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## The Information content of high-frequency traders aggressive orders: recent evidences

In this chapter, we answer Question 2: Do HFTs trade opportunely when they use aggressive orders? We present some evidence concerning the impact of aggressive orders on the price formation process and the information content of these orders according to the di erent order Øow categories (high-frequency traders, agency participants, proprietary participants and retail members). For this, we conduct a study on CAC 40 stocks data from September 2017 to November 2017. Over the analysed period, we use both trade and LOB data to describe the dynamics of the LOB accurately before and after each aggressive order. The whole dataset contains approximately 8 millions aggressive orders and 423 millions events (an event can be an order insertion, an order cancellation, an order modiÆcation or a transaction). We summarise in the following our main Ændings.

**Three di erent groups of aggressive orders**

We begin with looking at how the price impact of a single aggressive order varies according to the proportion of liquidity it consumes compared to that present at the best limit. This is why we split aggressive orders into three groups:

• Partial aggressive orders: they consume less than the quantity present at the best limit.

• Exact aggressive orders: they consume exactly the quantity present at the best limit.

• N-limit aggressive orders: they consume more than the quantity present at the best limit.

Partial (resp. exact) aggressive orders constitute approximately 50% (resp. 47%) of aggressive orders in number. Partial and exact aggressive orders are unequally distributed across the di erent order Øow categories. Indeed HFTs send more exact aggressive orders than partial ones: 63% of the exact aggressive orders are sent by HFTs, while only 39% of partial ones are sent by them. Furthermore, we Ænd that exact aggressive orders take place when the LOB is signiÆcantly unbalanced. The imbalance at time t, just before the aggressive buy (resp. sell) order takes place, is computed as follows:

I mbt = Qt1 +Qt2 ,

where Qt1 denotes the volume available at the best bid (resp. ask) at time t, and Qt2 that at the best ask (resp. bid) at time t when it is a buy (resp. sell) aggressive order. We Ænd that the value of the imbalance one microsecond before the exact aggressive trades (on average equal to 27%) is signiÆcantly higher than that (on average equal to 3%) before the partial aggressive trades.

**Price impact according to the groups of aggressive orders**

In order to measure the price impact related to each group of aggressive orders, we Ærst deÆne our price impact measure.

QuantiÆcation of the price impact The price impact of an individual buy (resp. sell) aggressive order, taking place at time t and evaluated at time t + h is denoted by P It+h and deÆned as follows: P It+h = BPt+hS° BPt° § si g nt

where BPt° denotes the best ask (resp. bid) one microsecond before the buy (resp. sell) aggressive order and S the average spread of the asset. For a given Øow of aggressive orders, and for a given time t + h, we compute the average of this measure weighted by the executed quantity across all aggressive orders. This is done from 17 minutes before till 17 minutes after the aggressive order. The results, relative to each group of aggressive orders are the following.

Result 4. As expected, one microsecond after the aggressive order, because of the mechanical impact, the price impact due to n-limit aggressive orders is higher than that of exact ones, which is higher than that of partial ones. One relevant question is whether this mechanical impact is permanent or not. We Ænd that the price impact of exact aggressive orders is permanent: it is above that of partial ones, over all time horizons, higher than two-thirds of the bid-ask spread. On the contrary, n-limit aggressive orders have a temporary component in their price impact: market participants tend to reÆll the LOB by submitting new orders in place of the consumed ones. Indeed, starting one second after the aggressive order, the price impact begins to attenuate. On a 17 minutes time horizon, the remaining mechanical impact of n-limit aggressive orders is quite equal to that of exact aggressive orders.

We then shed light on the interest of studying the price impact according to the traded share (relative to the volume available at the best limit) instead of the traded volume.

