Limit Order Book (LOB) shape modeling in presence of heterogeneously informed market participants
LLimit Order Book (LOB) shape modeling in presence ofheterogeneously informed market participants
Mouhamad DRAME FiQuant, Laboratoire de Mathematiques et Informatique pour la Complexite et les Systemes, CentraleSupelec,Universite Paris-Saclay,3 rue Joliot Curie, 91190 Gif-sur-Yvette
September 8, 2020
Abstract
The modeling of the limit order book is directly related to the assumptions on the behavior of realmarket participants. This paper is twofold. We first present empirical findings that lay the ground for twoimprovements to these models.The first one is concerned with market participants by adding the additionaldimension of informed market makers, whereas the second, and maybe more original one, addresses the racein the book between informed traders and informed market makers leading to different shapes of the orderbook.Namely we build an agent-based model for the order book with four types of market participants: informedtrader, noise trader, informed market makers and noise market makers. We build our model based on theGlosten-Milgrom approach and the most recent Huang-Rosenbaum-Saliba approach. We introduce a pa-rameter capturing the race between informed liquidity traders and suppliers after a new information on thefundamental value of the asset. We then derive the whole ‘static” limit order book and its characteristics-namely the bid-ask spread and volumes available at each level price- from the interactions between theagents and compare it with the pre-existing model. We then discuss the case where noise traders have animpact on the fundamental value of the asset and extend the model to take into account many kinds ofinformed market makers.
Keywords : Price formation, High frequency trading, Limit order book, Asymmetry of information, Adverseselection, Bid-ask spread, cross-impact, speed-bump
Contents θ = 0) . . . . . . . . . . . . . 133.3.1 The case without tick size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.2 The non-zero tick size case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Adding toxicity from noise traders (the case θ (cid:54) = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . 171 a r X i v : . [ q -f i n . T R ] S e p Extension of the model to heterogeneously informed market makers 185 Numerical examples of LOB shape 206 Some foundations of the framework from empirical market facts 237 Conclusion and prospects 24A Classification of market participants 25
Most modern financial markets are order-driven markets, in which all of the market participants display theprice at which they wish to buy or sell a traded security, as well as the desired quantity. This model is widelyadopted for stocks and futures, due to its superior transparency.A market for a security is ’liquid’ if investors can buy or sell large amounts of the security at a low transac-tion cost. Liquidity is a valuable characteristic of an asset because it allows investors to trade at prices close totheir decision price. Liquidity is generally supplied by market makers, who are willing to take the other side ofa trade for a premium relative to the current fundamental value. Traders, who are willing to pay this premiumin order to execute a trade immediately, demand liquidity.In an order-driven market, all the standing buy and sell orders are centralized in the limit order book (LOB).Orders in the LOB are generally prioritized according to price and then to time according to a FIFO (First-InFirst-Out) rule. Some exchanges prioritize according to a proportional rule and others use a mix of these tworules.With these electronic markets, trading strategies have become more and more important. In particular,market making - liquidity providing - strategies lay at the core of modern markets. Since there are no moredesigned market makers, every market participant can provide liquidity to the market, and the choice of quan-tity of shares to add -otherwise the LOB shape- is a question of crucial practical relevance.A trader may desire to trade immediately because he has some private information about the future valueof the asset or because he wants to rebalance his portfolio. The presence of traders with private informationexposes the market makers to adverse selection risk and consequently impacts the prices they quote. The spe-cific trading rules of the exchange also impacts the premium that the trader pays for liquidity. The interactionbetween trader information, the market makers and the trading rules is at the heart of many policy questionsto improve market quality. In order to answer questions about if large trading costs (i.e high spread and/orsmall market depth) are due to adverse selection costs or strategic market makers, it is necessary to considermodels that can incorporate these effects.Our starting point is to assess the empirical literature considering a generic group of market makers withhomogeneous level of information. Several models in the literature study the LOB assuming the presence of onetype of market makers -while it is generally admitted that asymmetry of information exist between specula-tors traders (considered informed) and pure liquidity traders (considered uninformed). For instance the papers[BG19; HRS19; GM85] consider all identical liquidity suppliers with no asymmetry of information between them.Our paper differs from these by introducing an asymmetry of information between liquidity suppliers. Weempirically bring evidence of this heterogeneous level of information. We consider for this purpose the “tradesignature”, a metric to empirically assess the level of information of both liquidity traders and liquidity providers.We identify different clusters of liquidity traders and liquidity providers based on different metrics and build onthis an agent-based model to derive the whole LOB shape. sometimes they must post limit orders for tactical reasons How informed are market participants?
Modern markets are two-sided financial markets with market makers quoting bid and ask prices on one side ofthe market and traders submitting market orders on the other side.The market makers buy low, and sell high, adjusting their bid-ask spreads in accordance with the adverseselection they face. Traders, on the other hand, benefit from market makers competing to offer the best quotes.The set of traders comprises informed traders (speculators) as well as noise traders (liquidity traders who tradefor reasons due to liquidity shocks unrelated to the asset value).Since Kyle, and Glosten and Milgrom [GM85; Kyl85], a common assumption in market microstructureliterature on asymmetric information is that while market-makers do not have superior information on marketfundamentals, some traders have private information. The canonical model of dealer markets is due to Glostenand Milgrom [GM85], who assume that traders are better informed than market makers.Empirical evidence show that market makers can be better informed than some traders in the informationacquisition game. [Moi10] analyzes potentially informed liquidity provision with hidden limit orders and showsthat informed market makers act on dark pools to benefit from their superior information, [Bia93] who considersmarket makers with private information about inventories.In the following, we present some statistical properties on market participants showing heterogeneously levelof information for both liquidity takers and liquidity providers. We define the trade signature at horizon k as: ST ( k ) = (cid:15) (cid:80) t Q t ( X t + k − P t ) (cid:80) t | Q t | where • X t denotes the asset’s efficient price at time t • P t denotes the effective trade price at time t • Q t denotes the trade quantity at time t (positive if the trade triggered by the liquidity taker is a buy,negative otherwise) • (cid:15) is equal to 1 for trade signature of aggressive traders and -1 for liquidity providers.The rationale behind the trade signature is that informed traders should win on average and thus havetheir signature positive at more or less short horizons while noise traders should lose on average and havetheir signature negative at intraday horizons. The trade signature is actually a normalized P & L for marketparticipants with respect to their efficient price.In the following we consider (unless formally specified) the micro-price as the efficient price. The micro-priceis defined as mp t = bid t V at + ask t V bt V at + V bt where bid t (resp ask t ) denotes the bid (resp ask) price at time t and V bt (resp V at ) the volume available atthe best bid (resp ask) limit in the book. We define also the mid-price as mid t = bid t + ask t buying when the price is going down or selling when it is going up, i.e the correlation between trade and delta price .1.2 Clustering methodologies and their rationale We cluster liquidity providers and liquidity takers based on many criteria related to order deposition time, ordermodification, order volumes and order posted price distance to efficient price. In the following we will denotepassive traders by PT and aggressive traders by AT.
