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Modelisation de carnet d’ordres et gestionde risque de liquidite

These de doctorat de l’Universite Paris-Saclaypreparee a l’Universite d’Evry-Val-d’Essonne

Ecole doctorale n574 Ecole doctorale de mathematiques Hadamard (EDMH)Specialite de doctorat : Mathematiques appliquees

These presentee et soutenue a Evry, le 27 Aout 2018, par

FLORIAN RASAMOELY

Composition du Jury :

Caroline HILLAIRETProfesseur, ENSAE Presidente

Areski COUSINProfesseur des Universites, Universite de Strasbourg Rapporteur

Idris KHARROUBIProfesseur des Universites, Universite Sorbonne Rapporteur

Giorgia CALLEGAROMaıtre de Conferences, Universite de Padoue Examinatrice

Guillaume BERNISIngenieur, Ostrum - ex-Natixis Asset Management Examinateur

Shiqi SONGPR, Universite d’Evry-Val-d’Essonne Directeur de these

Vathana LY VATHProfesseur des Universites, ENSIIE Co-directeur de these

Remerciements

Mes premiers remerciements vont naturellement à mon directeur de thèse, ShiqiSong. Je le remercie de m’avoir apporté son soutien constant ainsi que pour la rigueurmathématique qu’il m’a apportée. Je lui suis particulièrement reconnaissant pour sadisponibilité tout au long de la thèse.

Je remercie mon co-directeur de thèse, Vathana Ly Vath, sans qui cette thèsen’aurait jamais pu voir le jour. Je le remercie de m’avoir proposé de réaliser unethèse CIFRE lors de mon stage de fin d’étude. Je le remercie également pour sesnombreux conseils.

Je voudrais exprimer ma reconnaissance à mon second co-directeur de thèse,Etienne Chevallier. Je le remercie de m’avoir initié à la recherche dans le domainedes mathématiques financières et du contrôle stochastique.

Je remercie Charles-Michel Dieuzeide, mon ancien patron, qui m’a fait découvrirles joies des marchés électroniques et du trading chez Carpe Aleam.

Je suis également reconnaissant envers Sergio Pulido pour son aide et sa colla-boration sur le modèle de résilience stochastique.

Mes pensées vont enfin aux actuels et anciens thésards et post-doc du LAMME( Mhamed, Thomas, Anna, Mai, Ricardo, Mohammed, Igor), en particulier, pour labonne ambiance et surtout pour leurs amitiés. Mes remerciements vont égalementaux collègues de l’ENSIIE, Thomas, Massinissa et Anastase, mes amis Danya, Né-hémy, Mahay, Carole, Eric, Romain, Mathieu, Clément, Anne, Namja et StarBurst.

Je tiens tout particulièrement à remercier vivement mes parents pour tout ce queje leur dois ainsi que mon entourage proche pour avoir contribué à un environne-ment favorable à mes activités de recherche. Je remercie toutes les personnes ayantcontribué de près ou de loin à l’élaboration de cette thèse.

Amy, merci de m’avoir supporté et de m’avoir soutenu ces dernières années, jepense que je n’aurais jamais pu finir sans toi.

Table des matières

Introduction iii

1 State of the art 11.1 Financial Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Empirical observations in limit order book . . . . . . . . . . . . . . . 51.3 Challenges of modeling the dynamic of limit order book . . . . . . . . 121.4 Optimal liquidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

I Limit order book modeling with depth representation 23

2 Limit order book modeling 252.1 Orderbook’s representation . . . . . . . . . . . . . . . . . . . . . . . . 272.2 The dynamic of the order book . . . . . . . . . . . . . . . . . . . . . 312.3 More about the randomness specification . . . . . . . . . . . . . . . . 332.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 The fundamentals of the Markov chain N 513.1 The irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 The recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 The transience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4 A more realistic transient model . . . . . . . . . . . . . . . . . . . . . 563.5 The volume index in a bullish market . . . . . . . . . . . . . . . . . . 61

4 Estimations et calibrations du modèle 634.1 Présentation des données . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Estimations de paramètres . . . . . . . . . . . . . . . . . . . . . . . . 674.3 Calibration par rapport au problème de récurrence . . . . . . . . . . 704.4 Faits stylisés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.5 Critères de liquidité de marché . . . . . . . . . . . . . . . . . . . . . . 75

5 Optimal liquidation in a limit order book 795.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Analytical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3 Characterization of the optimal strategy for one step model . . . . . . 90

5

6 TABLE DES MATIÈRES

II Optimal execution in a one-sided order book with sto-chastic volume process 101

6 Optimal execution in a one-sided limit order book with stochasticvolume process 1036.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.2 Analytical properties of the value function . . . . . . . . . . . . . . . 1086.3 Viscosity Characterization of the value function . . . . . . . . . . . . 1176.4 Examples and numerical results . . . . . . . . . . . . . . . . . . . . . 125

Notations

N Set of all Decreasing integer-valued sequence stationary at 0n Element of Nn|i i-th coordinates of nb Bid price operator b : N → NbS Bid sell price operator bS : N × N→ NTS Sell operator TS : N × N→ NTC Cancel operator TC : N × N× N∗ → NTB Buy operator TB : N × N× N∗ → N∆ Difference operator ∆ : N × N∗ → Ns Sum operator s : N → Ns Partial sum operator s : N × N→ Ng Gain operator g : N × N→ Nδ Notation for random variables for event typeα Notation for random variables for event sizeβ Notation for random variables for event price placementP Transition probability on N , P : N ×N → [0, 1]N Markov chain defined by P on NL Discrete generatorΘcondition Set of parameters satisfying the condition

i

Chapitre 0

Introduction

Motivations

Les progrès technologiques ont profondément changé la structure des marchés fi-nanciers qui deviennent de plus en plus électroniques et compétitifs. Les transactionsdeviennent plus liquides permettant ainsi une meilleure gestion de risque. La majo-rité des marchés financiers s’éloignent des marchés avec des spécialistes ou marketmakers et optent pour le mécanisme avec carnet d’ordres. Cela facilite les échangeset permet une réduction importante des risques de liquidité. D’une part, le carnetd’ordres est l’ensemble des ordres d’achat ou de vente disponibles et non exécutésà un instant donné pour un actif financier. D’autre part, le risque de liquidité restel’un des problèmes les plus étudiés dans les risques financiers mais l’un des moinscompris. Nous pouvons cependant noter que le risque de liquidité est étroitementassocié au risque dû à l’impact de prix créé par de larges transactions. De plus, laliquidité peut être caractérisée par la résilience, qui correspond à la vitesse à la-quelle le prix revient à la normale après un choc aléatoire et sans information [44].Le fait de "revenir à la normale" implique l’existence d’un prix de référence que l’onpeut définir comme un prix d’équilibre déterminé par l’agrégation de l’opinion desdifférents investisseurs à un instant donné.

La modélisation du risque de liquidité, et plus précisément du carnet d’ordresest d’une importance capitale pour les intervenants des marchés financiers. Les faitsstylisés, c’est à dire des observations communes parmi des propriétés observées dansdes études statistiques sur différents marchés et différents actifs financiers, imposentdes restrictions très contraignantes à la modélisation de carnet d’ordres. Il est en effetnécessaire de comparer les différents faits stylisés et les critères de liquidité tels quel’impact de prix ou le profil moyen de la densité du carnet d’ordres entre les modèlessimulés et les données du marché afin de valider leur consistance. L’estimation deparamètres est donc un problème fondamental dans la calibration des modèles àpartir des données du marché.

La modélisation du carnet d’ordres permet naturellement aux investisseurs dedéfinir des stratégies d’exécution optimale afin de minimiser l’impact de prix desstratégies et optimiser leurs coûts. Les stratégies d’exécution optimale sont au cœurdes problématiques concrètes des acteurs des marchés financiers, en particulier les

iii

Introduction

stratégies optimales d’achat ou de liquidation. Lorsqu’un investisseur possède unnombre conséquent de volume d’un actif financier, le problème de liquidation opti-male correspond à la recherche d’un algorithme de vente optimisant son critère degain sous un certain nombre de contraintes.

Malgré une riche littérature, les deux problématiques des mathématiques finan-cières, c’est à dire, la modélisation de carnet d’ordres et les problèmes d’exécutionoptimale restent actuellement parmi les sujets plus étudiés.

Cette thèse étudie précisément ces deux sujets d’actualité. Dans la première par-tie de cette thèse, nous proposons et étudions un modèle de carnet d’ordres à traversl’étude d’une chaîne de Markov dans un espace d’états dénombrables dont les étatssont les densités cumulées de carnet d’ordres, autrement dit, la profondeur du car-net, voir Chapitre 2. Le recours à la profondeur du carnet dans la modélisation est ànotre connaissance une approche nouvelle. Cette nouvelle approche facilite la résolu-tion des problèmes de liquidation optimale. De plus, cette approche permet surtoutde relâcher l’hypothèse imposant que chaque transaction soit de taille constante.Ce relâchement rend notre modèle de carnet d’ordres plus réaliste et plus fidèle àla structure même des marchés financiers. Une fois le modèle fixé, nous spécifionset étudions les comportements de notre carnet d’ordres selon les lois des arrivéesdes différents types d’ordre. Chaque ordre limite est muni d’un triplet (prix, volumeet temps d’arrivée) alors qu’un ordre de marché est caractérisé uniquement par levolume et le temps d’arrivée. Nous étudions le comportement de notre chaîne deMarkov, c’est à dire l’évolution du carnet d’ordres, selon les hypothèses fixées surles lois des arrivés d’ordre (prix, volume, temps). Il est en particulier importantd’étudier les cas ou ces lois dépendent de l’état courant de la chaîne de Markov.Dans un premier temps, nous avons prouvé que l’ensemble des paramètres rendantla chaîne de Markov transiente ou récurrente est non vide, voir Chapitre 3. Les résul-tats théoriques les plus importants de cette partie sont toutefois la caractérisationde cet ensemble, et plus précisément sa description. Pour cela, nous utilisons uneapproche semi-martingale pour les différents termes du générateur qui dépendentfortement de l’état courant. Nous exhibons une condition sur les paramètres dépen-dant de faits empiriques. Pour compléter notre étude, nous étudions le problèmed’estimation de paramètres en confrontant notre modèle aux données de marché viales faits stylisés et des critères de liquidité. A l’aide des données de marché, nousdonnons de manière concrète une paramétrisation pour les différents problèmes detransience et de récurrence, voir Chapitre 4. Enfin, nous revenons au problème ini-tial de liquidation optimale dans le cadre de notre modèle. Le critère d’optimisationconsiste à maximiser le gain total sous la contrainte de liquidation, c’est à dire, devendre l’ensemble du volume initial détenu par l’investisseur.

La partie 2 de cette thèse approfondit l’étude des problèmes de liquidation sous

iv

Introduction

contrainte d’impact de prix. Contrairement à la première partie où l’évolution ducarnet d’ordres est représentée par une chaîne de Markov, nous supposons danscette étude que la forme du carnet est fixé. Dans ce modèle, l’impact de prix estnon-linéaire et modélisé par une forme de carnet d’ordres fixe et une résilience surle volume manquant par rapport à la forme du carnet d’ordre fixe. Afin de tenircompte de l’impact des autres investisseurs, la dynamique du volume manquant àla forme de carnet d’ordres fixe est supposée gouvernée par une EDS avec sauts.Nous avons recours au Principe de la Programmation Dynamique pour obtenir unsystème d’inéquations variationnelles d’Hamilton-Jacobi-Bellman. Nous caractéri-sons et montrons que la fonction valeur est l’unique solution de viscosité au systèmed’inéquations variationnelles d’Hamilton-Jacobi-Bellman associé. Nous utilisons uneapproche numérique basée sur une méthode itérative de Huang et al. [42] dont leschéma converge vers la fonction valeur grâce aux critères de monotonie, de consis-tance et de stabilité. Nous illustrons notre étude avec des résultats numériques etdonnons des interprétations aux stratégies selon différents types de forme de carnetd’ordres.

Principaux résultats et contributions

Dans la littérature, la liquidité peut être modélisée selon différentes approches.La première approche consiste à étudier une fonction d’impact de prix. Cette ap-proche ne rend pas compte de la structure et du caractère dynamique du carnetd’ordres. Une seconde approche considère une densité fixe de carnet d’ordres. Unedernière approche, plus difficile, est de modéliser la dynamique du carnet d’ordres.Cependant, à notre connaissance, aucun problème de liquidation optimale a utilisécette dernière approche.

0.0.1 Modélisation de carnet d’ordres

La modélisation de carnet d’ordres consiste à définir les interactions entre lesdifférents types d’ordres envoyés sur le marché.

Les premiers modèles [9], [49], [19] ont cherché à comprendre le mécanisme ducarnet d’ordres et ont retrouvé certains faits stylisés pour valider leur modèle. Ce-pendant, leurs modèles ne tiennent pas compte simultanément des 3 principauxtypes d’ordres. Dans [26], les auteurs proposent un modèle basé sur des processusde Poisson représentant les 3 types d’ordres. Le placement des ordres limites estreprésenté par une suite de variable aléatoire de loi uniforme. Cont et al. [24] pro-pose une famille de processus de Poisson indépendant à chaque niveau de prix pourles ordres d’annulation et les ordres limites avec. Abergel et Jedidi [2] propose une

v

Introduction

famille de processus de Poisson indépendant à chaque niveau de prix pour les ordresd’annulation et les ordres limites dans un cadre mobile.

Dans les modèles présentés, la taille des volumes des ordres est supposée constanteet égale à une unité, ce qui ne reflète pas le mécanisme réel du carnet d’ordres. Deplus, les modèles de carnet d’ordres dans la littérature ne sont pas orientés pour lesproblèmes de liquidation optimale.

Dans la partie 1 de la thèse, inspiré par le papier de Predoiu et al. [56] dans lecadre d’un problème de liquidation optimale, nous proposons un modèle de carnetd’ordres dont les états sont représentés par la profondeur.

0.0.1.1 Description du modèle

Nous notons N , l’espace dénombrable des profondeurs et n un élément de N .

N = (n|1, n|2, n|3, . . .) : n|i ∈ N, n|i ≥ n|i+1 et à partir d’un certain entier k, n|k = 0.

n|1 correspond au volume totale d’un état du carnet d’ordres. L’indice i ∈ N∗

représente les différents niveaux de prix et n|i−n|i+1 représente le volume disponibleau prix i. Nous pouvons définir des fonctions de N dans R. Le prix au bid peut êtredéfinit :

b(n) := maxk ∈ N∗ : n|k > 0 =∞∑k=1

11n|k>0,

Nous définissons les opérateurs modifiant l’état du carnet d’ordres.TB l’opérateur d’achat, représentant la modification du carnet d’ordres après un

ordre limite d’achat de taille a et au niveau de prix b :

TB(n, a, b)|k :=

n|k + a, k ≤ b,

n|k, k > b.

et TS : l’opérateur de vente, représentant la modification du carnet d’ordres aprèsun ordre marché de vente de taille a :

TS(n, a) := ((n|1 − a)+, (n|2 − a)+, (n|3 − a)+, . . .),

et l’opérateur d’annulation, représentant la modification du carnet d’ordres aprèsune annulation de taille a et au niveau de prix b :

TC(n, a, b)|k :=

n|k − a ∧ (n|b − n|b+1), k ≤ b,

n|k, k > b,

Nous nous plaçons dans un cadre temporel discret, les instants sont représentés

vi

Introduction

δn

Pas d’évènement

Ordre de vente marché de taille (αSn)

Ordre d’achat limite de niveau de prix et de taille (βBn , αBn )

Annulation d’ordre d’achat limite de niveau de prix et de taille (βCn , αCn )

δn= N

δn = S

δn = B

δn =

C

Figure 1 – A chaque instant n, nous simulons les variables aléatoires(δn, αSn , αBn , αCn , βBn , βCn )

par les entiers N. À chaque instant n ∈ N, nous choisissons :

— δn l’indicateur d’évènements prenant ses valeurs dans l’ensemble N,S,B,Creprésentant les 4 évènements,

— le niveau de prix d’un ordre d’achat ou d’une annulation d’ordre d’achat parβBn , β

Cn ∈ N∗,

— la taille de l’ordre par αSn , αBn , αCn ∈ N∗ correspondant respectivement à unordre de vente, à un ordre d’achat et une annulation d’ordre d’achat.

Nous avons une représentation des choix des variables aléatoires sur la figure 1.Soit (Ω,A,P) un espace probabilisé, où A est une σ-algèbre sur Ω et P est une

mesure de probabilité sur A. δn ,αSn ,αBn ,αCn ,βBn , βCn sont des variables aléatoiressur (Ω,A,P).

Soit N0 ∈ N soit un état initial de carnet d’ordres. Nous définissons de manièrerécursive :

Nn+1 =

Nn, if δn+1 = N ,

TS(n, αSn+1), if δn+1 = S,

TB(Nn, αBn+1, β

Bn+1), if δn+1 = B,

TC(Nn, αCn+1, β

Cn+1), if δn+1 = C,

(1)

pour n ∈ N.Le processus N = (Nn)n∈N représente l’évolution du carnet d’ordres.

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Introduction

Soit F = (Fn)n∈N la filtration naturelle générée par la suite de variables aléatoires(δn, αSn , αBn , αCn , βBn , βCn ) pour n ∈ N.

Après avoir construit le formalisme du carnet d’ordres, la modélisation consiste àchoisir les distributions de probabilité des variables aléatoires δn, αSn , αBn , αCn , βBn , βCnfor n ∈ N pour n ∈ N.

Dans cette thèse, l’évolution du carnet d’ordres est modélisée par une chaîne deMarkov. Plus précisément, nous supposons qu’il existe une probabilité Qn[A] de Nvers N,S,B,C × (N∗)5 tel que :

P[(δn+1, αsn+1, α

bn+1, α

cn+1, β

bn+1, β

cn+1) ∈ A |Fn]

= P[(δn+1, αsn+1, α

bn+1, α

cn+1, β

bn+1, β

cn+1) ∈ A |Nn]

= QNn [(δ, αs, αb, αc, βb, βc) ∈ A],

Cette hypothèse d’existence donne le caractère markovien de notre modèle. Noustravaillons dans ce cadre là. Caractériser une probabilité Qx est équivalent à carac-tériser les variables aléatoires δn, αSn , αBn , αCn , βBn , βCn .

Assumption 0.0.1. Pour tout x ∈ N , sous la probabilité Qx, la variable aléatoireδ est indépendante de αS, αB, αC , βB, βC .

Nous allons travailler avec l’hypothèse 0.0.1 par la suite. Nous pouvons alorsdéfinir

pn(x) := Qx[δ = N ], ps(x) := Qx[δ = S], pb(x) := Qx[δ = B], pc(x) := Qx[δ = C].

Les 4 termes définissent les probabilités d’arrivée d’un évènement.Nous définissons la probabilité de transition P sous l’hypothèse d’indépendance0.0.1 :

pour x et y dans N ,

P(x, y) := pn(x)11x=y + ps(x)Qx[TS(x, αs) = y]+pb(x)Qx[TB(x, αB, βB) = y] + pc(x)Qx[TC(x, αCβC) = y],

(2)

et nous définissons ainsi le drift pour une fonction F de N dans R+

LF (x) := ps(x)(Ex[F (TS(x, αs)]− F (x))+pb(x)(Ex[F (TB(x, αB, βB)]− F (x)) + pc(x)(Ex[F (TC(x, αCβC)]− F (x)).

(3)

0.0.1.2 Résultats théoriques

Homogénéité Dans les modèles existants de carnet d’ordres [24], [2], [9], [49],[19], les auteurs supposent que la taille des ordres est constante. L’intérêt de cette

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Introduction

hypothèse réside notamment dans la simplification des calculs pour le drift sur leproblème de récurrence positive. De plus, les auteurs [24], [2] n’étudient pas la notionde transience de leur chaîne de Markov. Dans notre modèle, nous n’avons pas besoinde l’hypothèse de taille constant pour le problème de récurrence. De plus, nousavons étudié la transience. Nous utilisons les conditions de Foster-Lyapunov pourétudier la récurrence et la transience faisant intervenir le drift. Lorsque nous enlevonsl’hypothèse de taille constant, le drift dépend fortement de l’état courant du carnetd’ordres. La notion d’homogénéité consiste à réduire la dépendance du drift parrapport à l’état courant.

Irréductibilité

Nous cherchons l’irréductibilité afin de simplifier l’étude de la récurrence et dela transience. Nous rappelons que l’irréductibilité d’une chaîne de Markov exprimel’existence d’un chemin d’état vers un autre, de probabilité non nulle. Soit x ∈N , nous définissons l’ensemble S(x) correspond aux niveaux de prix possédant desvolumes

S(x) := k ∈ N∗ : x|k > x|k+1

Theorem 0.0.1. Supposonsi. Qx[βb = b(x) − 1] > 0 si b(x) > 1, et Qx[βb = b(x)] > 0, si b(x) > 0, et

Qx[βb = b(x) + 1] > 0, pour tout x.ii. Qx[βc = k] > 0 pour tout k ∈ S(x).iii. Qx[αb = 1] > 0 et Qx[αc = 1] > 0, pour tout x.iv. pn(x) > 0, pb(x) > 0, pc(x) > 0 pour tout x.

Alors, la chaîne de Markov N est irréductible et apériodique

Nous définissons Θirr, l’ensemble des paramètres rendant la chaîne de MarkovN irréductible et apériodique. Nous nous plaçons dans cette ensemble par la suite.

Récurrence

Nous montrons que la chaîne de Markov N peut être récurrente. Soit V (x) :=b(x) +H(x)

Lemma 0.0.1. Supposons que les variables aléatoires αb, βb sont intégrables sous Qx

pour tout x ∈ N . Le drift LV (x) est bien définit. Si l’ensemble x ∈ N : LV (x) > 0est un ensemble fini, la chaîne de Markov N est récurrent.

Nous définissons Θrec l’ensemble des paramètres rendant la chaîne de Markovrécurrente.