Importance of analysing the price impact according to the traded share The average traded amount of partial aggressive order (13 k e) is close to the one of exact aggressive orders (11 k e), but their price impact is signiÆcantly di erent. This is a Ærst indicator that when analysing the price impact, one should not only look at the traded volume but also at the traded proportion relative to the quantity present at the best limits. We deepen our analysis by studying the price impact of partial orders according to the consumed propor-tion (we separate partial orders into 10 groups according to the consumed share). We Ænd that the magnitude of the price impact over all time horizons after the aggressive order is increasing with respect to the consumed share. We then investigate whether the consumed share rather depends on the quantity present at the best limit or on the traded amount. We obtain that the consumed part varies with the traded amount, but also depends signiÆcantly on the quantity present at the best limit. This means that the price impact does not depend only on the traded volume, but also implicitly on the volume present at the best limit. Hence the relevance of investigating the price impact according to the traded share.

Now that we know that partial aggressive orders should be studied separately from exact aggressive orders, we focus on analysing the informational advantage according to each order Øow category, distinguishing between partial and exact aggressive orders. We Ærst quantify our notion of informational advantage and then give our main results, also illustrated in Figure .3.

### Informational advantage

QuantiÆcation of the informational advantage To estimate the informational advantage of an agent, we compute the potential proÆt of a buy (resp. sell) aggressive order that a market participant can realise if he unwinds his position passively at time t + h, denoted by P Pt+h : P Pt+h = BPt+h ° Pt § si g n where BPt+h is the best ask (resp. bid) at time t + h, Pt the price per share obtained by the aggressive order, S the average spread of the asset and si g nt takes the value 1 (resp. -1) if it is a buy (resp. sell) aggressive order. For a given Øow of aggressive orders, and for a given time t + h, we compute the average of this measure weighted by the executed quantity across all aggressive orders. This is done from 17 minutes before till 17 minutes after the aggressive order.

Result 5. The HFT Øow stands out with the (signiÆcantly) highest potential proÆt in the case of partial aggressive orders, over all time horizons. One second after partial aggressive orders, HFTs have a potential proÆt 0.36 spreads higher than that of agency participants, and 0.29 spreads higher than that of proprietary participants, see Figure .3. In addition to this, we show that the aggressive orders of HFTs are less autocorrelated than those of other categories. This allows us to deduce that the high potential proÆt of HFTs is due to an informational advantage and not to an endogenous price impact. Although HFTs still obtain a better potential proÆt than other market participants with exact aggressive orders, the di erence between the categories is not much signiÆcant in this case.

Does the potential proÆt vary among members within the same order Øow category? We investigate the potential proÆt disparities between di erent members belonging to the same order Øow category for partial aggressive orders. We Ænd that over a short time horizon (until 2 minutes after the aggressive order), all HFTs belong to the 25% market participants realising the highest short-term potential proÆts. Over a longer time horizon, from two min-utes after the aggressive order, the proportion of HFTs with potential proÆt higher than the third quartile starts to decrease to the beneÆt of proprietary traders. This could be due to the fact that HFTs do not target long-term strategies, high-frequency trading being an activity where participants typically hold positions for very short times.

It turns out that the analysis of aggressive orders is useful to understand other features than price impact and potential proÆt. For instance, we propose a new classiÆcation of market participants based on our investigation of aggressive orders. We also show that we can access to a more granular classiÆcation by segmenting member code Øows according to the di erent connectivity channels they use, called SLEs (French acronym for Serveur Local d’Emission ). Finally, by observing the evolution of the price before the aggressive orders, we deduce the di erent strategies of member codes, such as mean reverting or trend following.

**From single aggressive orders to strategies**

A classiÆcation tool It is usual to consider cancelled orders to classify members as HFTs or non-HFTs. However, it seems also possible to classify participants by relying on aggressive order potential proÆts. We propose a new classiÆcation: those realising the highest short-term potential proÆts (one second after the aggressive order) can be considered as HFTs, and those realising the lowest as non-HFTs. It turns out that merging both approaches allows us to obtain a more complete classiÆcation of market participants, see Figure .4. Three di erent classes can be distinguished:

• Pure HFTs: they are characterised by a high short-term potential proÆt and a low lifetime of cancelled orders.