The idea of clustering traders based on the time they send their ordersis natural as one expect that informed traders will try to profit from their superior information as soon as theyhave a new information. Thus they should send market orders as soon as they see a new update in the efficientprice. In the same way informed market makers are expected to send their limit orders soon to gain good queuepositions.
We use many time metrics to cluster market makers.1.
Trade to add time:
We consider a market maker as informed (conditional on his order being executed) ifthe duration between the last trade before he adds his order and the moment he adds his order is lower thana given threshold and otherwise uninformed. We actually generalize this clustering method to take intoaccount heterogeneously level of information for market makers. Given the thresholds th < ... < th n − we create n clusters of passive traders P T , ..., P T n − where P T is the best informed passive trader and P T n − the less informed one. The rationale is that informed PT will act quickly after a trade to addquotes.2. Add to add time:
Another criteria to cluster liquidity provider is to consider the duration between thelast add time of an order at a level price and the moment he adds at the same level price (conditional onthe order being executed). We expect that uninformed traders add orders at a given price far from thelast add time at the same level price, while informed may add in a clustered way.
We cluster aggressive traders based on the duration betweenthe time they send their trade and the last trade time before their orders execution based on the idea thatinformed traders will rush for liquidity at the same time buckets (they generally use the same sources ofinformation). Given the thresholds th < ... < th n − we create n clusters of aggressive traders AT , ..., AT n − where AT is the best informed aggressive trader and AT n − the less informed one. We cluster AT based on the liquidity they take in the book.We assume that informed AT will send market orders that take a high proportion or deplete the limit theytrade with (or take exactly the quantity available at the best limit they consume). The criteria we consider isthus the ratio between the traded volume and the available volume at the considered limit before the trade.Informed traders are supposed to send market orders that consume exactly the best limit -in a pure utilitarianpoint of view as they are informed on the next efficient price jump they have interest of sending orders thatconsume all the available liquidity up to the jump and thus deplete exactly a limit order (or more if the jumpsize is greater than the first limit price).
We cluster LP based on the number of updates (modifications ofprice or size) of their order between the time they add it and the time it is executed. The motivation is thatinformed PT may update their orders to incorporate new information.
Our attention is restricted to US treasury bonds futures traded on CME exchange. We consider 5 US treasurybonds futures, namely UB(Ultra US Treasury Bond), ZB(US Treasury Bond), ZF(5-Year T-Note), ZN(10-YearT-Note) and ZT(2-Year T-Note) traded on CME for a period of 144 trading days starting from June 04,2018to Jan 02,2019. The trading hours (in US central time) go from Sunday to Friday each day between 5:00 pm to5:00 pm (with a 60-minute break each day beginning at 4:00 pm). We used the so-called market-by-order datafeed as opposed to the market-by-level data. The market-by-order contains all order book events including limitorder postings, trades, and limit order cancellations with their given ids. Market-by-order makes it possible tofollow each order between the time it is added in the book and the time it is executed or canceled with all themodifications of the order. Thus it is easy to compute all the clustering classifications for each order executedand to compute easily trade signature by clusters. It also allow to easily compute the realized gain of orderswith respect to the efficient price and to compare it with the theoretical gains as computed by the marketmakers prior to adding them (conditional on their execution right now).6igure 1:
Histograms for duration between last trade event and new add event for different USTreasuries Futures on 27/11/2018.
Top left: ZN, top right ZF, bottom left: ZB, bottom right: UB The characteristic peaks are consistent acrossdays. 7igure 2:
Histograms for duration between last trade event and the last trade before on the sameproduct for different US Treasuries Futures on 27/11/2018.
Top left: ZN, top right ZF, bottom left: ZB, bottom right: UBThe characteristic peaks are consistent across days. 8 .3 Analysis of trade signatures by market participants groups
We present some results on the clustering based on time and based on volumes in this part (the other resultsare relegated in appendix (A)).We precise that the different clustering are standalone clustering where it is not necessary to have the typeof the order flow categories of the agents (such as retail flow, high frequency traders, etc). The authors of[Meg+17] for example use a classification based on data set provided by the French regulator “Autorit´e desMarch´es Financiers”. So we do not need to have the so-called “client id”. Our different classifications are aposteriori clustering but one can for clustering depending on time do it online with appropriate market data(market by order data with order ids and all details of an order).We compute the different trade signatures at time horizons up to 1000 seconds.In Fig. 3 and Fig. 4 we present the results with a clustering in 2 groups for AT and for PT.As expected at the time of trade the trade signature for aggressive orders (resp passive orders) is negative(resp positive) due to the basic fact that liquidity takers start by losing money by crossing the spread.The trade signature remains constant on horizon up to more or less 100 ns depending on the considered assetdue to the fact that the efficient price does not move in this interval just after a trade. Then it increases (respdecreases) and becomes (resp stays) positive for informed aggressive traders (resp informed passive traders) whileit stays negative for uninformed aggressive traders (informed and uninformed depend on our classification) andthis is not perfectly symmetric for liquidity providers and liquidity takers as our classification does not enforceto have an informed passive trading against an informed aggressive.The different metrics we define allow a separation in different clusters based on market participants realized P & L .It is therefore important to consider that a market maker can be informed and to incorporate this in LOBmodeling. The results in appendix (A) suggest also that there are different level of information between informedmarket makers (the ones with positive signatures) and so one also should take this into account when modelingthe order book. 9igure 3: Trade signatures for different products based on trade to add time clustering for passivetraders and trade to trade time for aggressive traders.
