From orders to prices: A stochastic description of the limit order book to forecast intraday returns
TThe What, When and Where of Limit Order Books
Johannes Bleher ∗ University of T¨ubingen , Michael Bleher † Heidelberg University
Thomas Dimpfl ‡ University of T¨ubingen
Tuesday 28 th April, 2020
Abstract
We model the limit order book (LOB) as a continuous Markov process and developan algebra to describe its dynamics based on the fundamental events of the book:order arrivals and cancellations. We show how all observables (prices, returns, andliquidity measures) are governed by the same variables which also drive arrival andcancellation rates. The sensitivity of our model is evaluated in a simulation studyand an empirical analysis. We estimate several linearized model specifications basedon the theoretical description of the LOB and conduct in- and out-of-sample forecastson several frequencies. The in-sample results based on contemporaneous informationsuggest that our model describes up to 90% of the variation of close-to-close returns,the adjusted R still ranges at around 80%. In the more realistic setting where onlypast information enters the model, we still observe an adjusted R in the range of15%. The direction of the next return can be predicted, out-of-sample, with anaccuracy of over 75% for short time horizons below 10 minutes. Out-of-sample, onaverage, we obtain R values for the Mincer-Zarnowitz regression of around 2-3%and an RM SP E that is 10 times lower than values documented in the literature.
Key words:
Limit Order Book, Master Equation, Continuous Markov Pro-cess, High Frequency, Market Microstructure
JEL:
C58, D43, G12 ∗ Corresponding author. Johannes Bleher; Department of Econometrics, Statistics and EmpiricalEconomics, University of T¨ubingen. Sigwartstr. 18, 72074 T¨ubingen, Phone: +49 7071 29 78165. Email: [email protected] † Michael Bleher; Mathematical Institute, Heidelberg University. Mathematikon, Im Neuenheimer Feld205, 69120 Heidelberg. Email: [email protected] ‡ Thomas Dimpfl; Department of Econometrics, Statistics and Empirical Economics, University ofT¨ubingen. Sigwartstr. 18, 72074 T¨ubingen, Tel: +49 7071 29 78165. Email: [email protected] a r X i v : . [ q -f i n . T R ] A p r Introduction
Ever since Glosten (1994) raised the question whether ’the electronic open limit orderbook [was] inevitable’, limit order books (LOBs) have become the most important wayof trading, resulting in more than 75% of exchanges around the world (including therecently sprouting cryptocurrency exchanges) using order driven systems and relying (atleast partially) on limit order books (Jain 2003). Nevertheless, ’no comprehensive andrealistic models (either statistical or economic) exist’ (Hasbrouck 2007, Ch. 12, p. 118)which describe the deep-rooted mechanisms of limit order markets in their entirety. WhileHasbrouck’s statement is more than 10 years old and the literature has made significantprogress, a dynamic model which comprehensively describes the interaction of individualorders and is able to incorporate the (strategic) behavior of market participants has, tothe best of our knowledge, not yet been developed. Gould, Porter, Williams, McDonald,Fenn and Howison (2013) provide an overview about research on the dynamics of the LOBand identify (roughly speaking) two branches of research. One of them originates in thefield of physics and focuses mainly on idealized models to describe statistical features ofthe LOB system, focusing on dynamic order flows. The second branch is rooted in theeconomics literature which tends to treat order flows as static. According to Gould et al.(2013), economists primarily focus on the (strategic) behavior of traders, but neglect thedynamical structure. Of course, this reduction is too simplistic and there are multipleattempts to combine strategic behavior and the order book dynamics. For example, Parlour(1998) develops a stylized, dynamic model for the LOB and strategic order placement.Nevertheless, only few models approach the subject by heuristically incorporating statisticalregularities observed in market microstructure and incorporate trader interaction basedon these statistical observations. A notable exception is Hautsch and Huang (2012) whouse high-frequency cointegrated vector autoregressive models to shift the spotlight on theorder flow of incoming orders and therewith, based on empirical analysis, draw attentionto the intersection of LOB mechanics and strategic behavior of market participants. Theyshow that the revelation of trading intentions through limit order placements affects the1OB states. Large (2007) shows how trades of different size (marketable incoming orders)affect future states of the LOB. Paulsen (2014) derives macroscopic limiting models for amicroscopic LOB system in which bid, ask, and transaction prices drive the dynamics ofthe order flow. These limiting models serve as first order approximations of the stochasticprocesses that describe the system.This paper aims to link models of statistical dynamic order flow and those of strategicinteraction. Adapting ideas from Paulsen (2014) and inspired by Baez and Biamonte(2018), we relate the LOB dynamics to the modeling of reaction networks (developedby Baez and Pollard 2017) and present a simple but comprehensive description of themicroscopic order book mechanisms. Based on operator algebra, we construct the LOBdynamics bottom up from the elementary events of the book, namely the entry and exitof orders which enables us to model the LOB system by a Markov process. Furthermore,the Hamiltonian, i.e., the operator which governs the time dynamics of the system, canbe constructed from these elementary events. Similar to Paulsen (2014), our approachallows to incorporate both the dynamics of the fundamental LOB mechanisms as well asthe (strategic) behavior of market participants. Using the event log of the first quarterof 2004 from XETRA, we heuristically develop, simulate and empirically evaluate theimplications of our operator formulation. By describing the interaction of individual ordersin a purely statistical fashion, the state space of the order book system is worked out whichdepends only on the rates of arriving and canceled orders as well as on the current stateof the book. Based on a limited set of key variables, we show that the return dynamicscan be approximated by a linear model. This variable set also allows to linearly forecastreturns, arrival rates, and other measures such as order book imbalance and liquidity.Compared to prediction errors reported in the literature such as Zhou, Pan, Hu, Tang andZhao (2018), forecasts based on these variables reduce the root mean squared predictionerror (
RMSPE ) drastically by a factor of 1 /
10. The in-sample R as well as the R of aMincer and Zarnowitz (1969) type regression for the out-of-sample predictions show anextraordinary fit for intraday return forecasts.2imilar to our approach, Cont, Stoikov and Talreja (2010) explore the idea of modeling theorder book as a Markov process depending on the rates of arrivals and cancellations. Theywork out closed form solutions for the probability of an increase in the midprice, executionof an order at the best bid price (before a change of the best ask price), and executionof both a buy and a sell order at the best quotes before the price moves. Cont andde Larrard (2013) show that such a model can also be used to calculate the distributionof the duration between transactions. Unfortunately, as Cont and de Larrard (2012)state, these models are based on several assumptions which empirical research has shownto be incorrect (Bouchaud, M´ezard and Potters 2002, Hasbrouck 2007). In particular,the time intervals between order arrivals and cancellations are neither independent norexponentially distributed and orders are not equally sized. Another model which is basedon the Markov properties of the order book has been developed by Daniels, Farmer,Gillemot, Iori and Smith (2003) and extended by Smith, Farmer, Gillemot, Krishnamurthyet al. (2003). However, the model imposes broad restrictions on the functional structure,the parameters and the assumptions about the stochastic processes which govern the LOBdynamics. Again, some of their assumptions like equal order size or balanced order flow,order placement with uniform probability, among others are ’too simple to be literallytrue’ (Smith et al. 2003), but the resulting insights provide a useful foundation for LOBmodeling.The paper proceeds as follows. Section 2 sets out the description of a typical LOB by analgebra of operators. Simultaneously, we present some selected empirical characteristics ofthe order book which guide our model development. Section 3 presents the XETRA orderbook data. In Section 4, we report the results of a simulation study where we identify keyfactors/drivers of the order flow which determine the statistical distribution of what willhappen in the order book and also when and where it happens on short time horizons.Finally, Section 5 holds an empirical analysis of the XETRA LOB and Section 6 concludesour paper. 3 The Model
The limit order book is the place where traders’ orders meet. These orders carry informationabout a trader’s willingness to accept a certain price, the limit price, in exchange for thechosen number of instruments, or vice-versa. The price level at which two orders arematched is called the reference price . Within the LOB, we distinguish buy (ask) orderand sell (bid) orders. Perceiving these two order types as species which populate priceand order size levels inspired the use of the mathematical tools presented in Baez andBiamonte (2018). At any point in time the exchange keeps track of all orders within theLOB. Aside from the market side, the location within the LOB is defined by three keycomponents: the limit price up to which the trader wants to buy or sell, the number ofsecurities, and the time when the order arrives in the LOB.If traders require immediacy, they rely on market orders which can be thought to have aninfinite (bid order) or zero (bid order) limit price depending on the market side they wereissued from. These orders are matched immediately and, normally, do not reside in theorder book for an extended amount of time. They enter the LOB at the best price level ofall limit orders currently residing in the LOB on the opposite market side.The majority of orders, however, are designed to remain in the book for some time attheir specified limit price level. These are called limit orders which (as we will show inSection 3) make up for roughly 90% of all orders in the XETRA LOB while only 3% aremarket orders. Limit orders have a well defined location in the price dimension. If theirdesignated price location is behind the best price level of all limit orders which currentlyreside in the LOB on the other side of the market, they are matched (partially) beforethey can reach their designated limit price level. The smallest populated price level on thesell side of the market is the best ask and the highest price level on the bid side is the bestbid . We will refer to one or the other as the best quote .There are generally further order types that are only submitted to the market if certain4onditions are met, for example stop orders , which are inserted in the LOB contingently onthe reference price reaching or falling below a certain threshold price, or
XETRA BESTorders . But once such orders enter the LOB, they are effectively equivalent to market orlimit orders. As conditional orders can be perceived as more sophisticated versions of limitor market orders, they can in principle be incorporated within the framework presentedbelow. For the purpose of this paper, we restrict our considerations, thus, to plain marketand limit orders.
In the following, we describe order creation and cancellation in the LOB as determinedby the rules of a typical order book. For this purpose, we borrow the so-called
Dirac or Bra-Ket Notation from physics, where the state of a system is denoted by a ket ∣ ψ ⟩ . InSection 2.2, we will discuss the underlying notion of a state in detail. For now, we can referto any possible configuration of the order book with ∣ ψ ⟩ . Even further, we can also assignweights (probabilities) to each of these possible configurations and refer to such a weightedbundle of pure states by ∣ ψ ⟩ . Nevertheless, we start off with a very concrete state: theempty order book (or vacuum) ∣ ⟩ . From this vacuum state, more complicated order bookstates are created by successively acting on it with creation and annihilation operators.As we will see below, the rules of the LOB induce certain commutation relations in thealgebra of these operators. It will be convenient to also introduce the notation ∣ ⟩ = ∣ ∣ (1)which represents an empty ledger. Rule 1a (Ask Order Submission) . Traders can submit a limit ask order of quantity q at aspecified price level k . The order is represented by a creation operator a + k,q that acts on theorder book state from the right . This choice will become relevant in the context of price-time priority, see Rule 3. n ask orders are residing in the book, each with its associated limit price k i and size q i , i ∈ , . . . , n , the order book state is given by ∣ ∣ a + k ,q . . . a + k n ,q n . (2) Rule 1b (Bid Order Submission) . Traders can submit a limit bid order of quantity q at aspecified price level k . The order is represented by a creation operator b + k,q that acts on theorder book state from the left . Analogously, for m bid orders residing in the book with specified limit prices k j and sizes q j with j ∈ , . . . , m , the following string of operators describes the current state: b + k m ,q m . . . b + k ,q ∣ ∣ . (3)Note that, so far, the rules only describe the successive submission of orders. In particular,we do not yet have a rule that would allow us to reorder the queue of creation operators.Put differently, creation operators generally do not commute: a + k,q a + s,p ≠ a + s,p a + k,q . As a result,the strings of creation operators of ask and bid type are time-ordered. Rule 2a (Ask Order Cancellation) . Traders can cancel a previously submitted ask order.An ask order cancellation is represented by an annihilation operator a − k,q which acts fromthe right and satisfies ∣ ∣ a + k,q a − k,q = ∣ ∣ . (4)Clearly, the probability of a cancellation must be zero if there is no order in the book.This means that when an annihilation operator acts on the empty order book, it generatesa state with probability mass zero: ∣ ∣ a − k,q = . (5)6here is a standard argument, that there is always one more possibility to create andthen delete an object than deleting and then creating one. In terms of the operators thisargument is represented by the commutation relation [ a + k,q , a − k,q ] = [ A, B ] ∶= AB − BA denotes the commutator of two operators. In fact this relationdirectly follows from (4) and (5) ∣ ∣[ a + k,q , a − k,q ] = ∣ ∣( a + k,q a − k,q − a − k,q a + k,q ) = ∣ ∣ a + k,q a − k,q − ∣ ∣ a − k,q a + k,q = ∣ ∣ Furthermore, since the cancellation of an order a + k,q does not influence other orders a + s,p ,we also have [ a + s,p , a − k,q ] = s ≠ k and p ≠ q . We can summarize these algebraic relations as follows: [ a + k,q , a − s,p ] = δ sk δ pq (6)where δ ij is the Kronecker-Delta, defined on an index set I ∋ i, j by δ ij = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ i = j Rule 2b (Bid Order Cancellation) . Traders can cancel a previously submitted bid order.A bid order cancellation is an annihilation operator b − k,q that acts from the left. By analogy In physics, these commutation relations are known as canonical commutation relations . ith the ask cancellations, it satisfies b − k,q ∣ ∣ = [ b − k,q , b + j,p ] = δ kj δ qp . (8)In comparison to the commutation relation of ask orders, the order of annihilation andcreation operators is reversed since bid orders act from the left.When there are several identical limit orders, i.e., orders with the same price level andquantity, we can distinguish their position in the order queue by means of their time stamp.In contrast, for cancellation orders, an observer cannot predict which of the identical limitorders is supposed to be canceled. The algebraic formalism captures this uncertainty:up to normalization, the commutation relations lead to a stochastically mixed state thatcontains each possible cancellation, for example (∣ ∣ a + k,q a + r,s a + k,q ) a − k,q = ∣ ∣ a + k,q a + r,s + ∣ ∣ a + r,s a + k,q . Remark.
We also introduce the convention that arrivals and cancellations with size q = a + k, = a − k, = b + k, = b − k, = . Rule 3 (Price-Time Priority) . Orders are organized according to price-time priority.
The order book state is the result of successive order submissions and the correspondingstring of operators is strictly ordered by time. Hence, we get a price-time ordering byrearranging the operators into groups with identical price level whilst maintaining the timeordering within each group. This can be achieved by letting ask and bid orders commute8henever they have different price levels k ≠ s : [ a + k,q , a + s,p ] = , (9) [ b + k,q , b + s,p ] = . (10)Using these relations, the order book state can always be written in the price-time orderedform ∣ ψ ⟩ = b + k ,q ⋯ b + k n ,q n ∣ ∣ a + k n + ,q n + ⋯ a + k n + m ,q n + m , (11)where k i ≤ k i + . Whenever k i = k i + , the order nearer to ∣ ∣ was submitted first.Given a LOB state in price-time ordered form, the priority of an order is encoded by itsdistance to ∣ ∣ , where orders closer to ∣ ∣ have higher priority. Rule 4 (Order Matching) . Two orders from different market sides permit a transaction if they have highest priority and the bid price is bigger or equal to the ask price. When theLOB executes orders that permit a transaction, the quantities are matched up as far aspossible and unmatched quantities remain in the book. We write b k,q ∣ ∣ a s,p for a pair ofexecuted orders, such that for k ≥ s the matching procedure is captured by b + k,q ∣ ∣ a + s,p = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩∣ ∣ a + s,p − q if q > p ∣ ∣ if q = pb + k,q − p ∣ ∣ if q < p (12) or as an algebraic relation of creation operators b + k,q a + s,p = θ ( q − p ) b + k,q − p + θ ( p − q ) a + s,p − q here θ ( x ) is the Heaviside step function θ ( x ) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ if x > if x = else. Recall for the case q = p that orders of size 0 are equivalent to the identity operator.Since incoming bid (ask) orders always act on a state from the left (right), we need tocommute them through older orders until they reach their designated position in theprice-time ordered queue. This means orders automatically ’walk the book’ until theyreach their destined price level k . Along its walk, an order may encounter orders of theother market side and will then be executed as described by Rule 4. It follows that marketorders are described by creation operators a + k = ,q and b + k =∞ ,q , which will walk all the waythrough the book until they are completely executed.Let us stress that the price level k of an order a + k,q is not necessarily the transaction price at which the order will be executed. Instead, the transaction price is usually determinedby the price level of the ’settled order’ that is encountered by the ’walking order’. The’settled order’, however, may depend on the trading mode (see Section 2.5). Also notethat we did not yet specify when orders are executed and when a transaction will takeplace. The reason is that such rules do not add further algebraic relations. The questionwhen orders are matched is not relevant for the description of the current state of thebook. However, it is relevant for the time dynamics, i.e., if we examine time series of orderbook states. The question is whether matching occurs after each event, as continuoustrading dictates, or whether matching is only conducted hypothetically after each event toproduce indicative prices like in the pre-auction phase or at the end of an auction. Thesedifferent modes matter for the evolution of the book. A detailed discussion of transactions In the LOB literature, ’walking the book’ usually refers to an arriving, marketable order that isexecuted against several orders on the opposite market side. We borrow this notion of the walkingorder and extend it. In our case, every order ’walks through the book’, however, only marketable ordersencounter orders on their way to their destined limit price level.
As we have seen in Section 2.1, at any given time the state of the order book is given by aprice-time ordered string of creation operators ∣ ψ ⟩ = b + k ,q ⋯ b + k j ,q j ∣ ∣ a + k j + ,q j + ⋯ a + k n ,q n , where k i ≤ k i + for all i ∈ S = { , . . . , n } . Each operator in this string is specified by itsmarket side m ∈ { a, b } = M , the price level k ∈ K ⊂ R , and the order size q ∈ Q ⊂ R . Here K and Q are the price and quantity levels at which orders can be submitted to the LOB.These typically are discrete subsets of R . In principle, price and quantity levels can becomearbitrarily large, but in practice one can introduce a cutoff for both prices and quantitiesat a large enough value. As a result, K and Q , can be thought of as finite sets.It is convenient to introduce a partial ordering ≤ on creation operators via the followingset of relations: b + k ,q ≤ b + k ,q ≤ a + k ,q ≤ a + k ,q ⇐⇒ k ≤ k ≤ k ≤ k . Then any price-time ordered string of creation operators is equivalent to a monotonicallyincreasing map z ∶ S → M × K × Q i ↦ z i = ( m + i ) k i ,q i s.t. z i ≤ z i + from some finite set S ⊂ N to the set of creation operators. We denote the associated11tates by ∣ z ⟩ ∶= z ⋯ z j ∣ ∣ z j + ⋯ z n . The collection {∣ z ⟩} fully describes the possible configurations of the order book at anygiven moment. We may refer to these states as pure states . Note that S , M , K , and Q are countable sets, so the set {∣ z ⟩} is countable as well.Clearly, any prediction of the future state of the order book must be probabilistic. So, whilepure states are potentially observable, mixed states are not. Mixed states are composed ofseveral pure states in which each pure state is weighted with some probability mass. Forthis reason, mixed and pure states are elements of the vector space spanned by the purestates ∣ z ⟩ : H = { ∣ ψ ⟩ = ∑ ∣ z ⟩∈H p ( z ) ∣ z ⟩ ∣ p ( z ) ∈ R } . In reality, the order book must be in a stochastically normalized state, i.e., a state ∣ ψ ⟩ = ∑ ∣ z ⟩ p ( z ) ∣ z ⟩ where 0 ≤ p ( z ) ≤ ∣ z ⟩ and ∑ ∣ z ⟩∈H p ( z ) =
1. In this case, (thecoefficient) p ( z ) is the probability that the state ∣ z ⟩ will be realized. Remark.
The normalized states are a closed subset of the full vector space H . In particular,they do not form a vector space themselves. It is often easier to work with unnormalizedvectors and rescale the result to a normalized state at the end.In Section 2.3, we describe the time evolution of an initial state ∣ z ⟩ at time t . We willsee that (by construction) time evolution produces a state ∣ ψ ( t ) ; z , t ⟩ = ∑ z ∈H p ( z, t ∣ z , t ) ∣ z ⟩ . (13)In the literature, this object is usually referred to as generalized probability generatingfunction (see for example Weber and Frey 2017, Section 2). For brevity, we often write ∣ ψ ( t )⟩ and drop the reference to the conditional nature of the generalized probability12enerating function.It is customary to denote linear functionals by ’bra’ vectors ⟨ ψ ∣ ∈ H ∗ and introduce thedual basis ⟨ z ∣ , which satisfies ⟨ z ′ ∣ z ⟩ = δ z ′ ,z . In this notation, the conditional probability to find a state ∣ z ⟩ at time t in ∣ ψ ( t ) ; z , t ⟩ isgiven by p ( z, t ∣ z , t ) = ⟨ z ∣ ψ ( t )⟩ . (14) In this section, we introduce dynamics to the order book, i.e., we explain how an order bookstate evolves over time. Throughout this section, we closely follow Baez and Biamonte(2018), where the general theory of stochastic time evolution is laid out in great detail.The future state of the order book arises from acting on an initial state with the orderoperators introduced in Section 2.1. This means that we are automatically in the situationof a Markov process.The only issue is that the rate (probability) of incoming orders can depend on the historyof the order book. It is, however, not sensible to assume that the entire history of theorder book affects the properties of arrival and cancellation rates as old configurations ofthe LOB are usually not relevant for the decision process of market participants. Theyusually seek to maximize the probability of order execution based on the current state ofthe order book and possibly a very narrow history of preceding order book configurations.This only implies that arrival rates may be dependent on several preceding states of theLOB, which is not in contradiction to the Markov property per se. It would only meanthat the process governing the LOB dynamics might be of Markov order higher than 1.13owever, theoretically, by appropriately extending the state space, every Markov processof finite order can be expressed as a Markov process of order one. Thus, we assume thatthe dynamics of the order book follow a Markov process of order 1.As a continuous Markov process, the order book satisfies the Master equation (cp. vanKampen 1992, Weber and Frey 2017). In our notation, the Master equation is given by ∂∂t ∣ ψ ( t )⟩ = H ∣ ψ ( t )⟩ , (15)where the so-called Hamiltonian operator H encodes all information on the transitionprobabilities between order book states.A solution of the Master equation is provided by a stochastic time evolution operator U ( t, t ) via ∣ ψ ( t )⟩ = U ( t, t ) ∣ ψ ( t )⟩ . If the Hamiltonian is time-independent, the time evolution operator is remarkably easy: U ( t, t ) = e H ( t − t ) . (16)If the Hamiltonian is time dependent, the time evolution operator can similarly be writtenas U ( t, t ) = exp ( ∫ tt H ( τ ) dτ ) , (17)but the evaluation of this expression is typically more involved.We assume that other variables, like news from outside the order book, may impactthe rates of incoming orders. However, these variables are pre-determined outside themechanism of the order book. This may lead to time dependent arrival and cancellationrates. In the system description, this would mean that the Hamiltonian is time dependent.14evertheless, a time-independent approximation of such a system may still serve as agood approximation if the time intervals are small enough. In Section 5, we will take anindeterministic approach towards those other variables and regard them as predeterminedoutside the book. Again, this is not in contradiction to the Markov property of the LOBsystem which is the key assumption for the Master equation (15) to hold. At this point, itis also interesting to note that beyond the model presented in this paper, the LOB maybe an open Markov process, which can be described by relying on a compositional modelframework – in the sense of Baez and Pollard (2017) – and would allow to incorporatetrader behavior.Given the above considerations, the dynamics of the order book are fully described by thechoice of a Hamiltonian H . Baez and Biamonte (2018) show how H can be constructedfrom infinitesimal stochastic operators which describe the elementary transitions that cantake place in a system.In the LOB, there are four possible transitions for each price level k and each quantity q :entry of an ask order E Ak,q = ( a + k,q − ) entry of a bid order E Bk,q = ( b + k,q − ) cancellation of an ask order C Ak,q = ( a − k,q − N Ak,q ) cancellation of a bid order C Bk,q = ( b − k,q − N Bk,q ) where a ± k,q , b ± k,q are the creation and annihilation operators of Section 2.1. The numberoperators N Ak,q = a + k,q a − k,q and N Bk,q = b + k,q b − k,q return the number of active bid and ask orderson price level k and of quantity q when they act on a pure LOB state (see Section 2.4). Remark.
