.763 0.282 5.902 9.419 6.115 9.879PUBLICIS GROUPE PAR 7.6 2.600 1.251 0.704 3.221 0.390 3.851 4.462 3.925 2.509ATOS PAR 7.5 2.323 0.834 0.499 2.449 0.267 3.157 3.959 2.948 2.368JCDECAUX PAR 2.9 3.682 1.578 1.177 3.845 0.229 5.260 8.482 5.022 6.323EIFFAGE PAR 6.9 4.086 1.324 1.785 4.083 0.248 5.410 6.656 5.868 8.973GECINA PAR 7.8 2.988 2.394 2.004 2.928 0.321 5.382 8.370 4.932 7.860NATIXIS PAR 6.5 0.763 0.513 0.524 1.672 0.155 1.276 2.040 2.196 3.868SES FDR PAR 3.0 3.138 0.649 0.905 3.218 0.186 3.786 4.132 4.123 5.768SEB PAR 7.6 3.998 1.284 0.998 3.807 0.215 5.282 7.246 4.805 5.629UBISOFT PAR 10.2 2.814 0.870 0.578 3.635 0.204 3.684 5.136 4.213 3.409ALSTOM PAR 9.4 1.988 0.715 0.866 2.955 0.279 2.703 3.280 3.821 3.975TECHNIPFMC PAR 2.6 1.113 0.729 0.552 2.478 0.217 1.842 2.955 3.029 3.097ACCOR PAR 5.9 2.384 0.677 0.645 3.079 0.251 3.061 3.374 3.724 3.215VEOLIA PAR 9.8 1.476 0.838 0.484 2.598 0.267 2.314 3.791 3.083 2.296COLRUYT BRU 7.3 3.340 1.373 1.245 3.390 0.318 4.713 5.689 4.635 5.159AGEAS BRU 6.7 2.408 1.115 1.254 2.735 0.321 3.524 4.585 3.989 5.155SOLVAY BRU 8.0 1.866 0.856 0.440 1.241 0.193 2.722 4.589 1.681 2.725UMICORE BRU 8.9 2.428 0.834 0.486 2.003 0.314 3.262 3.490 2.489 2.033PROXIMUS BRU 5.2 2.618 0.845 0.666 2.820 0.271 3.463 3.964 3.486 3.126ABN AMRO BANK AMS 6.9 1.989 1.216 0.627 3.004 0.195 3.205 5.194 3.631 3.838CNP ASSURANCES PAR 7.3 3.899 1.615 2.046 2.756 0.381 5.514 5.855 4.802 7.418UNIBAIL RODAMCO AMS 5.7 1.983 1.368 1.052 1.947 0.390 3.351 4.876 2.999 3.750SODEXO PAR 8.8 3.559 1.852 1.696 3.668 0.245 5.411 8.970 5.364 8.629Average - 6.4 2.84 1.30 1.06 3.12 0.24 4.14 5.76 4.17 5.19 Table 2: Table of results, for the 50 stocks in our sample with the lower market cap. Thecolumn Exch. displays the exchange the stock is traded on (Amsterdam, Paris, Brusselsor Lisbon) and the column Mcap shows the market capitalization of the stock in billionsof euros (in October 2020). Then, a A and a B are the estimated tail exponents of selland buy limit order distributions, for the left (l) and right (r) tail. c is the estimatorfor the constant in equation (9) and a = a A + a B without market orders (no MO), and a = min( ( c +1) a A c , a A + 2 a B ) with market orders (MO), both displayed for left (l) andright (r) tails. 22 tock Exch. Mcap a A (l) a B (l) a A (r) a B (r) c a (l)no MO a (l)MO a (r)no MO a (r)MOAMUNDI PAR 12.3 2.066 1.544 0.816 2.496 0.110 3.610 5.676 3.311 5.807BIOMERIEUX ORD PAR 16.4 2.815 1.595 2.294 3.312 0.247 4.410 7.225 5.606 8.917NN GROUP AMS 10.9 3.187 1.051 1.010 3.797 0.376 4.238 3.847 4.807 3.698SARTORIUS PAR 28.9 2.928 1.069 1.449 2.474 0.280 3.997 4.892 3.923 6.397WORLDLINE PAR 13.2 2.386 1.001 1.131 2.902 0.241 3.386 5.146 4.034 5.819EDP LIS 18.0 3.236 0.710 1.043 3.683 0.341 3.946 2.793 4.727 4.106TELEPERFORMANCE PAR 16.1 3.869 1.862 1.256 4.010 0.134 5.731 9.601 5.266 9.276BOUYGUES PAR 11.7 2.302 1.124 0.953 2.452 0.331 3.425 4.518 3.405 3.833AHOLD DEL AMS 26.6 1.334 0.727 0.875 2.234 