The numbers indicated correspond to the number of points in each cluster during the considered period (144days)Top left: ZF, top right ZN, bottom left: ZB, bottom right: UBWe use a threshold of 10 ns for both PT and AT. 10igure 4: Trade signatures for different products based on volume clustering
The numbers indicated correspond to the number of points in each cluster during the considered period (144days)Top left: ZN, top right ZF, bottom left: ZB, bottom right: UBWe use a threshold of 0.1 of the ratio between traded volume and available volume at best limit at the time oftrade to separate the two groups 11
The model
We consider an asset whose underlying efficient price is composed of a jump component with Poisson arrivaltimes and a noise trader component as follows: P ( t ) = P + N t (cid:88) l =1 B l + N (cid:48) t (cid:88) j =1 θ (cid:0) Q uj (cid:1) ( X j − ρX j − )where P > • (cid:80) N t l =1 B l is a compound Poisson process and ( N t ) t ≥ is a Poisson process with intensity λ i and l=1 the( B l ) l> • (cid:80) N (cid:48) t j =1 θ (cid:0) Q uj (cid:1) ( X j − ρX j − ) where ( N (cid:48) t ) t> is a Poisson process with intensity λ u , ( Q uj ) j> the noise tradersvolume distributions and θ a function that measures the size of noise traders trade impact, X j is the sideof the j th trade of the noise traders(1 if noise trader’s trade j is a buyer initiated and -1 otherwise), ρ isthe first order auto correlation of the trade signs X . We assume that ( Q ) and ( B ) are independent.This part captures the efficient price move due to noise traders , particularly efficient price move dueto the surprise component of the noise traders flow -the surprise component of the noise traders flow isassumed to be X j − ρX j − (a buy order followed by another buy order has less surprise than a buy orderfollowed by a sell order). We follow an here an approach a la [MRR97] to capture the toxicity due to thenoise traders as it is not really exact to assume it null.Let’s define γ = P ( X j = X j − ) and ρ = E ( X j X j − ) V ( X i − ) , we can easily draw that ρ = 2 γ −
1. The case ρ = 0corresponds to independent trade signs whereas ρ > . We will assume that θ is a constant without loss of generality. One can make it dependent on the volumewith the approach suggested in [MRR97]: if we have k typical size ranges on can choose k parameters foreach volume bucket but this will add complexity to the problem while the qualitative conclusions remainthe same. We assume as discussed in the first part that there are four types of market participants: • One informed trader • Informed market makersThe informed market makers and the informed trader have superior information on the fair price dynamics.They have low latency infrastructure and can exploit the market inefficiencies. They are able to assessefficient price jumps before other participants by using different techniques(statistics, news etc). Weassume they receive the price jump just before it happens. Once there is a price jump, they compete toprofit from this information. We introduce a parameter f with 0 ≤ f ≤ One critic to this modeling is that correlation decay exponentially E (cid:0) X j X j + k (cid:1) = ρ k See [Bou+18; Gar+20] for a recentdiscussion Having information first is not enough and an asymmetry can appear here between the IT and the IMM. C.A.Lehalle andP.Besson [CA] made an experiment showing that easy to process news cause price shifts without trades while difficult to processnews cause price move with trades. In a word easy to process news allow informed market makers to update their price and thusis more beneficial for them while difficult to process news cause price moves with trades and so are more profitable to informedtraders(one can imagine that IT can not value quickly with accuracy the news but know that on average following it is successful andthus react and adjust later while market makers have no incentives to cancel as they can make profit with traders misinterpretingthe news). We will not consider this asymmetry in the following between the IT and the IMM. − f ). The introduction of the IMM and this parameter arethe main differences and bring the main effects compared to the conclusions drawn by [HRS19] • One noise trader: he sends market orders without intelligence. Usually he is a liquidity taker tradingfor liquidity shocks unrelated to the actual dynamics. We assume that theses trades follow a compoundPoisson process with intensity λ u . We denote by f u the density of the noise trader volumes and F u theircumulative distribution function. • Noise market makers: We call them noise market makers as opposed to the informed market makersbecause we assume they don’t receive the efficient price jump before it happens but right after it happens.We assume that they are risk neutral. We assume that they know the ratio of price jumps compared tothe number of events happening in the market.We denote by L the cumulative shape function. L ( x ) denote the available liquidity between prices P ( t ) and P ( t ) + x (We assume that there is no tick size for the moment and will relax this assumption later in the sameway that [HRS19]).In the following parts we derive the results for the ask side of the book (deriving them for the bid side canbe done in a similar way).Let’s recall our main assumption: at a time of efficient price jump, the informed market maker sends anorder to cancel his staying order in the limit order book on the side of the jump to avoid being adverse selectedby the informed trader (one can imagine that he cancels and send a market order to adverse select the noisetrader but for the framework this is the same than him canceling and the informed trader adverse selecting thenoise trader ). t the same time the informed trader sends a market order in a greedy way to hit all the availableliquidity in the LOB between P ( t ) and P ( t ) + B where B is the jump size (actually he sends marketable limitorders up to the level price B in order to avoid trading at a level price greater than B when the informed marketmakers has successfully canceled his order). With probability f the informed trader’s order is processed beforethe informed market makers one. The sizable limit order quantity sent by the informed trader is Q i = L ( B )where L is as already said the shape of the book. L is actually the sum of the books provided by the IMM andthe NMM. θ = 0 ) We derive the shape of the book in the case θ = 0 corresponding to noise traders not impacting the efficientprice (We relax this assumption in section 3.4) and consider a general f . The particular case when f = 1corresponds to the [HRS19] case and we will compare market quality parameters in this particular case and thegeneral case. We first consider the case without tick size. The general case with tick size is derived from this case in section3.3.2. We define G IMM ( x ) and G NMM ( x ) as the conditional average profit of a new infinitesimal order ifsubmitted at a price level x by respectively the informed and the noise market makers knowing that it is filledand without any information on the trade’s initiator.We start by computing the market makers expected gains. Definition 1 (Market makers expected gains) . We define G IMM ( x ) and G NMM ( x ) as the conditional averageprofit of a new infinitesimal order if submitted at a price level x by respectively the informed and the noise marketmakers knowing that it is filled and without any information on the trade’s initiator. To derive G IMM ( x ) and G NMM ( x ) we define the following :1. agg a random variable that is equal to IT if the trade is initiated by the informed trader and NT if it isinitiated by the noise trader One should think to the “self trade protection” mechanism in many exchanges that allow a market participant to not tradewith himself . G NTIMM ( x ) the gain of the new infinitesimal order submitted by the IMM knowing that it is filled by the NT3. G ITIMM ( x ) the gain of the new infinitesimal order submitted by the IMM knowing that it is filled by the IT4. G NTNMM ( x ) the gain of the new infinitesimal order submitted by the NMM knowing that it is filled by theNT5. G ITNMM ( x ) the gain of the new infinitesimal order submitted by the NMM knowing that it is filled by theIT6. r = λ i λ i + λ u the ratio of efficient price jumps compared to all the events happening in the market Proposition 1.