Creation and annihilation operators are not infinitesimal stochastic operators.This is why there are additional terms (− , − N ) in the operators corresponding to orderentry and cancellation. 15s mentioned above, the Hamiltonian of the LOB is a combination of elementary transitions H = ∑ k ∑ q E Ak,q α A ( k, q ) + E Bk,q α B ( k, q ) + C Ak,q ω A ( k, q ) + C Bk,q ω B ( k, q ) , (18)where each transition is weighted by its arrival rate α or cancellation rate ω , respectively.Also note that the arrival rates need to be scaled such that the time evolution operator U ( t, t ) is indeed stochastic and maps one stochastically normalized state to another.Generally, the arrival and cancellation rates in a LOB are observed to be time dependent.Intraday patterns of order flow have been documented for example by Biais, Hillion andSpatt (1995). Even for the very recent development of international Bitcoin markets, inwhich trading is possible 24/7, Eross, McGroarty, Urquhart and Wolfe (2019) documentactivity patterns related to the opening and closing of major markets. The observedclustering of transactions in time can be conceived as the result of time dependent arrivaland cancellation rates. These are usually modeled using Autoregressive ConditionalDuration (ACD) models (see Engle and Russell 1998, Fernandes and Grammig 2006).We may treat the arrival and cancellation rates, especially on small and intermediatetime scales, as mainly determined by the state of the order book, in the sense that thedistributions of the rates across k and q depend on the current state of the order book –for example via current best bid/ask or the spread. With this dependence on the currentstate, we allow for a quite general feedback mechanism between the current state of theorder book and arrival and cancellation rates. If the state of the LOB changes by anevent, the arrival and cancellation rates may change subsequently, as well. We investigateboth static and dynamic specifications of arrival rates. In Section 4, we will investigatestatic distributions using empirical unconditional frequencies and a uniform as well as atheoretical discrete Gaussian exponential (DGX) distribution for arrival and cancellationrates across price levels. The latter can be justified heuristically by the characteristicsfound in our data as described in Section 3, in particular Figure 2. We will, in onesimulation scenario, also allow the parameters of the assumed DGX distribution, for arrivaland cancellation rates across relative price levels, to depend on the spread. In Section 5, we16easure the arrival rates during fixed non-overlapping time intervals and therewith allowthem to vary over time. We also incorporate in our empirical analysis in Section 5 the ideaof conditional autoregressive arrival and cancellation rates and include lagged terms ofarrival rates, moments of the spread and the distance to the opposite best quote. Samplingthe LOB data on different time intervals, i.e., taking snapshots of the current state atdifferent frequencies (for example 1, 2, and 5 minutes), allows to relate the moments ofthe relative integer distance d l (as defined in Equation (28) in Section 3) and the quantityof incoming and canceled orders q to price changes and other observables of the system.Empirical tests of these implications can be found in Section 5.For now, we focus on the conceptual implications of these empirical findings and on howthey affect the set up of the time evolution of the LOB system. Thus, we denote thearrival rate of an order at price k and quantity q as α M ( k, q ) , M ∈ M . Since the distanceto the opposite market side d and the prevalent spread ∆ depend on the current state ofthe order book, the arrival rates must be considered to be operators. When α M ( k, q ) actson a pure state ∣ z ⟩ , it returns an arrival rate which depends on the values of d and ∆ thatare realized in the state ∣ z ⟩ : α M ( k, q ; z ) = ⟨ z ∣ α M ( k, q )∣ z ⟩ M ∈ { A, B } . A similar operator yields the distribution of cancellation rates corresponding to the currentstate of the order book: ω M ( k, q ; z ) = ⟨ z ∣ ω M ( k, q )∣ z ⟩ M ∈ { A, B } . In the Hamiltonian given in Equation (18) the operators α M ( k, q ) and ω M ( k, q ) act on thestate first, thus determining the rate of the corresponding transition E Mk,q that acts on thestate subsequently.
Remark.
Since the Hamiltonian H is linear in the transition operators, it can be decomposedinto smaller pieces that describe a subsystem of the LOB. For example, we can split up17he Hamiltonian into ask and bid Hamiltonians H = H A + H B H M = ∑ k,q E Mk,q α M ( k, q ) + C Mk,q ω M ( k, q ) , M ∈ M . Similarly, we could decompose H into the Hamiltonians for all price and quantity levels: H = ∑ k,q H k,q H k,q = E Ak,q α A ( k, q ) + E Bk,q α B ( k, q ) + C Ak,q ω A ( k, q ) + C Bk,q ω B ( k, q ) . While these decompositions are convenient in calculations, they also allow a different viewon the evolution of the book: In principle, one could argue that the time evolution should bebased on (groups of) traders, whose order submissions and cancellations can be describedby Hamiltonians H g where the index g may indicate a group of traders or individual traders.The notion of particular groups can be found quite often in the literature. For example,Foucault, Sraer and Thesmar (2011) group traders into institutional and individual traders,whereas Foucault, Kadan and Kandel (2005) distinguish patient and impatient traders.These subsystems sum up to an effective Hamiltonian H eff = ∑ H g which will necessarilybe of the form (18). The only difference is that now the rates α and ω become populationparameters in a fundamental model about traders. In this paper, we refrain from modelingtraders and instead estimate effective arrival rate distributions from LOB data.However, there is surely a trader induced clustering or autocorrelation in arrival rateswhich we cannot ignore. There are also patterns induced by general business activitythroughout the day. Additionally, when submitting orders to the LOB, traders often careabout the probability that their submitted orders are executed in due time. There isa trade-off between immediacy and a slightly delayed order execution. The probabilitythat an order is executed is directly linked to the arrival rates of orders in the LOB.Thus, traders may incorporate the history in their decision process, i.e., when, at whichlimit price, and with which quantity they want to submit their orders to the LOB and18gain induce autocorrelation into arrival and cancellation rates. The decomposition ofthe Hamiltonian, as discussed above, would allow for an explicit model and cover such ascenario. In general, the model above does not exclude the notion of autocorrelation inthe arrival rates. Especially in Section 5, however, we take a more indeterministic viewin that we allow prior arrival and cancellation rates of ask or bid orders to proxy currentarrival and cancellation rates. The idea that prior arrival rates determine current rates isalso the guiding notion for the ACD literature mentioned above. A specific configuration ∣ ψ ⟩ of the order book contains an enormous amount of information.Usually, the focus lies on selected descriptive quantities which can be extracted from theorder book at any state. We will call these quantities observables and describe them bythe action of an operator O on pure order book states ∣ z ⟩ . The value of O for a given state ∣ z ⟩ can be calculated as O ( z ) = ⟨ z ∣ O ∣ z ⟩ . More generally, given a state ∣ ψ ⟩ , the ν th conditional moment of the observable O is givenby E [ O ν ; ψ ] = ∑ ∣ z ⟩∈H ⟨ z ∣ O ν ∣ ψ ⟩ . (19)Similarly, we can calculate the expected value of sums and products of distinct operators.This gives rise to covariance and correlation measures, e.g.,Cov ( O , O ) = ∑ ∣ z ⟩∈H ⟨ z ∣ ( O − E [ O ])( O − E ( O )) ∣ ψ ⟩ . In quantum mechanics, a similar relation holds, known as the Born rule ⟨ Ψ ∣ ˆ O ν ∣ Ψ ⟩ . Since we workwith stochastic probabilities (and not with quantum mechanical amplitudes), ⟨ Ψ ∣ needs to be replaced bythe sum over all dual basis vectors ⟨ z ∣ . ∣ ψ ⟩ , we obtain the moments of anobservable’s probability distribution at time t ( t > t ) as E [ O ν ; ψ ( t )] = ∑ ∣ z ⟩ ⟨ z ∣ O ν e H ( t − t ) ∣ ψ ⟩ = ∑ ∣ z ⟩ ⟨ z ∣ O ν ( + H ( t − t ) + H ( t − t ) + . . . ) ∣ ψ ⟩ . (20)Note that the expected value in Equation (20) is a conditional expectation. It is conditionalon the state ∣ ψ ⟩ at time t . Later on, in Section 5, to make this conditioning clear, we willdenote conditional moments with E t [ O ν ] . The following example illustrates the rationalebehind the formula in Equation (20). Consider the operator β B which extracts the valueof the best bid order in a state ∣ z ⟩ (cf. Section 2.4). Furthermore, assume that at t = ∣ ψ ⟩ = b + k ,q b + k ,q ∣ ∣ a + k ,q . Clearly, since k < k < k , the best bid is currently at price level k . According to (20),the expected value of β B at time t is given by an infinite sum. To begin with, we considerterms for which the state does not change during the time period ∆ t = t − t . These includeof course the identity operator 1 in Equation (20). But H is build from terms of the form α ( p, q )( b + p,q − ) , so there are additional contributions at any order in H k . Together, theycontribute the following term to the expected value of the best bid price only for pricelevel k : k ( − ∑ k,q α ( k, q ) ∆ t − ∑ k,q ω ( k, q ) ∆ t − . . . ) . The expression in parenthesis is of course nothing but the probability that the state willnot change within ∆ t .For other price levels, these contributions are different. Therefore, we next investigate thelinear terms in H which describe the entry of a single order. There are only three cases inwhich the best bid changes. First, we may observe an entry of a limit bid order with a20rice level in between best ask and best bid. It’s contribution to the expected value is ∑ k < k < k ,q kα B ( k, q ) ∆ t . Second, the order residing on k , the current best bid price level,may be canceled. In this case the contribution to the expected value is k ω ( k , q ) ∆ t .Third, an arrival of an ask order above the best bid which exhausts the best bid order’squantity may arrive. In this case we get ∑ k ≥ k ,q ≥ q k α A ( k, q ) ∆ t . In all other cases ofincoming limit orders the value of β B remains at k .A similar analysis is possible at order H . This would entail interaction terms of twoorders entering the book: a + k,q α A ( k, q ) b + (cid:96),r α B ( (cid:96), r ) . There are again several different caseswhich depend on the type, price level and quantity of the incoming orders, each contribut-ing differently to the expected value. More generally, at order H n one encounters theprobabilities that n orders enter the book and influence the best bid during time period∆ t . For short time periods, such higher order contributions are negligible compared to thelinear contributions because they depend on products of arrival rates, which are typicallyvery small. However, for long time periods the powers ∆ t n will eventually dominate.The example above illustrates an important property of the model. According to Equa-tion (20), the moments of observables depend solely on the (current) distributions α M ( k, q ) and ω M ( k, q ) . These distributions, if they vary sufficiently slowly, may be measured ormodeled from the event stream of the book. Therefore, a testable hypothesis implied byour model is whether the distributional moments of k (or equivalently d l for that matter)and q (calculated by perceiving α M ( k, q ) and ω M ( k, q ) as their underlying probabilityfunctions) can be used to predict the expected value of observables, including price changesand inter-transaction duration.In the following, we present a selection of observables, which are important for our analysis.21 .4.1 Number and Volume Operators A basic observable is the number of active orders on price level k with size q . It can bedescribed for the bid and ask side by the number operators N Bk,q = b + k,q b − k,q ,N Ak,q = a + k,q a − k,q . These operators can be utilized to extract several other observables. In particular, thetotal number of active orders on price level k of ask or bid type M ∈ M = { A, B } N Mk = ∑ q N Mk,q , the quantity of active orders on price level k and the total quantity on each market side M ∈ M Q Mk = ∑ q qN Mk,q ,Q M = ∑ k Q Mk , or the volume of active orders at price level k and the total volume on each market side V Mk = kQ Mk ,V M = ∑ k V Mk . There are also operators that describe a global aspect of the configuration of an orderbook state ∣ z ⟩ , e.g. the best bid and best ask prices β M , M ∈ M . In the following, let thestate of the LOB be ∣ z ⟩ = b + k ,q . . . b + k j ,q j ∣ ∣ a + k j + ,q j + . . . a + k n ,q n . best bid and best ask operators act on ∣ z ⟩ as follows: β B ∣ z ⟩ = k j ∣ z ⟩ ,β A ∣ z ⟩ = k j + ∣ z ⟩ . Note that k on the right hand side is not an operator but the price level associated withthe best quote. Combining the two, one obtains the spread operator ∆ and mid priceoperator β mid as ∆ = β A − β B ,β mid = ( β B + β A ) . These operators also allow to extract more complicated measures from the book like theorder book imbalance, for example. It is a relevant quantity for order execution andof special interest to practitioners that design and develop trading algorithms (see, e.g.,Bechler and Ludkovski 2015, Lipton, Pesavento and Sotiropoulos 2014, Cartea, Jaimungaland Penalva 2015). In general, the literature relies on two measures to quantify order bookimbalance. First, Lipton et al. (2014) use the total number of ask and bid orders in themarket and calculate order book imbalance IQ as the relative deviation of standing askand bid orders as IQ = Q A − Q B Q = Q A − Q B Q A + Q B = Q A Q A + Q B − Q B Q A + Q B . (21)Second, Bechler and Ludkovski (2015) use the volume of active orders. Their measure ofmarket imbalance IV is given by IV = V A − V B V = V A − V B V A + V B = V A V A + V B − V B V A + V B . (22)23 .4.3 Liquidity Harris (2003) defines liquidity as ”the ability to trade large size quickly, at low cost, whenyou want to trade.” According to the same source, the notion of liquidity incorporates fourdimensions: immediacy of trade execution for a given size, depth, width, and resilience ofthe market. Therefore, the spread itself is used frequently as a liquidity measure in theliterature.There are multiple approaches to measure liquidity and we rely on the exchange liquiditymeasure (
XLM ) which is based on the concept of implementation shortfall, introducedby Gomber and Schweickert (2002). It covers three dimensions of liquidity: depth, width,and immediacy. The
XLM (also known as XETRA Liquidity Measure) is composed ofliquidity measures for the ask side (
XLM A ) and the bid side of the market ( XLM B ), XLM = XLM A + XLM B , (23)where XLM A = , ∑ ∞ k V Ak ∑ k Q Ak − β mid ∑ ∞ k V Ak ∑ k Q Ak , (24) XLM B = , β mid − ∑ ∞ k V Bk ∑ k Q Bk ∑ ∞ k V Bk ∑ k Q Bk . (25)The XLM depends on the volume weighted price which can be realized immediately oneach side of the market for a round trip order with a certain volume ¯ V , i.e., simultaneouslysubmitting marketable ask and bid orders with a total volume of ¯ V . In other words, the XLM measures the cost of a round trip order (in basis points).24 .5 Transactions
Up to now, we have deferred the discussion of transactions since, strictly speaking, they arenot necessary to set up the order book states. In this section we first discuss the tradingmodes of the XETRA order book and explain how one can augment the LOB states toalso record information about transactions. This will allow us to study the transactionprice and transaction rates, which were so far not available in the order book state.The XETRA order book is organized as continuous trading augmented by opening-,intraday-, and closing-auctions. Before stating the rules for these modes, we make a smallchange in notation: Instead of the symbol ∣ ∣ for the empty book, we record via ∣ T k,q ; t ∣ thelast price k , quantity q , and time t at which a transaction occurred. Rule 5a (Continuous Trading) . Assume an incoming order is assigned highest priorityand is such that it permits a transaction with its partner on the opposite market side.Then the orders will be executed at the price of the partner that was already residing in themarket and a transaction of the matched-up quantity will be issued at this price. For anarriving ask order, this results in ( ⋯ b + k,q ∣ ∣ ⋯) a + s,p = ⋯ b + k,q ∣ T k, min ( q,p ) ; t ∣ a + s,p ⋯ , (26) while for an arriving bid order, we have b + k,q ( ⋯∣ ∣ a + s,p ⋯) = ⋯ b + k,q ∣ T s, min ( q,p ) ; t ∣ a + s,p ⋯ . (27) Rule 5b (Auction) . Auctions consist of an outcry/call phase, during which incomingorders are collected and ordered by price-time priority as usual, but are not executed . Theexchange may provide an indicative pricing to market participants i.e., the price level atwhich the current order book state would settle if the call phase were to end immediately.Upon closing of the call phase the transaction price is determined according to the principle f highest traded volume. Subsequently, orders of highest priority are executed iteratively atthe previously determined transaction price. The transaction is recorded at the transactionprice and with the total traded quantity. A description of the matching procedure likein Equation (26) and Equation (27) is possible for concrete situations. The principleof highest traded volume makes a general formulation exhausting and is not particularlyilluminating. Therefore, we omit a general formulation at this point. The rules above are illustrated in Figure 1. We can now introduce the transaction price , transaction quantity , and transaction volume operators , which extract the correspondingnumbers from the last recorded transaction. Let the state of the book be ∣ z ⟩ = z . . . z i ∣ T k,q ; t ∣ z i + . . . z n . The operators are then defined as follows T K ∣ z ⟩ = k ∣ z ⟩ ,T Q ∣ z ⟩ = q ∣ z ⟩ ,T V ∣ z ⟩ = kq ∣ z ⟩ . Furthermore, we can extract the time at which the last transaction occurred via T t ∣ z ⟩ = t ∣ z ⟩ .This is the basis for an important observable, the inter-trade duration T ∆ t = t − t , i.e.,the time between two transaction. In our current setup, T ∆ t cannot be expressed as anoperator which only acts on the current state, while in practice we can calculate the timeintervals from remembering earlier transactions’ time stamps. The data used in the present paper were provided by Deutsche B¨orse in 2004 and havebeen previously used by Grammig, Heinen and Rengifo (2004). They consist of all recorded26 igure 1: Transaction matching
The figure illustrates the matching b k,q ∣ ∣ a s,p for s ≤ k and q > p . s k price a + s,p b + k,q a + s,p s k price b + k,q − p order book events of the XETRA system for trading of the 30 stocks that constituted theGerman stock market index, DAX between January 2 and March 26, 2004. Additionally,Deutsche B¨orse provided the open order positions in their books as of January 1, 2004,12 pm. The data allow for the full recovery of the order book. Over the three monthsperiod, 228,275,832 events were recorded. Additionally, 2,282 initial positions are availableat the beginning of the period. The data cover order arrivals, (partial) matches, changesand cancellations. The XETRA trading system allows for limit and market orders. Itis also possible to mix the two standard order types (market and limit orders) with amarket-to-limit order (MTL). An MTL order is filled on the best limit price in the book,either fully or partially. If an MTL order is matched only partially on the best ask or bidprice, it enters the LOB on the best limit price with the remaining order size. Additionally,iceberg orders (ICE) are allowed for which only a fraction of the total volume chosen bythe issuer is displayed to market participants.Market and limit orders can also have a specified stop price. They are then called stoporders. Different from the limit price, i.e., the price upon which the trader is willing totrade, the stop price specifies a price level from which onwards the trader is willing tosubmit an order. Hence, if the reference price exceeds (in case of bid order) or undercuts XETRA Release 7.0 A provides an overview of the distribution of the events in the LOBlog. It presents the total number of submitted limit, market, iceberg, and MTL ordersalong with their relative occurrences on the bid and ask side.In empirical investigations of LOB data, it is frequently observed that the distributionsof arrival and cancellation rates show a relatively stable connection with the currentdistance d = ∣ β M − k ∣ of the respective price level k to the best active price level on theopposite market side β M , M ∈ M = { A, B } (cp. Bouchaud et al. 2002). We find a similarphenomenon in the XETRA data. We set the distance d of all crossing, arriving orders to0 and define the logarithmic relative integer distance as d l = log (