0.326 2.061 2.960 3.108 3.562AKZO NOBEL AMS 17.7 2.678 1.921 1.190 2.669 0.292 4.599 7.276 3.859 5.268ASML HOLDING AMS 138.3 2.497 1.165 2.369 1.073 0.329 3.662 4.704 3.442 4.334DSM KON AMS 24.9 3.283 1.333 1.158 3.381 0.312 4.616 5.601 4.539 4.866HEINEKEN AMS 45.4 3.917 1.101 1.146 3.833 0.270 5.018 5.177 4.979 5.391ING GROEP AMS 24.7 1.344 1.067 0.419 2.333 0.139 2.411 3.755 2.752 3.428PHILIPS KON AMS 37.5 3.129 0.498 0.884 3.566 0.318 3.626 2.063 4.450 3.666UNILEVER AMS 138.7 2.929 0.619 1.417 2.879 0.322 3.548 2.543 4.297 5.818WOLTERS KLUWER AMS 19.3 3.473 1.417 1.823 2.997 0.244 4.891 7.235 4.820 7.817DANONE PAR 34.6 2.195 1.112 0.761 2.312 0.404 3.307 3.862 3.073 2.642BNP PARIBAS ACT.A PAR 40.2 1.326 0.771 0.636 2.133 0.223 2.098 3.424 2.769 3.496AXA PAR 36.1 0.740 0.698 0.548 1.878 0.321 1.438 2.178 2.426 2.253SOCIETE GENERALE PAR 10.2 0.815 0.923 0.383 2.051 0.160 1.738 2.553 2.434 2.777L’OREAL PAR 163.0 3.201 0.685 1.209 2.552 0.270 3.886 3.222 3.760 5.683SANOFI PAR 108.1 1.245 0.741 0.790 2.080 0.411 1.986 2.544 2.869 2.712SAINT GOBAIN PAR 20.0 1.242 0.936 0.402 2.014 0.237 2.178 3.420 2.416 2.099LEGRAND PAR 18.6 4.179 1.304 1.790 3.851 0.328 5.482 5.283 5.641 7.254TOTAL PAR 74.8 0.592 0.632 0.549 0.534 0.308 1.224 1.816 1.082 1.616HEINEKEN HOLDING AMS 20.2 3.198 1.244 2.004 3.506 0.333 4.442 4.982 5.509 8.025ESSILORLUXOTTICA PAR 50.8 2.433 0.555 1.040 2.691 0.347 2.988 2.155 3.731 4.037AB INBEV BRU 93.5 1.800 0.809 0.499 2.441 0.210 2.610 4.410 2.940 2.879DASSAULT SYSTEM PAR 41.4 3.381 1.439 1.144 2.779 0.341 4.819 5.653 3.923 4.495CHRISTIAN DIOR SE PAR 74.0 3.482 1.585 1.860 4.092 0.095 5.068 8.550 5.952 10.044ARCELORMITTAL AMS 13.4 0.450 0.610 0.566 2.015 0.138 1.060 1.511 2.580 4.595SAFRAN PAR 38.7 2.944 1.467 1.154 2.380 0.386 4.411 5.267 3.534 4.141ENGIE PAR 28.3 0.485 0.652 0.269 1.846 0.138 1.136 1.621 2.114 2.220EDF PAR 31.9 0.789 0.872 0.906 2.429 0.234 1.662 2.451 3.335 4.780CREDIT AGRICOLE PAR 21.2 0.642 0.780 0.670 1.729 0.193 1.422 2.065 2.399 4.128CAPGEMINI PAR 18.5 2.622 0.754 0.747 2.987 0.256 3.377 3.701 3.735 3.667AIRBUS PAR 50.4 2.921 1.232 1.320 3.268 0.116 4.153 7.075 4.588 7.856ORANGE PAR 25.3 0.480 0.925 0.515 2.162 0.250 1.405 1.885 2.676 2.577THALES PAR 13.9 2.656 1.241 0.865 2.351 0.317 3.897 5.161 3.216 3.598MICHELIN PAR 16.6 2.360 0.740 0.416 2.724 0.320 3.100 3.050 3.139 1.713KERING PAR 73.7 2.572 0.864 0.684 2.463 0.332 3.436 3.466 3.146 2.742PERNOD RICARD PAR 37.1 2.740 1.090 1.153 2.256 0.297 3.829 4.761 3.409 5.036SCHNEIDER ELECTRIC PAR 58.2 2.644 1.534 1.058 2.594 0.326 4.178 6.240 3.652 4.304PEUGEOT PAR 14.2 1.725 0.888 0.805 2.248 0.128 2.613 4.339 3.053 5.301ROYAL DUTCH SHELL AMS 83.0 0.677 0.529 0.350 0.753 0.137 1.206 1.883 1.103 1.856GBL BRU 11.9 2.952 1.323 1.095 2.457 0.194 4.275 7.227 3.552 6.009KBC BRU 18.5 2.124 0.702 0.716 2.881 0.217 2.826 3.944 3.597 4.018UCB BRU 17.9 2.607 0.760 1.136 3.077 0.228 3.367 4.101 4.213 6.126VIVENDI PAR 27.9 1.968 0.936 0.733 2.806 