The average profit of a new infinitesimal order if submitted at price level x by the IMM satisfies G IMM ( x ) = x − f r E ( B B>x )(1 − r ) P [ Q u > L ( x )] + f r P [ B > x ] (1)
Proposition 2.
The average profit of a new infinitesimal order if submitted at price level x by the NMM satisfies G NMM ( x ) = x − r E ( B B>x )(1 − r ) P [ Q u > L ( x )] + r P [ B > x ] (2)
Remark 1 (Case f = 1) . Note that the average profit for the NMM (2) corresponds to the case f = 1 for theIMM in (1) . In this case (f = 1), there is no difference between the IMM and the NMM as the IMM can notprofit from his knowledge to cancel his order and thus suffers from the same adverse selection than the NMM.Proof. We provide the proof in the case of the informed market maker, the case of the noise market maker canbe derived in a similar way or with the remark just above. We have: G IMM ( x ) = G NTIMM ( x ) P ( agg = N T | Filled) + G ITIMM ( x ) P ( agg = IT | Filled) G NTIMM ( x ) = x and G ITIMM ( x ) = x − E ( B | B > x )= ⇒ G IMM ( x ) = x − E ( B | B > x ) P ( agg = IT | F illed )= ⇒ G IMM ( x ) = x − rf E ( B B>x ) P (Filled) where we used conditional probability formula and the definition of conditional expectation . P (Filled) = r P (Filled | agg = IT ) + (1 − r ) P (Filled | agg = N T )= ⇒ P (Filled) = rf P ( B > x ) + (1 − r ) P ( Q u > L ( x )) which ends the proof. The main point in this proof is to keep in mind the fact that the IMM order on back of the queue being filledmeans that either there is an efficient price jump and he did not succeed to cancel before the IT market order’sexecution or the NT sends an order of size greater than L ( x )We now discuss how the IMM and the NMM build the LOB with respect to these gains. Market makerscompute the value of the shape functions in order to break even on average (conditional on being filled now).To do this the IMM and the NMM compute the respective values of the LOB shape function in order to have G IMM ( x ) = 0 and G NMM ( x ) = 0. From (1) and (2) one can see that they can not for the same shape havetheir gains equal to zero. Hence the following theorem. Theorem 1 (Cumulative LOB shape) .
1. The cumulative LOB shape for the noise market makers is smallerthan the one of the informed market makers. . The effective (visible) LOB shape is thus imposed by the informed market makers and is given by L ( x ) = F − u (cid:20) (1 − f ) + f − r − f r − r E (cid:18) max (cid:18) Bx , (cid:19)(cid:19)(cid:21) (3)
3. The shape function is decreasing with respect to f and with respect to r; it is increasing with respect to x.Proof.
Let’s denote by L i ( x ) and L u ( x ) the optimal LOB shape for the IMM and the NMM (the LOB shapethat allow them to break even on average).From (1) and (2) it is straightforward to see that the break even condition correspond to x = fr E ( B B>x )(1 − r ) P [ Q u >L i ( x )]+ fr P [ B>x ] for the IMM and x = r E ( B B>x )(1 − r ) P [ Q u >L u ( x )]+ r P [ B>x ] for the NMM.From the remark that E (cid:2) max (cid:0) Bx , (cid:1)(cid:3) = E ( B B>x ) x + P ( B < x ), we then derive that: L i ( x ) = F − u (cid:20) (1 − f ) + f − r − f r − r E (cid:18) max (cid:18) Bx , (cid:19)(cid:19)(cid:21) (4) L u ( x ) = F − u (cid:20) − r − r − r E (cid:18) max (cid:18) Bx , (cid:19)(cid:19)(cid:21) (5)Let’s remark that 1 − F u ( L i ( x )) = f (1 − F u ( L u ( x ))) and as 0 ≤ f ≤
1, we have F u ( L i ( x )) ≥ F u ( L u ( x )) andas F u is a non-decreasing function we have the first point of the theorem:1. L i ( x ) ≥ L u ( x )2. This point follows immediately from the precedent point.3. As F − u is a non decreasing function, it is sufficient to show that h ( f ) := 1 − f + f − r − f r − r E (cid:18) max (cid:18) Bx , (cid:19)(cid:19) is a non-increasing function with respect to f . The derivatives of h with respect to f is given by h (cid:48) ( f ) = − − r E (cid:0) max (cid:0) Bx , (cid:1)(cid:1) − r which is clearly non-positive.The other statements can be derived in a similar way.Let’s make some comments on this theorem. Each market maker computes its optimal LOB shape. Whenthere is a possibility for a future profit, the market makers will add liquidity in the market until they are intheir break even point with a vanishing gain. The interesting point is that while noise market makers will stopproviding liquidity in order to avoid negative gains, informed one can still continue adding while having a non-negative gain when f (cid:54) == 1. When the informed market makers become faster than the informed traders, thatis when f decreases the liquidity increases and and becomes eventually unbounded when f → x → L ( x ) is an increasing function of x, we can derive the bid-ask spread based on thisremark. Theorem 2 (Bid-ask spread) .
