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## Qualitative observations and context

High-frequency trading is the use of automated strategies to trade securities such as cash equities, currencies or derivatives, with the distinguishing feature that positions are held for a very short period of time, ranging from a few seconds to a few hours. The term encompass several distinct trading techniques, that are often associated with the use of highly quantitative or data-intensive decision methods, heavy technology infrastructure, and no overnight position.

However, due to the recent increased availability of electronic trading technologies, as well as regulatory changes, a large range of investors are now able to implement high frequency trading strategies. The main regulatory frameworks that recently impacted high-frequency trading are MiFID in Europe (Market in Financials Instruments Directive, implemented 1 November 2007) and RegNMS in the United States (Regulation National Market System, 2007). They both aim at fostering competition between marketplaces, and promoting fair price formation processes. The practical results of these framework is the development of alternatives marketplaces (such as BATS or Chi-X, for example), and coincidently new needs in liquidity provision, orders routing and arbitrage.

In most of modern public security markets, the price formation process, or price discovery, results from competition between several market agents that take part in a publicauction. In particular, day trading sessions, which are also called continuous trading phases, consist of continuous double auctions. High-frequency trading takes place in the continuous trading sessions, and therefore the precise study and modelling of actual mechanisms implementing this continuous double auction is of central importance when designing a high frequency trading strategy. This is precisely the subject of market microstructure: from [56], market microstructure theory is “the study of the process and outcomes of exchanging assets under a specific set of rules. While much of economics abstracts from the mechanics of trading, microstructure theory focuses on how specific trading mechanisms affect the price formation process.” In the next subsection, we present the main mechanisms involved in price formation process.

### The different types of limit order books

In this subsection, we present the mechanisms for order peering in the continuous trading phase, along with the general vocabulary that we will be using throughout this thesis. The continuous trading phase is implemented in the general setup of continuous dou48 Introduction ble auctions. This means that the marketplace (for example the London Stock Exchange) displays publicly at least partial information about offered selling and buying prices. The liquidity providers are market agents that offers those prices, waiting for a counterpart market agent to take their offer, therefore leading to a trade. Liquidity providers compete in an auction on both buy side (called the bid side) and sell side (called the ask side). Practionners often distinguish between price-driven markets and order-driven markets. Although the definition of those notions may vary depending on the author, the general distinction consist in the following:

• Price-driven markets are markets where liquidity providers offer a price for any transaction volume. Generally speaking, there is a small number of dedicated market agents that act as liquidity providers. In actual markets however, e.g. FX markets, the price offered by the liquidity providers often depends on the volume wanted by their counterparts. This microstructure can also be encountered on more rudimentary markets, as for example real-time online betting markets, where a monopolistic market-maker set prices for a bet game so that the number of bets is balanced on both side of the game.

• Order-driven markets are markets where liquidity providers offer a given quantity at a given price, either to buy or to sell. Contrary to the previous organization, any market participant is able to act as a liquidity provider, thanks to the use of limit order trading (see below). This mechanism is the most common microstructure on electronic financial markets, for example it can be found on European cash equities, commodities or interest rates derivatives. This is implemented by the use of a limit order book (LOB), an object that we will describe in the following paragraphs. In this thesis, we will focus on order-driven markets, since this is the mainstream market organization. Let us now define what is a limit order book, and examine two different orders peering rules.

We mention the complete survey article [33] about the limit order book, from where we adapted the following definitions. The role of a marketplace is to gather and to match the order to trade, originated from market participants, that can be submitted at any time during the continuous trading phase. They are of two types: Definition. A market order of size m is an order to buy (sell) m units of the asset being traded at the lowest (highest) available price in the market.

#### Issues faced in high-frequency trading industry

In this subsection, we sum up the main industrial issues where high-frequency trading applies. We focus on the strategic stakes of high-frequency trading, and we put aside the technology issues such as latency minimization, direct market access or hardware speed improvement, which are however crucial aspects of the high frequency trading practice.

Indeed, our aim in to provide coverage for several distinct use of high frequency trading strategies, which are listed and summarized below. Indirect trading costs minimization Indirect trading costs minimization consists in obtaining the highest possible price from a sell trade, or obtaining the lowest possible price for a buy trade. This problem naturally arises when the traded volume is large, due to finite liquidity offering in the LOB (see the above section) : indeed, a large single transaction at market price can desequilibrate the LOB by consuming several levels at once. For example, if an

investor sends a market order to buy e.g. 200 shares in the book represented in table 2.1, the result of that transaction is:

• 80 shares at 50.01.

• 53 shares at 50.02.

• 67 shares at 50.03.

therefore, the ask price at the end of this transaction is 50.03 with a volume offered of 14. Then, the Volume Weighted Average Price of this single transaction is (80 × 50.01 + 53 × 50.02+67×50.03)/200 = 50.0193 which is about one tick greater than the ask price before the transaction, which leads to a loss of 2 bp. This effect is known as market impact. To give a comparison point, a strategy that trades on a daily basis, and that is expected to make a 5% return a year, have a daily expected return of 2 bp, and this is wiped out by the market impact. Moreover, several other costs, as the cost of crossing the spread, the brokers’ fee or latency-related issues can penalize a single trade. Therefore we see that it is of crucial importance for portfolios managers to ensure the best possible execution of their trades.