• Pure non-HFTs: they are characterised by a high lifetime of cancelled orders and a small short-term potential proÆt.

• Intermediary agents: they are characterised by a small short-term potential proÆt and a low lifetime of cancelled orders.

We point out that, as expected, no member code has high short-term potential proÆts and high lifetime of cancelled orders. Moreover, note that all SLPs belong to the pure HFT category.

A more granular classiÆcation using the di erent connectivity channels Market mem-bers connect to Euronext via SLEs to convey their orders. Splitting the Øows issued by a same member code, and belonging to the same order Øow category according to these connectivity channels can in some cases bring up new information concerning the di erent activities fol-lowed by this member code. For example, we dissociate the Øow of a given member code who is an agency broker serving as intermediary for an HFT (among other clients) according to the di erent SLEs. We compute the potential proÆt of each of these Øows. We Ænd disparities in potential proÆts according to the di erent SLEs: one SLE is probably dedicated to the HFT client, while another is dedicated to other type of clients.

Di erent strategies By observing what happens before the aggressive orders, we Ænd that on average, HFTs and proprietary agents are mean reverting (they buy when the price de-creases and sell when the price increases). In contrast, agency members seem on average trend following: they buy when the price increases and sell when the price decreases. Analysing the potential proÆt by member code, we get that purely mean reverting strategies are not followed by all SLPs. Some of them carry out distinct strategies simultaneously: mean reverting, trend following and another strategy consisting in consuming new orders within the spread.

In Part 2, we focus on agent-based models with the objective of providing new relevant tools for regulators and exchanges. Also, we wish to take into account the empirical observations obtained in Part 1. We answer in this part to Questions 3, 4 and 5.

**From Glosten-Milgrom to the whole limit order book and applications to Ænancial regulation**

In Chapter III, we answer Question 3: How could we extend the Glosten-Milgrom model to the whole LOB? This means that instead of computing only best bid and ask quotes as in the seminal paper by Glosten and Milgrom, we want to be able to build from the interactions between agents the whole LOB: best bid, best ask and volume available at each limit of the LOB.

**E cient price and behaviour of the di erent market participants**

In our model, we assume the existence of an e cient price P(t). It is exogenous, independent of the order book dynamics and satisÆes P(t) = P(0)+Y (t), where Y (t) is a compound Poisson process of the form: Y (t) = NP(t) Bi , with jump rate ∏i and where the Bi are centred price jumps. i =1

We consider three di erent types of market participants as in the Glosten-Milgrom model:

• An informed trader: he receives the value of the price jump size B right before it happens. He then sends his trades based on this information. The informed trader trade size is denoted by Qi . This market participant can be assimilated to an HFT using aggressive orders when he is more informed than the rest of the market. This feature stems from what we saw in Chapter II when answering Question 2.

• A noise trader: he sends random market orders that follow a compound Poisson process with arrival rate ∏u . The noise trader order size is denoted by Qu and its cumulative distribution function is denoted by F∑u .

• Several market makers: they receive the value of the price jump size B right after it happens. We assume that they are risk neutral. They know the proportion of price jumps compared to the total number of events happening in the market which is denoted by r = ∏i∏+i∏u .

**LOB modelling with a zero tick size**

We start with the case where the tick size is assumed to be equal to zero. The obtained results help us to understand those when the tick size is positive. We write L for the cumulative LOB shape function on which we make no a priori assumption (for example it can have a singular part and discontinuities).

The informed trader computes his gain according to the future e cient price. If he knows that the price will increase (resp. decrease), which corresponds to a positive (resp. negative) jump B, he consumes all the sell (resp. buy) orders leading to positive ex-post proÆt. This is formalised in the assumption below.

Assumption 1. The informed trader sends his trade in a greedy way such that he wipes out all the available liquidity in the LOB till the level P(t) + B. Thus his trade size satisÆes: Qi = L(B°).