Nous avons prouvé qu’il existe un ensemble non nul de paramètres pour lequella chaîne est récurrente. La véritable question est de produire un ensemble concretde paramètres. En effet, nous prouvons que

ix

Introduction

Proposition 0.0.1. Soit x ∈ N et si αC à support fini dépendant de ∆βCx SoitΘrec∗ ⊂ Θrec l’ensemble des paramètres vérifiant :

Pour x|1 assez grand,pb(x)D ≤ pc(x)Ex[αC ],

avec pc(x) est une constante positive par hypothèse, D := Ex[αB] + Ex[βB]Alors la chaîne de Markov N est récurrente.

La condition veut dire que pour un niveau de volume x|1 grand, les annulationsd’ordre d’achat doivent être supérieurs aux arrivées d’ordre d’achat. L’ensembleΘrec∗ est plus explicite, nous utilisons cette ensemble dans la calibration du modèle.

Transience

Dans la recherche de modèles transients, nous allons considérer naturellement lesmarchés haussiers. Nous montrons que la chaîne peut être transiente.

Theorem 0.0.2. Supposons que les espérances Ex[(αs)2], Ex[(αb)2], Ex[(αc)2] sontuniformément bornées. Supposons que, pour une constante θ > 0, LH(x) ≥ θ uni-formément. Alors,

lim infn→∞ N|1nn> 0.

En particulier, la chaîne de Markov N est transiente.

Nous définissons Θtra l’ensemble des paramètres rendant la chaîne de Markov Ntransiente.

Proposition 0.0.2. Soit h, g ∈ N∗. Supposons que les espérances Ex[(αs)2], Ex[(αb)2],Ex[(αc)2] sont uniformément bornées.

Soit Θtra∗ ⊂ Θtra l’ensemble des paramètres vérifiant :

— Ex[αb] est une constante ab > 0

— βc est une variable aléatoire suivant une loi uniforme sur S(x)

— αc est une variable aléatoire suivant une loi uniforme sur 1, . . . , g ∧∆βcx

— αs est une variable aléatoire suivant une loi uniforme sur 1, . . . , h ∧ x|1

— ps(x), pb(x), pc(x) sont constantes et

θ := −ps1 + h

2 + pbab − pc1 + g

2 > 0. (4)

Alors la chaîne de Markov N est transiente.

La condition veut dire que l’arrivée des ordres d’achat doit être supérieur àl’arrivée des annulations et l’arrivée des ordres de vente. Cela va impliquer le prixb et le volume totale n|1 va tendre vers l’infini. Nous allons chercher des modèleshaussiers plus réalistes.

x

Introduction

Un modèle de transience plus réaliste

Nous définissons d(x) le prix le plus bas possédant des volumes par

d(x) =

inf S(x) si S(x) 6= ∅,

0 si S(x) = ∅.

Nous définissons pour tout x ∈ N , le processus support comme b(x)− d(x).Dans un marché haussier, le processus bid tend vers l’infini, c’est la raison pour

laquelle la chaîne de Markov N doit être transiente. Cependant, le processus support(b− d) ne doit pas explosé. Cela a un sens, les ordres limites achat disponibles dansle carnet d’ordres à des niveaux de prix éloignés du bid sont supprimés par lesinvestisseurs car la probabilité d’être exécuté est faible. De plus, certains marchéspeuvent avoir un mécanisme de suppression lorsque les niveaux de prix sont tropéloignés du bid. Nous cherchons un marché pour lequel les volumes sont contraintsdans une rangée de prix.

Theorem 0.0.3. Soit Θtra1 ⊂ Θtra l’ensemble des paramètres vérifiant les hypo-thèses suivantes :

— la variable aléatoire αs est bornée

— ps(x) = 0 si Qx[αs ≥ x|1] = 0, et ps(x) est une constante sinon.

— la variable aléatoire βb prend la forme :

βb = b(x)− (1− ε)βb,− + εβb,+,

où βb,− ∈ [0, b(x)], βb,+ ∈ N∗ sont indépendantes de ε ∈ 0, 1 sous Qx, etβb,−, βb,+ sont bornés par b− et b+ respectivement

— Sous Qx, βc est une variable aléatoire suivant une loi géométrique de paramètrep (le même p pour tout x) et conditionnée sur l’ensemble S(x).

— Pour une constante c > 0, Lb(x) ≥ c pour tout x.

— Pour des constantes g > 0, h > 0, c′ > 0, Lg(b − d)(x) ≤ −c′ for all x tel queb(x)− d(x) ≥ h.

Alors, le processus bid b(Nn) tend vers l’infini ( la chaîne de Markov N est tran-siente), alors que le processus support (b− d)(Nn) revient vers de valeurs inférieursà h, après une certaine durée.

Proposition 0.0.3. Soit Θtra1∗ ⊂ Θtra1 l’ensemble des paramètres tel que ps(x),pb(x), pc(x) vérifient :−ps(x)bs + pb(x)bp − pc(x)bc = c1 > 0, pour x.

−ps(x)bs + pb(x)bp − pc(x)(bc + dc) = c2 < 0 pour x tel que b(x)− d(x) > h.

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Introduction

avec

— le terme bs correspond à la borne uniforme de Ex[b(x)− b(TS(x, αS))],

— le terme bp := b+ correspondant à la borne de la variable aléatoire βB+,

— le terme bc := e−1

− ln q correspond à une borne pour le changement de prix b(x)−bo(x) pour une annulation,

— le terme dc := Qx[αC ≥ ∆d(x)x | βC = d(x)] est une constante indépendante dex et correspond à la probabilité d’annuler la totalité du volume au niveau deprix d(x).

Dans l’ensemble Θtra1∗, le bid b(Nn) tend vers l’infini ( la chaîne de Markov Nest transiente), alors que le processus support (b − d)(Nn) revient vers de valeursinférieurs à h, après une certaine durée.

Lorsque (b− d)(Nn) le processus support est trop grand ( il existe une valeur htel que b(Nn)− d(Nn) ≥ h ), le drift doit être négatif pour faire revenir le processusvers des valeurs plus petites. Le terme dc permet de rendre le générateur négatif. Ceterme correspond à l’annulation totale du volume au niveau de prix d(Nn), ce qui apour but de réduire (b− d)(Nn).

Nous approfondissons la recherche de modèle plus réaliste en restreignant leprocessus volume H et le processus support b− d.

Theorem 0.0.4. Supposons les hypothèses du théorème 0.0.3. les hypothèses duthéorème 0.0.2, sauf la condition LH(x) ≥ θ qui est remplacé par : pour toutconstante c′ > 0, h′ > 0, LH(x) ≤ −c′ lorsque x|1 > h′. Alors, la chaîne de MarkovN est transiente et le processus support (b− d)(Nn) et le processus volume H(Nn)reviennent vers de valeurs inférieurs à h′, après une certaine durée.

Proposition 0.0.4. Nous supposons que ab = Ex[αb], Ex[αs] sont des constantespositives, Nous ne modifions pas les variables aléatoires αs, βb,−, βb,+, βc fixé dans??. Pour les probabilités ps(x), pb(x), nous les choisissons tel que :

0 < ps(x) < 1 a constant,0 < pb(x) ≤ apc

ab, pour x tel que x|1 > h′.

avec apc := pc(x)Ex[αc] que nous pouvons prendre constant. Alors, la chaîne deMarkov N est transiente et le processus support (b− d)(Nn) et le processus volumeN|1n reviennent vers de valeurs inférieurs à h′, après une certaine durée.

0.0.2 Estimations et calibrations du modèle

Cette partie consiste à choisir les bons paramètres pour que le modèle reflètele marché. En particulier, nous travaillons ici avec des données du marché, venant

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Introduction

du contrat du Future Bund négocié sur la bourse EUREX. Les données ont uneprécision de l’ordre de la milliseconde et ont été enregistrées entre mars 2013 etseptembre 2013.

Les modèles de carnet d’ordres qui étudient les problématiques d’estimation etde calibration, se ramènent à un problème d’estimation. En effet, ils sont modélisésà l’aide de processus de Poisson et ils estiment les intensités des processus de Poissonà l’aide des données et d’une méthode de maximum de vraisemblance. Il n’y a doncaucun degré supplémentaire dans leur modèle et il n’y a pas de calibration à faire.

Notre problème de calibration consiste à trouver les paramètres dans l’ensembledes probabilités Q dans Θrec. Pour réduire l’ensemble des lois d’arrivées, nous allonsnous placer dans le cas d’une chaine de Markov récurrente. Il s’agit d’un choixartificiel mais nous justifions ce choix par la littérature existante [24] et [2].

Les faits stylisés et les critères de liquidité donnent des restrictions aux modèlesstochastiques de carnet d’ordres. Les bons modèles devraient être capable de cap-turer simultanément la plupart d’entre eux avec peu de paramètres. En se plaçantdans Θrec, nous étudions les faits stylisés et les critères de liquidité en imposant deslois.

Réduction de l’ensemble Θmarket par la récurrence Nous étudions dans cettepartie différentes lois pour αC et βC . Soit ΘαC l’ensemble des probabilités tel queαC dépend de l’état courant, c’est à dire αC possède la forme :

αc = 11εc=1∆βcx + 11εc=0ξc

avec εc ∈ 0, 1, ξc ∈ 1, . . . , 1 ∨ (∆βcx − 1), indépendant de βc.Lorsque αC ne dépend pas de l’état du carnet ou lorsque αC suit une loi uniforme

et αC dépend de l’état du carnet, la condition de récurrence n’est pas vérifiée.Lorsque αC suit une loi binomiale et αC dépend de l’état du carnet, nous vérifions

le caractère récurrent de la chaîne. Les deux ensembles sont disjoints et sont séparéspar une frontière. La frontière est différente selon le choix de βC (Figure 2a, 4.7a et4.7b). Il y a une indifférence au choix de βC dans la recherche de paramètres rendantla chaîne récurrente. Cette étude nous a donné des restrictions sur le choix de la loiαC et de l’importance de la dépendance par rapport à l’état courant.

Ainsi nous réduisons notre recherche de l’ensemble des probabilités dansΘαC=bin =ΘαC ∩ αC ∼ une loi binomiale

Recherche de l’ensemble Θmarket par les critères de liquidité Soit ΘαS l’en-semble des probabilités tel que αS dépend de l’état courant, c’est à dire αS possède

xiii

Introduction

(a) βC suit une loi binomiale, αC une loi bi-nomiale et Q dans ΘαC

(b) βC suit une loi géométrique, αC une loibinomiale et Q dans ΘαC

(c) βC suit une loi lognormale, αC une loibinomiale et Q dans ΘαC

Figure 2 – Représentation des positions qui vérifient la condition de récurrence :les parties noires sont les positions non atteintes par les données du marché,la partieblanche correspond aux positions ne vérifiant pas la condition, la partie grise corres-pond aux positions vérifiant la condition. En abscisse, nous avons le volume totale(x|b(x)−p − x|b(x)−p

min)/120 et en ordonnée, le prix du bid b(x)− bmin

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Introduction

(a) Impact de prix avec Q dans ΘαS avec∆t = 100

(b) Impact de prix avec Q dans ΘαS avec∆t = 1000

(c) Profil moyen avec Q dans ΘαS

Figure 3 – Représentation graphique de critères de liquidité

la forme pour un certain niveau de prix l :

αS := αS(nb(n)−l)

Cela veut dire que le support de la loi de αS doit être fini et dépendre de nb(n)−l.Dans le cas de l = 0, la loi ne dépend que du volume disponible au niveau du bid.

Nous définissons le profil moyen comme la moyenne temporelle des volumes àchaque niveau de prix.

M(p) = limn→∞

n∑k=0

∆b(Xk)−pXk.

Nous prenons comme référence le prix du bid.Dans le cas où αS dépend de l’état du carnet (Figure 4.11b), la courbe empirique

s’accorde bien pour les valeurs proches du bid avec le profil moyen calculé par lesdonnées du marché.

Nous définissons l’impact de prix, dans notre cas :

Iind(∆t, q) := E[r(t,∆t)|δt = S , qt = q]

avec δt = S l’événement à l’instant t est un événement de marché arrivée etqt = q, la taille du marché est q.

Nous obtenons une bonne concordance entre la courbe simulée et la courbe des

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Introduction

données de marché. Cela peut s’expliquer notamment par la sélection de liquiditédû aux traders. Les traders choisissent d’envoyer des ordres de marché de grandetaille lorsque la liquidité est élevée. En faisant dépendre αS de l’état courant, nouscaractérisons ce phénomène qui caractérise la concavité de la courbe de l’impact deprix. Nous retrouvons bien la concavité dans le cas αS dépendant. Cette étude nousa montré l’importance de la dépendance par rapport à l’état courant de αS.

Ainsi nous avons réduit notre recherche de l’ensemble des probabilités dansΘαC=bin ∪ΘαS .

0.0.2.1 Problème de liquidation optimale dans le modèle de carnet d’ordres

Une caractéristique commune dans les modèles pour les problèmes de liquidationoptimale est l’existence de l’impact de prix. Le marché peut être vu comme un jeucompétitif entre les apporteurs de liquidité envoyant des ordres limites et les preneursde liquidité envoyant des ordres marché. Chaque participant cherche à acheter ouvendre de manière optimale en se basant sur le comportement global des autresparticipants. Les problèmes de liquidation optimale étudient la stratégie optimalepour les preneurs de liquidité.

Bertsimas et Lo [13] ont supposé que l’impact sur les prix était linéaire avec lataille des ordres marché. La stratégie optimale est d’exécuter à une vitesse constantela même quantité tout le long de l’horizon T. Almgren et Chriss [8] ont modélisédeux types d’impact sur les prix, un permanent et un instantané. Ils ont utilisé uncritère de variance moyenne pour l’optimalité, et ont montré un compromis entrel’exécution rapide et lente.

En tradant à un rythme lent, le trader fait face à un faible impact sur les prixmais à une forte volatilité due au mouvement de prix de l’actif. Cependant, si iltrade très rapidement, il risque d’avoir un impact de prix très élevé sur les prix maisest alors exposé à une faible volatilité des mouvements de prix.

Obizhaeva et Wang [53] ont mis en place un modèle qui génère un impact sur lesprix à travers le carnet d’ordres, en particulier le profile moyen du carnet d’ordres.L’impact des prix résulte de la forme et de la profondeur du carnet. L’impact deprix n’est ni permanent ni instantané, mais décroît avec le temps. Ils ont trouvé lastratégie optimale où la profondeur du carnet d’ordres est constante.

Alfonsi et al. [7] et Predoiu et al. [56] ont généralisé le résultat. Ils ont trouvé lastratégie optimale sous le même critère et permettent à la forme du carnet d’ordresd’adopter des formes plus générales. Toutes les stratégies optimales de ces modèlesont un comportement singulier : il y a un pic au début et un autre à la fin de l’horizond’exécution, mais entre le vitesse d’exécution est constante.

Récemment, dans un problème connexe, Abergel et al. [1] étudie la stratégieoptimale pour les apporteurs de liquidité dans son modèle définit dans [2]. Ils ca-

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Introduction

ractérisent le problème à l’aide de la théorie des processus de décision markovienne. Ils utilisent une approche numérique et abordent le problème de grande dimensionpar une méthode de randomisation sur le contrôle et une méthode de quantification.Lorsque les intensités sont indépendantes de l’état courant, la stratégie optimale etla stratégie naïve, consistant à envoyer des ordres limites aux meilleurs prix, sontéquivalentes. Cependant dans le cas où les intensités dépendent de l’état courant, lastratégie optimale fonctionne mieux que la stratégie naïve. Ils trouvent aussi qu’au-toriser de placer des ordres limites au niveau du second meilleur prix donne demeilleurs résultats.

Dans cette thèse, pour le problème de liquidation optimale, nous utilisons deuxapproches très différentes : dans la première approche de manière similaire au pro-blème d’Abergel et al. [1], nous allons étudier le problème de liquidation optimaledans un modèle stochastique de carnet d’ordres et dans une seconde approche, nousétendons le modèle de Predoiu et al. [56] en autorisant la dynamique du volumemanquant a être stochastique.

Description du problème de liquidation optimale Dans cette étude, nousconsidérons la modélisation de carnet d’ordres définit dans la partie 0.0.1. Cepen-dant, nous devons définir l’influence du trader stratégique sur ce modèle.Pour le trader stratégique, une stratégie de contrôle Q = (Qk)k∈0,...,T est unprocessus prévisible et non décroissant où, pour tout k ∈ 0, . . . , T, Qk est unevariable aléatoire à valeurs dans N et Q0 = 0.Nous notons ∆Qk := Qk+1 − Qk la quantité d’actifs vendus au temps k, alors queQk est le nombre d’actions vendues par le trader stratégique jusqu’à l’instant k−.Á l’instant T , le trader stratégique n’a pas le choix, il doit vendre la plus grandequantité possible d’actifs pour obtenir le plus petit inventaire possible.

Lorsque le trader stratégique connaît l’état n, il peut interagir instantanément avecles ordres de vente. La dynamique contrôlée est définit par l’équation suivante :

N|ik+1 = N|ik+ − 11δk=S(αSk ∧ N|ik+) (5)

−11δk=C11βCk≤i(αCk ∧∆βC

kN|ik+) pour tout (i, k) ∈ N∗ × 0, . . . , T − 1

+11δk=B11βBk≤iα

Bk ,

où nous avons défini le processus représentant le carnet d’ordres juste après que letrader stratégique vend :

N|ik+ = (N|ik −∆Qk)+

Nous considérons également que le trader stratégique a l’obligation de respecter la

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Introduction

contrainte de liquidation. Le nombre d’actions vendues par le trader stratégiquedoit être le plus proche possible de X à l’horizon T . Cependant, la liquidité peutbaisser et le trader stratégique peut ne pas respecter la contrainte de liquidation.Pour k ∈ 1, T, nous imposons qu’une stratégie admissible Q devrait satisfaire :

Q0 = 00 ≤ ∆Qk ≤ N1

k−1

∆QT = N|1T ∧ (X −QT )( Contrainte de liquidation)

Les ordres de vente du trader stratégique induisent le même mécanisme que lespetits investisseurs qui vendent des ordres. Si, à l’instant k, l’état du carnet estNk = n ∈ N , et que le trader stratégique décide de vendre ck alors le nouvel étatinduit par le trader stratégique est TS(n, ck). Nous pouvons ensuite définir les pro-cessus d’état et leur dynamique.

Nous introduisons d’abord l’espace d’état :

Y := 0, . . . , T × N × 0, . . . , X

Nous définissons ensuite le processus d’état comme suit : Y := (τ,N, Q) où τ repré-sente l’instant et est tel que τk = k pour tout k ∈ 0, . . . , T, N est le processusassocié à la stratégie Q et sa dynamique est donnée par l’équation (5.1).

N|ik+1 = N|ik+ − 11δk=S(αSk ∧ N|ik+) (6)


kN|ik+) pour tout (i, k) ∈ N∗ × 0, ..., T − 1


Bk ,

Où nous avons défini N|ik+ = (N|ik − ∆Qk)+. Q est un processus représentant laquantité d’actifs vendus.Pour (k, n) ∈ 0, . . . , T fois Nr, nous notons Nk,n,Q la solution de l’équation (5.1)tel que Nk,n,Q

k = n.Pour (k, n, x) ∈ Y , nous sommes maintenant en mesure de définir l’ensemble desstratégies admissibles par :

A(k, n, x) :=Q = (Qj)j∈k,...,T+1 : Q est un F− processus prévisible et croissant s.t.

Qk = x, ∀j ∈ k, . . . , T − 1, 0 ≤ ∆Qj ≤ (Nk,n,Q)|1jet ∆QT = (Nk,n,Q)|1T ∧ (X −QT )

Pour (k, n, x) ∈ 0, . . . , T × N × [0, . . . , X], A(k, n, x) est fini.Soit k ∈ 0, . . . , T et n ∈ N l’état du carnet à l’instant k. Lorsque le trader

stratégique envoie un ordre de vente de taille q ≥ 0 à l’instant k, le trader stratégique

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Introduction

vend q ∧ n|b(n) au prix b(n).Si q < b(n), le gain est qb(n). Le trader stratégique n’a pas d’impact instantanée surle prix de l’offre dans ce cas.Si b(n)− bS(n, q) = j > 0, le trader stratégique vend au prix b(n)− p avec p allantde 0 à j . Pour chaque p allant de 0 à j − 1, il vend ∆b(n)−p mn et pour p = z, ilvend q − n|b(n)−z.

Par conséquent, nous sommes maintenant capables de définir la fonction de gain g

pour un ordre de vente de taille q pour un état du carnet n par :

g(n, q) :=∞∑i=1

i(q ∧ n|i − n|i+1)+

Notons que

g(n, q) :=∞∑i=1

i(n|i − n|i+1)−∞∑i=1

i(TS( mn, q)|i − TS(n, q)|i+1)

Une intégration par parties donne ∑∞i=1 i(n|i − n|i+1) = ∑∞i=1 n|i.

Par conséquent, si nous définissons s(n) := ∑∞k=0 n|i, nous pouvons réécrire g de la

façon suivante :g(n, q) = s(n)− s(TS(n, q))

Nous remarquons que g est évidemment lié à l’état du carnet d’ordres

Comme dans [56], le coût d’un vente dépend du carnet, c’est-à-dire de la diffé-rence entre le volume cumulé sur l’état du carnet avant la vente et après la vente.Cependant, dans [56], il y a une hypothèse de martingale sur le prix du bid et defortes hypothèse sont faites sur la forme du carnet d’ordres par rapport au prix dubid. Dans notre cas, le prix de référence de notre modèle de carnet d’ordres est 0.

Le trader stratégique a pour but de maximiser la valeur attendue du gain totalobtenu à l’instant final T. Nous considérons la fonction valeur suivante v défini surY par,

v(k, n, x) := supQ∈A(k,n,x)

JQ(k, n, x),

avec

JQ(k, n, x) := Ek,n,q[ T−1∑j=k

g(Nk,n,Qj , ∆Qs) + g

(Nk,n,QT , ∆QT

) ]

Rappelons que ∆QT = (X −QT ) ∧(Nk,n,QT

)|1.