100 max ( d, ) + ) . (28)Taking logarithms exposes the heavy tails of the distribution across price levels and itssimilarities with the DGX-distribution (see Appendix C for a short description and Bi,Faloutsos and Korn (2001) for further details). For our simulation study in Section 4,we use the DGX distribution to model order arrivals across d l . Figure 2 displays theempirical logarithmic frequencies of order arrivals and cancellations at the logarithmicrelative integer distance d l of their limit order price to the best ask or best bid price,respectively. The red line indicates a DGX distribution (truncated at 1). The logarithmicfrequencies of different types of marketable orders (limit, stop, iceberg and market orders)are displayed separately at d l = d l and the prevailing spread∆ = v A − v B in logarithms. Figure 3 presents scatter plots of d l against the logarithmicinteger spread log ( ) . For large spreads there is a stronger correlation between events29 igure 2: Frequency of order arrivals The graphs show the logarithmic frequency of arriving (a and b) orcanceled (c and d) limit orders (including stop and iceberg orders)for the stock VOW, respectively, on their logarithmic relative integerdistance to the best bid or ask prices. The logarithmic relative integerdistance is defined as d l = log (
100 max ( d, ) + ) . The red line is thelogarithmic probability of a truncated discrete Gaussian exponential(DGX) distribution for d l >
0, i.e., d > C ).The theoretical value for d l = d = d l = lll lll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lll llllllllllllll l ll − − − − − − Log relative integer distance
Log f r equen c i e s lllll (a) Ask arrivals lll lll llll lll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll − − − − − − Log relative integer distance
Log f r equen c i e s llll (b) Bid arrivals l lll lll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllll ll ll − − − − − − Log relative integer distance
Log f r equen c i e s (c) Ask cancellations l lll lll llll lll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllll − − − − − − Log relative integer distance
Log f r equen c i e s (d) Bid cancellations able 1: Correlations between d l and q The table lists the correlation coefficients r between the logarithmicinteger distance to the opposite best quote d l as defined in Equation (28)together with standard errors calculated as s.e. = √( − r )/( n − ) where n is the number of observations. The approximately standard normallydistributed z -score = ( r )/( s.e. ) is reported as well, complemented by its p -value. All values reported concern the MEO stock.Event r s.e. z -score p -valueask arrivals -0.1517 0.0060 -25.2 < < q associated with a certain eventand the logarithmic integer distance to the best opposite quote for the MEO stock. As canbe seen, the approximating assumption that the two variables q and d l are independentdoes not mirror reality exactly. Nevertheless, we will make the assumption several timesin this paper as it keeps the estimation and simulation manageable.31 igure 3: Relation between spread and relative price distance The graphs present the logarithmic relative integer distance to the bestbid or ask price of orders arriving (a and b) or being canceled (c andd) related to the stock ALT together with the prevailing logarithmicinteger spread. The logarithmic relative integer distance is defined as d l = log (
100 max ( d, ) + ) whereas the logarithmic integer is log ( ) with ∆ being the prevailing spread at arrival or cancellation. Even thoughcancellations smaller than the spread seem counter intuitive, they occurwhen orders are immediately canceled right after their insertion into theevent log. The red line indicates the bisecting line. l ll ll ll ll l ll lll ll ll llll lllllllll lllllllll l llllllll llll l l ll l ll l ll llll lllll ll lll ll l ll lll lll lll llll l lll lll ll lll lll l llll ll ll l lll llll ll ll ll l ll lll ll l ll lllll ll l l ll ll l ll lll l lll ll ll l lll l ll l ll ll ll llll l l l llll l llll llll lll l l llll lll llll lllllll lll lll l llllll ll lll l l ll ll ll l ll llll lll lllllll lll l llllll ll lllll lll ll lll llll l l ll llll lll llll ll ll ll llll ll l lll ll lll ll lll lll ll l ll l ll lll ll l l ll ll lll lll lll ll llll ll l l ll lll l lll l ll ll l lllll lll ll ll lll lll llll l ll lll lll ll l l ll lll ll l ll l l l ll ll l ll lll l l ll l lll llllll lll ll ll lllll l ll llll ll lll lll ll ll l lll lllllllll l ll lll llll llllll ll lll llllllllllllllllllllllllllll lll ll ll ll l l lll l lll l lll lll l llll lll ll lll l ll ll l lllll l ll ll ll l lll l ll ll l ll lllll l ll llll lllll llllllll lll ll l lll l l lll ll ll l ll ll ll l lll ll l l llll llll l ll ll ll ll lllll l ll ll ll ll lll ll l ll ll lllll l ll l ll lll ll ll ll lll lll ll l llllll l l l l lllll l lllll ll ll ll l lllllllll l llll lll ll l ll lll ll l ll lll ll lll lll l l ll ll l l ll l llll ll l ll l lll l ll ll ll lll ll ll l l lllll ll l ll ll l ll lll ll ll ll ll l llll lllll llllllll lll ll lll ll lll l lll l ll lll l l ll llll llllll lll l l llll l llll ll l ll l ll l ll llll l lll ll llll l ll ll lll ll ll llll lll l lll l l ll l l l lllll l ll ll lll ll l ll llll llll llll ll ll ll l ll ll ll l ll llllll l l llll lll l ll l ll ll lllll lll ll ll l lll lllll l lll ll ll llll llll ll ll lll l lllllll ll l llll lll ll ll l ll lll ll l l ll lll ll l ll ll lll lll l ll lllll l l lll lll l llll l l ll llll lllll ll lll ll ll lll ll l lll ll l ll l l ll ll l ll l ll lll l ll lll lllllllll lll ll lll l lll ll ll lll l ll l llll lllll l lll lll l ll ll l ll ll lll ll llll lll lll l llll l lll lll l l llll ll ll llll l ll ll l ll l lll lll l l llll l lll ll lll l llllllllllll ll lllllll ll lllll l ll l lll l lll l lll lll lll llll l l lll l l lll llll ll lll ll ll ll ll lll ll l lll ll ll l lll lll lll lll lll ll lll lll ll l ll ll l lll l ll ll l lll lllll lll ll l ll ll lll llll lll l llll lllll l l llll ll l ll llll l l lll llll lll ll lll lll ll lll l llll l ll ll lll l l lll ll l ll l ll lll lll ll lll ll ll ll l l ll ll lll llll l l llll ll lll llll l ll ll l ll ll lllllllll l l llll ll ll l lllll l ll ll l ll ll l ll lll ll ll llll l ll l l lll llllll l ll ll l l llllll llll l llll l l ll lllllll l l ll ll ll lll ll lllll l l ll l l lll l l llll l ll llllll llll l ll l lll lllll l ll ll ll lllllll l lll l ll ll l ll ll llll lll lll l ll l ll llll ll llllll llll ll l ll lll llllll llllll lll l lll ll ll ll l lll l l lll ll l ll llll l llllll ll ll llll lll l ll lll ll ll l ll lll lll lll ll ll llll ll ll l lll l lll ll lll ll ll l ll ll lll ll lll l ll l l ll llll ll lll ll l ll llll ll l ll ll l ll ll lll lllll llll l llll ll ll l lll l lll ll l ll lllll l lll l lll ll lll ll ll l ll lll llllll l l llll ll l lll l ll ll ll l ll l l lll ll llll lll lll l lll llll llllllllllllllll l lll l ll lll llllll llll l lll lll ll l lllll lll l ll ll lll ll ll l lll llll ll l ll l lll lll ll ll l l llllll l ll l l llll lll lll lllll lll lll ll lll llllll ll lll lll l ll ll l llll l l lllll lll ll ll ll ll ll l lllll llll l llllll ll llll lll ll l lll ll lll lll l l l ll ll ll ll l llll l ll lll l l ll l lll l lll ll l l llll l ll ll l l ll ll ll llll lllll ll l lllllllllll lll ll llll ll lll ll llll ll lll l ll l ll ll l l llll l ll ll l llllll l lll l lll ll lll lll l ll ll ll lllll l l lll ll lll ll ll l llll ll lll l ll llll lll ll ll lll ll l lll l lll ll ll lll ll ll l lll l ll llll ll l lll ll lll l l lll llll llllll l ll ll ll lllll ll ll l lll ll ll ll l l ll lll ll ll l lll l l ll lllll lll ll llllll l ll ll lll lll ll l ll l l lll ll lll llll lll ll l llll ll ll ll l lll ll lll ll l ll l ll ll ll lll l lll llll ll ll lll ll ll l ll l ll ll l ll ll lll ll l lllll ll lllllll ll ll l lll ll l lll lll lll l ll lll lllll l ll lll ll llll ll ll l ll ll l l lllll l lll lll ll ll lllllll lll l l l l ll lll l ll lll ll l lll l ll l ll ll lll l ll lllll lll llllll l ll ll lll lll ll lll l l ll ll ll l ll ll ll ll ll l ll l l llllllll l lll l l lll ll lll llll l ll ll ll llll lll ll l ll lll ll ll ll l lll l lll ll l lll lll l ll l lll ll ll ll ll ll lll lll llll l ll l lllllll llll lll llllll l l lll llll lll lllll llllll l l ll llll l ll llll l lll l l ll llll ll l ll lll ll ll lll lll l ll ll lll l l l ll lllll ll ll llll lll ll llllllllllllll llll ll l l lllll l ll ll ll lll ll l llll ll lll llll l lll l lll l lll ll l lll l ll ll ll lllll llll l lll llll lll l ll ll l l ll l llll l llll ll l lll ll ll l lll l l ll llll lllllllll ll l l ll lll lll lll lll lll l l lll l lll l lll l llll l llll lll ll l lll ll lll l l l llll l ll l l l lll lllll l llll ll ll lll lllll ll ll ll l ll l lllllll ll ll ll l l l ll l llll ll l ll lll l llll l l ll lllll ll l lll ll ll lll ll lll ll lll l l lll l ll lll llll ll llll ll ll ll ll lll l lll l l ll l lll l llllll ll lll ll lll llllll l l ll lllll ll ll lll l ll lll ll ll ll lll l l ll lll ll lll ll l ll l lll l l lll l ll ll l ll lllllll llll l l llllll ll l lll lll ll ll ll llll llll ll ll ll l ll ll ll llll ll lll ll l lll l lll ll lll ll llllll llll ll ll llllllll ll l ll lll ll l l lll ll lll l l ll ll ll l ll l l l ll l l lll lll lll ll lll lll l l ll ll l ll l lll ll ll ll ll lllll ll ll ll l lll lll l l lllllllll ll ll ll l lll ll llll l llll llll l llll lll ll l ll ll l lll lll l ll ll lll ll llll ll l l llllll lll l ll ll ll l lll l ll l l l ll ll l ll ll ll ll ll ll l llll ll lll l lll ll l lllll l llll ll lll ll ll l ll l lllll lll ll llll ll l ll llll lll ll l lll ll l l llll lll ll lll l lll lll llll lll l ll l ll l llllll lll lll lll l l lllll l llll l ll lll l ll lll ll ll lll lllll llll ll lll ll l l l llllllll l ll llll ll ll l l lll ll l llll lll ll l ll l ll ll lll ll l lllll llll ll ll l llllll l lll llll lll lll llll lll llll lllll l ll llll lll l ll ll l lll lll ll ll lll lll lll lll lllll lll lll l l ll l ll l lll l lll l ll ll ll l ll lll ll ll l ll ll l ll l lll l lll l l lll ll lll ll lll l lll ll l l ll ll lll lllll ll lll ll llllll lll l ll l ll l ll lllll ll ll l l lllll lll l llllll llll ll lllll l ll l ll l ll lll lllll lll ll ll llll ll llll l ll ll lll ll ll ll l lll lll l l ll lll ll lll ll l l lll l llll ll l ll l lllll l ll l llll l lll llll lll ll ll l l ll l llll ll ll lll l llll lll ll ll llll l ll ll llll lll ll l l ll ll lll l lll lll lllll l llll l ll lll l ll l ll llllll ll ll l l lll l lll ll ll l l lllll ll ll lll l l ll l lll l l lll lll lll l ll ll ll lll l l ll l ll lll l lllll ll l ll l l lll l ll l ll lll llll ll lll l lll ll ll ll lll l l lll lllll ll l llll l l l l ll ll l lll ll l l llll l l ll lll ll ll lll l ll lllll l ll l l lllll l ll lll l lll ll lll l l ll ll ll ll ll l l lll l ll llll l l ll llll llll llll ll l l ll lll l l ll Log distance to best bid
Log i n t ege r s p r ead (a) Ask arrivals ll ll lll lll ll lll ll lllllllllllll llllllllll lllll lllll ll l lll lll ll ll ll lll lllllllll ll lllllllllll lllllll ll l l lll lll lll llll ll llll lll l ll l lll l llll l ll lll llll llll lll ll lll l l l ll lll lll ll lll l ll l l l ll ll lll l lll ll lll ll l l l lll l lll l ll ll l l l lllll ll llllll l llll ll ll lll ll ll lll l ll l lll lll ll ll lll l l ll ll llll l ll ll l l ll ll ll ll lll ll llll llll ll l lll ll ll l lll ll l lll lllllll l ll ll ll llll l ll ll lll ll ll lll l ll ll ll lll l l l l ll l l ll lll ll llllll l llll l ll ll l l ll ll lll ll l l ll lllll l ll l lllll ll ll lll l ll l lll llll l lll ll lll ll lll ll lll l l ll ll lll ll l ll ll lll l lll l ll ll llll l lll l llll llll lll ll l l l ll lll ll l ll ll l ll ll l lll ll ll l lll l lll ll l ll lllll l lll l l ll ll llll lll ll l lll ll lll ll lll l l l l ll l ll ll lllll lll llll ll llll ll ll ll ll lll l ll ll l ll l ll lll ll l ll ll ll ll ll l ll ll ll l l ll ll ll l l ll ll ll l ll lll ll ll ll llll ll ll l l ll ll llll ll lllll ll ll ll ll ll llll ll ll ll lll ll l llll llllll ll llll l ll ll llll ll lll lll lll l ll ll l l l ll l l lll l l llll l lllll lll ll l ll l llll ll ll l l lll l lll l llll l llll ll lllll ll l llll l ll ll ll ll lll l lll l l ll ll lll l lll lll l l ll lll lll lll lll l lll ll ll lll ll l llll l l ll lll lll l ll lll ll ll lllll llllllllll ll l ll lll ll l lll l llll lll llll l ll ll ll l llll lll l lllll l llll ll lll l ll ll ll lll ll ll ll ll l ll ll l l ll lll l ll llll ll l l ll l l llll ll l ll ll lllll l l lll lll lll ll lll l ll l l ll ll lll ll ll l lll lll l ll l ll lllll ll ll l l llll lllll lll lll ll ll ll ll l l lllll ll lll l lll ll ll ll l lll ll llllll l ll lll l llll l ll lll ll lllll lll lll ll ll ll llll l ll l ll lll l ll l lll ll ll l lll lll llll llll l ll ll llll l ll ll ll llll lll l l lll l lll lll l llll l llllll l lll lllll l ll l lllll ll lll l ll lll l llll l l ll lll ll ll ll lll lll llll l lll ll lll ll ll ll lll l l ll ll llll l ll ll l lll lll l l ll l ll ll lll ll lll l lllllll l ll l ll lllllllllll ll lll l l ll l l llllll llllllll l ll lll ll ll l lll l ll ll ll l l lll ll l llll llllllllll llll ll lll ll lll llll l ll lll lll ll ll ll ll ll lll lll l ll l ll lll l l lll l llll ll ll ll ll ll llllllllllllllllllllll l lll l llll l lllll l l l lll l l ll l lllll l l lll ll ll l ll ll llll ll ll l llll ll l ll l lll l l lllll lllll ll ll lll ll llllll ll l llllll lll lll llllllll l l lll lll lll lll ll ll ll lllll l lll ll lllll lll lll l lll l lllll lll ll ll ll ll llll lll lll lll lll l l llll l l llll l lll l ll l llll ll lll ll ll ll ll l lllll lllll ll lll ll llllllllllll l l ll l l ll lll ll lll l lll ll ll ll lllll l lll ll ll ll lllll lll l lll l ll lllll l l ll llll l llll l lll l lll lll ll llll ll l lll ll lll llll ll lllll l l lll ll ll lll ll lll lll lll lll ll llll lllll llll l l lll ll ll ll ll l ll l lll llllll l l lll lll l lll ll ll ll lllllll ll l ll llll llll lll l ll lll lll ll l ll ll llll ll llll lllll llll ll ll ll lll llll ll ll lll lllll lllll lllll ll ll llll lllllll l lllllll ll llllll l llll ll l ll l l ll llll ll ll l ll ll l ll llllll l lll l lllll ll lllll lll lll l ll l ll ll ll ll l ll lll ll ll ll ll l lll l llllll ll l lll lll ll l l lll l llll lll l ll l ll llllll ll lll lll l ll lll ll lll ll ll llllllllllll ll l l l llll llll lllll ll llllll ll lll lll ll llll lllllll ll l ll lll ll lllll l llllll l l lll lll l l llll ll ll l l ll ll ll llllll l l l llll lll ll ll l ll ll l llllll l ll lll ll l llll l llll l llll lll lllllll l lll l l llll l lll l llllll llll l llllllll l ll lllll llll lll lll l llll lll ll lll lll l lll llll l lll ll llll ll lllll l llll lll ll lllll l ll lllll ll l lll l ll ll l ll l lllll l lll l l lll lll l lllllll ll lll llll lll ll ll l l ll llll l ll l llll l ll l ll l ll l lll ll ll l l ll lll ll l l l lll l ll l ll l lll llll ll lll ll lll llllll l ll l llll ll lll lllllll l ll ll ll lllllllllllll l lll lllll l l llllll ll lllllll l ll lll ll llllll llllllll l ll ll ll ll l lll llllllll lll llllllllllll ll ll llllllllllllllllllllllllll lll lll lll llll l llll ll ll l ll llll l ll l lll l llllllllll ll lll llllllll ll ll llll lll ll l lll ll llll lll ll l lllll l llll llll ll l lll l lll l lllll lll ll lll l lll lll ll l l l lll ll l lll ll lllll ll llll l l lll ll l l llll lll lll ll ll lll llllll ll llll lll l l ll llll ll lll l lll lll l lllllll llll l llllll lllllllll llll l ll l llllll lll ll ll lllllll ll ll ll l lllll l llll l ll lll l l ll l lll ll lll l l lll ll ll lll ll ll l lll lll ll lllll lll lll llll lll l l ll lll ll lll llllllllll l l l llll llll ll l l ll l lll llll llll l ll l lllllllll lllll ll l lll ll l lll llllll lll lll ll lll l lll llll lll llll l ll ll l lllllllll ll ll llllll ll l lll l llllll ll l lll lll l llllll llll l ll l l llllll l lll l ll lll l l l l lll l ll llll l lll ll l lll lllll llllllll ll ll lll l l ll l llll lllll ll l lllll ll ll lll lllll ll l ll ll l llllll llll l lllllll ll llll lll ll lll ll ll llll l l lllll ll lllllll lllllllll l llll l lll ll lll lllll llll lllll l ll l ll ll l ll l l ll l lll l lll ll ll lll ll ll llll lllll lll l ll lll l l lll ll ll lllll ll l l ll lllll llll l ll llll lll l lll ll ll ll l lll l l ll l lll ll ll llll ll l l lll lll l lll lll lllll lll lll ll l l l ll lll ll ll l lllll ll ll l ll lllllll lll llll lll ll ll l lll l ll llll ll lllll l ll llll ll ll lll ll l llll lll llll lllll lll l lll l ll lll lll ll lll llll llllllllllll llll l ll ll lll l ll lll llll ll l ll lll l lll ll ll lll l lll ll ll lll llll lll lll l lll ll lll llllll llll l ll lll lllll lll llll ll ll ll l ll l llll l ll ll lll lll ll ll ll llllll lllll lll ll ll lll lll ll ll ll llll llll l l llll l llll l ll lll lll ll lllll lll lll ll llll llll l lllllll lll l l llll ll lll llll lll l ll llll ll l l ll ll ll l lll ll lll ll ll llll lllll llllll ll ll lllll ll lll lll lll lll ll lll ll ll ll ll ll ll l ll lll l lllll ll ll lll llll l ll lll lll ll lll llll lll llll l l llllllll llllll ll lllll llllllll ll l ll l llll l l l ll l lll l ll l ll lll ll lll ll lll lll ll l lll ll llll l l l ll l ll ll ll l lll l ll ll l l l lll l ll lll l ll l l lll ll l l l ll lll lll ll llllllll l lll ll ll lll l lll l ll l lllll ll l lllll ll l lll l llll llll ll lllllllll lllll lllllll lll ll ll l ll l l ll ll lll llll ll lllll l ll lll l ll lll llll ll l llll lll l llllll llllll llllllllll llll llllll llll lllllllllll ll ll ll lllllll ll llllll lll lll ll l llllll ll ll l l ll ll ll l llllllllll l ll ll lllll lll llllllll llll ll lllllll l l ll l lll ll l ll llll ll llllll llll l ll lll l ll l lll lll lll l lll lll lll ll lllllllllllllllllllllllll ll ll llllll lll lll llll llllll ll ll lll l ll lll llllllllll ll llll ll ll ll ll llll ll llllll l l ll l lll ll ll lll ll ll l lll lll l lll l ll l llllll l ll l lll llll ll l lll ll llll lllllllllll l llllll l ll l ll ll lll lll llllllllllllllllllllllllllllll ll llll l lll ll ll ll ll llllllllllll l llll lll l llllll l lllllll llll ll lll lll l lllll l l l llll lll lllll l l l lllll lllllllll llllllllll l ll llll ll l ll l lllllllll ll l ll l ll l llllll ll l lll lll llll lllll ll ll ll lll ll l llllll ll ll l ll l l ll ll llll l lllll ll l ll l lll lll ll lll ll ll l ll l ll lll ll lll ll ll lll ll l lll l l ll ll l l ll l ll l l ll l l l llll ll l l lllll l lll lll lll l llllll l ll l ll ll llll l llllll lllll l ll ll l ll ll ll ll lll ll l lllllll ll ll ll ll lll l lllll lllll l ll l lllll lll ll ll llll l l ll ll l ll l lll lll ll ll lll lllll l l ll l l llll ll l ll l lll l ll ll l llll lllllllll l ll l llll lll lll lll lll llll ll lll ll l ll l ll lll l ll l l ll ll ll ll l llll lll l ll ll lll l llll l llll ll llll l l ll l l ll l ll ll ll ll ll l lllll lll lll l llll ll lll ll lll llllllllll lllll ll ll llll llll l l l ll llll l l ll l lll ll lllll l ll lllll l llll ll lll ll ll ll lll l lllll l lll lll lll l lll lllll ll llll ll llllllllllllllll lll ll ll ll l lll lll ll l lllllllllll l lll llll lll lllll ll lll ll llll l lll lllllllllllllllllllllllll lllllllllll ll ll ll l llllll ll l llll lll l l llll l l l lll ll ll ll l ll ll lll l lll lll l lll l ll l lll llll lll lll l ll l lll l lll lllllll ll ll ll llll l llll l l l lll ll l lllllllllll llllllllll ll l lll l lllll lll lllll ll l ll ll ll lll l l lll l llll ll l ll l ll lll l llll l l l ll lll lll l lll l lll llll l ll lll lll llllllll llll l lll l ll lll l lll ll l lll l lll ll l llll llll l ll ll l lll l lll l lllll lll llll ll l ll ll ll ll l lll lll l ll l l llll l ll ll l ll l l ll lll l lll l l ll l l ll llll ll lll ll lll ll l l ll l l ll ll llll l llllll lll llll lll lll lllll l lll l ll ll lllll ll l ll lll lll llllllll l ll ll lll ll ll ll l ll lll ll lll llll l lll llll lll ll ll l lll l lll l l lll ll ll lllll lll lll llll ll ll lll llll ll ll ll l lll ll ll ll lll lll ll l lll lllll llll llll ll l ll lll l l l llll llll l ll l ll l ll ll lll l llll l lllll l l ll ll lll lllll lllll ll lll ll ll llll ll l l lll ll ll lll ll ll llll ll ll ll ll ll lllll lll l lll lll ll lll l l llll l lll ll llll ll ll l ll lll ll lllll lll l llllllll l ll ll llll l l ll l l ll llll ll l ll l lllllll l ll lll lll l ll l ll ll l l l ll lll ll ll llll lll l l l llll ll lllll ll lllllll ll lll ll ll llll ll l l ll l ll ll l lll l l l lll ll l lll l ll l lll ll l llll ll lll llll lll ll l lll l ll l l lll l l lll lll l lll llll l llll ll ll l ll lll lllll l lll llll ll lllllllllll llll l llll lllll lll ll l l lll lll lll l llll lll lll ll lll ll llllll l l lll l l l ll ll ll ll lll ll lllll llll lllll l lllllll ll lll ll l lll llll ll ll lll ll l ll l lllllll ll ll l llllllll lll lll l ll llllllllllllll ll l llllll l lll ll lll ll lllllll l l l lllll l lll lll lllllllll ll l ll lll llll ll l ll ll lll l ll ll ll lll ll l l llll llll ll ll llll ll lll ll llll l ll l llll lll l ll ll l l lll llll ll lll ll llll l ll lll lll ll lll l lll lllll l ll llll ll lll lll l l ll l ll ll ll ll ll ll l ll ll l ll l ll lll ll l lll lll lll lll llll ll ll l ll l llllll lll llll lll ll l Log distance to best ask