1. The cumulative LOB shape satisfies L ( x ) = 0 for ≤ x ≤ φ and for x > φ it is increasing where φ is the unique solution of the following equation: E (cid:20) max (cid:18) Bφ , (cid:19)(cid:21) = 1 + 12 f (cid:18) r − (cid:19) (6) The bid-ask spread is equal to φ . . the bid-ask spread is increasing with respect to f and to r .Proof. We have already seen that the cumulative LOB shape is an increasing function of x, and as marketmakers add volumes in order to break even on average. The condition to add volumes for the informed marketmakers is then L i ( x ) > L u ( x ) > F − u is increasing and F − u ( ) = 0, wededuce the existence of φ such that L i ( φ ) = 0 with φ verifying the implicit equation(1 − f ) + f − r − f r − r E (cid:18) max (cid:18) Bφ , (cid:19)(cid:19) = 12 (7)and for x > φ L i ( x ) >
0. For 0 ≤ x < φ L i should be negative which means that market makers do notprovide liquidity at these level prices and thus L i ( x ) = 0 for these points. φ is thus the half bid-ask spread. Itis straightforward to derive 6 from 7.The second point is straightforward from the equation solved by the half bid-ask spread. Remark 2.
Actually the theorem tells us that the bid-ask spread is fixed by the informed market makers asthe spread for noise market makers (case f = 1 ) denoted by µ solves E (cid:104) max (cid:16) Bµ , (cid:17)(cid:105) = 1 + (cid:0) r − (cid:1) andconsequently φ < µ . The meaning is clear: after noise market makers stop adding volumes and fix their spread,the informed market makers still continue to add volumes near the efficient price. We remark (as expected)that the spread does not depend on noise traders distributions as we have assumed that they do not have anytoxicity for market makers. We recall that we do not make any assumption on inventory costs (or any kind ofcosts due to liquidity provision) that would let emerge the spread. The spread appears as a purely mechanismto balance asymmetry of information between market makers and informed traders.
In this section we study the effect of introducing a tick size in the same way as [HRS19]. We suppose that we havethe same efficient price than in the case without tick size. The cumulative LOB shape becomes now a piece-wiseconstant function as liquidity can be add only on the prices in the tick grid. Let’s define d = P alpha ( t ) − P ( t )where we define P alpha ( t ) as the smallest admissible price in the tick grid that is greater or equal to the currentefficient price P ( t ).We now consider the cumulative LOB shape L ( x ) as defined on the tick grid by L d ( i ) where i ∈ N ∗ is the i th closest price to P ( t ) in the ask book (we consider as in the previous section the ask side of the book, thederivation for the bid size is straightforward). We define L d ( i ) = L ( d + ( i − α )where α is the tick size. The quantity placed at the i th limit is defined by l d ( i ) = L d ( i ) − L d ( i − f . The informed trader sendsa marketable limit order of size Q i = L d ( i ) if B ∈ [ d + ( i − α, d + iα ] and the informed market maker sendsan order to cancel all his waiting orders at this limit.The computation of the informed market makers and noise market makers expected gains conditional onbeing filled now is quasi-exactly the same than in the case without tick size. We denote by G dIMM ( i ) and G dNMM ( i ) the conditional gains of a new infinitesimal passive order placed at the i th level for the IMM and theNMM. Proposition 3.
The average gain of a new infinitesimal order submitted by the informed market maker at levelprice i conditional on being filled now is G dIMM ( i ) = G IMM ( d + ( i − α ) = d + ( i − α − f r E (cid:0) B B>d +( i − α (cid:1) (1 − r ) P [ Q u > L d ( i )] + f r P [ B > d + ( i − α ] (8)16 roposition 4. The average gain of a new infinitesimal order submitted by the noise market maker at levelprice i conditional on being filled now is G dNMM ( i ) = G NMM ( d + ( i − α ) = d + ( i − α − r E (cid:0) B B>d +( i − α (cid:1) (1 − r ) P [ Q u > L d ( i )] + r P [ B > d + ( i − α ] (9) Theorem 3 (Cumulative LOB shape with tick size) .
1. The cumulative LOB shape for the noise marketmakers is smaller than the one of the informed market makers.2. The effective (visible) LOB shape is thus imposed by the informed market makers and is given by L d ( i ) = F − u (cid:20) (1 − f ) + f − r − f r − r E (cid:18) max (cid:18) Bd + ( i − α , (cid:19)(cid:19)(cid:21) (10)
3. The shape function is decreasing with respect to f and with respect to r (all things being equal)
The proof is identical to the case without tick size as we still assume market makers add orders to breakeven on average. The comments for theorem (1) still hold. In a word, when noise market makers stop providingliquidity (otherwise their gain becomes negative), the informed market makers can still continue to provideliquidity while having a non-negative gain. When the informed market makers become infinitely faster thanthe informed traders, the liquidity becomes unbounded. In any case an interesting point to notice from thisframework is that the cumulative LOB shape is strictly increasing and goes to infinity with i . So the bookis never empty and the noise trader can always find volume to trade with. This is due to the smoothnesshypotheses we made on the noise traders volume and on the efficient price distributions.Now our goal is to derive the spread when the tick size is not zero. Theorem 4.