Comme A(k, n, x) est fini, le supremum est atteint et il existe une stratégie optimale.Nous avons donc,

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Introduction

v(k, n, x) := maxQ∈A(k,n,x)

JQ(k, n, x)

Nous avons les conditions aux limites suivantes :

v(T, n, x) = g(n, x ∧ n|1T ) et v(k, n, X) = 0

Nous pouvons établir le principe de programmation dynamique,

Theorem 0.0.5 (Le principe de programmation dynamique).Soit (k, n, x) ∈ Y. Pour tout temps d’arrêt τ prenant ses valeurs dans k+1, . . . , T,nous avons :

v(k, n, x) = maxQ∈A(k,n,x)

Et,n,x[τ−1∑j=k

g(Nk,n,Qj , ∆Qj) + v(τ,Nk,n,Q

τ , Qτ )]

Résultats théoriques Nous définissons pour n ∈ N et c ≥ 0, la fonction bS parbS(n, c) := b(TS(n, c))

Definition 0.0.1. Nous définissons Ab l’ensemble des tailles des ordres marchévente qui conduit au même prix du bid, c’est à dire :

For b ∈ [0, b(n)],Ab(n, x) := c ∈ [0, x], bS(n, c) = b

En utilisant le principe de programmation dynamique pour k = K − 1, nousobtenons : pour n ∈ N , pour x ∈ N

v(K − 1, n, x) = maxq∈[0,...,x∧n|1]

g(n, q) + E[g(Nk+1, x− q)|Nk = TS(n, q)]

et nous définissons : pour n ∈ N , pour x ∈ N, pour q ∈ 0, . . . , x

J(K − 1, n, x, q) := g(n, q) + E[g(Nk+1, x− q)|Nk = TS(n, q)]

Le résultat principal de cette partie est le théorème suivant :

Theorem 0.0.6. Pour tout n ∈ N , pour tout q ∈ Ab avec bS(n, q), la fonctionJ(K − 1, n, x, .) est croissante en q. Alors, pour tout b ∈ [0, b(n)], pour tout q ∈ Ab,

arg maxq∈Ab

J(K − 1, n, x, q) = nB − 1

Concrètement, nous montrons que chaque terme est croissant en q. Par exemple,pour le terme lié à la vente, nous avons le lemme suivant :

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Introduction

Lemma 0.0.2. Pour tout n ∈ N , pour tout q ∈ 0, . . . , x, nous avons

JS(n, x, q) := psE[g(TS(TS(n, q), αS), x− q)− g(TS(n, q), x− q)] = A(n, x)−A(n, q)

avec A(n, q) := ∑i≤bS(n,q)

[HαS(n|i − 1− q) + FαS(n|i − 1− q) + E[αS]

]De plus, pour tout b ∈ [0, b(n)], pour tout q ∈ Ab, JS(n, x, q) est croissant en q.

Pour tout b ∈ [0, b(n)], pour tout q ∈ Ab, A(n, q) est décroissant pour q. Nouscalculons de la même manière JB et JC .

Le résultat nous montre que l’investisseur doit toujours vendre jusqu’à un certainniveau de prix B, la taille nB − 1 de telle manière à éviter le coût supplémentairelié à un changement de prix. La stratégie repose finalement sur la détermination duniveau de prix B.

0.0.2.2 Liquidation optimale dans un carnet d’ordres à un seul côté avecune résilience sur le volume

Description du problème de liquidation optimale Un agent financier veutvendre X lots d’un actif peu liquide sur l’intervalle [0, T ] ⊂ R. Soit (At)t≥0 le prixde référence de l’actif que nous considérons comme une P-martingale continu. Dansnotre modèle, en l’absence de trading, le volume disponible à l’instant t sur l’inter-valle de prix [At, At +x) est F (x). F est une fonction croissante et continu à gaucheassocié à la mesure infinie µ sur [0,+∞) de la manière suivante :

F (x) := µ([0, x)), pour tout x ≥ 0. (7)

Nous donnons 2 exemples de forme de carnet d’ordres sur les figures 5 et 4.

F

µ

a b x

y

Figure 4 – Représentation d’un carnet d’ordres à volume constant possédant ungap entre entre les niveaux de prix a et b. En abscisse, nous représentons le prix eten ordonnée, le volume.

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Introduction

F

µ

x

y

Figure 5 – Représentation d’un carnet d’ordres discret. En abscisse, nous repré-sentons le prix et en ordonnée, le volume.

At

Yt

At + ψ(Yt) Prix

Volume

Figure 6 – Représentation de volume manquant (région blanche) et du volumedisponible (région grise). La forme du carnet a pour référence At.

Les stratégies de l’agent. Les stratégies de l’agent sont données par des processuscroissant continu à gauche et adapté (Xt)0≤t≤T avec XT = X. Nous supposons queX0− = 0 et nous notons ∆Xt = Xt −Xt− le saut à l’instant t.

La dynamique du processus volume manquant (Yt). Nous supposons que lastratégie de l’investisseur a un impact sur le prix. Lorsque l’agent suit une stratégieX, nous supposons qu’à l’instant t, le prix de l’ask n’est plus le prix de référencemais est donnée par At + Dt où Dt := ψ(Yt) (voir figure 6, avec Yt représentantla dynamique du processus volume manquant définit par l’équation différentielle

xxii

Introduction

stochastique à sauts suivantes :

dYt = dXt − h(Yt−)dt+ σ(Yt−)dWt +∫RYt−q(Yt− , z)M(dt, dz);Y0− = y. (8)

la fonction réciproque de F continue à gauche ψ donnée par

ψ(y) := supa ≥ 0|F (a) < y, pour y > 0 et ψ(0) := 0. (9)

Coût de la stratégie. Nous définissons le coût de la stratégie X = (Xt)0≤t≤T

comme

C(X) :=∫ T

0(At + Dt−)dXc

t +∑

0≤t≤T[At∆Xt + (Φ(Yt)− Φ(Yt−))],

=∫ T

0ψ(Yt−)dXc

t +∑

0≤t≤T(Φ(Yt)− Φ(Yt−)) +

∫ T

0AtdXt.

Nous définissons la fonction valeur comme

v(t, x, y) := infX∈A(t,x)

E

∫ T

tψ(Y t,y,X

s− )dXcs +

∑t≤s≤T

(Φ(Y t,y,Xs )− Φ(Y t,y,X

s− )) , (10)

où Y t,y,Xs pour t ≤ s ≤ T dénote la solution de (6.4) avec Y t,y,X

t− = y et l’ensembledes contrôles admissibles A(t, x) est donnée par :

A(t, x) , X : X ; Xt− = x; XT = X. (11)

La fonction valeur à la date terminale T est donnée par :

v(T, x, y) = Φ(y +X − x)− Φ(y). (12)

et la fonction valeur satisfait au bord

v(t,X, y) = 0. (13)

Résultats théoriques

Theorem 0.0.7. La fonction valeur v est l’unique solution de viscosité continue surS de l’inégalité variationnelle :

max(−∂v∂t− Lv, −∂v

∂x− ∂v

∂y− ψ

)= 0, (14)

xxiii

Introduction

satisfaisant les conditions de croissance suivantes :

0 ≤ v(t, x, y) ≤ Φ(y +X − x)− Φ(y) on [0, T )× [0, X]× [0,+∞), (15)

et les conditions aux bords : v(t,X, y) = 0 and v(T, x, y) = Φ(y+X−x)−Φ(y),

Résultats numériques Nous présentons quelques résultats numériques obtenusde (6.25). Nous avons implémenté une méthode de différences finies pour un pro-blème de contrôle singulier pour une EDS à sauts basée sur [42].

Dans le cas du carnet d’ordre discret, la forme peut être déterminée par la fonc-tion suivante :

Φ(y) =∞∑k=0

k(y − 12k −

12)1l(k,k+1] = byc(y − 1

2byc −12)

(a) t=0 (b) t=250 (c) t=450

Figure 7 – Il s’agit de l’évolution de la stratégie à adopter à différentes périodessi le carnet d’ordres est discret. En abscisse, nous représentons le nombre de lots àliquider. En ordonnée, nous représentons le nombre de volume manquant. Dans larégion bleue, l’agent doit attendre. Dans la région jaune, l’agent doit acheter jusqu’àatteindre la région bleue. Le déplacement se fait suivant la direction x = y

Pour un carnet d’ordres discret, il y a des décalages de longueur de 1 ou plusieurs

xxiv

Introduction

ticks entre deux prix consécutifs. Donc, dès que l’agent a consumé tout le volume àun certain prix, il y a un saut dans le coût d’achat de nouveaux lots. L’agent préfèreattendre l’arrivée de nouveaux ordres limites vente pour remplir le décalage. Plusl’agent est proche de la maturité, moins elle doit attendre.

Organisation du document

Le manuscrit est organisé de la manière suivante :

— Le chapitre 1 vise à donner les concepts des marchés financiers auxquels nousfaisons référence tout le long du mémoire. Nous présentons les différents faitsstylisées et les critères de liquidité du marché. Puis nous donnons un pano-rama des modèles de carnet d’ordre en précisant les mérites et les manques.Nous finissons par une présentation des problèmes de liquidation optimale. Cechapitre est rédigée en anglais.

— Dans l’amorce du chapitre 2, nous motivons l’intérêt de considérer un modèlede représentation par profondeur. Nous faisons face à la question représenta-tion et unilatérale du carnet d’ordres. Nous sommes confrontés à la questionde la modélisation du carnet d’ordres qui découle du problème initial de laliquidation optimale sous un point de vue à haute fréquence. Pour cela, nousconsidérons l’espace de la représentation et unilatérale du carnet d’ordres N etnous décrivons les opérateurs de transition induits par les événements. Ensuite,nous construisons la chaîne de Markov sur l’espace N à travers les opérateursde transition après avoir spécifié le caractère aléatoire des événements. Nousspécifions et étudions l’influence des différents lois et des dépendances de l’étatcourant sur le prix et le volume du carnet. Nous simulons notre modèle. Cechapitre est rédigée en anglais.

— Dans le chapitre 3, nous étudions l’irréductibilité, la récurrence et la transiencede la chaîne de Markov que nous avons présenté au chapitre 2, à travers unoutil mathématique original en utilisant une approche semi-martingale. Cechapitre est rédigée en anglais.

— Le chapitre 4 est consacré au modèle d’estimation et la calibration à travers lesdonnées de marché. Nous considérons différentes arrivées d’événements dépen-dant de l’état et étudions leurs influences sur les faits stylisés et l’impact sur lemarché. Nous présentons une calibration concrète et donnons une simulationpour différentes classes de modèles de stabilité du chapitre 3. Ce chapitre estrédigée en français.

— Après avoir étudié et corrigé le modèle, le chapitre 5 est consacré au problèmede la liquidation optimale dans une modélisation LOB à haute fréquence quenous introduisons au chapitre 2. Ce chapitre est rédigée en anglais.

xxv

Introduction

— Le chapitre 6 s’attaque à un problème de liquidation optimal en basse fré-quence. Nous soulignons que ce chapitre est indépendant des autres. Nousintroduisons le processus d’impact du volume manquant et nous définissonsle problème optimal associé. Nous sommes amenés à étudier une équation deHamilton Jacobi Bellman. Nous prouvons l’existence et l’unicité de la solutionde viscosité. Enfin, nous proposons une approche numérique de la solutionbasée sur un schéma de différences finies et donnons une interprétation de lastratégie optimale. Ce chapitre est rédigée en anglais.

xxvi

Chapitre 1

State of the art

Contents1.1 Financial Market . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Organized markets or over-the-counter markets . . . . . . 21.1.2 Mechanism of limit order book . . . . . . . . . . . . . . . 2

1.2 Empirical observations in limit order book . . . . . . . . 51.2.1 Stylized facts . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1.1 Stylized facts on price increments . . . . . . . . 61.2.1.2 Stylized facts on order flow . . . . . . . . . . . . 8

1.2.2 Market liquidity criteria . . . . . . . . . . . . . . . . . . . 91.3 Challenges of modeling the dynamic of limit order book 12

1.3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Classical limit order book models . . . . . . . . . . . . . . 14

1.4 Optimal liquidation . . . . . . . . . . . . . . . . . . . . . . 181.4.1 Price impact model . . . . . . . . . . . . . . . . . . . . . . 191.4.2 Volume impact model with limit order book shape . . . . 201.4.3 Optimal execution with Market and Limit orders . . . . . 201.4.4 Optimal execution in limit order book . . . . . . . . . . . 21

This chapter aims at introducing financial market concepts and mathematicalbasic tools that will be applied along my thesis. One of our subject is the opti-mization of a financial asset trading program from a given position to zero in agiven time which can be called optimal liquidation problem. Thereby, the questionof financial asset dynamic modeling is primary and tackles at the same time theliquidity modeling. The liquidity can be study through different types of model :price model, volume model and limit order book model. Since we want to mimicas much as possible the financial reality, we validate the models by comparing thedistribution of limit order book characteristics from the chosen model and the data.

Firstly, we give the financial context by introducing the organized markets andover the counter markets (OTC) by emphasizing their differences. Then, we explainthe mechanism of a limit order book. We furnish financial interpretations on stylizedfacts and market liquidity criteria which involve the distribution of limit order bookcharacteristics. We discuss the main existing limit order book modeling. At the endof this chapter, we present several classes of optimal liquidation problems.

1

Partie , Chapitre 1 – State of the art

1.1 Financial Market

1.1.1 Organized markets or over-the-counter markets

Financial markets are complex organizations with their own economic and insti-tutional structures that play a critical role in determining how prices are determined.There are two basic ways to find and trade financial assets through a so-called over-the-counter (OTC) market or through an organized markets known as exchange.A market will refer to a set of financial assets and trading rules.

In OTC market, financial assets are non standardized. Most securities and deri-vatives involved in the financial crisis that began with a 2007 breakdown in the U.S.mortgage market were traded in OTC markets.

Exchanges, whether stock markets or derivatives exchanges, started as physicalplaces where trading took place. Some of the best known include the Chicago Boardof Trade (now part of the CME Group), which has been trading future contractssince 1851 or EUREX, which is the largest European futures and options market.An exchange centralizes bid and offer prices from market participants. Market par-ticipants can sell or buy at one of the quotes or reply with a different quote. Whentwo parties find an agreement, the price at which the transaction is executed, iscommunicated throughout the market. The result is a level playing field that allowsany market participant to buy as low or sell as high. The advances of electronic tra-ding has eliminated the need for exchanges to be physical places. The London StockExchange and the NASDAQ Stock Market and EUREX are completely electronic.Some derivatives exchanges such as the CME Group, maintain both old-style pitsand electronic trading.

OTC markets are less transparent and operate with fewer participants than ex-changes. We only consider the organized markets for the rest of the thesis. Moreover,traders refer to direct market participants.

1.1.2 Mechanism of limit order book

According to their future believes of market price and their goals, traders canbuy or sell financial assets through different orders.

For each financial asset, there is a price grid on which traders can place theirorders. The smallest step on the grid between two consecutive price levels is thetick value. All price levels are multiple of the tick value.

A limit order is an order characterized by the order sign (buy or sell), theorder size and the price level. A patient trader may have a belief on the futureprice of the assets and wants to buy a quantity at a specific price level. Thus, hewill send a buy limit order with specific size at specific price level.

2

1.1. Financial Market

askpricebidprice

midprice

spread

Figure 1.1 – Representation of a state of the limit order book. On the x-axis, wehave the price levels and the volume is the second coordinate.

The limit order book is the set of all available limit orders for a particularfinancial asset. The limit order book contains each buy or sell limit orders for allprice levels. For each price levels, there is a limit orders queue. For a specific pricelevel, the limit order queue is the set of all available limit orders and the volumeis the quantity of all available limit orders. The total volume is the quantity of allavailable limit orders for any price level. When a new limit order arrives at specificprice level, the order is appended at the end of the limit order queue. The limit orderbook keeps the sending timestamp of any limit order received by the exchange.

The bid price is the highest price available for buy limit orders. The ask priceis the lowest price available for sell limit orders. The bid price is always strictlysmaller than the ask price. The midprice is the average between the bid price andthe ask price. The spread is the difference between the ask price and the bid price.The difference is at least one tick, the smallest value between two consecutive prices.If the spread is strictly superior at one tick, trader can send buy (resp. sell) limitorder inside the spread, it will increase the bid price (resp. decrease the ask price).

Depending on the market and trader subscription, the exchange will typicallypublish the volume for each p price levels (typically p = 1, 5, 10, 20, . . .) deeper thanthe bid price or ask price.

A cancel order is an order characterized by the order sign (buy or sell), theorder size and the price level. A trader can cancel his own existing limit order inthe limit order book with a cancel order. Therefore, it will modify the limit orderqueue at this specific price level. Trader can modify their own existing limit orderby increasing or decreasing the limit order size or modifying the price level. It isthe same as considering simultaneous new limit order and new cancel order. Whentrader cancels their own limit order and the limit order is alone in the bid pricequeue (resp. the ask price queue), the bid price (resp. the ask price) decrease (resp.increase).

3


A market order is an order characterized by the order sign (buy or sell) andthe order size. An impatient trader can have a strong belief of the future price ofthe particular financial asset and he wants to sell a quantity without specific pricelevel. Thus, he will send a sell market order with the specific size.

A trade occurs when a market order is sent. A trade is characterized by theorder size and the last trade price. The last trade price is the bid price (resp. askprice) if the order sign is sell (resp. buy) for the market order. The order size for sell(resp. buy) market order will be executed against the volume in the bid price (resp.ask price). If it is not enough, the remaining sell market order size will be executedagainst the volume in the following price levels until all sell (resp. buy) market ordersize is filled. It will decrease the bid price (resp. ask price). The last trade price isoften the graphical representation of the "price". In the literature, "price" could referto the bid price, the ask price, the last trade price or the midprice.

An event is a limit order sending, market order sending or cancel order sending.There are two different sources of information of the state of the market :

— All of trades (market order sending) are notified with a delay depending onmultiple factors. A last trade price, a size and a timestamp are given.

— There is an update limit order book snapshot. Between, there could have beenmore than one event depending on multiple factors.

Multiple factors could be software location between the exchange, brokers, highactivity. There can be a delay between the trade notification and the limit orderbook modification by the trade notification.

Limit order book mechanism explain at the lowest level the price formation.

Order Matching Algorithm

The pro-rata order matching algorithm fills orders according to price level, ordersize and time. A market order size is multiplied by each resting order’s pro-ratedpercentage to calculate allocated trade size. All fills are rounded down to the nearestinteger ; if an allocated trade size is less than two, it is rounded down to zero. Anorder’s pro-rata percentage is calculated by taking order size divided by volumeat a certain price level. Excess size, which occur as a result of the rounding downof the original allocated trade size, may be allocated with First-in, First-out ordermatching algorithm.

The First-in, First-out (FIFO) algorithm uses price and time as the only criteriafor executing an order. The limit orders available in the ask price (or in the bid price)have execution priorities. At a specific price level, the limit orders with the oldestsending timestamp have execution properties. FIFO is often used (like EUREX).

We will consider FIFO order matching algorithm for the rest of the thesis.

4

1.2. Empirical observations in limit order book

1.2 Empirical observations in limit order book

In this section, we will discuss about statistical observations of the characte-ristics of a limit order book such as the price or the volume. Since one of ourmotivation is the limit order book modeling, we need to confront the distribution ofcharacteristics from financial data and the the distribution of characteristics fromsimulated model in order to validate the model.

Since we don’t have the theoretical quantities such as the distribution of priceincrements, we study the empirical quantities. In order to ensure that we can usedifferent periods in the financial data, we need to check the stationary of the timeseries. A time series (Xn) has stationary property if the joint distribution of anysubset of variables X1, X2, . . . , Xn is not affected by a shift in time. Moreover, ifwe want the convergence of the empirical quantity, we need to check the ergodicproperty of the time series. Given f , a measurable mapping of Ω into R, define thesample averages

〈f〉n = 1n

n∑i=1

f(Xi)

A time series will be said to have the ergodic property with respect to a measurablemapping f if the sample average 〈f〉n converges almost everywhere as n → ∞(Chapter 6 of [36]).

Many statistics exhibited power law tail [35] such as the distribution of priceincrements (Maslov [49], Cont [22]), the distribution of market order size (Maslov[49]) or the distribution of price level placement (Bouchaud [15], Gu [37]). A randomvariable X with fX probability distribution have a power-law tail with exponent αif fX(x) ∼ x−α as x→∞.

1.2.1 Stylized facts

Stylized fact is a term well known to physicists in the field of econophysics, theapplication of methods of physics to problems in economics and finance.

It was in 1961, the economist Nicholas Kaldor originally introduced the term“stylized facts” about theories of economic growth. He said that a theory should be-gin from a summary of the relevant and important facts which requiring explanation.It means the facts first and the theory after. Hence, Kaldor said that theorists shouldwork from “a stylized view of the facts”. They should “concentrate on broad tenden-cies, ignoring individual detail”. Broad tendencies are facts deserving attention, andestablishing such facts in finance and economics has been a primary achievementof econophysics. Many such facts have been established with appreciable precisiononly in the past 5− 10 years.

The first study of stylized fact dealt with the distribution of price increment were

5


done by Benoit Mandelbrot [48], in 1963, with only a few thousand data points byproposing to extend Gaussian law by a family of stable laws for the last trade priceincrements. Last trade price data where the first to be available, stylized facts onprice increments are more established than stylized facts on events information.

These stylized facts give restrictions on stochastic models who attempt to repro-duce the financial reality. Good models should be to capture simultaneously mostof them with few parameters. We will give the most studies stylized facts based onCont [22] and Gould [35].

1.2.1.1 Stylized facts on price increments

Without any precision about the price, the price will refer to last trade price,bid price, ask price or mid price. The price increment is the difference between theprice P evaluated at time t and at time t + ∆t, i.e. r(t,∆t) := Pt+∆t − Pt and ∆tbe a time window such as 100 milliseconds, 1 minute or 2 days.

Heavy tails of distribution of price increments The stylized fact is that dis-tribution of price increments are not normally distributed. The distribution of priceincrements has a narrower central part and fatter tails than the normal distribu-tion. This empirical regularity was noted already by Mandelbrot [48]. Since then, awhole range of different distributions have been suggested but there is no consensuson the exact form of the tails according to Chakraborti et al. [18]. Guillaume etal. [39] and Gopikrishnan et al. [34] used more than 200 million data points span-ning half a century to establish a strong case for an inverse cubic-power law tailof stock-market distribution of price increments, at least asymptotically for largevalues corresponding to big market movements.

At different time window ∆t, the unconditional distribution of price incrementsr(t,∆t) exhibit a power law tail with an exponent α > 2 that ensure a finite varianceaccording most of studies such as Maslov [49] and Cont [22].