Log i n t ege r s p r ead (b) Bid arrivals l llll l l lll ll l lll lllllllllll ll llll ll lllll ll l lll l ll ll llllllll lllll ll lll ll llllll lll l lll lll ll ll lll ll ll ll ll ll lll ll ll l l ll llll l llll l ll lll ll l ll l lll llll l lll l ll ll ll llll l ll llll llllll l lll llll ll lll l llll l lll ll lll lll l l ll l l lll lll lll l ll l lllll l lll lll ll lll l lllll lll llllll ll l l ll ll ll ll ll l l lll ll l lll ll llll ll ll l ll lllll l lllll lll lll ll ll lll lll lll lll lllll llllll l l ll ll lll lll ll l llll ll l ll lll ll l lll llll llll ll l lll l lll ll ll ll ll llll ll lll l ll llll llll lll lll lllll lll l l ll l ll ll ll l ll ll lll lllll ll lll l llll lll lll lll l ll lllllllllllllll lllll llll lllll lll ll ll ll llll llll l l ll lll ll l lll ll lll ll lll l lll ll l l lll l ll lll lllll ll lllll l ll ll ll llll ll l lll ll ll ll lll ll l ll l lll ll llllll ll lllllll ll ll llll llll lllllll l l l ll llll llll ll ll llll ll l lll lllll l llll lll l l ll lll ll ll lll lll ll ll ll l lll ll lllll ll lll ll ll lllll l ll ll l l lll l ll l lllllll l llll ll llll llll ll ll l lll l lll ll l lll llll l l lllll l lllll ll l lll ll l l ll l llll lllll l lll ll l lll l lll lll ll ll llll ll llll lll lll llll llll ll ll l l llll lll lll l l ll ll ll l ll ll l ll l lllll l ll lll ll l l ll ll ll ll ll l llllll ll ll ll l lll ll lllll ll ll l lllll lll ll lllll lll ll ll ll ll l l l lll ll lllllll ll lll llllllllllllllllllllll l l ll ll ll llllll ll l ll lll ll lllll ll lll l l lll l ll llll lll ll lll llll l lll lllll ll l lll l l ll l ll l ll ll l ll lll l ll ll l lll lll ll ll ll llll lll l lllllllll ll lllll lll ll l lll ll ll ll l ll lll lll l l ll lllll llllllll lllllllll lllll ll lll ll l l lllll l ll ll ll l ll l ll ll ll lll lll llll ll ll ll ll llll l lll l ll l ll ll lll lll ll lll ll ll lll llll ll lllll ll l ll llll llllll ll ll ll l ll ll ll llllll ll ll ll llll l ll ll l l ll ll l lll llll l ll ll l ll lllll llll ll l ll ll lll ll ll l lll ll llllll llll ll lll l l ll l ll l lll ll lll l lll ll lll lll l ll l l lllll llll l lllllll l lll l ll lll ll ll ll lll lll ll ll lll lll llll lllllllll ll lll ll lll ll ll lll llll l lllllll l l ll l ll l llllll l lllllll ll ll l l lll llll ll l ll ll lllll l ll lll l ll ll lllll l lll ll ll ll ll lll ll l l ll l ll ll lll l llll ll l ll l llll lll l ll ll ll ll llll llll l ll ll l ll l llll l l lll lll ll ll lll llll l ll lllll l lll lll l ll lll llll ll lll llllll l ll ll lll l lll ll lllll ll ll lllll llllll ll l lll l lllllllll lll lll ll llll lll l l llll lllllll ll l lll lll lllll lll ll ll ll ll ll ll l lll lll ll lll lll llll ll l ll l lll llll llllllllll l lll ll l ll lll llllll l lll llll l ll ll l lll l lllll l lll l lll ll ll ll lllllll l ll ll ll lll lll ll lll llll ll ll lllll lll l l ll lllllllll llllll l lll l ll l llll lll llll l ll llllllll ll llll ll llll ll ll l llllll ll lll lll l l lll l l lllllll ll lll ll ll l lllll lll ll ll l l l llll l llll l llllll l ll lll llll ll lll ll l l lllll ll llll l ll l ll llll l l lll llll l ll l ll l lll l l lll l lll ll ll l lll l l llll lll ll l lllll lll l ll ll llll lllllll lllllllllll llll lll lllll llllll ll llllll ll l l l lll lllll ll lll llll ll l ll ll ll ll lll l llll l ll ll lll l l ll ll ll ll ll lll ll ll ll l lll l ll lll l ll lll llll ll ll ll ll lll ll l lll lllll ll l lll lll l ll ll ll l ll ll ll ll lll lll llll lll lll ll llll ll ll l l ll l lll l l lll ll l l lll ll l ll lll lll lll ll lllll ll ll lll ll ll l ll ll lllll lll ll ll lll ll lll ll ll l lll llll lll ll lll lll ll l ll lll ll lllll l ll ll llllll lllllll l ll l ll ll llll lll l llll llll ll l l lll lllll ll ll ll ll ll l l lll ll l l llll l l llll l l ll l l ll llll ll l lllllll l lll ll l ll l llll llll llll ll llll llll lll ll ll ll lll lll l ll llll lll ll ll ll ll ll ll l l ll ll lll l llll lll ll ll ll lllll ll l ll lll llll lll lllllll l ll ll llll ll lll llll lll lllllll lllllll lll lll lll lll ll l ll ll llllll l ll lll l lll ll l llll ll l ll l llll lllllllllllllllll llllll l l llll ll l lll ll l ll llll l lll llllll ll llll llll ll ll l l ll ll llll ll lll l l l lll llll ll l ll ll lll ll ll ll l lllll l lllll ll lll ll llll l llll ll l ll l llll l llll l lll lll lll lll lll lll lll ll ll ll l l lll ll l ll ll lllll l ll llll ll ll lll l l llll ll l ll llll lll ll llll lll ll l lll lll ll l l ll llll ll lll l l lll l lll ll ll ll ll l llll lllll llll llll ll ll l lll ll l llll ll l l ll l l l llll llll ll lll lll ll ll lll ll ll lllll lllll llll l lll lll ll ll llll lll lllllll lllll l ll llllll l ll ll l lll lll ll lll l l ll ll l ll ll l ll ll l ll l lll ll lll ll lllll l ll ll ll ll l lllll ll lll ll ll ll l l ll llllll ll llll ll ll lll l lll lll ll l lll lll l ll ll ll l lll ll ll ll lll ll l ll lll llllllllll llll ll ll lll lll lllll l ll ll lll l ll l ll lllll llll l lll lll lll ll ll llllll l l lll ll lll ll ll l lll l lll ll ll l ll lll lll l llll l ll lll lllll l l ll lll ll llll ll lll ll l ll l ll l ll ll llll llll lll lll l lll ll llll l lll llll l l l lll l ll lll l llllll lll ll ll l ll ll l ll ll ll ll ll l lll ll lll ll lllll llllll l l ll llll llll lll l ll l l l ll llll l ll ll l l l l ll ll llll l ll ll ll ll l llllll lllll ll lll lll ll ll lllll lll ll ll lll lll ll ll ll l ll l l l l l ll ll llllllllll lll l ll l ll l lll ll ll lll l l ll l ll lllll l l lll llll lll lll ll ll l ll llll ll llll lll l lll llllll l l llll lll l l lllll llll ll ll lllllll ll l lll ll ll lll ll ll ll llll lllll ll ll ll ll lll lll lll lll lllll ll l ll ll ll ll lllll ll ll llll l llllllll llllll l lll ll l ll l ll ll llll llllll ll lll llll l lll lll l ll ll l l l ll llll l ll lll l ll lll ll ll ll l llll lll llll lll ll ll lll l lll l l lll lll lllll l lllllllll llll l l lll lll l ll ll ll ll l ll l ll ll llll l ll llll ll ll llll ll l llll llll lll l lllll l lll l lll lll lll l ll lll l lll l llll ll l l ll lll l lll ll l lll ll l lll ll lll lll ll l llll l l lll ll ll llll l ll lll ll l ll ll lll ll l lllll ll ll lll l ll lll ll lllll l lll l lll lll l llll lllll l ll lll ll ll l ll l ll lll lll lll l lllll llllllllllll ll ll ll ll llll ll Log distance to best bid
Log i n t ege r s p r ead (c) Ask cancellations llll lllll l lll lllllll lllllllll lllll ll llll lll l l ll llll lll lll l ll ll llll lll llllllll lll ll lll ll lllll lll ll lll ll ll lll l lll lll lll l llll l llllllllll l lll l llll llllll ll ll lll lll l ll ll l ll ll lll l lll l l lll lll lll l lll lllllll ll l llll lll l lll ll lll l ll ll lll lllll l lll l lll lll ll ll lll lllll lllll lll ll l ll ll lll ll l llllll l lll llll lllll llll ll lll ll lll llll ll l lll l l lllll l l l ll ll l lll ll lll lll ll ll lll ll lllll ll l ll l l ll lll ll lll lll ll lll ll l ll l llllll ll llll l llll ll l lllll l lll ll l lllll lllllll lll l llll llll llll ll ll l ll lll l ll ll lll l ll l ll ll llll lll lll lll l ll ll llll ll lll l l lll ll llllllll ll l ll lll ll ll lll l llll l l l l l l l ll llll ll lll ll ll lllll l ll llllll ll lll l lll ll ll l l l l lll ll lllll ll llll ll l ll lll l llll l lllll l llll l llll lll ll l llll ll l lll ll lll ll l ll llll l ll l ll l lll lll ll lll l ll l llll l l llllll l ll lll ll lll lll lllll lll ll ll l lllll l ll l ll llll lll l lll lll l ll lll ll l ll ll l llll lll ll ll llll l ll l ll lll llll ll ll ll ll ll ll l l ll l ll l l l ll l ll lll llll l l l ll l ll ll ll l ll ll ll ll l l ll lll ll ll lll ll l l lll l lllll lll l ll lll ll l lll ll l lllll l llll lll l lll ll lll ll ll lll lll ll ll ll lll lll l ll ll ll l ll ll llll l lll l lll ll llll ll ll lll ll l lllll l l l l lll llll ll ll ll ll ll ll ll lllllll l ll lll lllll l l llll llll ll llll ll lll lll lll ll lllll l ll l lll l llllll ll l l lllll ll ll ll lll ll llll ll lll l lll lllll ll ll llll llll ll l lll l l ll ll l llllllll l ll ll l ll l lll lllll ll ll llllll l l ll lll lll llll l ll ll lll l l ll l ll lll llll l ll ll lll lll ll llll l ll l l l ll l ll ll ll llll l lllllllllllllllllllll lll ll ll l lll l ll ll ll ll l lll l ll ll ll ll ll lll ll l l lll l ll l lll l l llll lll l lll ll lllll l ll l l ll l ll lllll l lllllllllll lllllll ll l lll llll l ll llll ll l ll ll llll ll lll ll lll ll ll lll llll lll l ll l l ll ll l lll lll l ll ll llll l l ll l llll lll ll l llllll llll ll l ll l lll l lll lll l ll llll llll l lll lll lll ll llll ll l llll ll l lll lll ll lll l ll llll ll ll ll l llll ll l llll ll lll l l ll ll lll llll ll lllll ll l lll ll ll l l l ll l lll l ll lll lll ll l lll l l ll ll ll lllll ll ll llll l lll l ll ll llllllll lll lll l l ll ll ll lll ll ll llll llll ll lll ll lllll ll ll l lll ll ll ll ll ll l lll ll lllll ll l ll l ll l ll l l ll ll l ll l ll l lllll ll lll ll ll l ll ll ll llllllllll l lll l lll l lll l lllll lll llll l ll l l llllll l ll l ll ll ll l ll l l lll lll ll l lllll l lll l ll lll l l ll ll l l ll ll l ll ll l lll lll ll lll llll llll l ll l l l ll ll ll l l llll ll l ll l ll ll lll ll l llll ll ll ll l l l lllll l ll ll ll ll l l ll l lll llll lll lll l ll ll ll ll l ll llll ll l ll ll lll l ll llll lllll ll ll lllllllllll lllll ll ll lll l lll llll l lll l lllll llll l lll lll lll l lll ll ll ll ll ll l l ll lll ll l l lll ll l l l ll lll lll lllll llllll ll ll llll l lll lllll l lll ll l l lll l ll llll l ll ll lll l llllll l ll l ll l l l l lll lll lll ll llll lll ll lllllllll l ll llll l llll ll ll llllll l lll l ll lll lll lllll llll l llllll ll lll ll ll lllll llllll ll llll ll lll l lll l lll l ll ll ll ll llll l ll l ll llll ll ll ll l llllllll ll llll l ll l l l l ll l ll ll l lllll llllll l l ll lllll lll l llllll lll lllll ll lll l l lll lll l l ll l llll l ll ll llllll llll lll l llll l ll l ll l ll l llll ll l lll ll l ll l lll l ll ll ll l l l ll l ll lll lll ll l ll lllll l lllll ll llll l lll ll ll lll ll ll lll lll l l ll ll ll ll lll l l ll lll l l llll ll lll llllll lll l llll lll ll ll l lll lll ll l l lllllllll l ll lll ll lll l l ll ll lllllll lll llllll ll ll lll ll lll lll l lll lllll l ll l lll l l l ll l l ll lll l ll ll l lll l llllll l ll ll llll lll ll ll lll lll ll ll l lll lll ll lll l l lll lllll l ll ll llll ll lll l l ll llll lll l l lll l ll l ll l l l l ll llll ll l llllll lllllllllll ll lll l ll ll ll llll lll ll l ll l ll lll l lll l lllll lll lll llll ll lll l ll lllll lll ll l ll ll l ll ll l ll l lll ll l lll lll l llllll ll ll ll llll lll lllll ll lll ll lll l llll ll l ll ll lll lll l l lll l llllll ll l ll ll ll lll l l llll l l ll lll l l ll lll l lll l ll l ll ll ll lllll l l lll llllllllllllll lllll l l ll lll ll l ll lll l ll l l lll lll ll ll lllllllllllllllll ll l lll l llllll llll ll l lll ll l l ll l ll l ll lll llll lll ll l lll lll ll ll llll l lll l ll lll l ll l lll ll l ll l ll l lll l ll lllll llll ll l lll l llll lll l ll llll l l l lll l l l ll l l ll lll lllll llll ll lll ll lll ll l ll ll ll ll llll ll l ll ll l lllll ll l ll ll l lllll ll l l l l llll l lll l lll l l ll l ll l lll l ll l ll llll ll lll ll l lll ll ll llll ll lll l ll llll l ll l l lll ll ll lll l lllllll ll ll l l lll l l l llllll l l l ll ll lll ll ll ll lll llll ll ll l l ll lll lllllllll ll ll ll ll ll lll ll ll lll lllll ll l ll ll ll lllll llll llll l ll ll ll ll l lll l l ll l llll l ll l l ll ll ll ll l l lllll l ll l lllllll ll llllll l lllll l l ll l ll l ll l ll ll ll l l lll llll lll ll ll ll ll lll ll lll lll llll l l ll ll lll l lll ll lllll ll lll l llll ll lll l ll l lllll l ll ll ll lll ll l ll lll ll ll ll ll lllll lll ll ll l ll ll l l l lll ll ll l ll l lllll llll llll ll lll ll ll llll llll llll l llllll ll lll l ll l l ll l lll lll lll l ll lll ll llll llll lll ll ll lll lll l lllll ll lllll l lllll ll l l llll ll ll ll ll ll l ll ll ll llll llllll ll l llll l l ll ll lll l ll l ll ll l ll llll ll l l lll lllllll ll ll lll l lll ll l lll llll ll llllll lll ll l ll lll ll lllll l ll lll ll lll lll ll ll lll ll lll lll ll ll l l lll lll ll lll ll l ll ll ll l ll ll l ll ll lll lllll l ll ll lll l ll ll l ll lll l llll llllll l l ll lll ll lll llll lll llll l ll ll l llll ll ll lll lll lll l lll ll ll ll ll ll lllllll ll l ll ll lll l ll lll l ll lll lll lllllll lll l l ll ll lllll ll lllll lll llll l ll l l l ll l lll llll lll llll ll ll l ll lll ll ll ll l ll ll l ll ll l lll l ll l l llll ll lll l ll ll lll ll lllll l lll l ll lll lll ll ll ll llll l ll l l l l lllll ll l ll llll ll ll lll ll lll l llll ll l l lll ll l ll l l lll l lll ll llll llll lll llll ll ll l ll l ll lllll l lll l ll ll l ll llll l ll lll ll ll l l lll l l l ll ll l ll l ll lll ll ll lll l l lll lll lll l ll lllll l l l lll lll ll lll lllll l lll l ll l llll l ll ll ll lll lll l ll ll l ll ll ll lll ll ll lllll ll lll l ll ll ll ll lll ll ll lll ll l l ll Log distance to best ask
Log i n t ege r s p r ead (d) Bid cancellations igure 4: Relation between order size and relative price distance The graphs show the decadic logarithmic relative integer distance tothe best bid or ask price of arriving (a and b) or canceled (c and d)orders related to the stock BAS against the decadic logarithm of thesize. The decadic, logarithmic, relative integer distance is defined as d l = log (
100 max ( d, ) + ) . For the arriving orders, the order sizedepicted is the original order size, while for the canceled orders, theorder size depicted is the actually canceled order size, not the originalsize at entry. ll ll lllll l ll llll lll ll ll l llll llll lll lll lllll l llll l lll ll l ll lll ll lll lllll l lll l llll l l ll ll ll l ll ll ll lll l l ll ll l ll ll ll llllll l llll l llll l l ll ll l lll ll llll ll l lll ll l lll lll ll ll l lllll lll lllll ll lllllll l lllll ll lll lll l lll lll l l ll ll ll ll ll ll ll lll l ll l l lll lll ll ll lll lllll ll llll lll l llll l ll l ll lllll l lll llll l l lll llll ll lllllll lllll ll lll lll ll ll ll ll ll lll lll ll ll lll ll l l lll l lllll llll ll lll lllll lll llll lll ll l l llll l lll ll ll l ll lll llll l lllll l ll ll ll l ll lll llll ll ll llll ll llll l lll ll lll lllll ll lll ll lll l lllll llllll lll lll lll llll lllll ll l ll l lll ll l lllll ll ll ll llll lllll l ll ll ll l llllllll lll ll lll ll llll llllllll l ll lll l ll lll lll l ll ll lll lll l ll ll l ll l lllll lll lll llll l ll l lll llll ll lllllll ll ll ll ll llll l lllll lll ll l lll lll l lll lll llllll llll l ll ll l lll ll lll lll ll l ll ll ll llll llll llllll ll l lllllll ll ll llll lll ll lll ll llll ll lll ll ll lll llllll lll ll l llllllll ll l ll llll ll lll l ll lll l ll ll ll l ll llll lll l lll ll llll ll l llllll lll ll l llll ll lll ll l l l lll ll lllll lll lll l llll l llll lllll llll lll l l ll lll llll ll lll l llllllll l llll ll llll ll lll ll l lll llll ll l l lll lll lllllllll llll l llll ll ll ll l lll lllll l ll ll lllll l ll ll l lll l lll l l lll ll l l llll l ll ll lllllllll ll lll lll lllll ll ll l lll lll lll ll llll llll lll lll ll l l llllll l ll lllll ll ll lllllll lllll l lll l llllllll lllll ll ll lllll ll l ll llllllll ll ll ll lll l ll ll l lll ll l l llll llllllll ll lll l lll lll l lll llll ll llll ll lll llll l lll ll lll llll ll lll lll ll lll l ll llll l ll l llll l ll ll lll llll ll l ll lll llll ll l llll lll l llll l llll l llll lll lll ll lllll llll ll l llll ll l lll ll ll l ll llllllllll l llllll llllll lllllll lll ll l llll llll ll lll ll ll l lllll llllllll ll l lll l llll ll ll llll l ll ll lllll lll l lll ll ll lll ll ll lll lll lllll l ll llll l lll ll ll ll lll l ll llll llll l lll l l lll lllll lll l ll ll l ll lll llll ll ll lllllll l lll lll ll lll l llllll l ll ll lll ll l ll l lllll lll l ll lllll ll lllll llll lllll ll llll ll l lll lllllll lllll ll llll ll llll ll ll lll lllll ll ll l ll l l lll l ll llll l lll ll lll lll ll lll ll lllll l lll ll lll llll l llll lllll ll ll lllll lllllll lll ll llll l l lllll l lllllll ll l ll llll ll lll lll ll ll lllll ll ll l ll lllll lll ll lllllll lllll lll l llll l lll ll lll llllllll ll ll l ll llll ll l ll lll l ll lll ll ll lll ll l ll llll llll llll ll ll llllll l l lll lllll lll l lllllll lllllll llllllll llllll l llll ll llll ll lll lll llll l ll ll l ll ll llll lll lll llllll lll l l llll l llll l l llll l l l llll lll lll lll lll l ll ll l l ll l llll lll ll l ll l llll ll lllll l ll llll llllllll llll ll ll llll ll lll ll lll ll ll lll llll l l ll llll llll l ll llll ll lll llll lll llll ll l lllll ll llll ll llll l lll lllll l lll l lll ll l lllll l ll ll ll l ll l lll ll ll ll ll llll l llll ll l l l ll ll l ll l l l lll ll l ll lll lll ll lll ll ll ll lll ll l l l l llll lllllll ll ll ll lll l lllll ll ll l l l l llll l lll l ll ll ll l ll lll ll ll l ll l ll ll llll lll ll ll ll ll llll ll l ll l llll lll l lllll l lllll ll lll lll llll l ll l lllll ll ll lll ll lll ll lll lll ll l l ll ll l llll lll lll ll l l lll lllllll lllll llll ll ll llll lll ll lll l lll ll ll lll l ll l l lll l llll lll lll lll llllll l lll l ll lll l lll l lll ll l lll ll ll l ll lll ll lllll l lll lll ll lll lll ll lllll ll ll lll ll ll ll ll lllll l lll lllll l ll lll lll llll lll ll ll llll lll l llll llll lll llll l ll ll lll l llll llll lll ll ll l lll lll l l ll lll lllll llll ll lllllll ll llll l l llll ll llll l lll l lll ll l ll lll lll l llllll ll ll ll lll l lll llll llll lllllll ll ll llll l ll ll lll ll llll lll ll llll ll ll l l lll l lll llll llll l l ll llll l llll lll ll ll l lllll ll lllll lll l llll llll ll lll lll ll lll ll ll l ll l lll l ll l ll l l ll ll llllll lll l ll lll lll ll l l lll lll l lll llll ll ll l ll lll ll llll ll l ll l lll ll lll lllllll llll ll lllll lll ll ll l ll lll l ll llll ll ll llll lll lll ll lll ll llll ll lll ll ll llll llll ll lll l llll llll l ll ll llll ll lll l ll lll ll l ll ll l lll l ll lll ll ll ll l ll ll lll lll lll lll l ll ll llll ll lll llll lll l llll ll ll lll lll l llll ll lll l ll ll l llll l llllll llll l lll l llllll lll ll ll l llll ll lllll llll ll llll llll l lll ll l llllll ll ll llll l ll lll llll ll lllll l lll l llll lll l ll lll llll ll ll llll ll llll llll l lll l lll ll lll l ll l ll lll llllllllll llllll lll l ll l lllllllll lll llll ll lll l lll ll l l l l ll lll l ll l l lll llll lll l llll lll lll ll lll ll ll llll lll llll l lll ll llllllll lll lll l l lll llll l lll ll ll ll lll ll lll lll ll l l llllll ll ll llll ll lllllllll ll l lll l l ll lll l llll ll ll l ll l ll ll l llll ll l ll ll ll lll ll lll l ll lll lllll lllll llll ll ll ll ll l l llllll l ll ll ll ll lll l l ll lllll lllll lllll ll l ll lllllllllllll ll ll l lll lll l lll ll l l ll l ll ll lll lll lll ll lllll ll l ll ll ll l lll ll l lll ll llll lll l ll lll l ll lll lll lll lllll l llll l l l lll lll lll l ll lllll l l ll ll ll l ll l lll l llll ll ll lll llll ll lll ll l llll lll llll ll ll lll lll ll lllllll l llll lll ll lll lll ll lllll l lll l ll lll lll lll ll lll l llll lllll ll ll ll lll ll lllll l l ll llll l ll llll llll lll ll lll llllll llll llll l lll l lll llll ll ll ll llllllll l l l lllll llll ll ll ll ll lll l l lll ll lll l ll l l llll lllllllll ll l l lll lll ll lll l lll lllll l ll ll lll llllll ll lll ll l ll lll l lll ll l ll lllll ll llll llll ll lll llll ll l ll l l ll ll lll lll ll ll lll l llllll l l lll l ll 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ll lll lllll ll ll lll ll ll ll ll l lllll ll ll ll l ll lll llllll llll lll l ll l lll ll l l lll llll l ll llll llll lll ll ll ll l ll llllll ll l l ll ll ll llll lllll ll llllll llllll ll lll ll ll l lll ll ll l l l ll lll ll ll l lll lllll l ll llll ll l ll ll ll llllll ll ll lll llll l lll l lll ll llll l llll lll l lll llll ll l lllll l lll lll ll ll l ll llll l lllll ll ll ll l lll lll ll lll l ll llll l ll lllll ll l lll lll lll l ll ll ll ll l ll ll lll l ll lll llll ll l l l llll llllll lllll ll ll llll l l lll lll llllll lllll ll llll l llll lll llll l llll ll l ll ll lll l ll lllll ll ll l ll llll l ll ll lllll llll ll l ll lll l ll lll llll l llllllll ll ll ll ll lllll l llll lllll lll l ll lll llll l llll ll ll lll l ll ll ll ll ll ll ll ll ll lll l l lll llllllll l ll lll ll ll lllll lll ll lll l l lll l lll l lll lll ll lll l llllll lll llllll llll l llll l ll lll llll llllll ll llll ll llllll llll l lll l ll ll l llll l ll l lll lll ll l lll ll l ll ll ll lllll lll 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l ll lll lll l ll lll ll lll lll ll lll lll l ll lll lll l l lllllll ll llll lll ll l ll ll ll lll ll l ll llll lll ll ll lll ll ll ll llll llllll ll lll ll lll ll l ll l ll llll ll ll l ll llll l l lll lll l llll llll ll l llll lll lll l llll ll ll ll lll ll ll ll ll l l ll lllll llll ll ll l lll ll llll ll ll ll llll lll ll ll ll llll lll ll lll ll ll l ll l ll ll llllll ll l lll l l ll lll llll ll ll ll llll lll ll lllll lll l ll l l llll l llll ll llllll lll llll l l lll lll lllll ll lll llll lll l l lll lllll l lllllll ll ll l lll lll ll ll lll lll ll ll lll llll lll llllll l llll ll lll lllll l ll ll lll llllll l l ll ll lll llll lllllll lll lll llll lll ll lll ll l ll lll llll lll ll l lllll ll l lll ll lll ll lll llll llll llll l ll lllll lllll lll lll lll ll l ll l ll lll llll ll ll ll lll llll l ll l ll ll l lll ll ll lll llll lllll llll ll l llllll llll lll ll ll lll l lll lll lll l l ll lll ll ll llll l l lll ll ll llll ll ll lll l llll lll ll l ll llll lll l ll lll lll l 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i v i ng v o l u m e (a) Ask arrivals ll llll lllll ll ll lll llllllllllll llll llll llllllllll llllll ll l ll llll ll llllll llll lll lllll lll lllll lll ll l lll l ll llllllll ll ll ll l lllll lllllll lll ll l lll lll lll l l l llll l lll l lll l ll lll lll lllll lll l lllllll llll ll ll lll ll ll llll llll lll llll llll ll ll lllllll lll ll ll lll ll l ll llll lll ll lllll ll ll ll l lll ll ll l llllll lllll l llll l l l lllll ll l lll ll llllll lll ll l lll l lll lllll ll l ll llll ll ll l ll lll ll ll llll ll l llllll llllll lll l llll ll l ll lll ll ll llll ll ll ll l ll l ll ll ll l lll l lll l lll l ll llll l l ll ll ll ll lll ll l ll ll lll ll llll ll lll lll llll l lll l ll llll l ll lll lll llll ll ll l lll ll lll ll l l llll ll ll l lll ll ll l l l ll llll lll ll ll ll lllllll l l l l l lll l l l l ll llll llllll l ll l ll ll ll l ll l l ll lll ll ll l l lll lll ll ll l ll ll lll llll ll ll ll l lll ll lll llll l llll ll l l lll ll llll ll l ll l ll lll lll ll lll ll lll ll 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ll lllllll lll ll l ll llllll l ll l ll ll l ll ll ll ll lll l ll l ll llll l lllll l ll lllll ll l ll l lll ll llll lll l llll ll l ll lll l l l lll llll lll l lll lllll l ll ll llll lll l lll ll ll lll lll ll l ll ll lll ll ll lll llll ll llll ll llll ll l ll l ll ll ll l l ll llll lll l l lll ll lll ll ll ll ll llll ll llll llllll l l ll lll l ll lllll ll lll l l ll lll ll ll ll ll l l ll l l lll ll l lllll l ll ll l ll lll ll l lll ll l lll l lll ll llll ll ll ll l llll l l ll lll lll ll llll ll ll llll lll l lll lll lll ll lll l ll llll l lll l ll lllll ll ll l ll lll l ll ll l lllll ll ll ll lll ll lll ll ll l l l l lllll l lll ll l lll l ll l ll l l l l ll ll ll l ll ll llll lll lll l l llll ll lll l lll ll ll lll ll ll l lll ll ll ll lll l llll l l lll ll l lll lll l lll llll l lll ll llll ll l ll ll l ll lllll lll lll llll llll ll ll l ll llllll lllll llll l