Regarding the market makers, they compute the conditional average proÆt of a new inÆnites-imal order if submitted at price level x knowing that Q > L(x) and without any information about the trade initiator. This quantity is denoted by G(x) and its value is given in the next proposition.

Assumption 2. For every x > 0 (resp. x < 0), if Lˆ(x) ∑ 0 (resp. Lˆ(x) ∏ 0), market makers add no liquidity to the LOB: L(x) = 0. If Lˆ(x) > 0 (resp. Lˆ(x) < 0), because of competition, the cumulative order book adjusts so that G(x) = 0. We then obtain L(x) = Lˆ(x).

Note that under our zero-proÆt assumption, market makers who place Ærst their orders in the queue, can still make proÆts. In this setting, we can show the emergence of the bid-ask spread and the LOB shape. We have the following result.

Equation (1) shows that the spread emerges naturally from adverse selection risk. Indeed, it is an increasing function of r . This means that market makers are aware of the adverse selection they face when the number of price jumps increases. As a consequence, they enlarge the spread in order to avoid this e ect due to the trades issued by the informed trader just before the price jumps take place. This in line with what we show in Chapter I: market makers withdraw from the market before the announcements because they are aware of the adverse selection risk.

**LOB modelling with a non-zero tick size**

Now we consider the case where the tick size Æ is non-zero. We denote by d the distance between the smallest possible price level that is greater than or equal to the current e cient price P(t); d 2 [0, Æ). We write ld (i ) for the quantity placed at the i t h limit and the cumulative volume at level i is denoted by Ld (i ):

= Ω L(d + i Æ) for i 0.

Ld (i ) L(d (i ° 1)Æ) for i > 0

Considering the same assumptions as in the case where the tick size is vanishing, and com-puting market makers proÆts, the bid-ask spread and LOB shape emerge in the following way.