It exhibits different shapes at different time windows ∆t and for ∆t big enoughit follow a Gaussian law called aggregational Gaussianity (Cont [22]).

By fixing the time window ∆t, we explain the behavior of the distribution ofprice increments. Then, we want to study the time dependency of price increments.Thereby we define a long-memory process X if in the limit τ →∞, X(τ) ∼ τ−γL(τ)where 0 < γ < 1 and L(τ) is a slowly varying function at infinity such as lim

x→∞L(tx)L(x) =

1, for all t ∈ R+. The degree of long-memory is given by the exponent γ, the smallerγ, the longer the memory. Long-memory is often discussed with the Hurst exponentH, H = 1− γ/2. Short-memory processes have H = 1/2.

6


Absence of significant value in autocorrelation of price increments Wedefine the autocorrelation of price increments,

C(s, τ,∆t) := corr (r(s+ τ,∆t), r(s,∆t)) .

The first and perhaps most obvious stylized fact of financial time series is that priceincrements are not significantly autocorrelated. Assuming that price increments weresignificantly autocorrelated, one could make correct predictions of price incrementsat least on average, which could be used by traders to make profitable investmentdecisions. When autocorrelations are exploited, market prices should equilibrate insuch a way that the autocorrelations disappear.

This stylized fact was discovered by Fama [28] and often called “efficient markethypothesis”. The efficient market hypothesis states that financial markets makesefficient use of all available information, thereby instantly incorporating any “hiddenvalue” of an asset into its price. If the efficient market hypothesis is strictly true,there is no chance of making money : the future price increments of exchange-tradedfinancial assets are always unpredictable.

Cont [22] claim that the autocorrelation of last trade price increments have smallnegative values, which it is due to the fact that the last trade price bounces betweenthe bid-ask spread over a certain period of time [17].

The absence of autocorrelation does not seem to hold systematically when thetime scale ∆t is increased : weekly and monthly returns do exhibit some autocor-relation. However given that the sizes of data sets are inversely proportional to ∆t,the statistical evidence is less conclusive and more variable from sample to sample.

Volatility clustering The absence of autocorrelation of price increments give apartial information of the behavior of the time dependency of price increments. Itgives the absence of linear dependency. We should at least look at the non lineardependency by studying autocorrelation of square price increments absolute priceincrements. We define the autocorrelation of square price increments

C2(s, τ,∆t) := corr(r2(s+ τ,∆t), r2(s,∆t)

)or absolute price increments

C0(s, τ,∆t) := corr (|r(s+ τ,∆t)|, |r(s,∆t)|) .

It is important because it says that price increments are not independently dis-tributed. In 1963, Mandelbrot [48] wrote that : “large price changes are not isolatedbetween periods of slow change” but “large changes tend to be followed by largechanges of either sign and small changes tend to be followed by small changes”. In

7


other words, the autocorrelation of absolute or squared returns is positive.This statistical fact has since been verified in many financial markets, and it has

been quantitatively refined. According to Cont [22], several authors have remarkedthat the autocorrelation in squared returns decays like a power law, with a coefficientβ ∈ [0.2, 0.4]. The quantitative results may be dependent on the time step ∆t.

This is sometimes referred to as a long memory effect or volatility clustering.There are several possible explanations for volatility clustering, including the arri-val of external news and the strategic splitting of orders by traders. The strategicsplitting of orders can be explained by optimal liquidation problem (see section 1.4).

In a nutshell, the price increment of a time series should have a power law taildistribution, should not have autocorrelation and are not independently distributed.

1.2.1.2 Stylized facts on order flow

Long memory on order flow We discuss the statistical properties of order flowby considering the time series of order signs. Specifically, we consider the symbolictime series (εt) obtained in event time by replacing buy orders with +1 and sellorders with −1, irrespective of the order size. We can do it for market orders, limitorders or cancel orders, all of which show very similar behavior.

We reduce these series to ±1 rather than analyzing the signed series of ordersizes directly in order to avoid problems created by the large fluctuations in ordersize. Fluctuations in order size are heavy tailed and have long-memory themselves,so statistical averages based on them converge only slowly. The essential behavioris captured by the series of order signs.

We denote the autocorrelations of order signs

Cε(s, τ,∆t) := corr (ε(s+ τ,∆t), ε(s,∆t))

This is sometimes interpreted as a sign of long-range dependence. In Lillo [47],the autocorrelation function decays roughly as a power law with an exponent of 0.6,corresponding to a Hurst exponentH = 0.7 observed in the London Stock Exchange.This implies that the signs of future orders are quite predictable from the signs ofpast orders. Bouchaud [16] measured in the Paris Stock Exchange, a larger intervalof γ ∈ [0.2, 0.7], H ∈ (0.65, 0.9).

Long memory is also observed if the signs of all orders, including both limit andmarket orders, are taken together. In contrast, if we assign a negative sign for a buycancel order, corresponding to the fact that price movements can be downward, thenthe combined sequence of signs for market orders, limit orders, and cancel order doesnot show long-memory.

8


1.2.2 Market liquidity criteria

Liquidity has many different aspects. Kyle [44] gives three criteria : tightness,depth, resilience. We focus on the distribution of the volume at the best price,price impact, resilience and depth represented bymean average depth profile.

Volume at the best price If we denote the volume available at the bid pricev at t and the distribution of v, fv, Bouchaud [15] found that fv follow a Gammadistribution with γ = 0.7.

Price impact The notion of price impact is something that all traders ask whenthey want to know the effect of the execution of a large sell or buy orders. There is noclear consensus on the definition of price impact. Bouchaud [14] gives a good reviewon the question of the price impact. We follow the explanation between individualand aggregate price impact. We give the notion of transient and permanent priceimpact.

Individual price impact Individual price impact can be understand as theexpectation price change after ∆t induced by a single transaction q. We can definethe price impact Iind as

Iind(∆t, q) := E[r(t,∆t)|δt = M , qt = q]

with δt = M the event at time t which is a market event arrival and qt = q, themarket order size is q.

If we assume that we can decompose the individual price impact as Iind(∆t, q) :=L(∆t)S(q), we can study the influence of ∆t and q separately.Instantaneous price impact S is the instantaneous price impact, it means theprice change just after the transaction q. By fixing ∆t, many studies found that Sis concave in q. Lillo [47] found a power law with exponent 0.5 for small size and0.2 for large size. For financial asset in London Stock Exchange, Farmer [29] foundthat a good fitting independently of q was an exponent 0.3. Weber [61] said thatthe concave shape of the function is very surprising : concave price impact wouldtheoretically be an incentive to make large trades as they would be less costly thanmany small ones. In contrast, a convex price impact would encourage a trader tosplit large trade into several smaller ones, which is what actually happens. Theconcave observation can be explain at least by the selective liquidity taking [31].Selective liquidity taking means that agents condition the size of their transactionson available volume in limit order book, making large transactions when availablevolume is high and small transactions when it is low. Concretely, we can write, for

9


∆t fixed,

R = E[r(t)|δt = M , qt = q] = P[r(t) > 0|δt = M , qt = q]E[r(t)]

At this point, we need to prove that P[r(t) > 0|δt = M , qt = q] is concave inq. In [14], they assume a correlation between the volume available at the bid pricev at t and the market order size q. The distribution of v is fv. If a trader wants tosell more that v, she will exactly sell v and if want less than v, she will sell q. Wecan rewrite the conditional probability with the distribution of v

P[r(t) > 0|δt = M , qt = q] =q∑i=0

fv(i)

Moreover, fv is decreasing in q (volume at the best price should follow a Gammadistribution) then P[r(t) > 0|δt = M , qt = q] is concave.

After talking about the influence of q on price impact, we talk about the influenceof ∆t.Lagged impact function : L is the lagged impact function. In order to explainthe lagged impact function, we use a simple model of the price impact by trade qkand a random shock which represent the price change due to limit and cancel order.

If E[r|εq] = εf(q), we define νk such as E[νk] = 0 and E[ν2k ] = σ2, a permanent

price impact model is defined as

Pn =∑k<n

εkf(qk, nk) +∑k<n

νk

Then :L(∆t) = E[εn(Pn+∆t − Pn)] = E[f ]

In the case of S(q) = Cq0.3 (Farmer et al. [29]), the individual price impact isIind(∆t, q) = E[f ]Cq0.3.

And a transient price impact model is defined as, with α ∈ (0, 1)

Pn =∑k<n

αn−k−1εkf(qk, nk) +∑k<n

νk

L(∆t) = α∆t−1E[f ]

In this case, Iind(∆t, q) = Cα∆t−1E[f ]q0.3

In [30], Farmer et al. claims that price impact is permanent but depends on thestate of the market but for Bouchaud et al., [16] price impact is state-independent,temporary and decays as a power law. Bouchaud et al. [14] answers that two modelsare equivalent in the case where the average impact function is history dependentby taking account the long memory of the order signs.

10


Aggregate price impact We define aggregate price impact. For a sequenceof N successive transactions beginning at time t, let QN = ∑N

i=1 εt+iqt+i be theaggregate volume and RN = ∑N

i=1 rt+i be the aggregate price increment. The averageprice impact conditioned on volume is

R(Q,N) := E[RN |QN = Q]

i.e. it is the expected price increment associated with a signed volume fluctuationQ. We write R(Q,N) to emphasize that this can depend both on the signed tradingvolume imbalance Q and the number of transactions N . Bouchaud et al. make thisstatements :

— the slope of the linear region decreases with N ,

— the shape and the scale of the aggregate price impact change with the aggre-gation scale,

— at short time scales, the function is significantly non-linear and

— at large aggregation scales the market impact becomes close to linear, and theslope of the impact decays with the aggregation scale.

Bouchaud et al. conclude that we can’t compare aggregate impact curves with dif-ferent scales.

Taking account on the question of arbitrage, Gatheral [33], the expectation oftrading costs should be positive in order to respect non dynamical arbitrage. Theymodel the price dynamics such as

Pt =∑i<t

S(qi)G(t− i) + noise

with the price impact such as a transient price impact model with S the instanta-neous price impact function and G the general decay kernel (G(t) = αt in Bouchaud[14]). Exponential decay kernel is only compatible with linear instantaneous priceimpact. In the case of S and G power law, they found a relation between the ex-ponent of market impact and the exponent of decay.

Cont et al. [23] study the price impact of all events i.e. limit orders, market ordersand cancel orders. They found a linear relation between price changes and bid-askvolume imbalance with a coefficient proportional to the inverse of the depth.

Resilience Kyle [44] propose a definition for resilience as “the speed with whichprices recover from a random, uninformative shock”. However, for Large [45] no shockis fully unexpected. He defines the resilience through the impact of aggressive marketorder (which change the best price) intensity of aggressive limit order (which changethe best price) arrival. Concretely, he defines the intensity of a point processes at

11


time t conditional on its natural filtration up to but not including time s ≤ t, Fs,as,

λ(t|Fs) = limδt→0

P[N(t+ δt) > N(t)|Fs]δt

,

where N(t) is the number of events to have occurred up to and including time t. Ifa shock of type m (aggressive market event) is time-invariant, denoted εm, happensat time s, then his impact on these intensities λr (limit aggressive event) at a latertime t can be defined as the function Grm

Grm(t− s) = λr(t|Fs, εm)− λr(t|Fs).

Grm measures the resilience. Large [45] propose a log-likelihood estimation methodfor the intensities.

The concept of resilience is widely use in optimal liquidation problem for priceimpact model such as [8], [53], [21], [4] and for the shadow order book model [5], [7],[6], [3]

Mean relative depth profile We will define the mean relative depth profile orthe average shape. We define the dt(i) the volume available at the level price bt− i.

The mean relative depth profile is the time averaging volume for each level pricebt − i :

d(i) = limK→∞

1K

K∑k=0

dk(i).

Through different studies on a wide range of markets, the mean relative depthprofile exhibit a hump shape. With “zero-intelligence” simple models, Maslov [50],Challet [19] and Bouchaud [15] reproduced this empirical observations. Ioane MuniToke [60] build an explicit formula for the shape of a continuous price order bookmodel. Even Weber [61] study the price impact through the mean relative depthprofile.

In some optimal liquidation problem [7], [3], [56], [32], they use the order bookshape in order to model the non linear price impact. In our knowledge, there are nostudies who link the order book shape and the mean relative depth profile.

1.3 Challenges of modeling the dynamic of limitorder book

What is a limit order book model ? We need to highlight the difference with aprice or volume model.

Under price model, financial assets are driven directly by a price process and theprice evolve without knowing the mechanism of order book directly. Black-Scholes

12

1.3. Challenges of modeling the dynamic of limit order book

model describe the price dynamic of a financial asset by a geometric Brownianmotion.

Under volume model such as Alfonsi et al. [7], Predoiu et al. [56], financial assetsare driven by a reference price process, a volume process and a fixed order bookshape. The reference price process can be modeled by a semi-martingale process.The volume process is the remaining volume related to the fixed order book shape.The order book shape links the best price evolution and the remaining volumeevolution.

Limit order book model describe the mechanism of limit order book and theprice dynamic is a consequence of event arrival and the mechanism of limit orderbook. In this case, the model is richer for example but more complicated.

The first difficulty is the complexity of the space state. In price model, there isone dimension which is the price scale as opposed to LOB model, there are the pricescale multiply by the volume scale.

Remark 1.3.1. More precisely, in order to explain completely the mechanism, weshould retain all limit orders with their size at each price.

The second difficulty is the complexity of the transition between the marketconfigurations induced by market orders and cancel orders which are mechanicallystate dependent. Since most financial assets with limit order book can always findcounterpart for market orders, it is weakly state dependent for the size of marketorder. Since, we can cancel orders which are attached to a limit order available in thelimit order book, we should retrieve the limit order associated. It leads to a modeldescribe in 1.3.1. In the case where all event orders size are constant and the samefor each event, we get around the problem. All previous models Maslov [49], Challet[19], Cont [24], Abergel [2] fall in this class of model. The third difficulty is relatedto the time structural of event properties such as the duration between 2 events.

1.3.1 Formalism

When we design a model with limit order book mechanism, we need to specify :How do we define the price level grid ? How are the main characteristics especiallythe best prices derived ? how do we model the event arrival randomness ?

The price level grid refers to the interval of the price and the reference price of thegrid. For example, in Bak’s model [9] the interval of the price is [0, n] ⊂ N and thereference price of the grid is 0. The main characteristics are the total volume and thebest prices. For example, in Cont et al. [24], they can derive the best prices with theactual state of the limit order book. As opposed to the previous one, Abergel et al.[2] derive the best prices with stochastic differential equations. The most importantpart is the event arrival randomness. We need to specify timestamp, event type, price

13


level and size randomness. In the case of the Poisson processes for each price level,as in Cont et al. [24], the duration between two consecutive events which determinetimestamp randomness, are modeled by exponential law. The price level randomnessis included in the fact that for each price level, there is a Poisson process. In thelimit order book modeling literature, size is constant.

After specifying the three questions, we need to validate the model by studyingthe simulated model with respect to the stylized facts and market liquidity criteria.This validation express that the characteristics of the limit order book evolve in thegood way with respect to the event arrival randomness. For example, we study theprice increment distribution or the average shape of the limit order book.

We will explain for each model how they design and how they check the modeleffectiveness.

1.3.2 Classical limit order book models

As our knowledge, Bak et al.[9] is the pioneer of limit order book modeling.

Bak (1997) They do a simple model in order to study the stylized facts. Thedimension is finite n <∞. The price position is relative to 0.

The number of sell (and buy) orders available remain constant to N . The totalvolume remains constant N can be a good approximation for a limit order book atequilibrium.

At each time t,

— Step 1 : We choose uniformly one order in 1, . . . , N.

— Step 2 : The chosen order, for example a sell order, at price p(t) is canceledand submitted at a new price p± 1 with probabilities 1±D

2 .

— Step 3 : If the new price of sell order takes place at the price p(t) of existingbuy order, there is a transaction at price p(t).

— Step 4 : In order to keep the total volume constant, there are one sell order sub-mitted uniformly at price between p(t), . . . , n and one buy order submitteduniformly at price 1, . . . , p(t).

The simulated autocorrelation of last trade price increments exhibit a Hurstexponent H = 1/4 < 1/2 at long time scale.

By modifying the after transaction rule, the sell order is submitted at the rangeof price p where Np 6= 0. This new rule gives some intelligence by assuming thattraders mimic others traders. In this case, the simulated autocorrelation of last tradeprice increments exhibit a Hurst exponent for short time scale H > 1/2 and at longtime scale H = 1/2 which is similar to a Brownian process. They found a heavy tailfor last trade price increment distribution.

14


They found that giving state dependency to the limit order placement fLβ givesbetter results on the last trade price increments at long time scale.

In this model, the drawbacks are :

— the non existence of market orders,

— the simultaneity of limit and cancel event,

— the constancy of total volume N ,

— the non focus in market liquidity criteria and

— the constancy of event sizes.

Maslov models Maslov [49] propose a simple model which can be seen as amultidimensional Markov Chain. The dimension is finite with n = 2×∆ dimensionswith ∆ the support of the fLβ . The price position is relative to the last trade price.

Traders can send (buy and sell) limit orders with a price placement probabilitydistribution fLβ .

With a simple model, they can exhibit 2 stylized facts such as fat tails of thedistribution of last trade price increments, the autocorrelation function last tradeprice increments exhibit a Hurst exponent H = 1/4, the autocorrelation function ofabsolute last trade price increments exhibit a Hurst exponent H = 3/4 (γ = 1/2).

The drawbacks are :

— the non existence of market orders,

— the price position relativity to the last trade price,

— the non focus in market liquidity criteria,

— and the constancy of event sizes.

Challet and Stinchcombe models Challet and Stinchcombe [19] propose a mo-del as an infinite-multidimensional Markov Chain with n =∞ dimensions. The priceposition reference is relative to 0.

At each time t, traders choose with constant probability δ send (buy or sell withthe same probability) limit orders with a price placement probability distributionwhich is centered Gaussian with variance σX(t) = Kg(t) + C with g(t) the spreadand K and C some constant.

They introduce cancellation event, limit order can be cancel with constant pro-bability ν and with a uniform cancel price. We made the remark that market ordersare modeled by limit orders which cross the opposite best price. The model has noover-diffusive behavior at short-time but under-diffusive with H = 1/4 and tends toH = 1/2 at large-time.

15


It is clear that the over-diffusive behavior at short-time that can be seen in allmarket can be produced by the phenomena of mimic in limit order placement likethe second model of Bak [9].

In this model, the drawbacks are the constant order size. They don’t modelmarket and they focus only on stylized facts and not in market liquidity criteria.

One year later, Challet and Stinchcombe [20] extend their model by testingdifferent type of changes of rate and separate in the model the market (rate α) andthe limit orders (rate δ). We will give some examples of type of changes in order tounderstand the underlying mechanism which give this behavior.

— At each time step, with probability p, all rates δ, α are random variable andare redrawn independently at random.

— Rates are defined with random variable ε who can takes values −1, 1 withprobability p, α(t) = α0 + α1ε(t) and δ(t) = δ0 − δ1ε(t).

By randomizing the rates, they find that the last trade price has an over-diffusivebehavior at short-time and a diffusive behavior at long time.

The drawbacks are :

— the constancy of event sizes and

— the non focus in market liquidity criteria.

Daniels, Smith, Farmer models In [26], [59], [29], they model the event orderflow as a six dimensional Poisson processes (sign × event type) with :

— arrival of a new market order with intensity µ with constant size,

— arrival of a new limit order with intensity α with constant size and uniformprice placement in the range of price b(t) < p < ∞,and similarly for bids on−∞ < p < a(t),

— cancellation of an existing limit order in the limit order book with intensity δwith size 1.

The dimension is infinite. The price position reference is relative to 0. The pricelevels are logarithm then price can be negative.

By taking another direction, the mean-field theory, they study some characte-ristics such as the depth profile, the spread, the slope of depth profile and the pricediffusion rate at short term and at long term. The drawback is the constancy ofevent sizes.

Preis 2006 Preis et al. [57] assume in their model that there are a fixed number oftraders. There are NA liquidity takers which send market orders modeled by Poissonprocesses with intensity µ. A fixed number of traders NA of liquidity providers send

16


limit orders which are modeled by Poisson processes with intensity α. Liquidity pro-viders can cancel their orders which are modeled by Poisson processes with intensityν.

— Price placement is simulated by uniform law and exponential law.

— When limit event happens, there is a probability qprovider (resp. 1 − qprovider)to have a buy (resp. sell) order.

— When market event happens, there is a probability qtaker (resp. 1 − qtaker) tohave a buy (resp. sell) order.

The event size is 1. They define the total volume recursively as :

N(t+ 1) = (N(t) + αNA)− (N(t) + αNA)δ − µNA

. At equilibrium, the total volume is : NeqNAδ = α(1 − δ) − µ. They show that a

market trend i.e. an asymmetric order flow of any type, leads to a Hurst exponentH < 1/2 for small time, H > 1/2 for medium time and converge to H = 1/2 at largetime for the price increment. Price increments don’t exhibit fat-tailed distributions.They randomize qtaker as a random walk and modify their price placement by takingaccount the randomness of qtaker. In this case, they find the stylized fact of fat tails.

The drawbacks are :



Cont (2010) In Cont et al. [24], the authors model the limit order book as acontinuous time Markov process Xt := (X1(t), X2(t), ..., Xn(t)), where −Xp(t) (resp.Xp(t)) is the total sell (resp. buy) volume at price p, for p ∈ 1, . . . , n. For all t > 0,the bid and ask prices are defined :

bt = supp,Xp(t) < 0 and at = infp,Xp(t) > 0

They assume that all orders are of unit size. The dimension is finite (= 2n+ 2). Theprice position reference is relative to 0.

The intensity of limit orders at level p is λ(p) := kpα

(follow a power law). Theselimit orders are canceled at rate θ(p) := θxp. Market orders arrive according to aPoisson process of intensity µ. Under these assumptions, the process (Xt) is Marko-vian and ergodic, and several quantities of interest such as transition probabilities forprices, distribution of duration are computed using Laplace transform techniques.The simulated average depth profile is almost the same as the data one.