lll lllll l lll llllll llllll ll ll ll llllll ll ll l lll l ll ll l l ll l ll l ll lll l ll llllll ll ll lll lll llll ll 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ll l ll lll ll ll l ll l lll l ll lllllll l ll l lll l lllll lllll l lll llll ll ll l lll ll ll llll l ll ll lll l l ll lllll l llll l ll llllllll l lll l lll ll lll lllll lllll ll llll ll lll l llllll ll ll lll lll llll ll ll l ll l llll lll ll ll ll llllll l ll l ll llll lll lll ll l ll l ll ll l l ll ll ll l llllll l lll lll l llllll llll lll lll lll lllll lllll lll ll lllllll llll ll l llll l l llll ll l ll ll lll lll ll l lllll ll lll l ll lllll lll ll ll ll ll l ll l ll ll lll ll l llll ll lll l lll ll lll l llll ll lll ll ll ll lll l lll l l llll l lll lll ll lllll lll ll l llll lll lll ll ll lll l ll l ll llllll ll ll ll llll llll ll lll l l ll llll l llll lll l l ll l ll l llll lll lll lll lllll ll lll ll ll l lll l lll llll ll lll ll ll ll llll l ll l ll lll ll l lll ll l ll ll lll ll l llll l lllll lll lll l llll ll ll ll ll lll ll ll ll l l lll l l lllllll ll ll l llll lll lll ll ll lll ll l lllll ll lll l ll ll l ll lll l ll l lll ll l l ll ll l ll llll l l l l l ll l ll l 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ll ll ll llll ll lll llll l llll llllll ll l lll lll lll llll lll ll ll lll l lll ll lll l lll lll ll lll lll llll ll ll ll lll ll llll lllll lll llllll llllll lllll l l lllll l lllllll l lll l lll l l ll l lll llll lll l ll ll l ll l lll ll lll l l ll llll ll llllll ll l llll l l ll ll ll lll lll ll l l llll l ll ll l lll l llll ll lll llllll l l ll ll ll l llll ll l llll llll ll l lll ll l ll l ll lll l ll ll l llll ll ll ll ll l lll ll llll lll ll ll l lllll l ll lllll lll ll ll lll l l l lllllll l l lllll lll ll ll ll l lll lllll lll l lll l lll lll lllll l lll lll l ll lllll lllll lll lll llllll llll lll llll l ll ll lll lllll ll lll llll lll l lll ll ll ll ll llll l ll l lll lll ll lll ll lll ll l ll lllll ll ll l l ll lll l ll ll lll lll lll lll l llll ll ll l l ll ll l llll ll lll l l lllll lll l ll ll llll l ll lll l ll lllll ll ll lll lllllll lll ll llll l ll llll l llll ll ll ll ll ll ll l lll l llll lllll ll llll ll ll lllll ll ll lll ll lll ll l lll l l llll lllll ll l ll 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ll ll l ll ll ll llll lll l l ll l ll lll ll l llll ll lll ll ll lllll l lll ll ll ll lll l l ll l l llll lll l l lll lll l lll ll ll ll llll l lll ll ll ll ll llll l l lll l lll ll l lllll l ll l ll llll ll ll ll ll l lllll lllll l l llll llll lll llll ll ll ll lll lll lll lll lll lll lll lll ll lll ll ll l l l lllll l l l ll l l ll ll ll lll lllll lll lll ll ll l llll ll llllll llll lll ll lll llll llll ll ll l ll lllll l l l ll lll l ll lll l lll l l lll lll l lll ll l l ll ll ll l lll lll ll ll ll ll lll ll l ll llllllllllll llll ll llll ll l l lll ll ll lll lll lllll ll ll l ll ll lll ll ll lll l llll lll l l ll ll lll l l ll lll lll ll l l l ll lll llllll ll ll l llll lll ll lll lll ll l lll l ll lll lll ll lll lll l ll lll l lll ll llll l ll l lll l ll ll llll ll l llll ll ll ll llll llll l ll ll l llll ll ll ll ll lll llll l lll l ll llll lll l ll ll ll ll l llll llll ll l ll ll l lllll l lll lllll lllllll l lll l ll l ll ll llll ll l l lll lll ll lll ll ll l lllll ll ll l ll 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ll lll ll l ll lll l ll ll lll lll ll l ll l ll ll lll l l l l l lll ll ll ll llll lll l l ll llll l ll llll lll lll llllllllll llllll lllll lllll l ll ll llll lll ll lll lll ll l llll llllll l lll lll ll llll lll lll ll ll l l llll ll ll ll lll llll lll l ll l llll l ll llll lll l lll lll lll lll llll l llllll ll ll lll l lll ll ll l lll ll llll ll lll ll l lll ll l llll lllll l lll l l ll llll lll l ll ll l llll lll llll ll l llllll llll l ll l llll lll ll ll ll l llll lll ll lll l ll ll l l ll lllll ll l ll lll ll l llllll ll ll llll lllll llll lll l l lll l ll l ll lllll lllllll l l llll lll llll ll ll llll llll lll llll llllll l l llllll ll lll l ll l lll ll lll ll llllll lll lll ll l l ll llll lll l ll ll ll lll lll ll lll l lll l ll ll llll ll lllll l llll l ll ll llllll ll ll lll llll lllll ll ll ll lll ll l ll ll lll ll lllll l l ll llll lllllll lllll llll ll l l lll lll l lll lll ll l ll lll lllll l ll lll lll l ll ll l lll llllllll ll lll llll lllll l lll l l lllll lll l ll 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lll ll ll lll lllll llll lll l llll l lll lll ll lll l ll ll llll l lll ll ll l l ll llllll llllll l lll ll ll l ll lll lll l lll l ll lll l l lll llll ll ll ll ll lll lll ll l lll ll ll l llll ll l lll llllll l ll ll ll lllll l l l lll lll l ll l lll lll l lll ll l l lll l lll l lll l ll l llll l ll ll ll lll l ll ll l lll lll ll l llll l lll ll l llll l llllll ll l ll lll l l ll lllll l ll ll ll ll lll lllll ll ll ll lll l llll lll lll llll ll ll lll lll l ll l lll lll lll lll ll llll lll llll ll ll lll lll l l lll l l ll ll llll ll ll ll l ll lll l ll l ll ll l l lll ll l llll l ll lllll lll l l ll l llll lll l lllll ll l l l l lll lll lll l lll l ll l l ll ll l lllllll ll ll ll l l ll ll ll l l ll l l ll l l l ll llll lll llll ll l ll l ll l llll lll l lll ll ll l l lllll l ll ll lll l ll l lll llll ll l llll ll l lll llllll lll llllll lll l ll l lllll ll l lll ll lll lll ll ll lll l l lll l llll lll ll lll lllllll lllllllll ll ll ll lll ll lll lll llllll l lll l lll l l ll l ll lll ll llll l ll lllll lllll lllll ll ll lll l lll llll llll l ll l l l lll lll lll l llll ll ll l l llll llll ll lll l lll llll l ll llll lllll ll ll ll llll l ll lll lll ll ll ll lll l lll llll l ll lllll llll ll l l ll ll l lll llllll ll ll llll l llll l llll ll ll lllll ll lll ll ll ll llll llllll llll l ll l lll ll l lll ll ll lll ll lll llll llll ll lll l ll ll l l lll lll ll l ll llll l ll llll ll lllll l l ll ll l l ll ll lll ll ll llllll l l ll llll llll l ll l ll ll l llll ll l llll ll l ll l llll l ll lll ll lll l ll ll l ll l llll ll ll l ll ll l l l l llll l l lll l ll ll llll lll lll lll l ll l ll llll l llll l ll llll l ll l l llll ll l l ll ll lllll lll l ll llll ll ll l ll l llll ll l l lllll l ll l l lll l ll llll l l l lll l ll l l ll l ll l ll lll l l lll ll l l ll l lll l ll lll l ll ll l l l ll l ll l ll llll l lll lllll ll ll l llll l ll lll l ll l ll ll ll ll lllll l l lll lll ll l lllll ll llll l l l l lll lll ll l lll ll ll l ll lll l ll l ll l ll ll ll l ll lll ll l l ll ll l llll ll lll ll lll lll ll l ll ll ll ll ll l lllll llll lll ll lllll l lll ll l ll llll l llll ll lll l ll l l ll lll ll l lll lll l l ll lll l llll lll ll lll lll ll l lll llll llllll ll lll l ll l lll l lll llll l ll lll ll l ll ll l ll l ll llll lll l lll l l lll ll l l l l llllll l llllllllll ll lll lllll lll lll l ll ll l llll ll ll lll llll l ll lll l ll lll ll lll ll lll llll lll lll ll lll lll l ll l ll lll ll l ll l ll l ll l lll lll ll ll lll llll l lllll l ll ll ll ll l l lllll lll l lll lll l lll l lllll ll l ll l ll l ll ll l ll ll lll lll lll lll lll ll lll l llll ll l ll llll ll lll lll ll l l llll llll lll l ll ll ll lllllll lll l llll l l lll lll l l lll ll lll llll lll l lll l lll lll l l lllllll lll l llll l l llll l ll l ll lll l lll lll lll ll ll ll ll l lll ll lllll l l ll ll ll l lll l ll ll ll ll ll l lllll ll ll l l ll ll ll ll ll l lllll lll ll ll l l lll lll lll l ll ll ll ll ll lllll ll l lll l ll lll lll ll lll ll ll l ll ll ll ll l ll llll l ll ll l lllll ll lll l l lll l l ll l lll l ll ll ll lll llll ll lll ll lllll ll l lll l l lll ll l l lll ll l ll ll ll l ll ll l ll ll lll ll llll l lll lll ll ll llll ll llll llll l llll l l ll llll l lll lll l ll ll lllll ll lll lllll lll l ll l lll ll ll l l l lll lllll lll llll l l lll l ll ll llll lll ll lll lll ll ll llllll ll l lll ll ll l ll ll ll lll l lll ll llllllll l l lllll ll lll ll ll l l lllll l lll lll l lllll l ll ll l l l ll lllllllll l lllll ll ll l llll ll lll lll l lll l ll lll ll l l ll lll l ll l llll lll ll l ll l ll ll l ll l ll ll l l lll llll l lll l lll lll ll ll ll l ll ll llll lllll l llllll lll ll lll lll lll llllll ll lll l l ll ll ll lll ll llll lllll l ll ll l llll l ll ll ll l ll llllll ll lll l ll ll l l lll lll ll ll ll ll ll llll l lll lll ll ll lllll l llllll l l ll lll ll ll lll l ll l l ll l ll lll ll llll ll l lllll l lll lll lllll lll l ll llllll lll ll lll l ll llll ll l lll l l ll l lll llll l ll ll l ll ll ll l ll l l lll l lll l l l ll llll ll l ll lllll l lll lll lllll llll ll lll llll ll l llll llll lll lllll ll ll llll ll lllll l lllll l lll ll llll lll l llll ll l ll llll llll ll l l l ll ll lll l lllll lll llll l lll lllll lll llll lll l llll l l ll lll l l l ll l lll lllll lll ll l lll ll l ll lll lll ll lllll ll ll ll ll ll ll lll l l ll llll ll ll lll ll lllllll lll lll lll ll ll llll ll ll l ll l llll ll lll ll ll ll l llll lll ll l lll l llll ll lll ll l ll llll lll lll lll lll l ll lll lll lll lll lll lllll ll l lll ll ll ll ll l ll lll lll l ll ll l ll l lll ll ll l l lll l lllll ll ll lllll l lll llll lllll l ll ll l lll ll l ll ll l ll lll ll ll ll lll llll lll l lll ll l ll ll l llll l l lll l l l ll ll ll ll ll l lllll ll ll lll l llllll ll llll l llll llll lll lllll ll lll l lll ll lllll ll l llll l lll lll l lllll l llllll lll l llll lllll l llll ll ll lll lll l lllll ll l l lllll ll lll l l l lll lll l ll lll llll l llll ll lll llll l ll llll l ll ll lll l lll ll l lllll l llllll llll ll llll l llll llll ll lllll l ll lll l l l llll ll l ll llll l llll ll ll ll ll lll lll llllll ll lll l l ll llll lll l lll l ll ll llll l llllll ll ll l lll ll l l lll lllll ll ll l ll ll lll lll llll ll ll ll l lll ll lll ll ll l ll lll ll ll ll lll l l lll l ll l ll llllll ll l ll ll l lll ll lll l llll ll l llllll ll ll ll ll l ll l lll lll llll ll ll llll lllll lll ll l lll lll ll lll lll llllll l lll llllll l ll ll llllll llll llll lllll ll ll ll ll l lll l ll lll l lll l lllll l ll l lll ll l l ll ll llll l ll lll ll lll l ll lll l l ll l llll ll l llll ll l lll ll l ll lll lll llll l llllll l ll ll lll l l lll l llll lll l l lll l ll llllll lll l lll l ll lll ll lll ll l llll ll lllll llll l ll ll l ll l ll ll llll lll ll llll lll l l lll llll llll lllllll llll lll l ll l lll ll l lll l llll l llll l l l llll ll ll ll llll ll ll ll l lll lll l llll lllll ll lll llllllll ll ll ll ll lll lll ll l lll lll ll lll ll lll ll lll l ll ll l llll ll ll l llll llllll l ll lllll llllll l l ll l ll l llll lll llll ll l ll lll lll lllll lll ll ll l lllll lllll l l ll ll l ll lll lll lllll lll ll lllll lllll lll ll llll lll l llll l lllll ll llllll l lllll ll l l ll ll lll l ll ll llll ll lllll llll lllll l l l lll llll ll ll l lll l l ll ll l lll l llll llll ll llll l llllll lll lll lll ll l llll l ll lll lll lllll ll l ll ll l ll ll l ll llll ll ll l llll ll ll ll ll llll ll l ll ll l l l ll lll ll ll l ll ll lll lll ll l ll lll l ll lll l llll l ll ll lll ll lll ll ll ll l ll lll ll lll ll ll lll lll l ll ll lll ll l l l ll lll ll l l ll ll l ll llll ll lll ll l ll l l lll ll lll ll ll l llll l l lll l l l llll lll ll ll l lllll l ll l ll l ll ll llll ll ll ll ll ll ll ll ll ll l lll ll lll lll lll ll l ll ll ll l ll ll l ll ll lll ll ll l ll ll lll ll ll ll l llll ll lll ll lll l ll lll lll lll l llll l l ll ll l lll lll l l llll l lll ll ll ll l l ll lll ll lll lll llll ll lll l llll ll ll ll lll ll lll lll lll l lllll l lll l llll llll ll lllll l llllll lll l lll llll ll ll l lll lll l llll ll ll ll l lll ll ll ll l ll lll llll lll ll ll l ll ll llll ll ll ll l lll ll lll lll ll lll lll ll l lll ll llll l lllll ll ll l ll llll lll lll llll l llll ll l ll lllll ll l l lll l l ll l llll ll ll l ll l ll l ll l ll ll lll llllll l lll l ll lll ll llll ll lll l lll lll ll l l ll ll lllll ll lll l lll lll l lll l ll ll ll l l ll l ll l ll ll ll l lll lllllll lll ll l l lll lll ll lll l ll l lll l lllll ll l ll ll ll lll ll ll ll lll ll ll ll ll ll llll lll lllll llll lll lll l lll ll ll ll llll lll lll l ll llll llll lll llll ll ll lll ll ll ll lll l ll ll llll ll ll lll ll lll ll ll l l ll l ll l l lll llll l ll ll llll l ll ll ll ll l l ll ll lll ll ll ll lll ll lllllll ll ll ll ll l llllllll lll l lll lll l ll lll l l ll ll lll l l lll ll lll l ll ll l lllll ll llll ll ll ll lll l lllll l ll lll llll l ll lll llll l lll l ll ll ll ll llll l lll l ll ll lll lll ll l llll ll l lll l llll ll ll ll lll lll l l l ll lllll lll l ll ll l l l lllllll ll l lllll l lll lllll l ll lllll llll llll lll l llllll ll lll l ll lll l lll ll ll lllllll lll llllll lll ll l ll l l ll lll lll ll lllll ll ll lll ll l l ll l ll l llll lll l l ll llll ll lll llll ll l ll ll l ll ll llll l lll lll ll ll ll ll ll lll lll ll l ll l ll ll ll ll l l l ll ll ll l ll l l lll l l lllll ll ll llll l llll l ll ll l l ll llll ll l llll llll ll lll llll l ll lll ll lll ll llllll lll l ll ll ll lll ll lllll lll lll llll ll lll ll llll ll l lll ll llll l l lll lll lll ll ll llll l l l llll l ll ll ll l lll l lll lll l lll ll l lllll lll llll l l ll lll l lll ll ll ll lll lll l lll lll l ll ll lll llll l ll lll l lll ll llllll l l l ll ll lll ll l ll ll ll llllll ll ll lll ll lllll l lll ll ll ll l lllllll l ll l llllll llll l lllll l l ll l ll llll ll l lll l llll l ll lll l ll l ll ll ll ll ll l lll lll ll lll l lll ll l lll lll l l ll ll l ll l l l ll ll l ll ll llll lll lll lll l lll lll ll l ll ll lll lll l lll lll llllll l l l ll lll l l lll lll l ll ll l ll ll lll lll l ll l l lll l ll ll lll ll l llll lll l lll lll l lll lllll l ll ll lll ll l lll llll lll ll ll llll ll lllll lll l ll l ll l ll ll lll ll ll ll ll llll l ll lllll l ll ll l lll l ll l lll l l lll llll lll ll lll ll ll lll llll ll l lll ll ll ll ll l lllllll lllll ll lll ll ll ll lllll lllll lll l ll ll ll l ll l ll l ll l lll lllll l l ll ll l lll ll l ll lll ll l ll ll llll l ll ll ll ll lll ll l l l lll l l lllll ll ll ll l lll ll lllll l l lllll lll l llll lll ll lll lll ll lll ll l lll lll l l l lll lll l lll l l ll ll l l lll lll ll lll ll l ll ll l ll ll lll lll l llll l lll ll llll l ll l l lll lll lllll llll ll l l l ll l lll lll l ll l l lll llll ll ll l lllllll llll l lll l lll ll ll ll ll lll l lll ll llll llll ll l ll lll ll lllll ll ll llll l ll lll ll ll llll ll ll lll l lllll l l lll ll llll lll ll ll l l llll ll ll l ll lllll lllll lll lll l ll l lll l lllll lll llll l lll ll ll l l lll l l llll l ll ll lllll lll l llll lll llll ll lllllll lll ll l ll ll l ll ll l lll ll l ll l lll lll ll lll lll l l l lllll lll ll ll l l ll l l lllll lll l l ll l ll llll l ll ll l l ll l l l ll l llll l l ll lll l l ll lll ll llll lll ll lllll llll llll lll ll l l lllll lll ll ll l llllll ll lll ll ll llllll llll ll llll ll ll ll l l l lll ll l lllll lll l ll l ll l ll ll lllll l lll ll lll l lll ll l ll l lll lll l lll ll l ll l llll ll lll ll ll lll llll lll lll lll ll l llll ll ll l lll ll llll ll ll lll llll lll l ll l ll ll ll lll ll llll ll ll lll ll l l llllll l lll ll lll l lll llll l ll lll llllll l ll l l ll ll l ll ll lll ll l l llll l lll ll ll lllll ll lll l ll ll lll lll lllll lll ll l llllll ll ll l l lll ll l ll llll lllllll ll lll lll ll l ll ll lll lll lll l ll ll ll lll ll lll ll l lllll l ll ll l llll lll l lll lll l llll lll lll ll l lll ll lll ll lll ll l l ll l l ll ll l l lll lll l l lll llll l lll ll ll l ll l l lll l lll l ll lll l ll llll lll lll ll l lllll l l lll lll ll l ll lll l ll ll l l ll l ll llll ll lll ll l l l ll l l ll l lll lll ll ll l llll l ll ll llll l lllll l llll lll l ll ll ll l ll l l lll lll ll l llll ll lll l l lll lll l lll lll lll l l lll ll llll l l l lllll ll lll ll ll lll l lll llll lll ll lll ll ll llll ll ll ll lllll l lll ll llll ll ll l ll l ll ll ll ll ll l ll ll ll lll lllll ll l ll l ll lll lll l ll l l l ll ll ll ll l lll ll ll l lllll l lll l ll lll lll lll llll lll llllll lll l ll ll lllll l lll llll l lll l l l l ll l ll l lll llllll l llll l lll ll l ll llll ll l lll lll lll ll lllll lllll llll l l l l lll l lll l lllll ll lll lll ll ll lllll lllll l lllllll ll llll lll ll ll ll l ll ll ll ll ll l l lll lll l l ll ll ll llll l lll l llllll lll l l ll ll ll l llll l ll ll llll ll ll l ll l l llllll lllllll ll l ll llll ll l lll lllll lllll lll ll lllll ll lllll lll l l ll lll ll ll l ll l lll ll ll ll ll l l lll l l ll l ll ll lll ll ll lll lllll ll ll ll lllll l l ll ll l ll l ll l llllll lll lll ll ll ll ll l ll lllll ll llll ll l ll l ll llll l ll ll llll ll lllll l l l ll l llll lll l ll l l ll llll l l l lll llll lll l ll ll ll l lll llll llll llll lll ll ll ll ll lll l lllll llll l l lll ll lll llll llll ll lll l ll ll ll ll llll ll llll ll llll lll lll ll ll ll lll l l llll llll ll llllll l llll lll ll l lll ll l l lll l l lll l ll ll ll ll l l lll ll ll llll lll l ll lll lllll lll l llll l ll ll lll l ll llllll ll ll llll l llll ll l l lll lll l llll ll lll lll l ll ll lll ll lll l l lll lll l ll l lll ll l ll l ll l llll lllll ll ll l llllll ll llll lll ll lll ll ll lll l ll ll l lll ll l lll l ll llll l ll lll lll l ll ll ll ll l lll l lll lll ll l ll ll ll llll ll l lll ll lll l llll l l l lll llll ll l llll ll ll llll l l lll lll ll lll ll ll ll ll l lll l ll ll ll l l ll lll l lll ll llll llll l lll l ll l ll l lll l lll lll lllll ll ll ll ll ll ll l l ll ll ll l ll ll ll l ll l llllll lll lll lll ll lll l ll l ll lll ll l lll ll ll l l ll lllllll lll ll lll ll lll lllll l ll ll l lllll l ll ll lll l ll ll ll ll l lllll ll l llll ll lll lll ll ll ll ll ll lll l ll l llll ll lllll l ll llll lll llll ll ll ll l ll ll l llll lll lll l ll ll lll ll ll ll ll llllll ll l ll l lll l lll l l lll ll llllll llllll lllll l l l ll l l l l ll ll l ll l llllllll ll lll l ll lll ll ll ll lll lll l llll l l ll l lll ll ll l lll l ll ll lll l llll ll ll ll ll lll llll l l llllll ll lll ll ll l ll llllll l l lll llll ll l lll lll ll l ll lll lll lll ll lll ll lllll ll lll l l llll llllllll ll ll ll lllll ll l ll l llll lll lll lll l ll ll l ll ll l ll lll lll ll l lll ll lll llll ll l lll lllll ll llll llll ll ll llll l lllll lllll lll ll l ll l l l lll ll llll l ll l ll lll ll lll l l llll l l lll l lllll ll l ll ll ll ll lll ll lll l ll ll lll lll lll lllll l ll lll ll lll llll l lll lll ll l ll llll ll l l l lll ll ll ll ll l lll l llllll lll lllll l l l l lll ll lll l ll lll l ll l lll l ll llllll l lll llll ll ll lll l ll ll ll l l lll ll llll lll l ll l ll ll l ll llll ll ll ll l llll ll l lll ll ll ll llll llll l l lll l ll ll lll lll l lll l ll l llll ll ll llll lll lll l l ll l lll l ll ll ll ll l llll llll l l l ll lll l lll l llll lll l ll lllllll l ll ll llllll l llll ll lll lll ll lllll lll ll l l l ll l llll lll ll llll ll l ll ll ll ll llll ll ll l ll lllll ll l lll lll lll llll ll l ll llll l lll l ll l ll lll ll ll l llll l ll l lll l llll ll ll lllll ll lll ll ll lll lll ll l ll ll ll ll l lll ll lll lllll ll ll ll ll ll lllll ll ll lllll l ll ll lll llll lll l ll lllll lll llll lllll lllll l lllll ll ll l llll ll lll l l lll lll lllllllllll llll ll ll l lllll lll lll l l llll l l llllll l llll lll lll ll ll l ll llll lll lllll ll l ll l lll l l lll l ll l ll l llll ll ll llll ll lll l llll llll lll ll ll ll lll l lll ll ll lll ll ll l l ll lll l ll ll lll l ll ll lllll lllllll ll ll l lll ll lll ll ll ll ll lllll ll ll l l lll ll l lllll l lll l l ll lll l ll l ll ll l ll l lll l ll l ll ll ll l l llll lll ll ll l ll l ll l l ll l lll lllllll lll ll ll llll ll l lll ll ll ll llll lll ll l llll ll ll l l ll lllllllll lll l llll l llll lll ll llll lll ll l ll lll ll l lll l ll ll l l ll ll l lll llll l lll l l lll lll l l l llll lll l llll l l l llll ll llll ll lllll l lll l l lllll lll ll ll l ll llllllll ll l lll l ll l lllllll ll lll l llll llll l lll ll ll ll l ll ll lll lll llll ll llll llll lll ll l ll ll l l lllll ll l ll lllll l ll ll llll ll l llll l ll ll llllll l lll ll lll ll ll ll llll lll llll llll lll l l lll lll l ll ll lll lll ll ll l ll ll l ll lll l l l l ll l l ll lll lll ll ll ll l lll l llll l llll llll l lll l lll lll lll ll lll ll ll l ll l ll l l l l lll lll lll lllll ll l ll l lll l l ll llll ll llll l lllllll ll lll lllll llll ll ll lll l lll l llll l lll ll lll l l ll lllll ll ll ll ll llll llll lll lll l ll ll l lll l l l llll ll lllll l ll ll ll l ll ll ll l lll ll ll lll ll llll l lll lllll lll lll ll lllll ll ll l lll l ll ll lll ll l lll lll ll lll ll l ll ll ll llll llll lll ll l ll l ll ll ll l l ll lll lll l ll lll lll ll l lll lllll ll ll l lll ll ll llll ll ll ll lll llllll llll ll lll ll ll lll ll l ll ll l ll ll ll ll ll l ll llll l l l ll ll l ll ll lllll lll ll llll l l ll l ll l lllll l lllll l l ll llll l ll l llll l l ll ll lll lll llllll llll lll l lll lllll ll ll lll ll ll l ll ll l l ll lll ll ll l lllll lllll l ll lll l l llll ll llll ll l l l lll l llll lll lll l lllll l l ll ll llll llll ll ll l l ll ll ll l l ll l ll l ll l ll llll ll lllll l lll ll ll lll lllll ll ll ll l ll l l ll ll lll llll lll lll ll llll ll l ll l l llll ll lll llll ll lll l l llll lll l lll ll ll lllll ll ll ll ll l ll l llll l lll l llll lll ll ll ll l ll lll lll l lll lllll ll ll l llll l lllll ll lll ll l llllll l lll lll llll l ll l lll ll l l ll lll ll llllllll llll ll lll ll ll ll l l l llll l llll lll l lll l llll ll ll lll lll lllll ll l ll llllll ll l l ll ll llll ll ll l lll ll ll lll lll ll llll llll l ll l ll lllllll llll l llll lll ll l l lll l lll ll ll lll lll lll l lll ll l lll l ll llll lll l lllllll ll ll l lll l lll lll ll lll lll l llll ll llll ll ll l lll lll lll lllll ll l lll l llll llll l ll llll l ll ll l llllll ll l llll ll l lll lll l Decadic logarithm of integer distance to best ask D e c ad i c l oga r i t h m o f c an c e l ed v o l u m e (d) Bid cancellations Simulation
The simulation algorithm presented in this section generates order book events by generatinganswers to the repeatedly asked question: when will what happen where next?