1. The LOB shape satisifies l d ( i ) = 0 for < i < k d where k d is determined by the followingequation: k d = 1 + (cid:100) φ − dα (cid:101) (11) where (cid:100) x (cid:101) denotes the smallest integer that is larger than x .The bid ask spread is given by φ dα = α (cid:18) (cid:100) φ − dα (cid:101) + (cid:100) φ + dα (cid:101) (cid:19) (12) Proof.
We showed in the case the tick size is zero that there exists φ such that for x < φ L ( x ) = 0 and for x > φL ( x ) >
0. This result remains true in the discrete LOB for k d where k d = min { k ∈ N ∗ | d + ( k − α > φ } , sowe have (11). The other results follow by a symmetry argument for the bid side. θ (cid:54) = 0 ) We come back to the general case with a non zero constant θ . we consider there is no tick size as the extensioncan be made in a very similar way than previously. we still consider the ask side of the book. Proposition 5.
The average profit of a new infinitesimal order if submitted at price level x by the IMM satisfies G IMM ( x ) = x − Θ + (cid:18) Θ − E ( B B>x ) P ( B > x ) (cid:19) rf P ( B > x )(1 − r ) P ( Q u > L ( x )) + rf P ( B > x ) (13) where
Θ := θ E ( X j − ρX j − | X j = 1) Proposition 6.
The average profit of a new infinitesimal order if submitted at price level x by the NMM satisfies G NMM ( x ) = x − Θ + (cid:18) Θ − E ( B B>x ) P ( B > x ) (cid:19) r P ( B > x )(1 − r ) P ( Q u > L ( x )) + r P ( B > x ) (14) where
Θ := θ E ( X j − ρX j − | X j = 1) 17 roof. We provide the proof in the case of the informed market maker, the case of the noise market maker canbe derived in a similar way. We have: G IMM ( x ) = G NTIMM ( x ) P ( agg = N T | Filled) + G ITIMM ( x ) P ( agg = IT | Filled) G NTIMM ( x ) = x − θ E ( X j − ρX j − | X j = 1) and G ITIMM ( x ) = x − E ( B | B > x ) The rest of the proof is as in (3.3.1).We now give the shape of the book and the bid-ask spread.
Theorem 5 (Cumulative LOB shape with θ (cid:54) = 0) .
1. The cumulative LOB shape for the noise market mak-ers is smaller than the one of the informed market makers.2. The effective (visible) LOB shape is thus imposed by the informed market makers and is given by L ( x ) = F − u (cid:20) rf − r xx − Θ − f r − r xx − Θ E (cid:18) max (cid:18) Bx , (cid:19)(cid:19)(cid:21) (15)
3. The shape function is decreasing with respect to f and with respect to r; it is increasing with respect to x.4. The cumulative LOB shape satisfies L ( x ) = 0 for ≤ x ≤ φ θ and for x > φ θ it is increasing where φ θ isthe unique solution of the following equation: E (cid:20) max (cid:18) Bφ θ , (cid:19)(cid:21) = 1 + 12 rf (1 − r ) φ θ − Θ φ θ (16)The proof can be derived in a very similar way than the case θ = 0 with the gain in the two previouspropositions (5) and (6). The first remark to make is that as expected φ θ > Θ as the left term in (16) is greaterthan 1 and in order to have the right term greater than 1 it is necessary to have φ θ > Θ.The second point to remark is that the spread in this case is greater than the spread in the case θ = 0 asthe right part of (16) is decreasing with Θ and the minimum is for θ = 0.The model tells us that adding toxicity from the noise traders just increase the bid-ask spread. Choosinga general impact function in this framework does not give a closed formula for the LOB shape but an implicitequation between the different parameters of the model. The qualitative conclusions do not change but to notovercharge this paper we decide to not add it. The link to the discrete LOB can be derived without difficultyas done in the previous part. Our goal now is to extend the baseline model in the case there are many sources of jumps in the efficient priceand for each kind of jump there are specialized informed market makers that can only capture this kind ofjumps. We still assume that there is a race for order insertion between informed market makers and informedtraders in the book as soon as there is jump. Our goal is to assess the impact of having competition betweenmany market makers with different type of signals. We will assume in the following that there is no toxicityfrom noise traders ( i.e θ = 0) as we have seen that adding this does not fundamentally change the conclusionsdrawn.Let’s denote by n the number of different sources of jumps (so the number of different types of informedmarket makers ). We will start by the case n = 2 and derive the explicit formula before giving the formula forthe general n -case. one can think about it as specialized types of market makers: for example some market makers have very good models tocapture information with lead-lag, others to process news, etc
18e consider an asset whose efficient price is given by P ( t ) = P + N t (cid:88) l =1 B l + N (cid:48) t (cid:88) j =1 B j where (cid:80) N t l =1 B l (resp (cid:80) N (cid:48) t l =1 B l ) is a compound Poisson process and ( N t ) t ≥ (resp ( N (cid:48) t ) t ≥ ) is a Poisson processwith intensity λ i (resp λ i ) and the ( B l ) l> (resp ( B l ) l> ) are i.i.d square integrable random variables. Wesuppose there are two types of informed traders (IT0 and IT1) and two types of market makers (IMM0 andIMM1). We recall that we assume that IT0 and IMM0 (resp IT1 and IMM1) can only capture the jumps of( B l ) l> (resp ( B l ) l> ). We keep our race assumption between IT0 and IMM0 (resp IT1 and IMM1) at a timeof efficient price jump with a race parameter f that we assume to be the same for all. We still assume thepresence of noise traders whose trade volumes are denoted as previously by ( Q uj ) j> .We define r := λ i λ i + λ i + λ u and r := λ i λ i + λ i + λ u We will derive the gain of a new infinitesimal order added by the market makers conditional on being filledright now.
Proposition 7.
For j ∈ { , } he gain of a new infinitesimal order submitted at price level x by the informedmarket maker j conditional on being filled is given by G IMM,j ( x ) = x − r j f E ( B B>x )(1 − r − r ) P ( Q u > L ( x )) + r j f P ( B j > x ) + r − j P ( B − j > x ) (17) Proof.