The drawbacks are :


17



Abergel (2013) Abergel et al. [2] consider the dynamics of a general order bookunder the assumption of Poissonian arrival times for market orders, limit orders andcancellations. The dimension is finite (= 2× 2K). It follows a finite moving frame

They assume that each side of the order book is fully described by a finite numberof limits K, ranging from 1 to K ticks away from the best available opposite quote.We will use the notation X(t) := (a(t); b(t)) := (a1(t), . . . , aK(t); b1(t), . . . , bK(t)),where a := (a1, . . . , aK) designates the ask side of the order book and ai the numberof shares available i ticks away from the best opposite quote, and b := (b1, . . . , bK)designates the bid side of the book. They adopt the representation of a finite mo-ving frame. They assume that all orders are constant size q. They impose constantboundary conditions outside the moving frame of size 2K. If the intensity of cancel-lation is state-dependent and non null, the continuous (resp. discrete) time processX is stable and converge to is stationary state at an exponential (resp. geometric)rate. If the intensity of cancellation is state-independent, the intensity of marketand cancellation are superior to the intensity of limit order, they have the sameconclusions. Empirically, they exhibit the first non negative lag and the fast decayfor the autocorrelation of price increments. The simulated average depth profile isalmost the same as the data one. By appending the last event e(t) to the order book(X(t)), the time process Y (t) := (X(t), e(t)) is stable with the same condition asbefore. By rescaling and centering the price increment processes, they prove a limittheorem on a Brownian motion by the functional central limit theorem.

The drawbacks are :

— the constancy of event sizes,

— the non focus in market liquidity criteria and

— the independence of (2× 2K) Poisson processes.

Huang [41] extend the model of Abergel [2] by adding a price reference andassuming the state dependencies of all intensities. In this model, under negativedrift conditions and boundness of the market intensity, the Markov chain is ergodic.They study empirically the dependence between the state by taking the third bestbid volume and the third best ask volume and the intensity.

1.4 Optimal liquidation

Optimal liquidation problem deals with the optimization of a trading programfrom a given position to zero in a given time interval [0, T ] with T ∈ R+ ∪ +∞.A trader takes the decision to buy or to sell a large number of shares, he needs toimplement his decision on trading platforms. If he trades too fast, he will suffer from

18

1.4. Optimal liquidation

market impact and liquidity costs on his price. But on the other hand if he tradestoo slow, he will suffer from a large risk penalization, the adverse selection (the “fairprice” will have time to change in the opposite way).

We need to optimize the trade schedule, how to split a large number of sharesover smaller number of shares to be executed in a given time interval. It requiresa cost function for immediate costs due to current trading decisions, transient costand permanent cost i.e. impact of current decisions on future prices. We need toconsider the different costs with respect to the relevant time scale. At long-timescale, we need to consider the permanent cost.

In order to solve the optimal liquidation problem, we need to model how thestrategy X impacts the price dynamic P and the strategy cost C which depends tothe price dynamic and the strategy. In the case of continuous time framework, thestrategy X can be decomposed with a discrete part and a continuous part. In thecase of optimal liquidation problem, a strategy is admissible if X0 = X > 0 andXT = 0 which means that the initial shares are executed in the given time interval.Optimal liquidation problem consists of :

Finding an admissible strategy X which minimize E[C([0, T ], P,X)]

with C([0, T ], P,X) the total cost i.e. the strategy cost between [0, T ]. In the discretetime framework, the instants are indexed by N. In this case, the strategy X is merelythe choice of the quantities x0, . . . , xT with ∑T

k=0 xk = X.

1.4.1 Price impact model

We will begin by some basic model of optimal liquidation with market orders. Inour knowledge, the pioneers were Bertsimas and Lo, in 1998 [13]. They model a linearpermanent price impact. They established the dynamical programming principle andfound in several cases, in a backward recursive way the optimal strategy. Almgrenand Chriss [8] add a linear temporary impact. In this case, the strategy cost isquadratic function on the execution rate.

They build an efficient frontier for the optimal execution problem by minimizingthe price impact and the variance. The phenomena of the resilience was addedby Obizhaeva and Wang [53]. They established the HJB equation of the optimalexecution problem and found an explicit value function quadratic on the remainingshare and the resilience. Recently, Alfonsi and Blanc [4] model the market ordersand cancellation orders by Hawkes Processes in order to capture the auto-excitationphenomena of market orders. They also found an explicit value function quadraticon the remaining share, the resilience and the intensity of Hawkes processes.

In [32], they model the linear permanent price impact with a time-dependent,

19


deterministic depth and an exponential resilience of the book. They determine op-timal portfolio liquidation strategies. In specific cases, they can state the optimalstrategy in closed form. In [21], they model the linear permanent price impact witha stochastic depth (qt) which follow a one dimensional finite Markov chain and aexponential resilience. The optimal order execution policy is solved by a MarkovChain Decision processes algorithm.

1.4.2 Volume impact model with limit order book shape

Under volume impact model, financial assets are driven by a reference priceprocess, a volume process and a fixed order book shape. The reference price processcan be modeled by a semi-martingale process. (At)t≥0 be the reference price of afinancial asset, which we assume to be a semi-martingale.

The volume process is the remaining volume related to the fixed order bookshape. The limit order book shape is a limit theoretical limit order book shapemean reverting after market order. He is defined relative to a theoretical referenceprice. The order book shape links the best price evolution and the remaining volumeevolution.

In [7], they introduce the notion of limit order book shape in order to introducethe dynamics of the limit order book at middle or low frequency for the optimalcontrol liquidation problem. Let the limit order book shape characterized by thecumulative function F or µ.

In the absence of trading, the total volume at time t in the price interval [At, At+x) is F (x). F is a non-decreasing and left-continuous function associated to aninfinite measure µ on [0,+∞) in the following way :

F (x) := µ([0, x)), for all x ≥ 0. (1.1)

In [7], the shadow order book density is price continuous. They found an explicitoptimal strategy. In [56], the density can be price continuous, discontinuous or pricediscrete. They show the existence of two types of optimal strategy according to theconvexity of a certain function depending on the density and the resilience.

1.4.3 Optimal execution with Market and Limit orders

Guilbaud [38] propose a framework for studying optimal market making policieswith limit and marker orders in a bid-ask spread modeling. They want to manageinventory risk, adverse selection risk and execution risk. They model the cash, thenumber of shares. The randomness which represent the other market order is mo-deled by Cox processes. They formulate the control problem and they characterizein terms of quasi-variational system by dynamic programming methods. They use

20

1.4. Optimal liquidation

numerical approach based in finite scheme. They exhibit distinct regions for thechoice of limit or market order according the spread and the number of shares. Theyfound that it is better to send market orders when approach the maturity.

Jacquier [43] propose a framework to study the optimal market making policieswith limit and marker orders in a Level 1 limit order book for large-tick stocks, withthe spread equal to one tick and unit size for orders. They formulate the controlproblem as a semi Markov decision process. The numerical approach for the dynamicprogramming, conclude that order limit is preferred when far to the maturity, lesslatency, price decrease and volume imbalance.

1.4.4 Optimal execution in limit order book

Abergel [1] propose a framework for studying optimal market making policieswith limit orders in the context of the limit order book. They characterize theproblem with Markov Decision Processes theory. For the numerical approach, a high-dimension problem arise and they tackle the problem with a control randomizationmethod and quantization method. When the intensity is state independent, theoptimal strategy and the naive strategy (which consists to put limit orders in bestprices) are equivalent. However in the case of state dependent, optimal strategyperforms better than naive. They find allowing to put in the second best priceperforms better.

21

Première partie

Limit order book modeling withdepth representation

23

Chapitre 2

Limit order book modeling

Contents2.1 Orderbook’s representation . . . . . . . . . . . . . . . . . 27

2.1.1 The characteristics of an order book . . . . . . . . . . . . 282.1.2 The basic transformations on an order book . . . . . . . . 29

2.2 The dynamic of the order book . . . . . . . . . . . . . . . 312.2.1 The evolution of the order book . . . . . . . . . . . . . . 312.2.2 The randomness specification : a Markovian setting . . . 32

2.3 More about the randomness specification . . . . . . . . . 332.3.1 A review of the literature . . . . . . . . . . . . . . . . . . 342.3.2 The homogenization . . . . . . . . . . . . . . . . . . . . . 39

2.3.2.1 The basic idea . . . . . . . . . . . . . . . . . . . 402.3.2.2 An analysis of the homogeneity of LV (x) . . . . 41

2.3.3 Statistics and calibrations . . . . . . . . . . . . . . . . . . 432.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Introduction

This chapter is devoted to the theoretical foundations of our limit order bookmodel.

Limit order book modeling consists to find a dynamical system which describeas much as possible the mechanism of limit order book (see the subsection 1.3.1 forexplanation).

One of the reason we use the depth representation is the optimal liquidation pro-blem in a limit order book modeling framework. One of a quantity we are concernedis the gain of a sell market order. The expression of the gain is easier to use in thecase of depth representation.

For example, suppose the limit order book configuration state denoted s in thebid side is : the bid price b is 100 and the volume v0 is 10, the second best price is99 and the volume v1 is 5, the thirst best price is 96 and the volume v2 is 10 andthere is no volume at other price.

25

Partie I, Chapitre 2 – Limit order book modeling

b v0 v1 v2 . . . . . .100 10 99 5 98 10 . . . 0

LOB

Figure 2.1 – Array LOB represents a configuration state s for each price level, thevolume associated

b n0 n1 n2 . . . . . .100 10 99 15 98 25 . . . 25

CLOB

Figure 2.2 – Array CLOB represents a configuration state s for each price level,the cumulative volume associated

The trader wants to sell with a market order with a size q = 20. The gain of thistransaction is : b× (q− v0) + (b− 1)× (q− v0− v1) + (b− 2)× (q− v0− v1) = 1985.If we consider the depth representation of the same limit order book configurationstate is : the bid price b is 100 and the cumulative volume n0 = v0 is 10, the secondbest price is 99 and the cumulative volume n1 = v1 + v0 is 15, the third best priceis 98 and the cumulative volume n2 = v2 + v1 + v0 is 25 and the cumulative volumefor other price is 25.

The gain of this transaction can be written in term of cumulative representation ele-ments (ni), (n0 +n1 +n2 + . . .+n99)−(n0−q)+−(n1−q)+−(n2−q)+− . . . (n99−q)+.The gain of a transaction with size q is the sum of the cumulative representationelements minus the sum of the cumulative representation elements after the tran-saction q. Moreover, the cumulative representation elements after the transaction qare just the positive part of the difference between the cumulative representationelements before the transaction q and the size q. For optimal liquidation probleminvolving shadow order book which is a density, they use the same consideration bytaking the cumulative density in order to compute the gain of a transaction [56].The gain operator is use in the chapter 5.

The second reason is the selective liquidity taking. According to [14], the priceimpact should be concave and one of the mean reason is the state dependency ofthe market order size to the actual limit order book state. A trader sell high whenthe liquidity is high and sell low when the liquidity is low [31]. The trader watchesthe first levels K + 1 of the state v0 + v1 + . . .+ vK which is nK .

We relax the constant order size assumption. The first reason is the mechanismof the limit order book which allow any order size. Since one of our subject isthe optimal liquidation problem, the trader should send any order size in order toincrease the set of his strategy.

26

2.1. Orderbook’s representation

We emphasize that we model one side of the limit order book, the bid side. Thus,the reference price is 0 and the interval of the price is N. The main characteristicsare derived through the actual state of the limit order book.

We present in the section 2.1, the mathematical definition of the depth repre-sentation space N , the characteristics such as the bid price, the different transitionoperators induced by sell order, buy order and cancel order. In the section 2.2, wedefine the Markov chain on N by constructing the transition probabilities. In thesection 2.3, we discuss about the distribution law since most of microstructure sta-tistical literature. Furthermore, the mechanism of limit order book gives constrainton cancel size and sell size, we talk about state dependence of the random variables.In the section 2.4, we give an algorithm for simulating trajectories of our modeland we talk about the influence of the bid price and the total volume against thedistribution law and the state dependence of the random variables.

2.1 Orderbook’s representation

As explained in the introduction, only the bid side of an order book, that wecontinue to call by abuse of language order book, is studied hereafter. In this case,the state of an order book at a given time is completely determined by the buy ordervolume at each of the price levels, or equivalently by the buy depth at the differentprice levels, which constitutes a decreasing sequence of integer numbers, called thedepth of market. We introduce the space N :

N = (a1, a2, a3, . . .) : ai ∈ N, ai ≥ ai+1 and for some index k, ak = 0.

The interpretation of the space N is clear. The index k ∈ N∗ represents the differentprice level, while any element ofN represents a particular situation of the order bookin form of the depth of market : the difference nk − nk+1, for k ∈ N∗, represents thevolume at the price level k and, in particular, n1 represent the total volume of theorder book.

Lemma 2.1.1. The space N is denumerable.

Proof : The space N is clearly imbedded into a subset of ∪∞k=1Nk.

The space N is very different from the usual infinite product space (N∗)∞. Tobe distinct, we employ a different system to denote the coordinates : for n ∈ N , wedenote by n|k the value of n at the coordinate k ∈ N∗.

27


2.1.1 The characteristics of an order book

The characteristics of order book are functions defined on the space N . The mostimportant characteristic of an order book is the bid price function b :

b(n) = maxk ∈ N∗ : n|k > 0 =∞∑k=1

11n|k>n|k+1,

with max ∅ = 0. Associated with the bid price function b, we introduce the secondbest bid function :

b(n) =∑

k<b(n)11n|k>n|k+1.

We define the support of order book :

S(n) = k ∈ N∗ : n|k > n|k+1, n ∈ N ,

and accordingly σ(n) = ]S(n). Notice that b(n) = max S(n). We will also considerthe scope of order book b(n)− d(n), where

d(n) = inf S(n) if S(n) 6= ∅, and d(n) = 0 if S(n) = ∅.

Notice that, contrary to the usual convention, here inf ∅ = 0.

We define the total volume of order book h :

h(n) = n|1

The elements n of N are decreasing functions on N∗, stopped at zero. For anyreal function f on N∗, the integral

∫∞0 f(s)dns = ∑∞

k=1 f(k)∆kn

is a well-defined function on N , where

∆kn = n|k − n|k+1, k ∈ N∗,

is the volume at the price level k. For example, if f(k) = k, the integral

s(n) =∫∞

0 sdns = ∑∞k=1 k∆kn

represents the total valuation of the order book.

28

2.1. Orderbook’s representation

b(n)Price level

Volume

b b(n)

a

Price level

Volume

Figure 2.3 – A representation of the transition between two states induced by abuy event at the price level b of a size a

2.1.2 The basic transformations on an order book

When a buy order at the price level b ∈ N∗ of size a ∈ N∗ is added, the orderbook changes from n ∈ N to a new state TB(n, a, b) (Figure 2.3) : for k ∈ N∗,

Tb(n, a, b)|k =

n|k + a, k ≤ b,

n|k, k > b.

We call Tb the buy operator on the space N . We employ the expression

n +b a := Tb(n, a, b).

Similarly, we define the sell operator TS :

TS(n, a) = ((n|1 − a)+, (n|2 − a)+, (n|3 − a)+, . . .),

n ∈ N , a ∈ N∗, representing the modification of the order book after a market orderof size a (Figure 2.4). Notice that one can want to sell with an arbitrary size a, butonly up to n|1 quantity can be executed. We introduce the notation

n−∗ a := TS(n, a).

Again, we define the canceling operator TC : for n ∈ N , a ∈ N∗, b ∈ N∗, for

29


a

b(n)Price level

Volume

b(Ts(n, a)) Price level

Volume

Figure 2.4 – A representation of the transition between two states induced by asell event of a size a

b b(n)

a

Price level

Volume

b(n)Price level

Volume

Figure 2.5 – A representation of the transition between two states induced by acancel event at the price level b of a size a

k ∈ N∗,

TC(n, a, b)|k =

n|k − a ∧ (n|b − n|b+1), k ≤ b,

n|k, k > b,

representing a cancellation order at the price level b of size a (Figure 2.5). Weintroduce the notation

n−b a := TC(n, a, b).

Notice that one can only cancel existing orders. From the view of modeling, a cancelorder can be formulated with an arbitrary size a, but only the quantity n|k − a ∧(n|b − n|b+1) of orders is canceled.

30

2.2. The dynamic of the order book

2.2 The dynamic of the order book

An order book changes along with the arrivals of new orders.

2.2.1 The evolution of the order book

Assumption 2.2.1. We suppose that the new orders arrive along a discrete time-line. At each point of the timeline, only one of the events can occur : nothing, sellorder, buy order or cancel order.

The points in timeline will be represented by the integers N. At each pointn ∈ N of the timeline, let δn be the event indicator taking values in the set of signsN,S,B,C representing the four events. When an order arrive effectively at timen, we denote the order size by αSn , αBn , αCn ∈ N∗ corresponding respectively to a sellorder, a buy order and a cancel order. The valuation levels of a buy order or a cancelorder are denoted respectively by βBn , βCn ∈ N∗. Recall that all sell orders consideredin this thesis are market orders without limit of price level.

The arrivals of the new orders are random events. We fix from now on a proba-bility space (Ω,A,P), where A is a σ-algebra on Ω and P is a probability measureon A. The quantities

δn, αSn , α

Bn , α

Cn , β

Bn , β

Cn

are henceforward random variables on the probability space (Ω,A,P).The new orders modify the order book. Let N0 ∈ N be the initial state of the

order book. We define recursively

Nn+1 =

Nn, if δn+1 = N ,

TS(Nn, αSn+1), if δn+1 = S,

TB(Nn, αBn+1, β

Bn+1), if δn+1 = B,

TC(Nn, αCn+1, β

Cn+1), if δn+1 = C,

(2.1)

for n ∈ N.The process N = (Nn)n∈N represents the evolution of the order book over the

time. Let F = (Fn)n∈N denote the natural filtration generated by the sequence ofrandom variables (δn, αSn , αBn , αCn , βBn , βCn ) for n ∈ N.

31


δn+1

No event

Sell order with size (αSn+1)

Buy order with price level and size (βBn+1, αBn+1)

Cancel order with price level and size (βCn+1, αCn+1)

δn+1= N

δn+1 = S

δn+1 = B

δn+1 =

C

Figure 2.6 – At each instant n, we simulated the random variables δn+1, αSn+1,αBn+1, αCn+1, βBn+1, βCn+1

2.2.2 The randomness specification : a Markovian setting

After having settled down the formalism of the order book, the modeling consistsin choosing the probability distribution of the family of the random variables δn, αSn ,αBn , αCn , βBn , βCn for n ∈ N. In this thesis, the evolution of the order book is simplymodeled by a Markov chain.

More precisely, we suppose that there exists a probability measure kernel Qn[A]from N to N,S,B,C× (N∗)5 (equipped with the discrete σ-algebra D) such that

P[(δn+1, αSn+1, α

Bn+1, α

Cn+1, β

Bn+1, β

Cn+1) ∈ A |Fn]

= P[(δn+1, αSn+1, α

Bn+1, α

Cn+1, β

Bn+1, β

Cn+1) ∈ A |Nn]

= QNn [(δ, αS, αB, αC , βB, βC) ∈ A],

for A ∈ D, n ∈ N, where (δ, αS, αB, αC , βB, βC) denotes (in form of the coordinates)the identity map on the space N,S,B,C × (N∗)5. Under this assumption, theprocess N is a Markov chain taking values inN . As usual, we introduce the transitionprobabilities

P(x, y) = P[Nn+1 = y |Nn = x], x, y ∈ N .

Note that the above conditioning is independent of n ≥ 0. We have

P(x, y) := Qx[δ = N ]11x=y

+Qx[δ = S, x −∗ αS = y] + Qx[δ = B, x +βB αB = y] + Qx[δ = C, x −βC αC = y].

We introduce equally the Markov chain generator (also called drift when applied

32

2.3. More about the randomness specification

to a function) L : for any real function F on the space N ,

LF (x) =∑y∈N

P(x, y)(F (y)− F (x)), x ∈ N , (2.2)

whenever the sum is absolutely convergent. We have

LF (x) = Ex[11δ=S(F (x −∗ αS)− F (x))] + Ex[11δ=B(F (x +βB αB)− F (x))]+Ex[11δ=C(F (x −βC αC)− F (x))],

where Ex denote the expectation under the probability distribution Qx.

In fact, we will not need to always work in the above general setting. Variouscomplementary assumptions can be introduced in different situations for differentdiscussions. Actually, we will suppose from now on the following assumption.

Assumption 2.2.2. For every x ∈ N , under the probability measure Qx, the randomvariable δ is independent of αS, αB, αC , βB, βC .

Let

pN(x) = Qx[δ = N ], pS(x) = Qx[δ = S], pB(x) = Qx[δ = B], pC(x) = Qx[δ = C].

Under Assumption 2.2.2, the transition function P(x, y) and the generator L takethe following forms :

P(x, y) = pN(x)11x=y + pS(x)Qx[x −∗ αS = y]+pB(x)Qx[x +βB αB = y] + pC(x)Qx[x −βC αC = y],

(2.3)

and

LF (x) = pS(x)(Ex[F (x −∗ αS)]− F (x))+pB(x)(Ex[F (x +βB αB)]− F (x)) + pC(x)(Ex[F (x −βC αC)]− F (x)).

(2.4)

2.3 More about the randomness specification

One must look the random variables δ, αS, αB, αC , βB, βC as parameters of themodel introduced in Section 2.2. This section is devoted to the question how theseparameters should be chosen. It is a second fundamental question, besides the de-finition of the model, on the order book modeling. We will demonstrate that thesimple model defined in Section 2.2 is rich enough to possess the various importantproperties of an order book.

33


2.3.1 A review of the literature

There are many empirical studies in the literature on the order book. Thesestudies reveal a number of features that characterize the order book. A good modelis one that reproduces the same properties. This requirement conditions the choiceof the random variables δ, αS, αB, αC , βB, βC in our model. This section is devotedto a discussion on the empirical properties of the order book and the issues. Weemphasize that some studies come from statistical point of view and other from orderbook modeling. Since most of studies use Poisson processes, the random variable δis the equivalent of the intensity of the Poisson processes.

In the subsection 1.2.1, we talk about the long memory of order signs. In thiscase, we need to add a time structure on orders flow. In order to mimic these stylizedfact, the literature consider multivariate Hawkes processes. In order to keep a simplemodel, we don’t add time structure on δ.