In order to simulate the LOB, several assumptions have to be imposed on the functionalstructure of the arrival and cancellation rates included in the Hamiltonian H of the modeldeveloped in Section 2. The following section presents the stochastic simulation algorithm(SSA) developed by Gillespie (1977) which can be used to simulate an artificial historyof the order book. The SSA is a direct consequence of the model and allows the exactsimulation of the system. Note that the assumptions made about the immanent functionalstructure of the arrival and cancellation rates are the crucial ingredients of the model.We therefore explore possible calibrations of our model with the subsequent simulationstudy. Our goal is not to fit the simulation results to an observed LOB history as closelyas possible. Instead, the simulation offers insights into the sensitivity of the order bookdynamics, especially the transaction price dynamics, to changes in the structure of arrivaland cancellation rates.The starting point of the SSA is based on the probability that within the next interval δτ no event occurs which we can denote in our notation as P ( δτ ) = ∑ z ∈H ⟨ z ∣ exp ( diag ( H ) δτ ) ∣ Ψ ( t )⟩ , (29)where the diagonal elements in H are obtained bydiag ( H ) δτ = ⟨ z ∣ Hδτ ∣ z ⟩ = − ∑ k,q,M α M ( k, q ; z ) δτ − ∑ k,q,M ω M ( k, q ; z ) δτ. ∣ z ⟩ .Gillespie (1977) shows how to formulate this probability for some event µ to happen duringthe interval τ without the operator algebra. First, the probability that an order arrivesduring the interval dτ is r µ dτ , where r µ is the rate corresponding to the event. In our case, r µ may be some rate from the set of arrival or cancellation rates, α M ( k, q ) δτ or ω M ( k, q ) δτ .In fact, we may label all possible events with integer numbers and let µ be a specificinteger denoting a specific event. Setting τ = δτ + dτ , the probability that given the state ∣ Ψ ( t )⟩ at time t the next reaction µ will happen during the next interval of τ , denoted P ( τ, µ ) , can be written as the product of the probability that nothing will happen during δτ and the probability that µ will happen during dτ : P ( τ, µ ) = P ( δτ ) r µ dτ. (30)From Equation (30), Gillespie (1977) deduces that the probability that nothing happensduring τ , can be formulated as P ( τ ) = P ( δτ ) ( − ∑ ν ≠ µ r ν dτ ) . (31)Noting that τ = δτ + dτ by definition, bringing all terms involving P to the left hand side,dividing both sides by dτ and taking limits for δτ →
0, yields a differential equation thatis solved by setting P ( τ ) = exp (− ∑ ν r ν ) . (32)Substituting Equation (32) into Equation (31), the probability that µ will happen duringthe next time interval τ is given by P ( τ, µ ) = r µ exp (− r τ ) = r µ ∑ z ∈H ⟨ z ∣ exp ( diag ( H ) τ ) ∣ Ψ ( t )⟩ , (33)35here in our case r = ∑ k,q,M α M ( k, q ; z ) + ∑ k,q,M ω M ( k, q ; z ) .From Equation (33), we may randomly generate the pair ( τ, µ ) , i.e., the time when anevent occurs τ and which event will happen µ . As we have set up the rates as price andsize specific, by generating the event µ we also specify the price location and the sizewhich are affected by the event. By noting that Equation (33) determines an exponentialdistribution with scale parameter r , we can first sample τ by drawing u from a uniformdistribution U ( , ) and calculating τ = r log ( u ) . Having determined when an event occurs, we may now ask the question what will happen.By numerically specifying the rates for all possible events r ν and drawing a secondrealization u from a uniform distribution U ( , ) , we may find the integer µ by solving µ − ∑ ν = r ν r < u ≤ µ ∑ ν = r ν r for µ . In other words, by drawing u and u , we can simulate an answer to the question when something will happen with u and, with u , what as well as where it will take place.In fact, we also draw a third realization from a uniform distribution u , to answer thequestion what size is affected (see Appendix B for details). Having drawn an event andit’s characteristics, the current state of the system can be updated. This may change therates r ν and their sum r . Note that by sampling the events in this fashion, the eventsare conditionally independent. They may not be independent as the rates are conditionalon the current state (and under the assumption of a higher Markov order also on finitelymany previous states) of the LOB.In order to simulate the LOB dynamics, we have to specify the rates of all possible eventsand how they depend on the current state. In our case, all possible events comprise theorder arrivals and order cancellations. Thus, we have to find a functional form for therespective rates α M and ω M . In our specifications presented in Appendix B , we let α M ω M be functions of the quantity q and the price level k (or more precisely of theinteger distance to the opposite best quote d l ). For simplicity, we will assume that allrates α M ( k, q ) and ω M ( k.q ) are separable in k and q such that α M ( k, q ) = α ,M ( k ) α ,M ( q ) and ω M ( k.q ) = ω ,M ( k ) ω ,M ( q ) . As the rates are proportional to the probability distribution of arriving (or canceled)orders across price and size, this means that the size of arriving (or canceled orders)is stochastically independent of the price level they concern. In Figure 4, we see thatfor lower distances to the opposite best quote the size of arriving and canceled order isequally spread out across possible size levels. A clear relationship between the price leveland the size level is not visible. In the absence of such a clear relationship, we find theapproximating assumption that the size and price level are stochastically independentjustifiable.We decompose the arrival rates further by setting the general intensity of events for eachmarket side ¯ r ,M,i to the average event rate over the entire sample of stock i , where ¯ r ,M,i is defined as ¯ r ,M,i = ∑ k,q,j α M,i ( k, q ) + ω M,i ( k, q ) which is calculated as the number of events on one market side divided by the total numberof events. Note that since we have several order types, the arrival rates may be splitinto market orders as well as marketable and non-marketable limit orders. The empiricalfrequencies for ¯ r ,M,i are reported in the last column in Table 5.Hence, the arrival and cancellation rates for limit orders can be described by the partitioningof the average event rate ¯ r ,M,i,j,a across price levels k and order sizes q : α M ( k, q ) = ¯ r ,M,i,j,a p K,M ( k ; θ M,a ) p Q,M ( q ; φ M,a ) ,ω M ( k.q ) = ¯ r ,M,i,L,c p K,M ( k ; θ M,c ) p Q,M ( q ; φ M,c ) , (34)37here ¯ r ,M,i,j,a is the rate for an order of type j (market or limit order) for stock i to arriveand ¯ r ,M,i,L,c is the rate for a limit order (i.e. j = L ) to be canceled. p K,M ( k ; θ M,a ) denotesthe discrete probability mass function of order arrivals across the integer price levels k given some parameter set θ M,a and similarly p Q,M ( q ; φ M,a ) is the discrete probability massfunction of order arrivals or cancellations across order sizes. The index a indicates theparameters for order arrivals, c the parameters for cancellations. The index M denotesthe market side.In our simulation, we consider three theoretical probability mass functions for p K,M (⋅) :The uniform distribution (uni), a discrete log-normal distribution with fixed parameters(fix), and a discrete log-normal distribution which depends on the prevailing spread (dyn).For p Q,M (⋅) we only consider a power law distribution (pow). The power-law distributioncaptures the heavy tails of the volume distribution. The distribution of order size and theheavy tails can be seen in Figure 5 which depicts the frequencies of order arrivals andcancellations.Appendix B lists all the functional specifications as well as a description on how marketorders are incorporated in the distributional setup. Iceberg orders, stop orders or fill-or-killrestrictions are neglected in the simulation study, as the events marked by these ordertypes only make up for less than 1% of all events in our data set.Additionally, we also investigate cases in which p K,M (⋅) and p Q,M (⋅) are described by theempirical univariate frequency distributions in our sample across k and q , respectively(emp). We also utilize the joint frequency distribution of the observed pairs ( k, q ) in onescenario (emp,emp). Note that although we use the empirical frequencies, the rates arefixed over the entire simulation run. Thus, in the scenario ’emp’, no dynamic feedbackbetween the state of the book and the arrival and cancellation rates is introduced.For all combinations of these distributional specifications (in total 8 scenarios ) for each The scenarios are (uni,pow), (uni,emp), (fix,pow), (fix,emp), (dyn,pow), (dyn,emp), (emp,pow) and(emp,emp) where the list of pairs utilizes the introduced abbreviations and states the distribution across k in the first coordinate and the one across q in the second coordinate. igure 5: Distribution of size The figure presents the logarithmic frequencies of logarithmic order sizesof the MEO stock for arriving (a and b) or canceled (c and d) orders.For incoming orders, the logarithm of the original order size is used,whereas for order cancellations, the actually canceled remaining ordersize is utilized. l l l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lll lllll l − − − − − − Log of incoming order sizes
Log . f r equen c i e s (a) Ask arrivals l l l llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l − − − − − − Log of incoming order sizes
Log . f r equen c i e s (b) Bid arrivals l l l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l − − − − − − Log of canceled order sizes
Log . f r equen c i e s (c) Ask cancellations l l l llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l − − − − − − Log of canceled order sizes
Log . f r equen c i e s (d) Bid cancellations able 2: Mean and standard deviation of simulated price changes For the values reported in this table, for each of the 30 stocks, the time seriesmean and standard deviation of the transaction price changes (in event time)across the 200 simulations has been calculated. The table reports the averageacross the 30 means ¯ µ and standard deviations ¯ σ multiplied by 10 .Scenario ¯ µ ¯ σ dyn emp -0.04 0.56dyn pow -0.05 0.52emp emp -0.23 1.57emp pow -0.29 4.85fix emp -0.24 0.77fix pow -0.20 0.77uni emp -0.20 3.75uni pow -0.15 3.13stock, we simulate 200 realizations of LOB evolutions over half a trading day (4 hours).The state at the beginning of our sample, i.e., after the opening auction on January 2,2004 at 9h00 CET, serves as a starting point for the simulation. We first turn to the results from the uni scenario which are depicted in Figure 6. In thisscenario, a lot of simulation runs ended with an empty order book. We also see that thevariance of transaction prices, which is induced by the uniform distribution, is rather high,especially, when using the empirical volume distribution. This is reported in Table 2 whichshows the average mean and standard deviation of the simulated transaction price changesacross all simulation runs.Note that for these simulations, the rates ¯ r ,M,i,j, ⋅ in Equation (34) are the same as inthe case of the fixed and dynamic arrival and cancellation rates. We may associate theuniform distribution across price levels with somewhat uninformed traders who, regardlessof the price, randomly submit orders in the vicinity of the current best quote. With theuniform distribution, the mean and variance of the price level is rather high compared tothe DGX specifications in other simulation scenarios. Throughout all simulation scenarios,40 igure 6: Scenario: Uniformly distributed arrival and cancellation rates The graphs show 200 simulated paths of transaction prices (in red) using thescenario in which the arrivals and cancellations of orders follow a uniform distri-bution. The starting point of each simulation is the LOB position of the MEOstock on January 2, 2004. The true history of transaction prices during the firsthalf of that day are depicted in black. In 6a, the empirical order size distributionis taken to generate the samples. In 6b, a power law is assumed to generate ordersizes. Paths that end earlier than 12h00 result in an empty order book.
Time on January 2, 2004 P r i c e (a) Empirical Time on January 2, 2004 P r i c e (b) Power law we see that a higher mean and variance in the distribution across price levels are relatedto a higher variance in transaction prices. This result is interesting since arrival rates maybe linked to trader’s behavior. So, if traders are uniformed on how the asset should bevalued and constantly shift their valuation with no clear tendency and/or if traders areindifferent between immediate execution and delayed execution, transaction prices becomehighly volatile.Second, the results for the fixed DGX distribution across price levels are presented inFigure 7. Interestingly, the small difference in the parameters µ and σ between the askand the bid side of the market lead to a detrimental effect on transaction prices. Thisobservation consistently holds across all simulations across all stocks. The parameter µ ofthe DGX distribution for incoming and canceled orders on the bid side has been specifiedslightly higher (at µ B,a = .
766 and µ B,c = . µ A,a = . µ A,c = . igure 7: Scenario: Fixed DGX distribution for arrival and cancellationrates The graphs show 200 simulated paths of transaction prices (in red) using thescenario in which the arrival and cancellations of orders follow a fixed DGXdistribution across price levels. The starting point of each simulation is the LOBposition for the MEO stock on January 2, 2004. The true history of transactionprices for the first half of that day are depicted in black. In 7a, the empiricalorder size distribution is taken to generate the samples. In 7b, a power law isassumed to generate order sizes. . . . . Time on January 2, 2004 P r i c e (a) Empirical . . . . Time on January 2, 2004 P r i c e (b) Power law parameter σ has also been set slightly higher for the bid than for the ask side. Hence,we can conclude that, ceteris paribus, limit order distributions with a high probabilitymass in the vicinity of the best quote in one market side push transaction prices in thedirection of the opposite market side. We can also conclude that the power law very rarelyinduces large jumps in transaction prices due to the extremely large order sizes that arepossible under this distributional scheme. In general, however, differences between thevolume distributions are not obvious, neither in the ’uni’ scenario nor in the ’fix’ scenario.Third, the simulated transaction prices resulting from the scenario with dynamical shiftingand scaling DGX distributions which depend on the prevailing spread, are shown inFigure 8. They hint at a dampened downwards trend compared to the scenario with a fixedDGX distribution. However, the volatility of transaction prices increases. Furthermore,the large jumps induced by the power law distribution are rather pronounced.42 igure 8: Scenario: Dynamical DGX distribution for arrival and cancel-lation rates The graphs show 200 simulation paths of transaction prices (in red) using thescenario in which the arrival and cancellations of orders follow a dynamicalDGX distribution across price levels. In the dynamical DGX distributions theparameters µ and σ are functions of the prevailing integer spread. The startingpoint of each simulation were the LOB positions for the MEO stock on January2, 2004. The true history of transaction prices for the first half of that day aredepicted in black. In 8a, the empirical order size distribution is taken to generatethe samples. In 8b, a power law is assumed to generate order sizes. . . . . Time on January 2, 2004 P r i c e (a) Empirical Time on January 2, 2004 P r i c e (b) Power law r ,M,i,j, ⋅ , the results presented in Figure 9 deviate from empirical stylized facts. Also,the fit is weak. The imbalance between ask and bid order arrivals and cancellationsexhibited by some stocks, together with the observation that on average ask orders arrivecloser to the best bid than bid orders to the best ask, drags transaction prices even furtherdown. The smaller distance to the best quote, on average, of ask orders can be seen inFigure 2 as well as in Figure 10. It seems that in our sample, sellers tend to seek quickerorder execution by placing their orders close to the buy side. Buyers on the other hand,test their fortunes and patiently wait for a good deal to occur deeper in the book. Thiscan be clearly seen in Figure 10: For the same average spread, arriving bid orders areplaced on average further away from the opposite market side than arriving ask orders.Also the variability of the price level of arriving bid orders is higher. When using theunconditional frequency, this tendency is the main factor which drives prices down in oursimulations. Market imbalances, i.e., a higher probability of the bid or ask side to move, donot affect the development of transaction prices as much. Nonetheless, in any simulationwithin the empirical scenario, the simulated transaction prices never entail the true pricehistory. This is somewhat expected as we use fixed unconditional frequencies over theentire simulation run to model conditional, constantly changing arrival rates. In our empirical analysis, we focus on three variables that we deem most important tomarket participants. The first is the logarithmic return which could be achieved basedon a buy-and-sell strategy in subsequent intervals. The second is the return of a sell-and-buy strategy. The third one is the exchange liquidity measure (XLM) as introduced inSection 2.4, calculated with a round-trip of EUR 100.000. To implement them, we samplethe data in intervals of fixed length ∆ t (1, 2, 5, 10, 15, 30, 45, 50, 60, 120, 240 minutes).Let t denote the last point in time of some arbitrary interval and t − p t,b ) and a sell-and-buy44 igure 9: Scenario: Empirical frequency distribution for arrival andcancellation rates The graphs show 200 simulation paths of transaction prices (in red) using thescenario in which the arrival and cancellations of orders follow a dynamicalDGX distribution across price levels. In the dynamical DGX distributions theparameters µ and σ are functions of the prevailing integer spread. The startingpoint of each simulation were the LOB position for the MEO stock on January2, 2004. The true history of transaction prices for the first half of that day aredepicted in black. In 9a, the empirical order size distribution is taken to generatethe samples. In 9b, a power law is assumed to generate order sizes. Time on January 2, 2004 P r i c e (a) Empirical Time on January 2, 2004 P r i c e (b) Power law igure 10: Estimated DGX parameters and average spread The scatter plot depicts the average spread of DAX components in the samplinginterval (first quarter of 2004) during continuous trading of the XETRA orderbook against the estimated parameters of a DGX-distribution fitted to theunconditional frequencies of order arrivals (and cancellations) across price levels.The parameters were estimated using the log-likelihood method described in Biet al. (2001). ll lll ll l lllll l llll l ll l ll llll ll . . . . . Logarithmic average integer spread m ^ i lllll ll l lllll l llll l ll l ll llll ll ll lll ll l lllll l llll l ll l ll llll ll lllll ll l lllll l llll l ll l ll llll llllll Ask arrivalsBid arrivalsAsk cancellationsBid cancellations (a) ˆ µ i against average spread ll lll ll l lllll l llll l ll l ll llll ll . . . . . . . Logarithmic average integer spread s ^ i lllll ll l lllll l llll l ll l ll llll ll ll lll ll l lllll l llll l ll l ll llll ll lllll ll l lllll l llll l ll l ll llll ll llll Ask arrivalsBid arrivalsAsk cancellationsBid cancellations (b) ˆ σ i against average spread strategy (∆ p t,s ) are given as ∆ p t,b = log ( β A,t ) − log ( β B,t − ) , ∆ p t,s = log ( β B,t ) − log ( β A,t − ) . For the
XLM the last observation in the respective interval is taken.We have shown in Equation (20) that the moments of some observable O of the LOBsystem can be expressed as E t [ O ν ] = ∑ ∣ z ⟩ ⟨ z ∣ O ν e H ( t − t ) ∣ ψ ⟩ . We can perceive the right hand side of this equation as an intricate function of the arrivaland cancellation rates. These rates depend on the event rates ¯ r ,M,i,j,e of arrivals andcancellations of the various order types, the relative logarithmic integer price level d l ,46he order size q , the spread ∆, and possibly other variables. Thus, we can formulate alinear approximation for the expectation of any observable in the moments of exactly thesevariables.By repeated Taylor series approximations of the terms in Equation (34) in the variables d l , q , and ∆ around their respective mean, collecting terms, taking expectations andneglecting terms with an order higher than four (see Appendix D ), we get for the expectedvalue of some observable a linear approximation of the form E t [ O i,t ] = γ ,i + ( δ i, + δ i, E t [ ∆ t ] + ∑ v = δ i,v E t [( ∆ t − µ ∆ ,t ) v ]) ×∑ M,j,e ρ ,M,i,j,e E t [ ¯ r ,M,i,j,e,t ] ×( κ M,i,j,e, + κ M,i,j,e, E t ,M,i,j,e [ d l,t ] + ∑ v = κ M,i,j,e,v E t ,M,i,j,e [( d l,t − µ d l ,t ) v ]) ×( ξ M,i,j,e, + ξ M,i,j,e, E t ,M,i,j,e [ q t,M,j,e ] + ∑ v = ξ M,i,j,e,v E t ,M,i,j,e [( q t,M,j,e − µ q,t ) v ]) + ε i , (35)where µ x,t = E t ,M,i,j,e [ x t ] . Note that the indices in the subscript of the expected valuesindicate their conditioning set as outlined in Section 2.4. Hence, the tuple of subscripts ( t , M, i, j, e ) indicates that the conditional expectation is formed with information availableat time t for stock i given the market side M , order type j , and event type e . As we are notinterested in γ ,i , ρ ,M,i,j,e , κ M,i,j,e,v , ξ M,i,j,e,v , and δ i,v , we can collect all possible productsin Equation (35) in the parameters γ ,i , γ ,i , . . . γ ,R . In total, this yields 1.971 parameters.Hence, the estimation of the parameters is sensibly feasible using ordinary least-squares upto non-overlapping intervals with a length of 15 minutes. For non-overlapping 15 minuteintervals, we can get 2.164 observations from the 64 trading days in our sample. Increasingthe interval length beyond 15 minutes makes the use of overlapping intervals and rollingvariable calculation necessary. While this is in principle feasible, we restrict the analysis forthe specification in Equation (35) to non-overlapping intervals and sampling frequencies47elow 15 minutes.To increase the length of the intervals, we use three alternative specifications which entailless parameters. In the first alternative, the moments of the spread are only includedadditively: E t [ O i,t ] = γ ,i + δ i, + δ i, E t [ ∆ t ] + ∑ v = δ i,v E t [( ∆ t − µ ∆ ,t ) v ]+∑ M,j,e ρ ,M,i,j,e E t [ ¯ r ,M,i,j,e,t ] ×( κ M,i,j,e, + κ M,i,j,e, E t ,M,i,j,e,t [ d l,t ] + ∑ v = κ M,i,j,e,v E t ,M,i,j,e,t [( d l,t − µ d l ,t ) v ]) ×( ξ M,i,j,e, + ξ M,i,j,e, E t ,M,i,j,e,t [ q t,M,j,e ] + ∑ v = ξ M,i,j,e,v E t ,M,i,j,e,t [( q t,M,j,e − µ q,t ) v ]) + ε i . (36)This specification entails the estimation of 399 parameters. This enables us to estimatethe model on interval lengths of up to one hour.In the second alternative, the moments of the spread and the moments of the order sizeare included additively: E t [ O i,t ] = γ ,i + δ i, + δ i, E t [ ∆ t ] + ∑ v = δ i,v E t [( ∆ t − µ ∆ ,t ) v ]+∑ M,j,e ρ ,M,i,j,e E t [ ¯ r ,M,i,j,e,t ] ×( κ M,i,j,e, + κ M,i,j,e, E t ,M,i,j,e,t [ d l,t ] + ∑ v = κ M,i,j,e,v E t ,M,i,j,e,t [( d l,t − µ d l ,t ) v ]) + ξ M,i,j,e, + ξ M,i,j,e, E t ,M,i,j,e,t [ q t,M,j,e ] + ∑ v = ξ M,i,j,e,v E t ,M,i,j,e,t [( q t,M,j,e − µ q,t ) v ] + ε i . (37)48his reduces the number of parameters to 95 and makes interval lengths of up to 4 hourspossible.In the third alternative, we employ a completely additive structure: E t [ O i,t ] = γ ,i + δ i, + δ i, E t [ ∆ t ] + ∑ v = δ i,v E t [( ∆ t − µ ∆ ,t ) v ]+∑ M,j,e ρ ,M,i,j,e E t [ ¯ r ,M,i,j,e,t ] + κ M,i,j,e, + κ M,i,j,e, E t ,M,i,j,e,t [ d l,t ] + ∑ v = κ M,i,j,e,v E t ,M,i,j,e,t [( d l,t − µ d l ,t ) v ] + ξ M,i,j,e, + ξ M,i,j,e, E t ,M,i,j,e,t [ q t,M,j,e ] + ∑ v = ξ M,i,j,e,v E t ,M,i,j,e,t [( q t,M,j,e − µ q,t ) v ] + ε i . (38)This reduces the number of parameters which have to be estimated further to 43.For each of the four specifications in Equations (35) to (38), we use a formulation in whichthe moments on both sides of the equations are estimated contemporaneously, i.e., at thesame time t . Naturally, this is only possible in-sample. Additionally, we also investigatespecifications of Equations (35) to (38) in which the moments on the right hand side areestimated at t − For the in-sample evaluation, we consider the adjusted and unadjusted R as well as theroot mean squared error ( RM SE ). Furthermore, following Zhou et al. (2018), we also use49he direction prediction accuracy (
DP A ) defined as
DP A = T T ∑ t = max ( , ∆ p ⋅ ,t ⋅ ∆ˆ p ⋅ ,t ) ∆ p ⋅ ,t ⋅ ∆ˆ p ⋅ ,t . (39)The in-sample results are depicted in Figures 11 to 14. In each figure, the contemporaneousmodels are depicted in subfigures (a) and (b) for returns and (e) for the XLM measurewhile the results for the specifications which use only past information to model the currentstate are presented in subfigures (c) and (d) for returns and (f) for the
XLM measure.Every line in the graph represents the results for one stock. The highlighted thicker line isthe average of the respective measure across all stocks. There are a few noteworthy results.First, the DPA as well as the R measures (adjusted and unadjusted) allow us to rejectthe hypothesis that the contemporaneous as well as the lagged models have no significancein explaining the data. Our results suggest that the extensive model in Equation (35)overfits the data with growing sampling frequency and, hence, less observations. This canbe seen by the drastically increasing R and DP A values when the sampling frequencyis increased. The adjusted R should account for this effect, and indeed remains ratherstable. However, when the degrees of freedom of the model become sparse, the adjusted R is not able to correct the full extent of the overfitting. Nonetheless, the high values atthe highest frequencies indicate that the contemporaneous model describes high-frequencyreturns very well. In all measures, the contemporaneous specification in Equation (35)(blue) turns out to be superior to the specifications in Equation (36) (red), Equation (37)(green) and Equation (38) (orange) , i.e., it has a higher direction prediction accuracy,better fit in terms of higher R values, and results in a lower root mean squared error.Subfigures (c), (d), and (f) in Figures 11 to 14 present the results using the laggedspecifications of Equations (35) to (38), i.e., they compare their out-of-sample forecastperformance. Recall that these specifications rely heavily on the assumption that thearrival rates stay constant for some (very) short time horizons. Therefore, the resultsregarding the performance of the different models turns out to be different compared tothe in-sample evaluation above. Now, the specifications in Equations (37) and (38), which50se far less interaction terms and have a rather small number of parameters, capture thedynamics of returns almost as well as the other two specifications in Equations (35) and(36) which becomes apparent, for example, when looking at the DP A in Figure 11. Onlyconsidering the adjusted R (Figure 12), the specification in Euqation (35) has a betterperformance. It also slightly decreases the RM SE .For the
XLM measure, the performance of the highly parameterized models is better. Inour opinion, this is due to the following two reasons. First, the
XLM measure changeswith each event, so that on a high frequency the volatility of the
XLM measure is higherthan the volatility of returns. The second reason is rooted in the definition of the
XLM measure as a fraction of observables which in general requires a higher polynomial degree toarrive at a sensible approximation. The more complex models in Equations (35) and (36)are, thus, better suited to provide such an approximation. Note also that we take thelast observation for the
XLM within an interval which is highly volatile. From a tradingperspective, the average
XLM over the interval may be better suited to describe liquidityof the market during the interval and might be a less volatile measure. Nevertheless, wechose the more volatile measure for our analysis. Our results should therefore pose a lowerbound with respect to the modeling accuracy.Nevertheless, for intraday returns, the relatively parsimonious models in Equations (37)and (38) perform remarkably well. Compared to other studies like Zhou et al. (2018) whouse high-dimensional neural networks (without contemporaneous information), we find adecisively smaller
RM SE . For example, the out-of-sample forecast error reported by Zhouet al. (2018) for their best model (GAN for minimizing forecast error loss and directionprediction loss with training sets with a length of 20 days and test sets with a length of 5days) is 0.0079 with a DPA of 69% on a 1 minute interval. Even though the in-sampleresults in our case are not exactly comparable, we can note that for the models with laggedinformation on 1 minute intervals an average
RM SE of 0.0010 can be achieved togetherwith a DPA slightly above 85% on average. We also see that, on high frequencies, themodels with lagged information all result in similar
RM SE and
DP A values. Again the51 igure 11: DPA
The figures report the in-sample direction prediction accuracy (DPA) (as definedin Equation (39)). The in-sample DPA is reported for the estimated modelequations specified in Equation (35) (blue), Equation (36) (red), Equation (37)(green) and Equation (38) (orange) for the sampling frequencies 1, 2, 5, 10, 15,20, 30, 45, 60, 120 and 240.