0.318 2.904 3.874 3.539 3.033Average - 39.7 2.27 1.02 1.00 2.61 0.26 3.29 4.25 3.61 4.59 Table 3: Table of results, for the 50 stocks in our sample with the higher market cap.The column Exch. displays the exchange the stock is traded on (Amsterdam, Paris,Brussels or Lisbon) and the column Mcap shows the market capitalization of the stockin billions of euros (in October 2020). Then, a A and a B are the estimated tail exponentsof sell and buy limit order distributions, for the left (l) and right (r) tail. c is theestimator for the constant in equation (9) and a = a A + a B without market orders (noMO), and a = min( ( c +1) a A c , a A + 2 a B ) with market orders (MO), both displayed for left(l) and right (r) tails. 23 Conclusions
In this paper we study the tails of closing auction return distributions, both from atheoretical and empirical point of view, focusing on large closing price fluctuations. Usingthe stochastic call auction model of Derksen et al. (2020a), we relate tail exponents oforder placement distributions and tail exponents of the return distribution. Empiricalanalysis supports the model’s predictions. In theory, large market orders could be acause of large closing price fluctuations, but this potential effect is cancelled by limitorders that are submitted against the direction of the market order imbalance. Instead,limit order placement appears to be the primary cause of observed heavy tails in closingauction return distributions.
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A Proofs
Proposition
Under assumption 1, we have P ( X > M | N A , N B ) ∼ N B T B ( M ) T A ( M ) N A , as M → ∞ . (7)25 roof The expression for the conditional distribution of X in equation (6), implieslim M →∞ P ( X > M | N A , N B ) T B ( M ) T A ( M ) N A = N A (cid:88) k =0 N B (cid:88) l = k +1 lim M →∞ (cid:18) N A k (cid:19)(cid:18) N B l (cid:19) (cid:18) − F A ( M ) T A ( M ) (cid:19) N A (1 − F A ( M )) − k × (cid:18) − F B ( M ) T B ( M ) (cid:19) (1 − F B ( M )) l − = N A (cid:88) k =0 N B (cid:88) l = k +1 lim M →∞ (cid:18) N A k (cid:19)(cid:18) N B l (cid:19) (cid:18) − F B ( M )1 − F A ( M ) (cid:19) k (1 − F B ( M )) l − − k = N A (cid:88) k =0 N B (cid:88) l = k +1 lim M →∞ (cid:18) N A k (cid:19)(cid:18) N B l (cid:19) (cid:18) T B ( M ) T A ( M ) (cid:19) k (1 − F B ( M )) l − − k = N B , where we exchange limit and sum by dominated convergence, and the last line followsbecause all terms are 0, except when l = 1 , k = 0. (cid:3) Theorem
Under assumption 1we have P ( X > M ) ∼ CT A ( M ) T B ( M ) , as M → ∞ , where C = (cid:80) Nn =1 np N A ,N B (1 , n ) = E [ N B { N A =1 } ] > . Proof
The result of proposition 2.5 implieslim M →∞ P ( X > M ) CT A ( M ) T B ( M ) = lim M →∞ E N B T B ( M ) T A ( M ) N A CT A ( M ) T B ( M )= lim M →∞ (cid:80) Ni =1 (cid:80) Nj =1 p N A ,N B ( i, j ) jT B ( M ) T A ( M ) i CT A ( M ) T B ( M )= N (cid:88) i =1 N (cid:88) j =1 jp N A ,N B ( i, j ) C lim M →∞ T A ( M ) i − = (cid:80) Nj =1 jp N A ,N B (1 , j ) C = 1 , where sum and limit are exchanged by dominated convergence and the last line followsby the fact that all terms in the sum are 0, except for i = 1. (cid:3) Proposition