Specification of δ : arrival event time Since most study in limit order bookmodeling are Poisson processes (continuous timeline), the duration between twoconsecutive events with the same type, follows exponential law [24], [2].

In our model (discrete timeline), assuming that for all x ∈ N , pN(x) := pN ,pS(x) := pS, pB(x) := pB, pC(x) := pC , the duration ∆t between two consecutiveevents with the same type, follows a geometric law. In the case of two consecutivesell events, we have :

P[∆t = j] = P[δk+j = S|δk+j−1 6= S, . . . , δk+1 6= S, δk = S] = (1− pS)j−1pS

Then, Poisson processes model are equivalent to our model since geometric lawis the discrete version of exponential law.

Specification of αB : limit order size In limit order book modeling, most studyconsider constant size [49], [59], [24], [2]. In Abergel et al. [2], they use log-normaldistribution for the simulation.

However, statistical study find power law tail distributions for limit order size.Maslov [50] find a power law tail and the log-normal distribution can be a good fit.Bouchaud et al. [15] get a uniform law in log size.

In order to explain the influence of the limit order size distribution and the shapeof the limit order book, Muni [60] consider for the discrete case a geometric law andthe continuous case an exponential law.

34


Assumption 2.3.1 (Specifications of αB). We will look for αB with a log-normaldistribution, binomial distribution or a geometric distribution.

Specification of βB : limit price placement Moreover, they use independentPoisson processes for each price level. In Cont [24], the intensity is a function of thedistance between price level and Bid Price (or the Ask Price). However in Abergel[2], they consider Poisson processes with constant intensity.

In Smith [59], they use one Poisson process for limit order and they considerUniform Law for order placement respect to the bid price and the ask price. Instatistical studies, Bouchaud [15], [62], Gu [37] found a power law tail for the orderplacement. It can be understand as the optimistic belief that large price movementcould happen.

But Bouchaud et al. [15], in the last part of his paper, propose an a prioridistribution based on empirical data.

Most of reference on order book modeling use the opposite best price for thelimit price level. As we consider bid part of the order book, we don’t have the askprice. We will use the bid price as reference for the buy price level.

Since there is no constraint on the limit order size, we propose αB is a r.v ∈ N∗.βB is a bid price dependent random variable. The event βB ≤ 0 should be

independent of the spread and the event βB > 0 is constraint by the spread, weseparate the event. Furthermore, the event βB > 0 increase the bid price.

Assumption 2.3.2 (Specifications of βB). We propose : εB ∈ 0, 1 and βB− ∈0, . . . , v ∧ (b(n)− 1) and βB+ ∈ N∗,

βB := b(n)− 11εB=0βB− + 11εB=1β

B+.

We will look for εB with a Bernoulli distribution and for (βB−, βB+) a log-normaldistribution, a binomial distribution, a geometric distribution.

The random variable εB explains the choice of increase the bid price. βB− (resp. βB+) gives the distribution of price placement inside the book (resp. inside thespread).

Specification of αS : market order size In limit order book modeling, like limitorder event, most study consider constant size ([49], [59], [24], [2]). In Abergel [2],they use log-normal distribution for the simulation.

However, statistical study find power law tail distributions for limit order size.Maslov [50] find a power law tail and the log-normal distribution can be a good fit.Bouchaud [15] get a uniform law in log size. Challet [19] find sizes are clustering onround amounts such as 10, 100, and 1000.

35


The previous studies are unconditional laws. As we said in the subsection aboutprice impact, the selective liquidity taking is one of reasons about the concavity ofthe instantaneous price impact with respect to the market order size. In this case,we need to add some state dependencies on the size distribution. In this case, weneed to consider random variable with finite support. For example, if we considerthat random variable which represent market order size follow a uniform law andall traders consider the l first levels of depth when sending market orders then thesupport of the random variable is J1, n|b(n)−lK.

Assumption 2.3.3 (Specifications of αS). We propose two specifications for αS :

— In the case of state independence of αS, αS is a r.v. ∈ N∗ and we will lookfor αS with a log-normal distribution, a binomial distribution, a geometricdistribution.

— In the case of state dependence of αS, αS is a r.v. ∈ 1, . . . , 1 ∧ n|b(n)−l. Wewill look for αS with a uniform distribution ∈ 1, . . . , 1∧n|b(n)−l or a binomialdistribution with parameters (n|b(n)−l, p).

Specification of βC : cancel order placement Most of reference on order bookmodeling [24], [2] use the bid price as reference. So we will follow these idea. ThenβB is state dependent random variable.

Assumption 2.3.4 (Specifications of βC). We propose βC− is a r.v. ∈ 0, . . . , v ∧(b(n)− 1),

βC(n) := b(n)− βC−

with a log-normal distribution, a binomial distribution or a geometric distribution.

Specification of αC : cancel order size In limit order book modeling, like limitorder event, most study consider constant size [49], [59], [24], [2]. In [2], they uselog-normal distribution for the simulation.

Most of studies don’t study the cancel order size law, they studies instead therate of cancellation since cancel event arrival are modeled by Poisson processes.Concretely, in the case of the rate of cancellation dependent of the state, Cont et al.[24] estimate the rate such as :

λ(i) = Nc(i)TQi

Sc

with T : total trading time in the sample, Nc(i) : the number of times of cancelevent at a distance of i to the bid price, Sc : the average size of cancel order and

36


Figure 2.7 – Histogram of cancel size unconditionally to the state, the x−axis isthe size and the y−axis is the occurrence in the data

Qi : the average volume at a distance of i to the bid price. They need to estimatethe average shape of the order book. Abergel et al. [2] used the same approach forthe rate of cancellation estimation.

Bayer et al. [12] considered a stochastic model for the dynamics of limit orderbook. They describe the model as a continuous time models with state-dependentprice dynamics. The cancel order sizes are described by a sequence of i.i.d. randomvariable (ωCi )i∈N. The random variables (ωCi ) take values in [0, 1] and describe theproportions of cancellations. The change after a cancellation of buy volume is state-dependent, have (ωCi )v(t, x) at instant t and v is the volume at price x.

We want to assume a sequence of random variable (ξCi ) such as :

αCi (n) := ξCi ∆bn

At this point, we didn’t mention the data we use in the chapter 4. But for thissubparagraph, we use the data for the approach (we refer to the chapter 4 for moreinformation on financial market data). With data, we draw histogram of cancel sizeunconditionally to the state and histogram of volume at the bid. In order to havethe sample, we extracted all size (αC)i cancel event happen to the bid.

With the same data, we draw histogram of cancel size conditionally to the volumeat the bid. In order to have the sample, we extracted all size (αC)i cancel eventhappen to the bid and all bid volume ∆bx just before the cancel and we computethe variable :

ξCi := αC

∆bxWe notice a non negligible frequency for ξCi = 1. This value means that ∆bx is

entirely canceled. It probably suggest that this value come from another variable.From our analysis, the probability that ∆βCx is entirely canceled should be an

independent parameter from the other aspects of αC .

37


Figure 2.8 – Histogram of volume at the bid, the x−axis is the volume at the bidprice and the y−axis is the occurrence in the data

Figure 2.9 – Histogram of cancel size conditionally to the volume at the bid price,the x−axis is the ratio between size and the volume at the bid price, the y axis isthe occurence in the data

38


Assumption 2.3.5 (Specifications of αC). We propose two specifications for αC :

— In the case of state independence of αC, αC is a r.v. ∈ N∗ and we will lookfor αC with a log-normal distribution, a binomial distribution or a geometricdistribution.

— In the case of state dependence of αC, suppose that, under Qx, there existindependent r.v.’s εC ∈ 0, 1, ξc ∈ 1, . . . , 1∨ (∆βCx−1), independent of βC,such that αC is decomposed as follows

αC = 11εC=1∆βCx + 11εC=0ξc

We will look for εC with a Bernoulli distribution and for ξC an uniform dis-tribution ∈ 1, . . . , 1∨ (∆βCx− 1) or a binomial distribution with parameters(∆βCx − 1, p).

To conclude the specifications, we look for level price and size (in the case of stateindependent) random variables with log-normal distribution, binomial distributionor geometric distribution. In the case of state dependent, we look for size randomvariables with uniform distribution or binomial distribution.

2.3.2 The homogenization

The following general result on discrete Markov chain, which links the asymptoticof a function on N with its drift, will be applied in the next chapter.

Lemma 2.3.1. Let F be a function such that LF is well-defined. Let

Xn = (F (Nn)− F (Nn−1)− LF (Nn−1)),e(Nn−1) = E[X2

n |Fn−1], vm = ∑mk=1 e(Nk−1) = ∑m

k=1 E[X2k |Fk−1].

Suppose that the Xn are square integrable.i. Suppose that there exists a constant 0 < c <∞ such that e(Nn−1) ≥ c for alln. Then,

lim infn→∞ F (Nn)vn

= lim infn→∞ 1vn

∑nk=1 LF (Nk−1),

lim supn→∞F (Nn)vn

= lim supn→∞ 1vn

∑nk=1 LF (Nk−1),

almost surely.ii. Suppose that ∑∞n=1

1n2E[X2

n |Fn−1] <∞. Then

lim infn→∞ F (Nn)n

= lim infn→∞ 1n

∑nk=1 LF (Nk−1),

lim supn→∞F (Nn)n

= lim supn→∞ 1n

∑nk=1 LF (Nk−1),

almost surely.

39


Proof : LetM = ∑n

k=1Xn,

which is a square integrable martingale, whose predictable quadratic variation isthe process (vm)m∈N (v0 = 0). Under the conditions of the lemma, the strong lawof large number theorem for martingales (cf. Hall and Heyde Theorem 2.18 [40]) isapplicable. We have respectively

either Mn

vn→ 0 or Mn

n→ 0

almost surely. The lemma is proved, because F (Nn) = F (N0)+Mn+∑nk=1 LF (Nn−1).

2.3.2.1 The basic idea

Lemma 2.3.1 is a useful result, because the drift sequence (LF (Nn) : n ∈ N)can be much less random than the initial sequence (F (Nn) : n ∈ N) is. The lemmatranscribes a random data into a functional representation.

That said, the application of Lemma 2.3.1 depends on the computation of thedrift function LF (x), which may appear laborious. Actually, the space N is a newsubject in the literature of order book. No general analysis theory exists yet for thespace N and little is known with the three transformations x−∗ a, x−b a, x +b a. Onemay feel ineffective in dealing with the computation of LF (x) and one may questionthe benefit of a model based on the space N .

This section is devoted to a detailed discussion on that question. Clearly, thecomputation of a drift function in general can be ineffective. However, we noticethat the drift functions are expectations under Qx. The model defined in Section 2.2possesses a so flexible system of parameters that, for a given function F , it is verylikely to find a specification of the parameters δ, αS, αB, αC , βB, βC which makes thedrift function LF (x) = c constant with respect to x, a case where the computationof LF (x) becomes trivial.

The idea is therefore to leave out the general question, and instead, to ask thequestion for some specific functions F and to look for parameters δ,αS,αB, αC , βB,βC which provide the drift functions LF (x) with desired properties. Recall that theultimate criterion of a good model is its relevance with respect to the applicationsin practice. The functions F are determined by the context and the search of theparameters respects the market reality.

A drift functions LF (x) is decomposed into several parts, each of which is a func-tion of x and has its own economical meaning. See (2.4). They should be consideredseparately. At the question of a simple and clear expression LF (x), the idea is tohave the aforementioned parts to become constant functions, whenever it is possible

40


regarding to the economic meaning and the model specification. We call such anidea a homogenization.

The homogenization aims to reduce the erratic state, while to focus at the es-sential characteristics of the model. This idea has always existed in the literature oforder book under various different forms. The best example is, instead of a directmodeling of the bid price b(Nn), to model it synthetically by a continuous Itô pro-cess. For other examples, we can mention the assumption of shadow book density, orthat of a unit size for all new orders. Indeed, the point is to question the relevanceof the homogenization with respect to the order book reality. In the case of ourmodel, the homogenization inherits the economic interpretation of the functions F ,and even, put some new insight on the order book.

To illustrate the idea, let us consider the drift function LV (x) for the functionV defined by

V (x) = x|1 + b(x), x ∈ N .

This function V will be used in the proof of the recurrence of the Markov chain Nin the next chapter, where one should estimate if the set x ∈ N : LV (x) > 0 is afinite set.

2.3.2.2 An analysis of the homogeneity of LV (x)

We compute the drift LV (x) with (2.4) and look at its dependence on the variablex.

LV (x) = pS(x)(Ex[(x|1 − αS)+ − x|1] + Ex[b(x −∗ αS)− b(x)] )+ pb(x)(Ex[αB] + Ex[b(x) ∨ βB − b(x)] )+ pC(x)(−Ex[αC ∧∆βCx] + Qx[βc = b(x), αC ≥ x|b(x)](b(x)− b(x)) ).

(2.5)The drift LV (x) contains three terms : one of them is positive and the two othersare negative. To simplify our discussion, let

αS = x|1 − (x|1 − αS)+ = αS ∧ x|1,αC = αC ∧∆βCx,

(2.6)

so that

LV (x) = −pS(x)(Ex[αS] + Ex[b(x)− b(x −∗ αS)] )+pb(x)(Ex[αB] + Ex[b(x) ∨ βB − b(x)] )−pC(x)(Ex[αC ] + Qx[βC = b(x), αC ≥ x|b(x)](b(x)− b(x)) ).

(2.7)

A perfect homogeneity of LV (x) would be the case where LV (x) is a constant func-tion on N outside of a finite set. This can be possible from the mathematical view-point. But the corresponding parameterization may correspond to no economical

41


reality. We now analyze each of the terms in (2.7) to find reasonable setting ofδ, αS, αB, αC , βB, βC to make LV (x) homogeneous.

i. We begin with the easiest one : If the random variable αB has the same lawunder Qx for any x, which is reasonable setting, the term Ex[αB] will be inde-pendent of x.

ii. For the term Ex[b(x)∨βB−b(x)], we can suppose, by definition, that βB takesthe form :

βB = b(x)− (1− εB)βB− + εBβB+,

where the random variables βB− ∈ [0, b(x)], βB+ ∈ N∗ are independent of therandom variable ε ∈ 0, 1 under Qx. Then,

Ex[b(x) ∨ βB − b(x)] = Qx[ε = 1]Ex[βB+].

Hence, if the random variables εB, βB+ have the same law under Qx for any x,the term Qx[εB = 1]Ex[βB+] will be independent of x.

iii. If αS is a bounded random variable (which is a natural assumption with respectto the market reality), for x|1 big enough, Ex[αS] = Ex[αS], which is a constantif the probability distribution of αS is defined independent of x.

iv. It is not illegitimate, with respect to the market reality, to define the quantity

Qx[βC = b(x), αC ≥ x|b(x)](b(x)− b(x))

independent of x, whenever x 6= 0. In fact, the probability Qx[βC = b(x), αC ≥x|b(x)] is the probability of the total cancellation of the bid, for which empiricalstudies exist, and it depend naturally the gap (b(x)− b(x)).

v. Clearly, each of the above quantities

Ex[αS], Qx[βC = b(x), αC ≥ x|b(x)](b(x)− b(x)),

etc., constitutes a statistic test that can be calibrated from the market. Noticethat the gap (b(x)− b(x)) is mostly equal to one unit in a liquid market.

vi. However, whatever we do, the terms such as Ex[b(x) − b(x −∗ αS)] or Ex[αC ]depend always on x. For example, Ex[αC ] can vary from 1 to infinity, accordingto x.Hence, to make the homogeneity, we should adjust the probabilities pS(x),pB(x), pC(x) so that, for example,

−pC(x)(Ex[αC ] + Qx[βC = b(x), αC ≥ x|b(x)](b(x)− b(x)) )

becomes constant for all but a finite number of x. Such adjustment is possibleunder reasonable setting. Actually, as Ex[αS] ≥ 1,∀x 6= 0, for any negativevalue −1 ≤ a ≤ 0, for any x 6= 0, we can find pS(x), pB(x), pC(x) to makeLV (x) = a. The idea of such an adjustment is not completely absurd, becausetraders, who have viewed the market configuration x, can modify their strate-gies, that leads to the modification of the probabilities pS(x), pB(x), pC(x).

Look at the value Ex[b(x)− b(x −∗ αS)] in particular. In an absolute sense, thisvalue can be very big. But, the big value happens only when the market has a

42


singular configuration (for example, a market with a very small total volume x|1

and a big gap b(x)− b(x)). The following result gives some ideas about the size ofEx[b(x)− b(x −∗ αS)].

Proposition 2.3.1. If there exists a m > 1 such that Ex[(αS)m] is bounded bya constant c > 0, then, for any positive real m′ such that m

m′> 1, the quantity

Ex[(b(x)− b(x −∗ αS))m′ ] is bounded on N .

Proof : Actually,

Ex[(b(x)− b(x −∗ αS))m′ ]= Ex[(∑b(x)

i=1 11αS≥x|i)m′ ] ≤ Ex[(αS)m](∑b(x)

i=11

(x|i)m/m′ )m′ ≤ c

∑∞i=1

1im/m

′ .

2.3.3 Statistics and calibrations

Before a thorough study of calibration in a later chapter, we make a quick dis-cussion about the model statistics. Under suitable specification, the law of largenumber theorem holds for the model, which lays the theoretical basis of the statis-tics. Consider the quantity Ex[b(x)− b(x−∗ αS)], which represents the prediction ofthe price impact after a sell order αS.

Proposition 2.3.2. Suppose the above homogeneity assumption on βB in Sec-tion 2.3.2.2. Suppose the assumption on αS in Proposition 2.3.1. Suppose thatQx[βC = b(x), αC ≥ x|b(x)](b(x) − b(x)) is constant for all but a finite numberof x. Suppose constant probabilities pS(x), pB(x), pC(x). Then, the average value1n

∑nj=1 ENj [b(Nj)− b(Nj −∗ αS)] converges, as soon as b(Nn)

nconverges.

Proof :We give only a sketch of the proof. In the case of a positive recurrent Markovchain N the result is immediate. Consider the case of a non positive recurrent Markovchain. We can then apply Lemma 2.3.1 to link the limit of b(Nn)

nto that of the average

of the drift sequence. The drift part of b(Nn) is divided into three parts : the onethat we consider in this proposition, the one corresponding to "buy", and the onecorresponding to "cancellation". Clearly the "buy" part converges correctly. Theergodic theorem implies that the occupation time at each x is negligible face ton ↑ ∞. Hence, the "cancellation" part converge also.

Notice that, in a liquid market, the average value 1n

∑ni=1 ENi [b(Ni)−b(Ni−∗αS)]

is expected to be very small. With the same consideration, the gap b(x) − b(x) isvery often equal to 1 in a liquid market.

Now, consider another quantity Ex[αC ]. This quantity can not be homogeneousin general. Let us compute it under concrete specifications. In the case where αC is

43


uniform on 1, . . . , ∆βCx, we have

Ex[αC ] = Ex[1 +∆βCx2 ].

If moreover βC is proportional to ∆x, we have

Ex[αC ] = 12 + 1

2x|1∞∑i=1

(∆ix)2.

In the case where βC follows the geometric distribution(p) from the level 1 to theinfinity, we have (q = 1− p)

Ex[αC ] = 12 + 1

2p∞∑i=0

qi∆i+1x.

In the case where, instead, βC follows the geometric distribution(p) from the leveld(x) to the infinity, we have

Ex[αC ] = 12 + 1

2p∞∑i=0

qi∆i+d(x)x.

Or in the case where βC follows the geometric distribution(p) backwardly from thelevel b(x) to the −∞, we have

Ex[αC ] = 12 + 1

2pb(x)−1∑i=0

qi∆b(x)−ix.

In any case, Ex[αC ] is a function of the ∆ix which is too erratic to be homogenized.That said, in spite of the non homogeneity, the parameter calibration about αC

remains possible. As an example, suppose that the bid price tends to the infinityand βC follows the geometric distribution(p) from the level d(x) to the infinity. Then,we can write an informal identity

limn→∞1n

∑nj=1 ENj [αC ] = 1

2 + 12p∑∞i=0 q

i limn→∞1n

∑nj=1 ∆i+d(Nj)Nj.

This formula links the statistic on αC to the statistic limn→∞1n

∑nj=1∆i+d(Nj)Nj,

which has been examined in the literature and has been proved to exhibit a simpledistribution profile.

To end this section, notice that, instead of considering Ex[αC ], we can considerthe product value

apc := pC(x)Ex[αC ] (2.8)

which may be reasonably homogeneous. Hence, the apc becomes an autonomous pa-rameter corresponding to the average cancellation : average in the expected volume

44

2.4. Simulations

and also average in the time.

2.4 Simulations

We propose the algorithm 1 based on the subsection 2.3.1. For the specification ofeach random variable, we invite to read for αS the assumption 2.3.3, αC the assump-tion 2.3.5, βC the assumption 2.3.4, αB the assumption 2.3.1 and βB the assumption2.3.2. We emphasize that this section is only dedicated to the simulation. The lawparameters are arbitrarily chosen. We refer to the chapter 4 for the calibration andthe estimation.

We begin by initialize the depth n, the maximal price and the set of law para-meters ppp := (pppE,pppS,pppPC ,pppV C ,pppPB,pppV B).