Length of sample period in minutes D PA Length of sample period in minutes D PA Length of sample period in minutes D PA (a) Contemporaneous Buy Length of sample period in minutes D PA Length of sample period in minutes D PA Length of sample period in minutes D PA (b) Contemporaneous Sell Length of sample period in minutes D PA Length of sample period in minutes D PA Length of sample period in minutes D PA (c) Lagged Buy Length of sample period in minutes D PA Length of sample period in minutes D PA Length of sample period in minutes D PA (d) Lagged Sell Length of sample period in minutes D PA Length of sample period in minutes D PA Length of sample period in minutes D PA (e) Contemporaneous XLM
Length of sample period in minutes D PA Length of sample period in minutes D PA Length of sample period in minutes D PA (f) Lagged XLM igure 12: Adjusted R The figures below report the in-sample, adjusted R for the estimated modelequations specified in Equation (35) (blue), Equation (36) (red), Equation (37)(green) and Equation (38) (orange) for the sampling frequencies 1, 2, 5, 10, 15,20, 30, 45, 60, 120 and 240. . . . . . . . . . . Length of sample period in minutes A d j u s t ed R . . . . . Length of sample period in minutes A d j u s t ed R . . . . . Length of sample period in minutes A d j u s t ed R . . . . . . . . . . . . . . . . . . . . (a) Contemporaneous Buy . . . . . . . . . . Length of sample period in minutes A d j u s t ed R . . . . . Length of sample period in minutes A d j u s t ed R . . . . . Length of sample period in minutes A d j u s t ed R . . . . . . . . . . . . . . . . . . . . (b) Contemporaneous Sell . . . . . . . . Length of sample period in minutes A d j u s t ed R . . . . Length of sample period in minutes A d j u s t ed R . . . . Length of sample period in minutes A d j u s t ed R . . . . . . . . . . . . . . . . (c) Lagged Buy . . . . . . . . Length of sample period in minutes A d j u s t ed R . . . . Length of sample period in minutes A d j u s t ed R . . . . Length of sample period in minutes A d j u s t ed R . . . . . . . . . . . . . . . . (d) Lagged Sell . . . . . . . . . . Length of sample period in minutes A d j u s t ed R . . . . . Length of sample period in minutes A d j u s t ed R . . . . . Length of sample period in minutes A d j u s t ed R . . . . . . . . . . . . . . . . . . . . (e) Contemporaneous XLM . . . . . . . . . . Length of sample period in minutes A d j u s t ed R . . . . . Length of sample period in minutes A d j u s t ed R . . . . . Length of sample period in minutes A d j u s t ed R . . . . . . . . . . . . . . . . . . . . (f) Lagged XLM igure 13: R The figures report the in-sample R for the estimated model equations specified inEquation (35) (blue), Equation (36) (red), Equation (37) (green) and Equation (38)(orange) for the sampling frequencies 1, 2, 5, 10, 15, 20, 30, 45, 60, 120 and 240. . . . . . . . . . . . . Length of sample period in minutes R . . . . . . Length of sample period in minutes R . . . . . . Length of sample period in minutes R . . . . . . . . . . . . . . . . . . . . . . . . (a) Contemporaneous Buy . . . . . . . . . . . . Length of sample period in minutes R . . . . . . Length of sample period in minutes R . . . . . . Length of sample period in minutes R . . . . . . . . . . . . . . . . . . . . . . . . (b) Contemporaneous Sell . . . . . . . . . . Length of sample period in minutes R . . . . . Length of sample period in minutes R . . . . . Length of sample period in minutes R . . . . . . . . . . . . . . . . . . . . (c) Lagged Buy . . . . . . . . . . Length of sample period in minutes R . . . . . Length of sample period in minutes R . . . . . Length of sample period in minutes R . . . . . . . . . . . . . . . . . . . . (d) Lagged Sell . . . . . . . . . . . . Length of sample period in minutes R . . . . . . Length of sample period in minutes R . . . . . . Length of sample period in minutes R . . . . . . . . . . . . . . . . . . . . . . . . (e) Contemporaneous XLM . . . . . . . . . . . . Length of sample period in minutes R . . . . . . Length of sample period in minutes R . . . . . . Length of sample period in minutes R . . . . . . . . . . . . . . . . . . . . . . . . (f) Lagged XLM igure 14: RM SE
The figures report the in-sample
RM SE for the estimated model equationsspecified in Equation (35) (blue), Equation (36) (red), Equation (37) (green) andEquation (38) (orange) for the sampling frequencies 1, 2, 5, 10, 15, 20, 30, 45, 60,120 and 240. . . . . . . Length of sample period in minutes R M SE . . . Length of sample period in minutes R M SE . . . Length of sample period in minutes R M SE . . . . . . . . . . . . (a) Contemporaneous Buy . . . . . . Length of sample period in minutes R M SE . . . Length of sample period in minutes R M SE . . . Length of sample period in minutes R M SE . . . . . . . . . . . . (b) Contemporaneous Sell . . . . . . . . Length of sample period in minutes R M SE . . . . Length of sample period in minutes R M SE . . . . Length of sample period in minutes R M SE . . . . . . . . . . . . . . . . (c) Lagged Buy . . . . . . . . Length of sample period in minutes R M SE . . . . Length of sample period in minutes R M SE . . . . Length of sample period in minutes R M SE . . . . . . . . . . . . . . . . (d) Lagged Sell + + + + + + + + Length of sample period in minutes R M SE + + + + Length of sample period in minutes R M SE + + + + Length of sample period in minutes R M SE + + + + + + + + + + + + + + + + (e) Contemporaneous XLM + + + + + + + + Length of sample period in minutes R M SE + + + + Length of sample period in minutes R M SE + + + + Length of sample period in minutes R M SE + + + + + + + + + + + + + + + + (f) Lagged XLM R show that overfitting is a problem for lower sampling frequencies. Theseonly differ in R (adjusted and unadjusted). Nevertheless, the size of the adjusted R ofaround 15% on 1 minute intervals is remarkable.All models perform better at intervals of 45 minutes than on other frequencies. This isrooted in an above-average precision of the prediction of the return in the last intervalof the day. When using 45 minutes intervals, the last interval of the day is only 30minutes long. Hence, estimated moments based on the observations within the previous45min interval are used to predict the last, shorter, 30min-long interval. If these shorterend-of-day intervals are not excluded from the data, the model performs better comparedto other frequencies. This suggests another avenue for further research: Can the modelperformance be improved by using longer intervals to estimate sample moments to thenmake predictions on smaller time horizons. Put differently, is it in general beneficial tocalculate moments on the last hour of data to forecast the next ten minutes? The result forthe 45min interval suggest there may be some benefits. However, answering the questionwould entail a rolling calculation of means on overlapping intervals which is beyond thescope of the current paper. We also evaluate our models based on out-of-sample predictions using rolling windows.Depending on the size of the non-overlapping intervals, we vary the number of intervalsincluded in the rolling window. Table 3 presents the lengths of these windows for thefrequencies considered. In addition to the root mean squared prediction error (
RM SP E )and the out-of-sample
DP A , we also use the R of a Mincer-Zarnowitz regression (Mincerand Zarnowitz 1969) to evaluate the model. Recall that when we talk about using non-overlapping intervals , we mean the procedure to use theobservations associated to the last interval of the rolling window to predict the observation of the nextinterval. Alternatively, one could use overlapping-intervals , i.e., update the observation at each new eventto predict the next interval. However, this procedure is computationally very demanding and, therefore,out of the scope of the current paper. able 3: Rolling windows The table lists the number of intervals (and thus observations) withinthe rolling windows used to fit the model. The third column lists theapproximate number of trading days over which the rolling window isspanned. For each window, we conduct an out-of-sample prediction. Inthe last column, the potential total number of non-overlapping inter-vals, i.e., the number of available observations is reported. The actualnumber of observations for which an out-of-sample forecast is produceddepends on the availability of the necessary moments for the estimationof Equations (35) to (38).
Frequency Intervals Days Total R rather high and the RM SP E rather low given that we intend to predictfinancial returns on a ultra-high frequency. For the
XLM , results are somewhat different. The precision of the forecast is poor which isin line with the in-sample results. Again, this is due to the high variability of the measure For the stock ADS, we also observe a 100% accuracy when predicting the direction of the next pricechange. Also the R of the Mincer-Zarnowitz regression is around 50%. However, it needs to be mentionedthat for this stock, only 19 out-of-sample predictions are made in total. Since, based on 1 minute intervals,some moments that enter the right hand side of Equations (35) to (37) cannot be calculated, we dropthe observations for these intervals from our sample. In effect ADS has 10.019 valid observations on a1 minute frequency. Due to such missing values also out-of-sample 1 minute results for DB1, FME andHEN3 are not reported since less than 10.000 observations are valid for these stocks. This is why we donot include the 1-min out-of-sample results for ADS in the figures. able 4: Out-of-sample results: 1 and 5 minute intervals The table presents the out-of-sample results for the 1 and 5 minute intervals forthe stocks ALV and FME. The model alternatives in Equation (35), Equation (36),Equation (37) and Equation (38) are referred to in the rows A1 - A4 respectively.For each model, the
RM SP E , the fit of the Mincer-Zarnowitz regression ( R MZ )and the direction prediction accuracy ( DP A ) are reported, both for the returnsof the buy strategy ∆ p t,b and the ones of the sell strategy ∆ p t,s . ALV is the mostliquid stock with the highest number of events in the sample period while FMEis one of the less liquid stocks. ∆ p t,s ∆ p t,b XLM ∆ p t,s ∆ p t,b XLMRM SP E
A1 0.00103 0.00104 146989 0.00202 0.00199 216247A2 0.00095 0.00096 161087 0.00183 0.00174 220908A3 0.00094 0.00094 164625 0.00180 0.00170 224286A4 0.00094 0.00094 165198 0.00179 0.00170 224677 R MZ A1 0.0101 0.0071 0.2546 0.0357 0.0150 0.1076A2 0.0131 0.0099 0.0802 0.0307 0.0316 0.0372A3 0.0146 0.0128 0.0396 0.0321 0.0378 0.0079A4 0.0145 0.0137 0.0213 0.0323 0.0393 0.0050
DP A
A1 77.02 77.75 47.11 74.66 72.96 51.58A2 80.22 80.45 46.56 78.28 79.19 41.18A3 80.67 80.80 45.94 80.32 79.75 34.16A4 80.72 80.92 46.32 80.54 79.86 32.13and its structure. The
XLM changes with each event and is a highly nonlinear functionin the arrival rates (see Equations (23) to (25)). Therefore, the linear approximation maybe poor and the approximation for longer time horizons may be especially poor. In thisline of argument, it is worth mentioning that the model with the highest complexity inour considerations (specified in Equation (35)) performs best in all evaluation measures.The results for all stocks and all frequencies are presented in Figures 15 to 17. As we cansee the smaller the interval, the better the forecasting ability of all our linear models. Theextensive linear approximation in Equation (35) predicts the direction of the returns verywell on small intervals. The R MZ of the Mincer-Zarnowitz regression of above 2% for the sell58trategy is above what we had expected for returns on ultra-high frequencies. On intervalslonger than 10 minutes, the predictive ability of all three linear approximations is, however,poor. It can also be noted, that the sparse model formulations in Equations (37) and (38)perform just as well, or even better in some situations, than the heavily parameterizedformulation in Equation (35) in the case of the return series. This is not true for the XLM .For the
XLM , the more complex formulations in Equation (35) and (36) perform betterin all measures. Especially, the constant
RM SE and the increasing
DP A and R MZ up to5-min intervals are remarkable. The variance of the DP A in Figure 15c shows how noisythe
XLM and the associated forecasts are and by how much the more complex model isable to reduce this variability.
In the paper, we have shown that the limit order book (LOB) can be described as acontinuous Markov process. The description is based on the operator algebra which weborrow from physics. Our model closely describes the reality of the order book andidentifies the arrival and cancellation rates as the key ingredients of the book’s dynamics.Via a simulation study, we show that the distribution of order arrival rates across pricelevels determines the shape of the book and, as a consequence, the transaction priceevolution. By varying the type and shape of arrival and cancellation rates across pricesand volume, we find that the moments of price levels and quantity levels of incoming andcanceled orders are important determinants for the evolution of the book.In an empirical study which is based on a linearized version of our model, we estimatethree different specifications on non-overlapping intervals of various frequencies. As wehave the entire record of the XETRA order book for 3 months at our disposition, we caninclude a large number of parameters in the estimation such that an evaluation of themodel becomes feasible. In-sample, all considered models exhibit a good fit in terms of R , RM SE and direction prediction accuracy (DPA). Our fully parameterized model seems to59 igure 15: DPA
The figures report the out-of-sample direction prediction accuracy (DPA) (asdefined in Equation (39)). The out-of-sample DPA is reported for the estimatedmodel equations specified in Equation (35) (blue), Equation (36) (red), Equa-tion (37) (green) and Equation (38) (orange) for the sampling frequencies 1, 2,5, 10, 15, 20, 30, 45, 60, 120 and 240 minutes, estimated with a rolling windowone-step ahead forecast. The respective window lengths are listed in Table 3.
Length of sample period in minutes D PA Length of sample period in minutes D PA Length of sample period in minutes D PA (a) Buy Length of sample period in minutes D PA Length of sample period in minutes D PA Length of sample period in minutes D PA (b) Sell Length of sample period in minutes D PA Length of sample period in minutes D PA Length of sample period in minutes D PA (c) XLM igure 16: Adjusted R The figures report the R MZ of the Mincer-Zarnowitz regression Mincer andZarnowitz (1969) based on the out-of-sample one-step ahead rolling windowforecast. The R MZ is reported for the estimated model equations specified inEquation (35) (blue), Equation (36) (red) , Equation (37) (green) and Equa-tion (38) (orange) for the sampling frequencies 1, 2, 5, 10, 15, 20, 30, 45, 60, 120and 240 minutes. The respective window lengths are listed in Table 3. . . . . . . . . . . Length of sample period in minutes M i n c e r − Z a r no w i t z R . . . . . Length of sample period in minutes M i n c e r − Z a r no w i t z R . . . . . Length of sample period in minutes M i n c e r − Z a r no w i t z R . . . . . . . . . . . . . . . . . . . . (a) Buy . . . . . . . . . . Length of sample period in minutes M i n c e r − Z a r no w i t z R . . . . . Length of sample period in minutes M i n c e r − Z a r no w i t z R . . . . . Length of sample period in minutes M i n c e r − Z a r no w i t z R . . . . . . . . . . . . . . . . . . . . (b) Sell . . . . . . . . . . Length of sample period in minutes M i n c e r − Z a r no w i t z R . . . . . Length of sample period in minutes M i n c e r − Z a r no w i t z R . . . . . Length of sample period in minutes M i n c e r − Z a r no w i t z R . . . . . . . . . . . . . . . . . . . . (c) XLM igure 17: RM SP E
The figures below report the out-of-sample
RM SP E for the estimated modelequations specified in Equation (35) (blue), Equation (36) (red), Equation (37)(green) and Equation (38) (orange) for the sampling frequencies 1, 2, 5, 10, 15,20, 30, 45, 60, 120 and 240. The predictions are based on the out-of-sampleone-step ahead rolling window forecast. The respective window lengths are listedin Table 3. . . . . . . . . . . Length of sample period in minutes R M SE . . . . . Length of sample period in minutes R M SE . . . . . Length of sample period in minutes R M SE . . . . . . . . . . . . . . . . . . . . (a) Buy . . . . . . . . . . Length of sample period in minutes R M SE . . . . . Length of sample period in minutes R M SE . . . . . Length of sample period in minutes R M SE . . . . . . . . . . . . . . . . . . . . (b) Sell + + + + + + Length of sample period in minutes R M SE + + + Length of sample period in minutes R M SE + + + Length of sample period in minutes R M SE + + + + + + + + + + + + (c) XLM R range for the minute-by-minute intervals around or over 10%whereas the direction is correctly predicted in around 70% of all cases.To evaluate the robustness of our results we also conduct an out-of-sample test of themodel. We use one-step-ahead forecasts on various frequencies and evaluate the accuracywith the R of a Mincer-Zarnowitz regression, the DP A as well as the
RM SP E . We findthat our model predicts 1 minute returns very well. Again, we find an unexpected fit forultra-high returns as the R is generally above 2%. In addition, on low frequencies the DP A is around 80%, we can predict the directional change of the next return very well.The time varying estimates for the parameters as well as the short forecasting horizonmake the return prediction astonishingly accurate. We also try to predict liquidity at theend of each interval with the
XLM measure. The measure cannot be forecasted well forlonger time intervals with adequate accuracy. Also on very short time horizons, the bestfitting model is barely able to predict the direction of the next change of the
XLM inmore than 50% of the cases. This has to do with the very volatile nature of the
XLM andhow the
XLM is defined in the first place.On the basis of the event log of XETRA for the first quarter of 2004, we have, nevertheless,shown that our model describes the LOB data well, both in- and out-of-sample. The datarequirements are rather high as knowledge about price and quantity levels of incomingand canceled orders are required. This sort of data is usually not available. Even thoughreturns may be predicted, market impact of actual trading strategies as well as order costsmay hamper profitability of a trading strategy based on our model. Nevertheless, we areconvinced that our empirical analysis provides the lower limits of forecast accuracy, as wehave made several approximating decisions in the course of this paper. In addition, fortime horizons beyond 1 minute other variables may possibly help to predict returns or anyother measure in the order book. 63 cknowledgments
The authors acknowledge support by the state of Baden-W¨urttemberg through bwHPC.64 eferences
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Table 5: Number of events related to order type and market side
The table lists the total number of events related to limit (
Ticker
ADS 1,129,682 16,976 4,749 1,969 B 48.4 55.1 49.7 41.4 48.5S 51.6 44.9 50.3 58.6 51.5ALT 1,094,414 16,785 9,552 1,412 B 46.4 50.1 60.6 39.9 46.6S 53.6 49.9 39.4 60.1 53.4ALV 4,237,243 68,446 39,416 2,105 B 48.3 55.3 44.4 51.2 48.4S 51.7 44.7 55.6 48.8 51.6BAS 2,585,776 31,450 36,082 1,885 B 49.5 50.7 58.7 45.3 49.7S 50.5 49.3 41.3 54.7 50.3BAY 2,199,894 34,187 29,943 1,721 B 49.6 50.9 42.9 46.5 49.5S 50.4 49.1 57.1 53.5 50.5BMW 2,087,167 28,557 34,625 1,707 B 48.6 60.5 47.0 45.0 48.7S 51.4 39.5 53.0 55.0 51.3CBK 1,676,325 23,601 24,753 1,413 B 49.8 48.7 44.2 41.7 49.7S 50.2 51.3 55.8 58.3 50.3CONT 1,130,309 15,866 9,641 1,419 B 48.5 49.2 51.0 43.6 48.6S 51.5 50.8 49.0 56.4 51.4DB1 936,959 16,205 16,381 1,315 B 48.5 50.8 49.2 39.4 48.5S 51.5 49.2 50.8 60.6 51.5DBK 3,339,752 48,314 47,215 2,278 B 49.3 46.2 50.9 45.8 49.3S 50.7 53.8 49.1 54.2 50.7DCX 2,711,327 44,301 57,847 2,003 B 50.1 41.8 56.0 49.5 50.1S 49.9 58.2 44.0 50.5 49.9DPW 1,001,394 26,360 28,571 1,468 B 47.6 50.1 50.5 40.2 47.8S 52.4 49.9 49.5 59.8 52.2DTE 2,349,138 87,942 49,129 3,581 B 49.7 51.8 48.1 39.0 49.7S 50.3 48.2 51.9 61.0 50.3EOA 2,701,672 35,484 34,695 2,106 B 51.2 54.3 53.2 47.9 51.3S 48.8 45.7 46.8 52.1 48.768 icker
FME 801,834 11,156 4,159 1,223 B 48.9 50.4 56.2 39.8 49.0S 51.1 49.6 43.8 60.2 51.0HEN3 1,101,152 11,405 3,565 1,492 B 47.1 51.7 39.3 39.5 47.1S 52.9 48.3 60.7 60.5 52.9HVM 1,482,520 29,616 41,955 1,392 B 50.4 54.1 50.9 41.6 50.5S 49.6 45.9 49.1 58.4 49.5IFX 1,594,470 50,125 63,497 1,584 B 48.8 54.1 42.4 40.7 48.7S 51.2 45.9 57.6 59.3 51.3LHA 1,169,415 23,026 28,737 1,570 B 48.8 48.6 50.3 42.2 48.8S 51.2 51.4 49.7 57.8 51.2LIN 1,157,591 13,496 7,384 1,807 B 48.3 48.6 63.2 46.0 48.4S 51.7 51.4 36.8 54.0 51.6MAN 1,023,998 14,952 15,252 1,697 B 47.3 49.0 60.2 39.1 47.5S 52.7 51.0 39.8 60.9 52.5MEO 1,144,291 16,064 15,028 1,460 B 48.8 52.0 52.8 39.3 48.9S 51.2 48.0 47.2 60.7 51.1MUV2 2,896,094 46,036 37,068 1,908 B 49.2 57.0 43.7 45.4 49.3S 50.8 43.0 56.3 54.6 50.7RWE 2,061,625 31,746 35,398 2,014 B 51.6 44.6 53.3 47.5 51.5S 48.4 55.4 46.7 52.5 48.5SAP 2,800,569 36,907 20,332 1,530 B 49.6 49.4 57.1 42.9 49.7S 50.4 50.6 42.9 57.1 50.3SCH 1,312,153 24,385 17,908 1,439 B 48.3 50.3 51.6 41.6 48.4S 51.7 49.7 48.4 58.4 51.6SIE 3,444,640 58,410 54,186 2,172 B 48.3 49.5 53.8 48.0 48.4S 51.7 50.5 46.2 52.0 51.6TKA 1,130,506 23,019 19,060 1,797 B 48.5 52.8 45.2 41.1 48.5S 51.5 47.2 54.8 58.9 51.5TUI 970,118 21,965 15,737 1,269 B 48.7 53.8 40.5 40.4 48.7S 51.3 46.2 59.5 59.6 51.3VOW 1,966,460 26,478 42,894 1,715 B 48.6 49.3 53.0 44.4 48.7S 51.4 50.7 47.0 55.6 51.3TOTAL 55,238,488 933,260 844,759 52,451 B 49.1 51.2 50.2 43.5 49.1S 50.9 48.8 49.8 56.5 50.969 ppendix BSimulation Specification
In order to simulate the order book, several probabilities and other conventions have to bespecified. Therefore, we go through the terms in Equation (34) and present how we havechosen to specify α M ( k, q ) and ω M ( k, q ) . For convenience, recall Equation (34) as α M ( k, q ) = ¯ r ,M,i,j,a p K,M ( k ; θ M,a ) p Q,M ( q ; φ M,a ) ω M ( k.q ) = ¯ r ,M,i,L,c p K,M ( k ; θ M,c ) p Q,M ( q ; φ M,a ) . Figure B1 illustrates the components of Equation (34).Recall that we chose three theoretical scenarios for the distributions across price levels p K,M (⋅) : First, the uniform distribution (uni), second, a discrete log-normal distributionwith fixed parameters (fix), and third, a discrete log-normal distribution with dynamic pa-rameters where the parameters depend on the prevailing spread (dyn). For the distributionacross order sizes, we only consider one theoretical specification: a power law distribution.Additionally, we also consider the unconditional empirical frequencies of incoming andcanceled orders as observed in the first quarter of 2004, both across price and size levels.