Without loss of generality we give the proof for IMM1. G IMM ( x ) = G agg = NTIMM ( x )[1 − P ( agg = IT | Filled) − P ( agg = IT | Filled)]+ G agg = IT IMM ( x ) P ( agg = IT | Filled)+ G agg = IT IMM ( x ) P ( agg = IT | Filled) P ( agg = IT | Filled) = P ( agg = IT , Filled) P (Filled) = r f P (cid:0) B > x (cid:1) (1 − r − r ) P ( Q u > L ( x )) + r f P ( B > x ) + r P ( B > x ) P ( agg = IT | Filled) = P ( agg = IT , Filled) P (Filled) = r P (cid:0) B > x (cid:1) (1 − r − r ) P ( Q u > L ( x )) + r P ( B > x ) + r P ( B > x ) G agg = NTIMM ( x ) = x , G agg = IT IMM ( x ) = x − E (cid:0) B | B > x (cid:1) and G agg = IT IMM ( x ) = x − E (cid:0) B | B > x (cid:1) From this it is easy to get the result of the proposition .To derive the LOB shape let’s remark that each of the three types of market makers will derive his optimalshape based on his gain. The main results of the first part do not change. The informed market makers will puta supplementary volume compare to the noise market makers. We give in the following theorem the cumulativeLOB shape for each informed market maker.
Theorem 6.
1. For j ∈ { , } he optimal cumulative LOB shape for informed market maker j is given by L j ( x ) = F − u (cid:20) r j f + r − j − r − r − − r − r (cid:18) r j f E (cid:18) max (cid:18) B j x , (cid:19)(cid:19) + r − j E (cid:18) max (cid:18) B − j x , (cid:19)(cid:19)(cid:21) (18)
2. The effective shape of the book is given by L ( x ) = F − u [max ( F u ( L o ( x ) , F u ( L ( x ))] (19)
3. When r r > and when f = 0 the shape function is not infinite as in the case with only one informedmarket maker. f → f near 0 does not allowto have unbounded liquidity (all other parameters being fixed) This is in a sense logical. When there is just oninformed market maker, he is willing to provide a smaller spread when a trader (noise) requests quotes, allowinga better market quality (smaller spreads and greater volumes) for liquidity traders.Let’s now give the general formula when there are n types of informed market makers capturing each one aparticular kind of jump. We assume that all the market makers and informed traders share the same common f . Theorem 7.
1. For k ∈ I , n − J the optimal cumulative LOB shape for informed market maker k is givenby L k ( x ) = F − u r k f + (cid:80) j (cid:54) = k r j − (cid:80) j r j − − (cid:80) j r j r k f E (cid:18) max (cid:18) B k x , (cid:19)(cid:19) + (cid:88) j (cid:54) = k r j E (cid:18) max (cid:18) B j x , (cid:19)(cid:19) (20)
2. The effective shape of the book is given by L ( x ) = F − u (cid:20) max k ( F u ( L k ( x ))) (cid:21) .
3. When f → and ∃ k (cid:54) = j, r k r j > the shape function is not infinite as in the case with only one informedmarket maker. The proof is straightforward by computing the gain of an infinitesimal order added by an informed marketmaker conditional on a fill and using the break even condition. The results induced by this theorem are in linewith the results discussed in the case with just two informed market makers. We could use a more realisticmodeling by allowing informed market makers to capture more than one kind of jump (and introduce differentrace parameters between to have a more realistic framework) but the conclusions would essentially remain thesame as long as we do not suppose that there is one perfectly informed market maker that can access all theefficient price jumps.
We give some numerical examples of LOB (ask side) for different parameters r , d and f . We take a normaldistribution for noise traders trade volumes with standard deviation 10 and a Pareto distribution for the absolutevalue of the efficient price jumps with shape 3 and scale 0.005. We take a tick size of 0.01.20igure 5: LOB shapes with the corresponding parameters r = 0 . , f = 90 and different values of d We take for the noise traders volume distributions a normal distribution with an arbitrary standard deviationof 10 and for the efficient price jumps a Pareto distribution with shape parameter of 3 and scale parameter of0.005. We take a tick size of 0 .