— Time step N = 100000

— Maximal price K = 400

— Initial price P = 200

— Depth ∆kn = 300 for k ∈ 1, . . . , 200

We simulate the random variables δ, αS, αC , εC , βC , αB, εB, βB−, βB+.We begin by plotting some paths of the Bid price b and the total volume h with

simple parameters.Law parameters

— αS is state independent,

— αC is state independent,

— (αS, αC , αB) follow a binomial law with parameters (40, 0.5),

— (βB−, βC) follow a binomial law with parameters (10, 0.2),

— (βB+) follow a binomial law with parameters (5, 0.05)

— Q[δ = C]Q[εC = 0] = 0.04,

— Q[δ = C]Q[εC = 1] = 0.3,

— Q[δ = B]Q[εB = 0] = 0.3,

— Q[δ = B]Q[εB = 1] = 0.04,

— Q[δ = S] ∈ 0.06, 0.065, 0.07

Empirically, we find that by changing Q[δ = S] by amount in the order of0.05, the bid price and the total volume change the direction. It means that themodel is really sensitive to the parameters. By fixing all parameters and letting freeQ[δ = S], it’s not easy to find neutral bid price path and total volume path. WithQ[δ = S] = 0.65 in figures 2.11a and 2.11b, the bid price path and volume price

45


Data: N : time step, K : maximal price, δ : Simulate the vector δi (size N),n : initial vector of depth (size K)

Result: B : vector (size N), B∗ : vector (size N), V : vector (size N)initialization : N[0]← n, B[0]← b(N[0]), V [0]← v(N[0]);for i ∈ J1, NK do

switch the value of δi docase N

N[i]← N[i− 1];case B+

a← Simulate αB;g ← Simulate βB+;b← Compute βB with g and B[i− 1] ;N[i]← Compute the new state with a,b and TB;

case B-a← Simulate αB;g ← Simulate βB−;b← Compute βB with g and B[i− 1] ;N[i]← Compute the new state with a,b and TB;

case C+g ← Simulate βC−;b← Compute βC with g and B[i− 1] ;a← Simulate αC ;

case C-g ← Simulate βC−;b← Compute βC with g and B[i− 1] ;a← Compute ∆bn;N[i]← Compute the new state with a,b and TC ;

case Sa← Simulate αS;N[i]← Compute the new state with a and TS;

case S depa← Simulate αS with n|B[i−1]−l;N[i]← Compute the new state with a and TS;

endswB[i]← Compute the Bid Price of N[i];V [i]← Compute the the total Volume of N[i];

endAlgorithm 1: Bid Price and the total volume simulation

46

2.4. Simulations

(a) Bid price path (b) Total volume path

Figure 2.10 – Path simulation for Q[δ = S] = 0.06





47



Figure 2.13 – Path simulation for law parameters 2

path seems to be neutral. The choice parameters to have a neutral bid price pathand volume price path are linked to the drift Lb and Lh.

Moreover, in figures 2.10a and 2.10b, 2.13a and 2.13b, the monotony of the bidprice path and volume price path seems to be linked. Q[δ = S] is involved in thedrift Lb and Lh.

However, by writing the drift Lb and Lh,

Lb(x) = −pSEx[b(x)− b(x −∗ αS)]+pBQ[εB = 1]Ex[βB+]−pCQ[βC = b(x), εC = 1](b(x)− b(x)).

Lh(x) = −pSEx[αS] + pBEx[αB]− pCEx[αC ].

We notice that in order to unbind the evolution of the bid price and the totalvolume, we can choose Q[εB = 1]Ex[βB+] big and Ex[αB] small. For example, Lawparameters 2

— αS is state independent,

— αC is state independent,

— αC follow a binomial law with parameters (40, 0.5),

— αB follow a binomial law with parameters (10, 0.5)

— βC follow a binomial law with parameters (10, 0.2),

— βB+ follow a binomial law with parameters (20, 0.05)

— Q[δ = C] = Q[δ = B],

— Q[δ = S] = 0

But we noticed that the choice of parameters is not realistic since we choosepS = 0. In the next chapter 3, we discuss about the choice of parameters for abullish market (the bid price increase) and the total volume should not explode.

48

2.4. Simulations

Conclusion

We designed a limit order book model with one side of the limit order book, thebid side. Moreover, the reference price is 0 and the interval of the price is N. Themain characteristics are derived through the actual state of the limit order book.We represented the state of the limit order book by the depth n. We constructeda theory around the space of depth N by defining operators of N . Through theMarkovian framework, we described the transition probabilities and the Markovchain generator. We specified the space of the random variables δ, αS, αB, αC , βB,βC by using the empirical studies in the literature on the limit order book. Weexplained the concept of homogeneity and gave a lemma 2.3.1, we use in chapter 3.

49

Chapitre 3

The fundamentals of the Markovchain N

Contents3.1 The irreducibility . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 The recurrence . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 The transience . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.1 The volume function . . . . . . . . . . . . . . . . . . . . . 55

3.4 A more realistic transient model . . . . . . . . . . . . . . 56

3.4.1 The bid b(Nn) and the scope b(Nn)− d(Nn) . . . . . . . 56

3.4.2 A result about the scope in a transient market . . . . . . 58

3.4.3 A discussion about the assumptions in Proposition 3.4.1 . 60

3.5 The volume index in a bullish market . . . . . . . . . . . 61

Introduction

The model introduced in the previous chapter possesses all the features and theflexibility for the study of limit order book. This chapter deals with the fundamentalsof the Markov chain.

Concretely, we study the usual properties of a Markov chain. Particularly, westudy the long term behavior, i.e. the classification of Markov chain. The classifica-tion of Markov chain on general state is well studied in Meyn and Tweedie [52]. Inour case, we refer to the book of Menshikov [51] for definitions and properties aboutdiscrete-time denumerable state Markov chain.

The question of positive recurrence is well studied in the limit order book mode-ling literature. In Abergel [2], the Foster-Lyapunov approach is used for proving thepositive recurrence of the Markov chain and the exponential rate of convergence tothe stationary state. The Lyapunov functions used are : V (n) = zn|1+q with z > 1and q is the change quantity per event. In the limit order book modeling literature,the question of transience is never mentioned. In our model, we emphasize that westudy different cases of transience.

51

Partie I, Chapitre 3 – The fundamentals of the Markov chain N

In the section 3.1, we characterize the space of parameters which lead to theirreducibility of the Markov chain. In the section 3.2, we prove that the Markovchain can be recurrent and we exhibit a partial characterization of the space of pa-rameters which leads to the recurrence. In the section 3.3, we prove that the Markovchain can be transient and exhibit in the same fashion a partial characterization ofthe space of parameters. The naive characterization of transience gives that maincharacteristics of limit order book especially the bid price b and the total volume h

tends to infinity when time goes to infinity. In the section 3.4, we introduce the scopewhich is the difference between the bid price and the smaller price which containsnon zero volume. We give another characterization of transience by bounding thescope process. In the same manner in the section 3.5, we bound simultaneously thevolume process and the scope process in order to give a more realistic transientmarket.

3.1 The irreducibility

Theorem 3.1.1. Supposei. Qx[βB = b(x) − 1] > 0 if b(x) > 1, and Qx[βB = b(x)] > 0, if b(x) > 0, and

Qx[βB = b(x) + 1] > 0, for any x.ii. Qx[βC = x] > 0 for any x ∈ S(x).iii. Qx[αB = 1] > 0 and Qx[αC = 1] > 0, for any x.iv. pN(x) > 0, pB(x) > 0, pC(x) > 0 for any x.

Then, the Markov chain N is irreducible and aperiodic.

Proof : The Markov chain N is aperiodic, because pN(x) > 0 for all x. As for theirreducibility, we can prove that every x communicates with the null element of N ,and vice versa. Actually, for any non null element x, the conditions of the theoremallow, with a positive probability, chaining the cancellations till the annihilationof the total volume. Inversely, starting from a null element, beginning with thefirst valuation level, by suitable combinations of buys and cancellations, we canconstruct any element x. For example, to have (1, 1, 0, 0, 0, . . .), we combine, withpositive probability, the sequence of the following orders :

i. buy 1 size at value 1,ii. buy 1 size at value 2,iii. cancel 1 size at value 1.

The theorem is thus proved.

Remark 3.1.1. In fact, for the purpose of order book modeling, we need not alwaysto have the irreducibility on the whole space N .

52

3.2. The recurrence

3.2 The recurrence

The recurrence is a frequently asked question about the order book models. Oneof the reasons of studying the recurrence is that the limit theorems under recurrenceprovide the theoretical basis for the eventual calibration methods and for the studyon the stylized facts. For the model N, it is so flexible that there is no difficulty forN to have the recurrence. We will illustrate this feature by some examples.

We consider the function V introduced in Section 2.3.2.1 :

V (x) = x|1 + b(x).

Clearly, for any integer N > 0, x ∈ N : V (x) ≤ N is a finite set. The next resultfollows from Menshikov Theorem 2.5.2 [51]. It is the Foster-Lyapunov criterion.

Lemma 3.2.1. Suppose that the random variables αB, βB are integrable under Qx

for any x. The drift LV (x) is well-defined. If the set x ∈ N : LV (x) > 0 is a finiteset, the Markov chain N is recurrent.

Recall the formula (2.7) :

LV (x) = −pS(x)(Ex[αS] + Ex[b(x)− b(x −∗ αS)] )+pB(x)(Ex[αB] + Ex[b(x) ∨ βB − b(x)] )−pC(x)(Ex[αC ] + Qx[βC = b(x), αC ≥ x|b(x)](b(x)− b(x)) ).

With this formula, it is easy to find model specifications which make the set x ∈N : LV (x) > 0 finite. For example, we can simply define the random variablesunder Qx so that

pB(x) = 0, whenever V (x) becomes big. (3.1)

The condition (3.1) erases the only positive term of LV (x). It is not very realistic.Look at now another model specification :

i. Under Qx, the random variable (αB, βB) is uniformly bounded and has a pro-bability distribution independent of x, so that the quantity Ex[αB] + Ex[βB]is a constant D.

ii. The probabilities pS(x), pB(x), pC(x) also are positive constants.iii. UnderQx, the random variable αC is uniform on 1, . . . , ∆βCx and the random

variable βC distributed proportionally to ∆x.Under these conditions, the Markov chain N is essentially recurrent. Actually,

under the conditions, the reduced space x ∈ N : b(x) ≤ K, where K is the upperbound of βB, is absorbing for N. Moreover, according to the discussion in Section2.3.2.1,

Ex[αC ] = 12 + 1

2x|1∞∑i=1

(∆ix)2.

53


Hence, for x|1 big enough,

pB(x)D ≤ pC(x)(

12 + 1

2x|1∞∑i=1

(∆ix)2),

(recalling that pC(x) is by assumption a positive constant,) which implies that theset x ∈ N : LV (x) > 0, b(x) ≤ K is finite. Applying Lemma 3.2.1 to N as aMarkov chain in the reduced space, we obtain its recurrence in the reduced space.

We can change to another scenario.

i. The random variable βB is bounded.ii. There is a constant K such that the probabilities pS(x), pB(x), pC(x) also are

positive constants, whenever b(x) ≤ K. Otherwise, pS(x) = 1 (which defines amarket resistance level).

iii. UnderQx, the random variable αC is uniform on 1, . . . , ∆βCx and the randomvariable βC distributed proportionally to ∆x.

Again, the Markov chain N can be proved essentially recurrent in this scenarioin its value space.

To end the discussion on the recurrence, we notice that the random variables suchas αC , βB, etc., have simple economical meaning, whose probability distributionscan be calibrated directly from market data. The various limit theorems under therecurrence ensure the validity of the calibration.

3.3 The transience

As for the recurrence, the Foster-Lyapunov criterion (Theorem 2.5.8 [51]) canbe applied to establish the transience . However, for the proof of transience, theFoster-Lyapunov criterion is neither convenient nor intuitive. In this section, we willfollow a different, a more direct, and somewhat more intuitive approach. We applyLemma 2.3.1.

Corollary 3.3.1. Let F be a function on N , which satisfies the second conditionin Lemma 2.3.1. Suppose in addition that N is irreducible and, for some constant0 < θ <∞, LF (x) ≥ θ uniformly in x. Then, the Markov chain N is transient.

Proof : By Lemma 2.3.1,

lim infn→∞

F (Nn)n

≥ θ > 0,

which can not happen, if the Markov chain N is recurrent.

54

3.3. The transience

3.3.1 The volume function

We now consider the volume function h(x) = x|1. Recall (2.6) for the definitionsof αS and αC . If the random variables αB is integrable under Qx for any x, the driftLH(x) is well-defined and

Lh(x) = −pS(x)Ex[αS] + pB(x)Ex[αB]− pC(x)Ex[αC ]. (3.2)

Following Lemma 2.3.1 and Corollary 3.3.1, we consider the process

N|1n = N|10 +∑nk=1(N|1k −N|1k−1)

= N|10 +∑nk=1(−11δk=Sα

Sk + 11δk=Bα

Bk − 11δk=Cα

Ck )

= N|10 +∑nk=1(−11δk=Sα

Sk + 11δk=Bα

Bk − 11δk=Cα

Ck − LH(Nk−1))

+ ∑nk=1 LH(Nk−1).

The process is divided into two parts. The first part is a martingale :

M = N|10 +∑nk=1Xk, where

Xk = (−11δk=SαSk + 11δk=Bα

Bk − 11δk=Cα

Ck − LH(Nk−1)).

As in Lemma 2.3.1 lete(Nn−1) = E[X2

n |Fn−1].

Theorem 3.3.1. Suppose that N is irreducible. Suppose that the expectations Ex[(αS)2],Ex[(αB)2], Ex[(αC)2] are uniformly bounded. Suppose that, for a constant θ > 0,Lh(x) ≥ θ uniformly. Then,

lim infn→∞ N|1nn> 0.

In particular, the chain N is transient.

Proof : We compute

e(Nn−1) = E[(−11δn=SαSn + 11δn=Bα

Bn − 11δn=Cα

Cn − LH(Nn−1))2 |Fn−1]

= Ex[(−11δ=SαS + 11δ=BαB − 11δ=CαC − LH(x))2]for x = Nn−1,

= Ex[((−11δ=S(αS − Ex[αS ]) + 11δ=B(αB − Ex[αB])− 11δ=C(αC − Ex[αC ]))+ (−(11δ=S − pS(x))Ex[αS ] + (11δ=B − pB(x))Ex[αB]− (11δ=C − pC(x))Ex[αC ]))2]

= Ex[(−11δ=S(αS − Ex[αS ]) + 11δ=B(αB − Ex[αB])− 11δ=C(αC − Ex[αC ]))2]+ Ex[(−(11δ=S − pS(x))Ex[αS ] + (11δ=B − pB(x))Ex[αB]− (11δ=C − pC(x))Ex[αC ])2]

= pS(x)Ex[(αS − Ex[αS ])2] + pB(x)Ex[(αB − Ex[αB])2] + pC(x)Ex[(αC − Ex[αC ])2]+ Ex[(−(11δ=S − pS(x))Ex[αS ] + (11δ=B − pB(x))Ex[αB]− (11δ=C − pC(x))Ex[αC ])2].

55


By the assumption of the theorem, e(Nn−1) ≤ D for some constant D > 0. Hence,Corollary 3.3.1 is applicable, which proves the theorem.

Consider the following example. Let h, g ∈ N∗. Suppose the assumptions inTheorem 3.1.1 and Theorem 3.3.1. Suppose, in addition, that Ex[αB] is a constantab > 0 and that, under Qx for x 6= 0, βC is a uniform random variable on S(x) andαC is a uniform random variable on 1, . . . , g ∧∆βCx, and αS is a uniform randomvariable on 1, . . . , h ∧ x|1. Under this condition,

Lh(x) = −pS(x)1+h∧x|12 + pB(x)ab − pC(x)11x 6=0

1σ(x)

∑k∈S(x)

1+g∧∆kx2

≥ −pS(x)1+h2 + pB(x)ab − pC(x)1+g

2 ,

where σ(x) denotes the number of elements in S(x). Suppose that the probabilitiespS(x), pB(x), pC(x) are constant and are so that

θ := −pS1 + h

2 + pBab − pC1 + g

2 > 0. (3.3)

Then, the Markov chain N is transient.

3.4 A more realistic transient model

The study of the recurrence and the transience of the Markov chain N is moti-vated by the natural idea that the recurrence corresponds to a flat or bear market,while the transience corresponds to a bullish market. In a bullish market, the bidb(Nn) tends to the infinity, that is why the Markov chain N must be transient.However, even in a bullish market, all the market indicators do not go to the theinfinity. For example, the scope process b(Nn) − d(Nn) should not explode. (Seesection 2.1 for the definition of d(x).)

This section is devoted to a study of the parameterization which makes N a morerealistic market model.

3.4.1 The bid b(Nn) and the scope b(Nn)− d(Nn)

Recall

S(x) = k ∈ N∗ : x|k > x|k+1,d(x) = inf S(x) if S(x) 6= ∅, and d(x) = 0 if S(x) = ∅.

Let in addition

σ(x) = ]S(x),d(x) = inf(S(x) \ d(x)) if S(x) \ d(x) 6= ∅, and d(x) = 0 otherwise.

56

3.4. A more realistic transient model

Our intention is to find a parameterization of the Markov chain N so that the bidb(Nn) tends to the infinity, while the scope b(Nn)− d(Nn) remains, in some sense,recurrent. We write the drift

Lb(x) = −pS(x)Ex[b(x)− b(x −∗ αS)] + pB(x)Ex[b(x) ∨ βB − b(x)] (3.4)

− pC(x)Qx[βC = b(x), αC ≥ x|b(x)](b(x)− b(x)).

As for the function d(x), we have

d(Nn+1) =11δn+1=S,αS<x|1d(x) + 11δn+1=Bd(x) ∧ βBn+1

+ 11δn+1=C11βCn+1=d(x),αC≥∆βC

xd(x) + 11δn+1=Nd(x)

so that

Ld(x) =− pS(x)Qx[αS ≥ x|1]d(x) + pB(x)Ex[d(x) ∧ βB − d(x)] (3.5)

+ pC(x)Qx[βC = d(x), αC ≥ ∆βCx](d(x)− d(x)).

Putting together the two, we obtain the drifts

L(b− d)(x) = −pS(x)(Ex[b(x)− b(x −∗ αS)]−Qx[αS ≥ x|1]d(x))+pB(x)(Ex[b(x) ∨ βB − b(x)]− Ex[d(x) ∧ βB − d(x)])−pC(x)(Qx[βC = b(x), αC ≥ x|b(x)](b(x)− b(x))

+Qx[βC = d(x), αC ≥ ∆βCx](d(x)− d(x))).

(3.6)

Notice that the last term writes also as

−pC(x)(Qx[βC = b(x) > d(x), αC ≥ x|b(x)](b(x)− b(x))+Qx[βC = d(x) < b(x), αC ≥ ∆βCx](d(x)− d(x))).

In the next section, we will consider the increments

(b− d)(Nn+1)− (b− d)(Nn)= 11δ=S(b(x −∗ αS)− b(x))− 11δ=S11αS≥x|1(0− d(x))

+11δ=B((b(x) ∨ βB − b(x))− (d(x) ∧ βB − d(x)))+11δ=C(11βC=b(x),αC≥x|b(x)(b(x)− b(x))− 11βC=d(x),αC≥∆d(x)x(d(x)− d(x)))

≤ 11δ=S(b(x −∗ αS)− b(x))− 11δ=S11αS≥x|1(0− d(x))+11δ=B((b(x) ∨ βB − b(x))− (d(x) ∧ βB − d(x)))+11δ=C11βC=b(x)>d(x),αC≥x|b(x)(b(x)− b(x))−11δ=C11βC=d(x)<b(x),αC≥∆d(x)x(d(x)− d(x)) ∧ g)

=: Fg(x, Φ)(3.7)

57


with x = Nn and g > 0 a constant, where the δ denotes δn+1, etc., and Φ denotesthe set of all these random variables. Taking the expectation of Fg(x, Φ) under Qx,we obtain the function

Lg(b− d)(x) := −pS(x)(Ex[b(x)− b(x −∗ αS)]−Qx[αS ≥ x|1]d(x))+pB(x)(Ex[b(x) ∨ βB − b(x)]− Ex[d(x) ∧ βB − d(x)])−pC(x)(Qx[βC = b(x) > d(x), αC ≥ x|b(x)](b(x)− b(x))

+Qx[βC = d(x) < b(x), αC ≥ ∆βCx](d(x)− d(x)) ∧ g),(3.8)

considered as a truncation of the drift L(b− d)(x).

3.4.2 A result about the scope in a transient market

We want to fix parameterizations which make Lb(x) and L(b− d)(x) positive ornegative. This can be easily done, if we impose no restriction on the probabilitiespS(x), pB(x), pC(x). But we should not let these probabilities depending too muchof x, because this dependence will make impossible the calibration. Hence, in oursearch of the parameterization, we try to keep these probabilities as homogeneousas possible.

i. Suppose the assumption of Theorem 3.1.1.ii. αS is bounded.iii. pS(x) = 0 if Qx[αS ≥ x|1] = 0, and pS(x) is constant otherwise.iv. βB takes the form :

βB = b(x)− (1− ε)βB− + εβB−,

where the random variables βB− ∈ [0, b(x)], βB− ∈ N∗ are independent ofε ∈ 0, 1 under Qx (cf. Section 2.3.2.2), and βB−, βB− are bounded.

v. Under Qx, βC follows the law of a geometric(p) (same p for all x) randomvariable conditioned on the set S(x).

Notice that under the above conditions,

Ex[b(x) ∨ βB − b(x)] = Qx[ε = 1]Ex[βB−],Ex[d(x) ∧ βB − d(x)] = −Qx[ε = 0]Ex[(βB− − (b(x)− d(x))+].

Theorem 3.4.1. Suppose the above conditions on the random variables αS, βB, βC.Suppose also the followings :

i. For a constant c > 0, Lb(x) ≥ c for all x.ii. For some constants g > 0, h > 0, c′ > 0, Lg(b− d)(x) ≤ −c′ for all x such that

b(x)− d(x) ≥ h.

Then, the bid b(Nn) tends to the infinity (so that the Markov chain N is transient),while the scope process (b−d)(Nn) turns back to values smaller than h, after whatever

58

3.4. A more realistic transient model

long period.

Proof : Notice that, by Theorem 3.1.1, N is irreducible. There are two processesb(Nn) and (b− d)(Nn) to be studied. They can be both studied in the line fixed byLemma 2.3.1 and Corollary 3.3.1. For the bid process, we have in fact

0 < lim infn→∞

b(Nn)n≤ lim sup

n→∞

b(Nn)n

<∞. (3.9)

But we will not give the proof of (3.9).

We now consider the scope process

(b− d)(Nn) = (b− d)(N0) +∑nk=1((b− d)(Nk)− (b− d)(Nk−1))

≤ (b− d)(N0) +∑nk=1 Fg(Nk−1, Φk)

according to (3.7). Consider the martingale increments

Xk = Fg(Nk−1, Φk)− Lg(b− d)(Nk1).