**Table of contents :**

**Introduction **

Motivations

Outline

**1 Part 1: Empirical analysis of high-frequency traders behaviour **

1.1 Chapter I – The behaviour of high-frequency traders under di!erent market stress scenarios

**1.2 Chapter II – The Information content of high-frequency traders aggressive orders: recent evidences **

**2 Part 2: From empirical observations to agent-based modelling **

2.1 Chapter III – From Glosten-Milgrom to the whole limit order book and applications to Ænancial regulation

2.2 Chapter IV – From asymptotic properties of general point processes to the ranking of Ænancial agents

2.3 Chapter V – Market makers inventories and price pressure: theory and multi-platform empirical evidences

2.4 Multi-platform empirical analysis

**I The behaviour of high-frequency traders under di!erent volatility market stress scenarios **

1 Introduction

2 Data description, HFTs identiÆcation and volatility metrics

2.1 Description

2.2 HFT identiÆcation

2.3 Volatility metrics

3 Liquidity provision by HFTs

3.1 Liquidity metrics

3.2 Preliminary statistics

3.3 Day-to-day analysis

3.4 Intraday analysis

4 Trading activity of HFTs: Amounts traded and aggressiveness

4.1 Metrics used

4.2 Preliminary statistics

4.3 Typical behaviour of market makers

4.4 Day-to-day analysis

4.5 Intraday analysis

5 A more quantitative analysis around the 4 p.m. announcement

5.1 Analysis of HFTs market share in terms of market depth

5.2 Analysis of HFTs aggressive/passive ratio

5.3 Analysis of HFTs share in amounts traded

5.4 Summary

6 Detailed analysis of two events

6.1 Focus on the 3rd of December 2015

6.2 Focus on the 24th of June 2016 (Brexit announcement)

7 Conclusion

I.A Raw OLS regressions of Section 5

**II The information content of high-frequency traders aggressive orders: recent evidences **

1 Introduction

2 Data description and HFTs identiÆcation

2.1 Data description

2.2 The di!erent order Øow categories

2.3 HFTs identiÆcation

3 QuantiÆcation of the price impact and the informational advantage

4 Analysis of aggressive orders with respect to consumed share

4.1 Three di!erent groups of aggressive orders

4.2 Some preliminary statistics

4.3 Relationship between imbalance and aggressive order group

4.4 Price impact according to the groups of aggressive orders

4.5 Focus on partial and exact aggressive orders

5 Potential proÆts according to the di!erent order Øow categories

5.1 Potential proÆts after partial aggressive orders

5.2 Potential proÆts after exact aggressive orders

5.3 Does the potential proÆt vary among members within the same order Øow category?

5.4 Disparities in potential proÆts for a same member code according to the di!erent order Øow categories

6 From single aggressive orders to strategies

6.1 Autocorrelation of aggressive orders according to the di!erent order Øow categories

6.2 A classiÆcation tool

6.3 A more granular classiÆcation using the di!erent connectivity channels

6.4 Di!erent strategies

7 Conclusion

II.A Generalisation to all HFTs

**Part 2 From empirical observations to agent-based modelling **

**III From Glosten-Milgrom to the whole limit order book and applications to Ænancial regulation **

1 Introduction

2 Model and assumptions

2.1 Modelling the e$cient price

2.2 Market participants

2.3 Assumptions

2.4 Computation of the market makers expected gain

2.5 The emergence of the bid-ask spread and LOB shape

2.6 Variance per trade

3 The case of non-zero tick size

3.1 Notations and assumptions

3.2 Computation of the market makers expected gain

3.3 Bid-ask spread and LOB formation

3.4 Variance per trade

3.5 Queue position valuation

4 First practical application: Spread forecasting

4.1 The tick size issue and MiFID II directive

4.2 Data

4.3 Prediction of the spread under MiFID II and optimal tick sizes

5 Second practical application: Queue position valuation

5.1 Data

5.2 Pareto parameters estimation methodology

5.3 Queue position valuation

III.A Proofs

III.A.1 Proof of Proposition 1

III.A.2 Proof of Theorem 1

III.A.3 Proof of Theorem 2

III.A.4 Proof of Proposition 1

III.A.5 Proof of Theorem 1

III.A.6 Proof of Corollary 1

**IV From asymptotic properties of general point processes to the ranking of Ænancial agents **

1 Introduction

2 Market modelling

2.1 Introduction to the model

2.2 Order book dynamic

2.3 Market reconstitution

2.4 Some speciÆc models

3 Ergodicity

3.1 Notations and deÆnitions

3.2 Ergodicity

4 Limit theorems

5 Formulas

5.1 Stationary probability computation

5.2 Spread computation

5.3 Price volatility computation

5.4 An alternative measure of market stability

6 Numerical experiments

6.1 Database description

6.2 Computation of the intensities and the stationary measure

6.3 Ranking of the market makers

IV.A Market reconstitution

IV.B Proof of Remark 5

IV.C Proof of Theorem 1

IV.C.1 Preliminary results

IV.C.2 Outline of the proof

IV.C.3 Proof

IV.D Proof of Theorem 2

IV.D.1 Preliminary result

IV.D.2 Uniqueness

IV.D.3 Speed of convergence

IV.E Proof of Propositions 2 and 3

IV.F Stationary distribution computation

IV.G Proof of Proposition 5

IV.H Proof of Remark 19

IV.I Supplementary numerical results

**V Market makers inventories and price pressure: theory and multi-platform empirical evidences **

1 Introduction

2 Model

2.1 Buy and sell investors

2.2 Multiple market makers

3 Case of identical market makers

4 Case of heterogeneous market makers

4.1 Comparison with the case where market makers are homogeneous (i1 = i2)

5 Data description and preliminary statistics

5.1 Data description

5.2 Market makers activity and identiÆcation

6 Market makers aggressiveness

6.1 Preliminary statistics on market makers aggressiveness

6.2 Market makers aggressiveness according to the inventory

7 Price pressure and market makers inventories

7.1 Measure of the price pressure

7.2 Empirical analysis of the impact of market makers inventories on prices

7.3 Regression analysis

**Bibliography**