Actors involved in the indirect trading costs optimization are both investors such as large hedge funds or investment banks, that develops their proprietary solution to this problem, and brokers, that typically have a large daily volume to trade on behalf of their clients.

The brokers are moreover bound by the MIFiD regulations in Europe, and RegNMS act in the US, that force them to operate best execution algorithms. Some estimates that about 70% − 80% of the european equities [34] traded volume is done by execution algorithms, and other algorithmic trading. Classical solutions to this problem can be classified around two central ideas: the spaceoptimization methods, and the time-optimization methods.

**Optimal high-frequency trading with limit and market orders**

In chapter 5, we move to another important aspect of high-frequency trading, the marketmaking strategies. Market-making is the action of continuously providing liquidity to the market by trading with limit orders. In this work, we consider an investor who is able to trade with limit orders, but also with market orders, and therefore we consider a slightly larger class of strategies than strict market-making. The investor’s objective is to maximize the utility of their profit over a finite time horizon. Our goal is to obtain a simple and tractable market model, with a precise modelling of the underlying microstructure. We chose the context of the price/time microstructure, which is the most standard market microstructure, and can be encountered on most cash equities, for example. We propose an easy to calibrate model that reflects some crucial elements of the price/time microstructure: in particular, we are able to fit very general behaviour for the bid/ask spread, and we also take into account the fact that the market can react the investor’s actions, thanks to a control-dependent modelisation of the trades intensities. We represent this situation as a mixed stochastic control problem, that we study by dynamic programming means, and we provide a fast numerical scheme to solve it, thanks to a dimension reduction technique. We prove that this scheme is convergent, and we provide detailled numerical results along with precise performance analysis.

**Optimal high-frequency trading in a pro-rata microstructure with predictive information**

In chapter 6, we investigate a mixed market-making strategy in a exotic microstructure, called the pro-rata microstructure. This microstructure can be encountered for example on short-term interest rates futures. Here again, we consider the situation of an investor willing to maximize their terminal profit over a finite time horizon, who is able to trade with limit and market orders. We adopt the perspective of inventory management, which means that the investor primary objective is to keep their position on the risky asset close to zero at all times, in order to avoid being exposed to market risk. In this particular microstructure, we are able to define and address two other types of risk: the overtrading risk, which is the risk of large variations in the investor inventory, due to the fact that they do not control the quantity they trade at limit ; and the adverse selection risk, which is the risk of market reacting unfavorably to the investor quotes. For this last purpose, we introduce a new state variable, that we interpret as a predictive price indicator, that allows us to balance our position before the price changes. This last feature also provides an extra performance on our empirical tests.

**Table of contents :**

**1 Introduction g´en´erale **

1.1 Objectifs et motivations

1.2 Observations qualitatives et contexte

1.2.1 Pr´esentation g´en´erale

1.2.2 Les diff´erents types de carnets d’ordres limites

1.2.3 Les enjeux rencontr´es dans l’industrie du trading haute fr´equence .

1.3 Synth`ese des principaux r´esultats

1.3.1 Le probl`eme de l’ex´ecution optimale

1.3.2 Trading haute fr´equence optimal avec des ordres limites et au march´e

1.3.3 Trading haute fr´equence optimal dans une microstructure au prorata avec information pr´edictive

**2 Introduction **

2.1 General objectives and motivations

2.2 Qualitative observations and context

2.2.1 General presentation

2.2.2 The different types of limit order books

2.2.3 Issues faced in high-frequency trading industry

2.3 Thesis outline and main results

2.3.1 Optimal execution problem

2.3.2 Optimal high-frequency trading with limit and market orders

2.3.3 Optimal high-frequency trading in a pro-rata microstructure with predictive information

**3 Literature survey: quantitative methods in high-frequency trading **

3.1 Introduction

3.2 Costs optimization strategies

3.3 Market-making and mixed strategies

**4 Numerical methods for an optimal order execution problem **

4.1 Introduction

4.2 Problem formulation

4.2.1 The model of portfolio liquidation

4.2.2 PDE characterization

4.3 Time discretization and convergence analysis

4.4 Numerical Algorithm

4.5 Numerical Results

4.5.1 Procedure

4.5.2 Test 0: Convergence of the numerical scheme

4.5.3 Test 1: A toy example

4.5.4 Test 2: Short term liquidation

4.5.5 Test 3: Sensitivity to Bid/Ask spread

**5 Optimal high frequency trading with limit and market orders **

5.1 Introduction

5.2 A market-making model

5.2.1 Mid price and spread process

5.2.2 Trading strategies in the limit order book

5.2.3 Market making problem

5.2.4 Parameters estimation

5.3 Optimal limit/market order strategies

5.3.1 Value function

5.3.2 Dynamic programming equation

5.4 Numerical scheme

5.4.1 Mean criterion with penalty on inventory

5.4.2 Exponential utility criterion

5.5 Computational results

**6 Optimal HF trading in a pro-rata microstructure with predictive information **

6.1 Introduction

6.2 Market model

6.3 Market making optimization procedure

6.3.1 Control problem formulation

6.3.2 Dynamic programming equation

6.3.3 Dimension reduction in the L´evy case

6.4 Numerical resolution

6.4.1 Numerical scheme

6.4.2 Convergence of the numerical scheme

6.4.3 Numerical tests

6.5 Best execution problem and overtrading risk

6.6 Conclusion