01. 21igure 6:
LOB shapes with the corresponding parameters r , f , d Some foundations of the framework from empirical market facts
We will discuss the foundations of our modeling giving some insights on the different facts justifying the differentchoices we made. Let’s start by our assumption on the efficient price dynamics. One can think about the moregeneral case with correlated assets. Let’s assume we have two assets sharing some common dynamics with twocorresponding books. In book 1 we assume there are NMM1 (Noise Market Makers 1), NT1 ( Noise Trader 1)and in book 2 NMM2 and NT2. We assume there are IMM and IT that act in the two books (the same)-theidea behind being that informed market makers in one book can act as informed traders in the other as soon asthey get executed and get insight on the efficient price. Namely we assume that the dynamics of the two assetsare given by P t − P = (cid:88) t i ≤ t (cid:15) t i + (cid:88) t i ≤ t B t i + (cid:88) t i ≤ t θ u, (cid:16) Q u, t i (cid:17) + β (cid:88) t i ≤ t θ u, (cid:16) Q u, t i (cid:17) P t − P = (cid:88) t i ≤ t (cid:15) t i + (cid:88) t i ≤ t B t i + (cid:88) t i ≤ t θ u, (cid:16) Q u, t i (cid:17) + β (cid:88) t i ≤ t θ u, (cid:16) Q u, t i (cid:17) The first term in each equation is the idiosyncratic component of each asset (that does not impact the efficientprice of the other asset). The second term is the same for the two and represents the fundamental commondriver of the two assets (one can think on the way news affect two futures in the same underlying with differentmaturities). The third term of each asset represents how the noise trader impacts the efficient price and the lastterm captures how the efficient price is influenced by trades in the other asset with β a coefficient between 0and 1. One can see beta as the cross-impact coefficient of the 2 assets independently of fundamental jumps. Itcaptures for example in the case of the same asset in two exchanges how liquidity traders route their orders indifferent venues. Note however that the true correlation of the two assets is not β but depend on the differentjumps parameters and 2 nd -order moments of the different terms. This case is general as it can take into accountthe different situations in which market participants trade: different assets with common underlying or the sameasset traded in different exchanges. Thus our framework discussed in this paper takes into account this moregeneral case and it is straightforward to derive the LOB shape of the two books. Notice that this frameworkenforces the existence of many kind of efficient price jumps but this is actually relevant with respect to whathappens in markets. For example market makers can receive confirmation of trades in which they are involvedbefore it is publicly displayed as the channels in which these information pass are different.One should still not forget that LOBs drawn are “static” LOBs. All the gains we computed are conditionalon having a trade right now. Since when there is a large queue, the life of the order will not end with thenext trade (and traders will not cancel and resubmit their limit orders after every single trade) the order willprobably move in the queue if not executed by the next trade. As a result a dynamical model is needed to takeinto account the dynamics of the queue. We let this for further works or refer to [MY14] for a model capturingthe dynamical part of the queue position value (the authors solve this just for the best limit assuming this limitdoes not move which is restricting). We let this direction for further research.Let’s now discuss the motivations in introducing the race parameters between informed market makers andinformed traders. We recall that we introduce a parameter f , 0 ≤ f leq − f ). The introduction of this parameter is one of the key point of themodel. This parameter makes actually sense in most of the exchanges as there is a non-zero probability that2 orders sent quasi-simultaneously are executed in the opposite sense. On CME exchange-which is one of themost liquid exchanges for futures and options in the world- for example for two orders sent in a delta time ofless than 10 ns the probability the last sent order is executed before the first one is 0 . ns delta time isan approximation of the time necessary for 100 bits to reach the switches in the L1 layer of CME GLink. Thisso-called ‘Glink 10G” uses a 10 Gbps debit and in order to transmit 100 bits the necessary time is roughly 10 ns Empirical studies show evidence of considerable cross-impact effects in stock markets [HS01; PV15; WSG16; Ben+17]
23y a simple calculation). Another interpretation of the f parameter is the randomizer some exchanges addedto resort the different incoming orders or the so-called ‘speed bumps” that allow market makers to cancel theirorders before the order sent by liquidity takers are executed. In this paper, we assess the existing literature considering one generic group of market makers. We bringempirical evidence showing that we can classify market participants based on different parameters such as themoment they insert their orders, the relative volume they trade with respect to the available volume at the bestlimit, the number of updates of their orders etc.We introduce an agent-based model for the LOB. Built on the Glosten-Milgrom approach and the recentHuang-Rosenbaum-Saliba, we use a zero-profit condition for the different types of market makers which enablesus to derive a link between the proportion of different types of traders, the race parameter between informedmarket makers and informed traders, and the LOB shape. We discuss the effect of introducing a tick size.We then extend the model to take into account toxicity from noise traders auto-correlated trades andcompetition between different types of informed market makers. We discuss how taking into account the noisetraders trades signs auto-correlation would increase the bid-ask spread by a mechanical part due to this auto-correlation.We discuss also how the race parameter could impact the liquidity and the bid-ask spread but did notdiscuss the interesting retro-effects of this race parameter that is the following: when all liquidity providerscancel simultaneously their orders, some liquidity deterioration will be caused.We then show the model is realistic with respect to market participants motivations to trade with a discussionon how a ‘multi assets” model is straightforward to plug-in the framework.We bring some justifications to the race parameter we introduce and relate it to ‘speed-bumps”. Neverthelesswith a speed-bump, an informed trader who has access to private information will likely be filed 100% of thetime . We do not discuss the feedback effects of speed-bumps which bring market makers to become incentivizedto traders speeds increment, reducing the marginal cost of getting faster for HFT and consequently failing intheir intended purpose of protecting market makers. We refer to [Aoy18] for a more detailed description of thiseffect.One of the main limitations of our model is that it does not capture well the market feasibility that is howa trade can occur. A market is not admissible if no one wants to be the trader or the market maker: a traderwill likely not trade against a market maker if he is aware that the market maker is informed and vice versa .Indeed we do not address the mechanical substitution that would occur between traders and market makers:who would not be the informed market maker in this context ?We do not discuss the intricate question on how inefficiencies n one market are linked to the price discovery inother markets, in particular we do not address the accuracy and transparency of data provided by intermediariesmarkets -which is one of the goals of regulators. In the US for example while the regulation NMS pushes tofragmentation of order flow, the improvement effect on price discovery is not really clear. Some studies showfragmentation into the dark pools raises effective spread and increases price manipulation [AHH15; CC12].In Europe the Paris-based regulators group ESMA (European Securities and Markets Authority) aims for aregulatory regime with more transparency with the recent MiFID II -Markets in Financial Instruments DirectiveII- that is promoting the adoption of more transparent system and has for example capped the percentage ofshares for a given product that can be traded in dark pools. In this context where regulators seem to care aboutthe efficiency of the price discovery, adding speed-bumps would bring a supplementary loss of transparency.Finally in our approach, we do not consider any inventory management effect and in practice it is a keypoint that impact liquidity (in particular when there are more informed traders than noise traders). Extending ‘Trading with a noise trader” s a private information in many real words cases: bilateral trading or any exchange which iseither non transparent -Reuters and EBS where prices are sent periodically in a high-low fashion without volumes- or delayed -some dark pools where trade reporting is delayed. Acknowledgments
The author warmly thanks Cothereau A., Hattersley M. and Laffitte P. for their fruitful mentoring and discus-sions.
A Classification of market participants
Figure 7:
ZN trade signature based on add to add duration with a threshold of ns ZN trade signature based on update existence between order insertion and order exe-cution
Figure 9:
ZN trade signature based on trade to add duration with thresholds (10 , , )26igure 10: ZN trade signature based ration volumes traded and available volume at best limit attrade time with thresholds (0 . , . , .
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