Under assumptions 1 and 2, we have P ( X > M | N A , N B , ∆) ∼ K ( N A , N B , ∆ − T B ( M ) (∆ − − T A ( M ) N A − (∆ − + , (8) as M → ∞ , where K ( N A , N B , ∆) = (cid:0) N A ∆ (cid:1) if ∆ > (cid:0) N B − ∆ (cid:1) if ∆ ≤ . roof Suppose first that ∆ − >
0. Thenlim M →∞ P ( X > M | N A , N B , ∆) T A ( M ) N A − (∆ − = N A (cid:88) k =0 N B (cid:88) l =max( k − ∆+1 , lim M →∞ (cid:18) N A k (cid:19)(cid:18) N B l (cid:19) (cid:18) − F A ( M ) T A ( M ) (cid:19) N A − (∆ − × (1 − F A ( M )) ∆ − − k (1 − F B ( M )) l = N A (cid:88) k =0 N B (cid:88) l =max( k − ∆+1 , lim M →∞ (cid:18) N A k (cid:19)(cid:18) N B l (cid:19) T B ( M ) l T A ( M ) k − ∆+1 = (cid:18) N A ∆ − (cid:19) , where the last line follows because all terms are 0, except when k = ∆ − , l = 0.Now let ∆ − <
0, thenlim M →∞ P ( X > M | N A , N B , ∆) T A ( M ) N A T B ( M ) − ∆ = N A (cid:88) k =0 N B (cid:88) l = k +1 − ∆ lim M →∞ (cid:18) N A k (cid:19)(cid:18) N B l (cid:19) (cid:18) − F A ( M ) T A ( M ) (cid:19) N A − k × T A ( M ) − k (cid:18) − F B ( M ) T B ( M ) (cid:19) l T B ( M ) l +∆ − = N A (cid:88) k =0 N B (cid:88) l = k +1 − ∆ lim M →∞ (cid:18) N A k (cid:19)(cid:18) N B l (cid:19) T B ( M ) l +∆ − T A ( M ) k = (cid:18) N B − ∆ (cid:19) , where the last line follows because all terms are 0, except when k = 0 , l = 1 − ∆. (cid:3) Theorem
Under assumptions 3 and 4, there exists a constant
C > , such that P ( X > M ) ∼ CM − a , as M → ∞ , where a = min (cid:18) ( c + 1) a A c , a A + 2 a B (cid:19) . (10) Proof
Under assumptions 3 and 4, proposition 2.8 transforms into, P ( X > M ) ∼ N (cid:88) n =1 N (cid:88) m =1 N (cid:88) d = − N K ( n, m, d − M − ( a A ( n − d +1)+( a B − a A ) max( − d +1 , p ( n, m, d ) , as M → ∞ . Here, we used that max( − x, − max( x,
0) = − x , for all x ∈ R . By notingthat K ( n, m, d ) is bounded from above and below (by (cid:0) NN/ (cid:1) and 1), we see that the27tatement of the theorem holds true, for a = min n,d : p ( n,d ) > ( a A ( n − d + 1) − ( d − a B − a A ) { d< } ) , where the minimum is taken over all n, d such that p ( n, d ) = (cid:80) m p ( n, m, d ) >
0. Nownote that the function F ( n, d ) := a A ( n − d + 1) − ( d − a B − a A ) { d< } is increasingin n , for every d . So the minimum is attained for the lowest n with positive probability.Recall that we assumed ∆ = c ( N A − N B ) and N B ∈ { , . . . , N } , so p ( n, d ) = 0 for n < dc + 1, so the lowest n with positive probability is ˆ n ( d ) = max( dc + 1 , d .Inserting into F leads to F (ˆ n ( d ) , d ) = a A − ( d − a B if d ≤ − a A (( c − d + 2) if d ≥ , which is minimal for d = ±
1, proving that a = min (cid:18) a A (cid:18) c + 1 (cid:19) , a A + 2 a B (cid:19) = min (cid:18) ( c + 1) a A c , a A + 2 a B (cid:19) . (cid:3)(cid:3)