B.1 Rates of Order Types ¯ r ,M,i,j,e The first element of Equation (34) is ¯ r ,M,i,j,e , the rate for an arrival ( e = a ) or a cancellation( e = c ) of order type j on market side M for stock i . We first need to specify the ordertypes that we include in the simulation. In Figure 2, we have depicted limit orders andmarket orders across relative integer distances to the best quote to show that there is asomewhat stable distribution across price levels when the best quote is used as a fix point.At the zero level, we have plotted the marketable orders split up into different types.Table 7 shows the percentages of the different types of marketable orders in detail. Ingeneral, approximately 10% of all incoming orders (cancellations excluded) are marketable.In fact, about half of those marketable orders are arriving on the best quote i.e. with d =
0. Around a quarter is due to market orders with no limit price d < −∞ and anotherquarter are marketable limit orders i.e. with d <
0. Marketable iceberg and stop order aretiny in comparison. The inverse of the last column are the limit orders that are submittedbefore the best quote. As depicted in Figure B1, in our simulation scenarios, we only treatmarket and limit orders separately. We do not distinguish iceberg and stop orders, since70 igure B1: Simulation event tree
The Figure depicts a decision tree to visualize the components of Equation (34).Taking the subtree marked by the red box, each of the leafs that originate thered box has a different rate ¯ r ,M,i,j,e where the subscript e refers either to a anorder arrival or c an order cancellation. Also each of the nodes inside the blueand green box has a different p K,M (⋅)
Event occursAsk side movesArrivalMarketChose k Chose q LimitChose k Chose q CancellationChose k Chose q Bid side movesArrivalMarketChose k Chose q LimitChose k Chose q CancellationChose k Chose q ¯ r ,M,i,j,e p K,M (⋅) p Q,M (⋅) they are market and limit orders with some additional features. Thus, when we use theunconditional empirical frequencies for ¯ r ,M,i,j,e , we calculate¯ r ,M,i,j,e = n M,i,j,e ∆ T , where n M,i,j,e is the number of arrivals ( e = a ) or cancellations ( e = c ) of order type j onmarket side M for stock i observed during the entire first quarter of 2004. ∆ T refers to thetotal trading time during this period. In our case, ∆ T is specified to be 64 trading days. Aswe restrict the simulation to continuous trading, we only include events during the 8h28mof continuous trading to calculate the frequencies. In our sample, option settlement isconducted in three dates. On these three days, further 3 minutes have to be subtracted fromthe continuous trading phase. In total, we have ( ⋅ ( + / ) ⋅ − ⋅ ) ⋅ = , , s ofcontinuous trading time in our sample. Note that we can also decompose the unconditionalempirical rates according to¯ r ,M,i,j,e = n ⋅ ,i, ⋅ , ⋅ ∆ T ⋅ n M,i, ⋅ , ⋅ n ⋅ ,i, ⋅ , ⋅ ⋅ n M,i,j, ⋅ n M,i, ⋅ , ⋅ ⋅ n M,i,j,e n M,i,j, ⋅ , (B1)where n refers to a number of events and the indices specify which characteristic is relevantfor counting. n ⋅ ,i, ⋅ , ⋅ means that only the index i (referring to the event concerning stock i )is relevant to determine the number of events. Categories marked with a ⋅ in the indexare summed over. In other words, n ⋅ ,i, ⋅ , ⋅ denotes the number of events concerning stock i .71 able 6: Event rates for order types The table lists the order arrival and cancellation rates imposed in the scenarios ’dyn’, ’fix’ and’uni’. The separation between marketable limit orders is only used for the ’uni’ scenario. Inthe scenarios ’dyn’ and ’fix’, we only distinguish between market orders (incl. marketable limitorders) and limit orders. The rates have are given in the unit [orders/second].
Order type Market Side RateLimit order(non-marketable) arrival ask 0.12bid 0.12cancellation ask 0.10bid 0.10Limit order(marketable) arrival ask 0.0025bid 0.0025Market order arrival ask 0.0025bid 0.0025In the empirical scenarios, all elements of Equation (B1) can be observed. In theory, wecan craft theoretical scenarios to investigate, ceteris paribus, the sensitivity of the LOBdynamics to changes in just one conditional frequency in Equation (B1). In this paper, wechose to focus on the sensitivity of the order book dynamics to changes in the distributionacross price and quantity levels.In the scenarios that entail a theoretical distribution, we do not use the empirical valuesobserved in our sample. We also chose to focus on the distribution of arrival rates acrossprice and size levels. Thus, we set the values summarized in Table 6. The rates arespecified in the unit [orders/second]. They approximately mirror the observed values inreality but we fix them to parity, so that the two sides of the market are symmetric andbalanced.One peculiarity in the theoretical scenarios ’fix’ and ’dyn’ is that we treat marketable limitorders below or above the best quote as market orders. Marketable limit orders on thebest-quote, i.e., with d =
0, are modeled together with the rest of the limit orders as theyapproximately seem to fit into the discrete logarithmic distributions across price levels (cp.Figure 2). In the scenario ’uni’, we separate the market orders and the marketable limitorders (strictly) below or above the best quote up to d = − able 7: Marketable orders by type The table reports the share of marketable orders of all incoming orders in percentages across allstocks in the XETRA data. The column %L( d <
0) shows the share of marketable limit ordersbehind the best quote, whereas the column %L( d =
0) gives the share of all marketable limitorders directly at the best ask or bid. The column %M contains the percentages of marketorders. %I tables the share of marketable iceberg orders and %T those of stop orders. Thecolumn %all is the total share of all marketable orders.
Ticker Buy/Sell %L( d < ) %L( d = ) %M %I %T %all ADS S 5.40 1.73 1.50 0.14 0.02 8.79B 5.70 1.97 1.93 0.00 0.03 7.69ALT S 6.61 1.61 1.69 0.03 0.05 9.99B 7.54 2.06 1.76 0.00 0.06 9.67ALV S 5.09 2.59 1.99 0.02 0.05 9.74B 5.73 3.07 2.58 0.00 0.04 8.85BAS S 6.67 2.11 1.47 0.03 0.07 10.35B 6.63 2.15 1.46 0.00 0.10 8.88BAY S 7.15 1.93 1.90 0.01 0.10 11.08B 7.76 2.27 1.92 0.00 0.09 10.13BMW S 6.87 1.87 1.22 0.03 0.13 10.12B 6.97 2.10 2.00 0.00 0.12 9.19CBK S 6.10 1.34 1.81 0.01 0.11 9.37B 6.28 1.56 1.59 0.00 0.09 7.93CONT S 6.50 1.37 1.57 0.02 0.07 9.53B 6.75 1.53 1.35 0.00 0.06 8.34DB1 S 7.14 1.93 1.98 0.03 0.13 11.21B 7.66 2.08 1.98 0.00 0.14 9.88DBK S 7.01 2.90 2.07 0.04 0.06 12.08B 7.37 2.95 1.65 0.00 0.08 10.40DCX S 7.90 2.49 2.50 0.02 0.12 13.02B 7.85 2.52 1.56 0.00 0.15 10.53DPW S 8.24 1.80 3.10 0.04 0.21 13.40B 9.56 1.85 3.22 0.00 0.25 11.67DTE S 12.56 3.20 5.02 0.21 0.13 21.12B 13.09 3.34 5.43 0.00 0.11 16.54EOA S 6.97 2.42 1.51 0.04 0.06 11.00B 6.65 2.37 1.69 0.00 0.06 9.09FME S 5.34 1.56 1.37 0.01 0.03 8.31B 5.58 1.72 1.22 0.00 0.04 7.34HEN3 S 4.20 1.31 0.98 0.05 0.01 6.56B 4.44 1.68 1.03 0.00 0.01 6.1373 icker Buy/Sell %L( d < ) %L( d = ) %M %I %T %all HVM S 8.94 1.76 2.23 0.01 0.17 13.12B 9.32 1.93 2.59 0.00 0.17 11.41IFX S 11.40 2.29 3.91 0.05 0.37 18.01B 12.96 2.94 4.54 0.00 0.30 16.20LHA S 8.36 1.42 2.51 0.05 0.19 12.53B 8.67 1.50 2.28 0.00 0.20 10.37LIN S 5.59 1.19 1.34 0.08 0.03 8.23B 5.83 1.30 1.09 0.00 0.05 7.18MAN S 7.55 1.41 1.63 0.10 0.11 10.81B 8.06 1.53 1.50 0.00 0.12 9.71MEO S 7.75 2.01 1.35 0.04 0.07 11.22B 7.91 2.22 1.33 0.00 0.09 10.22MUV2 S 6.41 2.98 1.72 0.02 0.07 11.20B 6.90 3.32 2.39 0.00 0.06 10.29RWE S 7.67 2.16 2.26 0.04 0.11 12.24B 7.17 2.04 1.45 0.00 0.12 9.33SAP S 5.57 2.53 1.67 0.01 0.04 9.83B 5.79 2.58 1.61 0.00 0.05 8.43SCH S 7.57 1.67 2.31 0.03 0.09 11.67B 8.32 1.99 2.32 0.00 0.11 10.41SIE S 7.31 2.61 2.22 0.03 0.08 12.25B 8.08 3.02 2.25 0.00 0.10 11.20TKA S 7.54 1.89 2.42 0.10 0.14 12.09B 8.12 1.64 2.68 0.00 0.13 9.89TUI S 6.82 1.50 2.62 0.01 0.16 11.12B 7.67 2.07 2.96 0.00 0.15 9.88VOW S 8.55 2.85 1.55 0.01 0.16 13.13B 9.09 3.16 1.43 0.00 0.19 12.44
B.2 Order Distribution Across Price Levels p K,M (⋅)
For the probability distribution of order arrivals across price levels specified in the factor p K,M (⋅) , we distinguish three theoretical scenarios and one scenario using unconditionalempirical frequencies. 74 able 8: Parameters of probability distribution across k Order type Market Side µ σ
Limit order(non-marketable) arrival ask 1.726301 0.674654bid 1.765909 0.711773cancellation ask 1.619866 0.620127bid 1.674366 0.650024
B.2.1 Uniform Distribution (uni)
The easiest approach to define the arrival rates across price levels is a uniform distribution.In this scenario, we assume that the arrivals of orders are concentrated on the first 90integer price levels before the best quote of the opposite market. Additionally, marketablelimit orders are also allowed to cross the best quote up to 10 price levels. In essence, thismeans that the arrivals of bid and ask orders are concentrated on 100 price levels aroundthe best quote of the opposite market where the arrival rate on each price level is 0.0012orders per second.For the cancellations, we distribute the probability for an order cancellation uniformlyamong the occupied price levels.
B.2.2 Fixed Probability Distribution (fix)
Empirical frequencies of (non-marketable) limit orders across relative price levels exhibitpronounced probability mass at the tails of the distribution. For the distribution acrossprice levels, in the scenario ’fix’, we use a discrete Gaussian exponential distribution (DGX)as presented by Bi et al. (2001). The distribution is especially useful in cases where therandom variable to be modeled is discrete and has pronounced probability mass at thetails. It is particularly nice that the DGX reduces to the generalized Zipf distributionwhen µ → −∞ . Thus, it is flexible enough to incorporate situation where the probabilitydistribution is a straight line in log-log-plots and cases in which it exhibits some curvature.A short summary of the DGX distribution is given in Appendix C .In the simulation scenario with a fixed probability distribution, we chose to set the valuesas outlined in Table 8. The values are the empirical mean and standard deviation acrossincoming orders of a random sample over several stocks. Note that the mean of arrivals isslightly higher on the bid side of the market, i.e., orders are more likely to arrive deeper inthe book. Also the variance of order arrivals is higher. The same holds for cancellations.75o while there are more arrivals deeper in the book, slightly more orders deep in the bookare also canceled. B.2.3 Dynamical Probability Distribution (dyn)
Similar to the case in which we use a DGX distribution with fixed parameters µ and σ ,in the simulation scenario with a dynamical distribution across price levels, we also usethe DGX distribution as the fundamental distribution. However, in this case we specifythe parameters of the distribution to depend on the prevailing spread. The functionalrelationship we use is the following: µ ( ∆ ) = log ( ⋅ ∆ ) + ,σ ( ∆ ) = √ ⋅ log ( ⋅ ∆ ) + . . The functional relation is inspired by a scatter plot of ˆ µ i and ˆ σ i estimated on the uncon-ditional frequencies of order arrivals (and cancellations) across price levels against theaverage spread ∆ i for each stock i . This scatter plot is depicted in Figure 10. Note also,that we have switched the scale of standard deviation and expectation as observed in thedata on purpose. In that way, we hope to get an impression on how an increase in varianceand a decrease of the mean may affect the characteristics of the order book evolution.The ’dyn’ scenario is theoretically also motivated by the quest to study the sensitivity ofthe LOB system to feedback reactions between the state of the book and traders’ ordersubmission behavior.
B.2.4 Empirical Distribution (emp)
We also simulate one scenario where we take the empirical frequencies observed acrossprice levels into account. The empirical log-frequencies are depicted in Figure 2.
B.3 Order Distribution Across Size Levels p Q,M (⋅)
For the distribution across volume, we employ two different specifications. In one specifi-cation, we use a power law distribution. Even though, in our data at hand, we find that a We are aware of the fact that the expectation of the functional relationship between DGX parametersand spread is not the same as a function for the log-likelihood in dependence of the expectation of thespread i.e. E t [ µ ( ∆ )] ≠ µ ( E t [ ∆ ]) . p ( x ; λ ) = ( λ − ) x λ − ∀ x ∈ N . We fix the parameter in all simulations at λ = . u ∈ [ , ] , we can generate an integer realization ˜ x from the power law distribution bycalculating ˜ x = ⌊( − . )( − u ) − / λ + / ⌋ − , where ⌊⋅⌋ signify the floor operator which cuts off the decimal places of the argument.Note that since we assume independence of p Q,M (⋅) and p K,M (⋅) , and each arriving ordersurely has to be assigned a size, we simply generate a third realization from a uniformdistribution
U ( , ) to determine the volume. In other words, before randomly generatingwhat size is affected with u , we answer the question when something will happen with u and, with u , what as well as where (i.e. at which limit price level) it will happen, asdescribed in Section 4. B.3.1 Empirical Distribution (emp)
We also simulate one scenario where we use the empirical frequencies observed acrossquantity levels to simulate the LOB evolution. The empirical log-frequencies are depictedin Figure 5.
B.4 The Fully Empirical Scenario (emp,emp)
For the case where both the distribution across price levels and the distribution across pricelevels are sampled from the empirically observed frequencies, we take the joint frequencies77not the product of the marginal frequencies) to sample both size and price of an incomingor canceled order. 78 ppendix CDiscrete Gaussian Exponential Distribution (DGX)
In the simulation, as described in Appendix B , we use the DGX for the simulation of orderarrivals and cancellations across price levels.The probability mass function of the distribution can be defined according to Bi et al.(2001) as p ( x = k ; µ, σ ) = A ( µ, σ ) k exp (− ( log ( k ) − µ ) σ ) , ∀ k ∈ N , where the normalizing constant A ( µ, σ ) is defined as A ( µ, σ ) = { ∞ ∑ k = k exp (− ( log ( k ) − µ ) σ )} − . In Figure 2, we use a slightly modified version of the DGX by truncating the distributionat 1 to show how the DGX can be fit to the data. The truncated DGX can be derivedfrom the truncated (log-)normal distribution for continuous values and has the followingprobability distribution function: p ( x = k ; µ, σ ) = − φ (− µσ ) A T ( µ, σ ) kσ exp (− ( log ( k ) − µ ) σ ) , ∀ k ∈ N , The normalization factor A T ( µ, σ ) is similarly defined by A T ( µ, σ ) = σ ( − φ (− µσ )) { ∞ ∑ k = k exp (− ( log ( k ) − µ ) σ )} − . The parameters µ and σ can be estimated using a maximum likelihood specification asdescribed in Bi et al. (2001). 79 ppendix DTaylor Series Expansion of Linear Models In this appendix the Taylor Series expansion that justifies the model specifications inEquations (35) to (38) is derived.Starting point of the derivation is the decomposition of the event rate in Equation (34).First, for the sake of brevity, we introduce the intensity of events at the price level k withaffected order size q given a prevailing spread ∆. r M,i,j,e ( k, q ∣ ∆ ) = α M,i,j ( k, q ∣ ∆ ) + ω M,i,j ( k, q ∣ ∆ ) where the index M denotes the market side (either bid or ask), i indicates the instrument, j denotes the order type (limit or market) and e denotes the event type (arrival orcancellation). The right hand side follows when writing arrival α and cancellation rates ω separately.As we have seen in Section 2.3, the Hamiltonian H is directly constructed from the eventrates. As a direct consequence, the conditional probability to find the system in state ∣ z ⟩ given that it has been in ∣ z ⟩ at time t can be expressed as described in Equation (14) orin more detail as p ( z, t ∣ z , t ) = ⟨ z ∣ ψ ( t )⟩ = ⟨ z ∣ U ( t, t ) ∣ ψ ( t )⟩ = ⟨ z ∣ exp ( ∫ tt H ( τ ) dτ ) ∣ ψ ( t )⟩= ⟨ z ∣ ∞ ∑ w = ( ∫ tt H ( τ ) dτ ) w w ! ∣ ψ ( t )⟩ (D2)As one can see, by construction the conditional distribution is polynomial in the (timeintegral over) arrival and cancellation rates, and, depending on the succession of orders in ∣ z ⟩ (see e.g. Equation (9)), further combinatorical factors have to be introduced (whichinclude the factorial w !). Only regarding the terms up to order one the conditionalprobability can be written as p ( z, t ∣ z , t ) = ⟨ z ∣ ψ ( t )⟩ + ⟨ z ∣ ∫ tt H ( τ ) dτ ∣ ψ ( t )⟩ This first order approximation may fit the conditional probability for short time hori-zons well, however, for longer time horizons the interactions between order arrivals andcancellations may become the more important factor.80evertheless, with this approximation, we may view the conditional expected value ofsome observable O given the state of the order book at time t of stock i at some futuretime t > t as E t [ O i,t ] = ∑ z ⟨ z ∣ O ∣ ψ ( t )⟩ + ∑ z ⟨ z ∣ O ∫ tt H ( τ ) dτ ∣ ψ ( t )⟩= O i,t + ∑ z ⟨ z ∣ O ∫ tt ∑ q ∑ k α M,i,j ( k, q ) E M,i,j + ω M,i,j ( k, q ) C M,i,j dτ ∣ ψ ( t )⟩ where E and C denote the order entry and cancellation operators laid out in Section 2.3and O i,t is the realization of the observable at time t .We may generalize this notion for the conditional expected value to some function thatdepends on the intensity of events, and, thus, again on order size q , the price level k andadditionally further variables that determine arrival and cancellation rates, e.g. the spread.Making the very crude assumption that the expected value of some observable is linear inthe intensity of events, we could formulate the approximation as E t [ O i,t ] ≈ γ ,i + γ ,i E [ r M,i,j,e ( k, q ∣ ∆ )] . Decomposing r M,i,j,e ( k, q ∣ ∆ ) as done in Equation (34) and additionally assuming that thethe average intensity ¯ r ,M,i,j,a ( ∆ ) is some function of the prevailing spread yields E t [ O i,t ] ≈ γ ,i + γ ,i E [( α M,i,j ( k, q ∣ ∆ ) + ω M,i,j ( k, q ∣ ∆ ))]≈ γ ,i + γ ,i E [ ¯ r ,M,i,j,a ( ∆ ) p K,M ( k ; θ M,a ) p Q,M ( q ; φ M,a )+ ¯ r ,M,i,L,c ( ∆ ) p K,M ( k ; θ M,c ) p Q,M ( q ; φ M,c )] (D3)Now, expanding each term by a Taylor series expansion around the respective expectedvalue we have the following expansions for p Q,M ( q ; φ M,e ) p Q,M ( q ; φ M,e ) ≈ p Q,M ( E [ q ] ; φ M,e ) + dp Q,M ( E [ q ] ; φ M,e )( q − E [ q ])+ d p Q,M ( E [ q ] ; φ M,e ) ( q − E [ q ]) + . . . Taking expectations with respect to p Q,M ( E [ q ] ; φ M,e ) yields an approximation of theexpected value in moments of order 2 and higher E [ p Q,M ( q ; φ M,e )] ≈ p Q,M ( E [ q ] ; φ M,e ) + d p Q,M ( E [ q ] ; φ M,e ) [ q ] + . . . Using then, again, the first order Taylor approximation of p Q,M ( E [ q ] ; φ M,e ) around 081eintroduces the first moment E [ p Q,M ( q ; φ M,e )] ≈ p Q,M ( φ M,e ) + dp Q,M ( φ M,e ) E [ q ] + d p Q,M ( E [ q ] ; φ M,e ) [ q ] + . . . Thus, we may write the expected value of p Q,M ( q ; φ M,e ) as a linear function of the moments E [ p Q,M ( q ; φ M,e )] ≈ ξ M,i,j,e, + ξ M,i,j,e, E [ q ] + ξ M,i,j,e, var [ q ] + . . . (D4)Changing variables in p K,M ( k ; θ M,e ) by considering the relative distance d l to the opposingbest quote instead of the absolute price level k , the same can be done for E [ p K,M ( k ; θ M,e )] E [ p K,M ( k ; θ M,e )] ≈ κ M,i,j,e, + κ M,i,j,e, E [ d l ] + κ M,i,j,e, var [ d l ] + . . . (D5)Last but not least, we may model the expected value of the event specific intensity byits expected value shifted by a event specific factor ρ ,M,i,j,e and the expected value of anevent unspecific function f ( ∆ ) solely dependent on the spread E [ ¯ r ,M,i,j,e,t ( ∆ )] = E t [ ¯ r ,M,i,j,e,t ] ρ ,M,i,j,e E [ f ( ∆ )] . (D6)Approximating E [ f ( ∆ )] by an expansion in moments as above yields E [ f ( ∆ )] = δ i, + δ i, E t [ ∆ t ] + ∑ v = δ i,v E t [( ∆ t − µ ∆ ,t ) v ] . (D7)Reinserting the Taylor expansions in Equations (D4), (D5) and (D7) up to order fourtogether with Equation (D6) in Equation (D3) yields Equation (35) E t [ O i,t ] = γ ,i + ( δ i, + δ i, E t [ ∆ t ] + ∑ v = δ i,v E t [( ∆ t − µ ∆ ,t ) v ]) ×∑ M,j,e ρ ,M,i,j,e E t [ ¯ r ,M,i,j,e,t ] ×( κ M,i,j,e, + κ M,i,j,e, E t ,M,i,j,e [ d l,t ] + ∑ v = κ M,i,j,e,v E t ,M,i,j,e [( d l,t − µ d l ,t ) v ]) ×( ξ M,i,j,e, + ξ M,i,j,e, E t ,M,i,j,e [ q t,M,j,e ] + ∑ v = ξ M,i,j,e,v E t ,M,i,j,e [( q t,M,j,e − µ q,t ) v ]) + ε i ,,