We introduce

u1 = (b(x −∗ αS)− b(x)),u2 = (b(x) ∨ βB − b(x))− (d(x) ∧ βB − d(x)),u3 = 11βC=b(x)>d(x),αC≥x|b(x)(b(x)− b(x))− 11βC=d(x)<b(x),αC≥∆d(x)x(d(x)− d(x)) ∧ g,

and

w1 = Ex[b(x −∗ αS)− b(x)],w2 = Ex[b(x) ∨ βB − b(x)]− Ex[d(x) ∧ βB − d(x)],w3 = Qx[βC = b(x) > d(x), αC ≥ x|b(x)](b(x)− b(x))

−Qx[βC = d(x) < b(x), αC ≥ ∆βCx](d(x)− d(x)) ∧ g,

so that wi = Ex[ui] and

Lg(b− d)(x) = pS(x)w1 + pB(x)w2 + pC(x)w3.

With x = Nk−1, we have :

E[X2k | Fk−1] = Ex[(11δ=Su1 − pS(x)w1 + 11δ=Bu2 − pB(x)w2 + 11δ=Cu3 − pC(x)w3)2].

Notice that, under the conditions of the theorem, the expectations Ex[(ui)2] areuniformly bounded. (For the computation of u3, see next section.) (cf. Hall and

59


Heyde Theorem 2.18 [40]] is applicable. We obtain

lim supn→∞

(b− d)(Nn)n

= lim supn→∞

1n

n∑k=1

Lg(b− d)(Nn−1).

If the values (b − d)(Nn) ≥ h > 0 for all but a finite number of n ∈ N, by theassumption of the proposition, we would have

0 ≤ lim infn→∞

(b− d)(Nn)n

= lim supn→∞

1n

n∑k=1

Lg(b− d)(Nk−1) ≤ −c′,

an obvious paradox. The theorem is proved.

3.4.3 A discussion about the assumptions in Proposition3.4.1

Let us show in this section that the conditions in Theorem 3.4.1 are realistic.Suppose the conditions of Theorem 3.4.1. Look at the drift Lb(x) :

−pS(x)Ex[b(x)− b(x −∗ αS)] + pB(x)Ex[b(x) ∨ βB − b(x)]−pC(x)Qx[βC = b(x), αC ≥ x|b(x)](b(x)− b(x)).

The first term is bounded because of Proposition 2.3.1, as well as the second term.Look at the last term Qx[βC = b(x), αC ≥ x|b(x)](b(x)− b(x)) :

Qx[βC = k] = pqk∑i∈S(x) pqi

11k∈S(x), (3.10)

and consequently, when b(x) > 0,

Qx[βC = b(x), αC ≥ x|b(x)](b(x)− b(x))= Qx[αC ≥ x|b(x) | βC = b(x)] qb(x)∑

k∈S(x) qk (b(x)− b(x))

≤ Qx[αC ≥ x|b(x) | βC = b(x)] qb(x)

qb(x)+qb(x) (b(x)− b(x))

= Qx[αC ≥ x|b(x) | βC = b(x)] 11+qb(x)−b(x) q

b(x)−b(x)(b(x)− b(x))≤ Qx[αC ≥ x|b(x) | βC = b(x)]× e−1

− ln q ≤e−1

− ln q .

We see now that it is easy to choose pS(x), pB(x), pC(x) so that Lb(x) > 0 uniformly.

Consider next the three terms

− pS(x)Qx[αS ≥ x|1]d(x), pB(x)Ex[d(x) ∧ βB − d(x)],

− pC(x)Qx[βC = d(x), αC ≥ ∆βCx](d(x)− d(x)) ∧ g.

60

3.5. The volume index in a bullish market

The first term pS(x)Qx[αS ≥ x|1] ≡ 0. For the second term, as βB takes the form

βB = b(x)− (1− ε)βB− + εβB−,

the expectation

Ex[d(x)− d(x) ∧ βB] = Ex[(d(x)− βB)+]= Qx[ε = 0]Ex[(d(x)− b(x) + βB−)+] ≤ Qx[ε = 0]Ex[(βB− − h)+],

when b(x) − d(x) ≥ h. When the constant h is large enough, Ex[(βB− − h)+] = 0.Finally, for the third term, as βC follows the conditioned geometric distribution(3.10), we have, when d(x)− d(x) > 0, g ≥ 1,

Qx[βC = d(x), αC ≥ ∆βCx](d(x)− d(x)) ∧ g= Qx[αC ≥ ∆d(x)x | βC = d(x)] qd(x)∑

k∈S(x) qk (d(x)− d(x)) ∧ g

≥ Qx[αC ≥ ∆d(x)x | βC = d(x)],

which can be chosen uniformly bounded below.Notice that no constraint has been imposed on pC(x) when b(x) > d(x). After

the above analysis, it is easy to define the value of pC(x) to make Lg(b−d)(x) ≤ −c′

for some c′ > 0 when b(x)− d(x) ≥ h.

3.5 The volume index in a bullish market

As in Section 3.4, we discuss the adaptation of our model to real market si-tuations. A sufficient condition of the transience of N has been given in Theorem3.3.1 in term of the volume index h(Nn) which tends to the infinity. However, in areal market situation, the bid process can rise continuously, but the volume shouldremain limited. In this section we search the parameterization of N which modelssuch a market situation. We have the following theorem.

Theorem 3.5.1. Suppose the assumptions in Theorem 3.4.1. Suppose the assump-tions in Theorem 3.3.1, except the condition Lh(x) ≥ θ which is replaced by : forsome constant c′ > 0, h′ > 0, Lh(x) ≤ −c′ whenever x|1 > h′. Then, the Markovchain N is transient, while the scope process (b − d)(Nn) and the volume processN|1n turn back to small values, after whatever long period.

Proof : This theorem can be proved in the same way as Theorem 3.4.1 has beenproved. Notice only that, and the assumption of the theorem imply that, for someh′ > 0, Lh(x) ≤ −c′ whenever x|1 > h′, and consequently, the volume process N|1ncan not remain long time above the level h′, because of Lemma 2.3.1.

61


The assumptions in Theorem 3.5.1 can be satisfied by reasonable parameters. Infact, it is enough to refine the conditions given in section 3.4.3.

Only the term

Lh(x) = −pS(x)Ex[αS] + pB(x)Ex[αB]− pC(x)Ex[αC ]

is to be considered. Notice that, as βC takes values in S(x), Ex[αC ] ≥ 1 wheneverx|1 > 0. We suppose that pC(x), αC are so that

pC(x)Ex[αC ] = apc

is a positive constant. We suppose that ab = Ex[αB] is a positive constant, as wellas Ex[αS]. We will not modify the random variable αS, βB−, βB−, βC fixed in section3.4.3. As for the probabilities pS(x), pB(x), we choose them so that

0 < pS(x) < 1 a constant,0 < pB(x) ≤ apc

ab, for x such that x|1 > h′.

Conclusion

We found that our model is enough flexible for letting the Markov Chain to berecurrent or transient. The recurrent case is well studied in the literature and weuse the recurrent parameters set for the calibration in the next chapter 4.3. For thetransient case, the introduction of the scope process is an original way to expressthe fact that all limit orders are not not too far to the bid price. With the fact thatthe volume process is bounded, the total volume H doesn’t explode to infinity whichis the financial reality. We proved in all cases, the existence of parameters and wederived in particular cases the set of the parameters which lead to the properties.

62

Chapitre 4

Estimations et calibrations dumodèle

Contents4.1 Présentation des données . . . . . . . . . . . . . . . . . . . 64

4.1.1 Traitement des données . . . . . . . . . . . . . . . . . . . 65

4.2 Estimations de paramètres . . . . . . . . . . . . . . . . . . 67

4.3 Calibration par rapport au problème de récurrence . . . 70

4.4 Faits stylisés . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5 Critères de liquidité de marché . . . . . . . . . . . . . . . 75

Introduction

Ce chapitre est consacré au traitement du problème de calibration et d’estimationde paramètres du modèle.

Dans le chapitre précédent, nous avons abordé le problème de classification denotre modèle et nous avons exhibé un ensemble de paramètres pour laquelle lachaine de Markov est récurrente, transiente. Dans ce chapitre, nous confrontonsnotre modèle avec les données du marché. Il s’agit de la validation du modèle.

Dans la littérature, afin de valider les modèles, ils étudient, de manière empiriqueles faits stylisés. Les modèles de carnet d’ordres qui étudient les problématiques d’es-timation et de calibration, se ramènent à un problème d’estimation. En effet, ils sontmodélisés à l’aide de processus de Poisson et ils estiment les intensités des processusde Poisson à l’aide des données et d’une méthode de maximum de vraisemblance. Iln’y a donc aucun degré supplémentaire dans leur modèle et il n’y a pas de calibrationà faire.

Notre problème de calibration consiste à trouver les paramètres dans l’ensembledes probabilités Q dans Θrec. Pour réduire l’ensemble des lois d’arrivées, nous allonsnous placer dans le cas d’une chaine de Markov récurrente. Il s’agit d’un choixartificiel mais nous justifions ce choix par la littérature existante [24] et [2].

Les faits stylisés et les critères de liquidité donnent des restrictions aux modèlesstochastiques de carnet d’ordres. Les bons modèles devraient être capable de cap-

63

Partie I, Chapitre 4 – Estimations et calibrations du modèle

turer simultanément la plupart d’entre eux avec peu de paramètres. En se plaçantdans Θrec, nous étudions les faits stylisés et les critères de liquidité en imposant deslois.

Dans la section 4.1, nous étudions le traitement et les statistiques descriptivesdes données. Les données du marché, provenant du future Bund, sont séparées en2 sources : une possédant l’ensemble des transactions et une possédant les états ducarnet. Cependant, les deux sources ne possèdent pas le mime horodatage et entredeux états successifs du carnet, plusieurs évènements peuvent se dérouler. Nousproposons une méthode de reconstruction. Cela donne, par la mime occasion, uneinformation sur la qualité des données. Dans la section 4.2, nous étudions le pro-blème d’estimation de paramètres et nous comparons les faits stylisés et des critèresde liquidité entre le modèle simulé et les données du marché. Nous utilisons desméthodes d’estimation de maximum de vraisemblance. Nous donnons une paramé-trisation concrète des problèmes de classification dans la section 4.3. Nous étudionsles faits stylisés dans la section 4.4 ( définit dans la sous-section 1.2.1) et les critèresde liquidité dans la section 4.5 ( définit dans la sous-section 1.2.2)

4.1 Présentation des données

Nous utilisons les données de niveau 2 du contrat du Future Bund négociées surle marché de l’EUREX. Les données ont une précision de l’ordre de la millisecondeet ont été enregistrées entre mars 2013 et septembre 2013. Un contrat Future est uncontrat autorisant l’achat d’un sous-jacent (pétroliers, actions ou devises) à une datefuture et à un prix déterminé à l’instant initial du contrat. Il permet aux traders desécuriser le prix du sous-jacent qu’ils devront délivrer physiquement ou recevront àla date de livraison. Par conséquent àchaque instant, il peut exister un nombre infinide contrats, un pour chaque date de livraison. Sur un marché comme l’EUREX, lesdates de livraison sont standardisées, tous les trois mois (mars, juin, septembre etdécembre) et généralement peuvent être négociés en mime temps.

Dans nos données, nous avons 2 types de fichiers : "Quote Data raw" et "TradeData Raw".

Le fichier "Trade Data Raw" est une liste de transaction décrit par un horodatage,le prix de la transaction, la taille de la transaction. L’horodatage de la transactionest l’instant de la réception et de la validation de l’ordre par le marché. Il y a un prixunique de transaction sur chaque ligne. Lorsqu’un ordre de marché est exécuté contredes ordres limites possédant différents niveaux de prix, l’exécution est reportée surplusieurs transactions.

64

4.1. Présentation des données

Timestamp LTP LTV06 :01 :18.767 145.96 106 :01 :18.908 145.97 1

Trade Data Raw

Figure 4.1 – Représentation d’un fichier Trade Data Raw avec la colonne Times-tamp correspondant à l’horodatage de la transaction, la colonne LTP correspondantau niveau de prix de la transaction et la colonne LTV correspondant à la taille dela transaction

Timestamp bid 0 Volume bid 0 Ask 0 Volume Ask 0 . . . Ask 9 Volume Ask 906 :01 :20.517 145.96 48 145.97 41 . . . 146.06 5406 :01 :20.766 145.96 53 145.97 46 . . . 146.06 54

Quote Data Raw

Figure 4.2 – Représentation d’un fichier Quote Data Raw avec la colonne Times-tamp correspondant à l’horodatage de la réception de l’état du carnet d’ordres, lacolonne bid0 correspondant au meilleur niveau de prix à la vente et la colonne LTVcorrespondant au volume disponible au au niveau de prix bid0

Le fichier "Quote Data Raw" est une liste d’états du carnet d’ordre décrit parun horodatage, une liste fini (de taille K) de couple (prix, volume) qui sont les Kmeilleurs prix à l’achat et à la vente associées aux volumes correspondant. Les étatsdu carnet d’ordre dans les données sont des représentations partielles des états ducarnet d’ordre dans le marché. Pour des actifs financiers peu liquides et émergents,la liste des K meilleurs prix sont souvent non contigües. En regardant deux lignesconsécutives, il est possible de remarquer des différences de volume sur plusieursniveaux de prix. Cela implique qu’entre deux états consécutifs, il peut y avoir plusd’une arrivée d’évènements.

Il y a un décalage entre l’horodatage de la transaction et l’horodatage de lanotification de la modification de l’état du carnet d’ordres par la transaction.

Nous expliquons les procédures pour traiter les données puis nous donnonsquelques statistiques descriptives des données.

4.1.1 Traitement des données

Roulement (Rolling) Les futures possèdent plusieurs maturités simultanément.Nous traitons ce problème en gardant, pour chaque journée, la maturité enregistrantla plus grand somme de tailles de transaction, en éliminant les autres maturités. Nousconcaténons les données de transaction en appliquant le traitement et nous faisonscorrespondre les données des états de carnet d’ordre.

65


Timestamp bid 0 Volume bid 0 Ask 0 Volume Ask 0 . . . Ask 9 Volume Ask 906 :01 :20.517 145.96 48 145.97 41 . . . 146.06 5406 :01 :20.766 145.96 53 145.97 46 . . . 146.06 47

Figure 4.3 – Quote Data Raw

Fusion de transactions Lorsqu’un ordre marché traverse plusieurs niveaux deprix, nous rappelons que dans les données de transaction, cela crée plusieurs tran-sactions. Si nous gardons les données originales de transaction, certains paramètresseront mal estimés. En effet, lorsque nous voulons estimer la variable aléatoire re-présentant la taille d’un ordre marché lorsqu’elle dépend de l’état, alors un ordrede marché ne pourra jamais dépasser le meilleur niveau de prix du carnet. Nousagrégeons toutes les transactions possédant le mime horodatage et le mime prix detransaction, en sommant la taille des transactions et en gardant le meilleur prix.

Hypothèses sur les données Dans le fichier de données "Quote Data Raw",plusieurs évènements peuvent arriver entre deux lignes consécutives représentantdeux états du carnet d’ordres. Nous faisons l’hypothèse qu’au mime niveau de prixentre deux lignes consécutives, il ne peut y avoir qu’un seul évènement qui a produitle changement. Par exemple dans la représentation 4.3, au niveau de prix 146.06, ladifférence de volume est négative, nous considérons alors qu’il y a eu une annulationde taille 7 au niveau de prix 146.06. Au niveau de prix 146.06, la différence de volumeest positive, nous considérons qu’il y a eu un ordre limite de taille 5.

Nous randomisons les évènements entre deux états afin d’éviter une possibledépendance. Nous construisons ainsi les données d’évènements.

Distinction entre une annulation aux meilleurs niveau de prix et une tran-saction Lorsque nous construisons les données d’évènements, dans le cas d’unedifférence de volume négative au niveau du bid, nous voulons savoir si l’évènementest une annulation ou une transaction. Pour décider, nous utilisons le fichier de don-nées "Trade Data Raw". Nous prenons les horodatages que l’on note t du fichier de"Trade Data Raw" et nous cherchons dans une fenêtre [t−∆t, t+∆t] dans le fichierd’évènements, s’il y a un évènement aux meilleurs niveaux de prix, possédant unedifférence de volume négative et la différence est égale à la taille de la transaction.Cette méthode possède plusieurs buts :

— Cela permet de différencier les annulations et les transactions.

— Cela permet de donner le signe des ordres marchés (une transaction n’a pasde signe). En effet si la transaction coïncide avec une différence de volume auniveau du bid (resp. de l’ask) alors l’ordre marché est une vente (resp. achat).

66

4.2. Estimations de paramètres

— Cela permet aussi de connaitre la granularité des données donc la qualité desdonnées.

Nous faisons coïncider 72% des transactions avec les données d’évènements.

4.2 Estimations de paramètres

Pour la spécification de chaque variable aléatoire, nous invitons à lire αS leparagraphe 2.3.3, αC le paragraphe 2.3.5, βC le paragraphe 2.3.4, αB le paragraphe2.3.1 et βB le paragraphe 2.3.2. Nous nous référons au livre de Lejeune [46] pourl’estimation de paramètres.

Nous rappelons la fonction de vraisemblance d’un paramètre θ selon n obser-vations (x1, . . . , xn) indépendants et identiquement distribuées d’une loi lθ. Nousvoulons estimer θ dans le cas discret :

L(x1, . . . , xn, θ) :=n∏i=1

Pθ[X = xi]

Pour trouver l’estimateur par la méthode du maximum de vraisemblance, nous cher-chons le maximum de la fonction L par rapport à θ mais cela suppose d’imposer dansnotre cas, une loi à priori. Si nous posons Xn = 1

n

∑ni=1 xi et Sn = 1

n

∑ni=1 x

2i alors

dans le cas où la distribution est log-normale (µ, σ), l’espérance est E[X] = eµ+ 12σ

2 etla variance est V [X] = e2µ+σ2(eσ2−1)). il y a deux paramètres à estimer la moyenneµ et l’écart-type σ, nous trouvons :

µ = log(Xn)− 12 log

(SnX2n

+ 1)

et σ =

√√√√log(SnX2n

+ 1),

Dans le cas binomiale (n, p) (E[X] = np et V [X] = np(1− p)), il y a un paramètreà estimer p, p = Xn avec le paramètre n donné.

Dans le cas géométrique p (E[X] = 1−pp

et V [X] = 1−pp2 ), il y a un paramètre à

estimer p, p = 1Xn

.Dans un premier temps pour les cas indépendants de l’état courant des variables

αS et αC , nous calculons les estimateurs des paramètres des variables aléatoiresαB, βB+, βB−, αS, βC−, αC pour une loi log-normale, une loi binomiale et une loigéométrique. Nous pouvons calculer les intervalles de confiance asymptotique auseuil α de nos estimateurs à l’aide de ICα = [θ − xα√

nI(θ), θ + xα√

nI(θ)] avec xα le

(100 − α) percentile de la loi normale et I(θ) la matrice d’information de Fisherpour θ (cf chapitre 7 section 3 de [46].

Afin de choisir une des lois, nous réalisons des tests d’adéquation du khi-deux.

67


Log-normale µ, σ Binomiale p Géométrique pαS 2.44, 1.32 (0.023, 0.043) 0.025 (2× 10−5) 0.036 (6× 10−4)αB 1.229, 1.552 (0.008, 0.018) 0.008 (1.3× 10−6) 0.088 (4× 10−4)αC 1.041, 1.647 (0.010, 0.024) 1.352 (1.4× 10−6) 0.091 (5× 10−4)βC− 0.835, 0.753 (0.005, 0.005) 0.340 (9× 10−6) 0.327 (1.6× 10−3)βB− 0.623, 0.837 (0.005, 0.005) 0.294 (7× 10−6) 0.378 (1.6× 10−3)βB+ 0.001, 0.060 (0.002, 1.9× 10−4) 0.001 (3× 10−5) 0.997 (0.002)

Figure 4.4 – Calcul des estimateurs des paramètres et du terme intervenant dansl’intervalle de confiance asymptotique ( xα√

nI(θ)) donné entre parenthèses

(a) pour la variable aléatoire αB (b) pour la variable aléatoire αS

(c) pour la variable aléatoire αS (d) pour la variable aléatoire βB−

Figure 4.5 – Diagramme Quantile-Quantile contre la loi log-normale de paramètresdonnés dans 4.4

— Pour αB, αS, αC , nous ne pouvons pas rejeter l’hypothèse nulle (égalité entreles données et la loi log-normale avec les paramètres estimés).

— Pour βB− et βC−, nous ne pouvons pas rejeter l’hypothèse nulle (égalité entreles données et les lois log-normale et géométrique avec les paramètres estimés).

— Pour βB+, nous ne pouvons pas rejeter l’hypothèse nulle (égalité entre lesdonnées et la loi binomiale avec les paramètres estimés).

Les résultats des tests d’adéquation semblent cohérents car les variables αB, αS,αC sont de même nature, ils représentent des tailles d’ordres. Il en est de mêmepour βB− et βC− qui correspondent aux placements d’ordres qui ne modifient pasle prix du bid. La variable aléatoire βB+ correspond aux placements d’ordres dansle spread. L’adéquation avec la loi binomiale, dont le support est fini, est en accordavec la bornitude du spread.

68

4.2. Estimations de paramètres

Pour estimer δ, nous devons introduire une fenêtre ∆t pour considérer l’évène-ment δ = N. Notre modèle est à temps discret et deux temps d’arrivée consécutifsdes évènements sont espacées par ∆t constant. Ainsi, nous calculons la fréquencefδ=A d’avoir un évènement de type A = S,C,B,N dans une fenêtre ∆t :

fδ=A = Occurrence de l’évènement de type A ×∆tT

avec T la durée de l’échantillon considérée.Pour estimer les paramètres des variables aléatoires εB et εC , nous supposons que

les variables εB et εC suivent des lois de Bernoulli. Nous donnons le terme intervenantdans l’intervalle de confiance pour les estimateurs. Nous calculons les fréquences (laproportion) en comptant le nombre d’occurrence de l’évènement

— d’avoir un ordre limite possédant un niveau de prix inférieur ou égal au prixdu bid (εB = 0),

— d’avoir un ordre limite possédant un niveau de prix supérieur au prix du bid(εB = 1),

— d’avoir une annulation totale au niveau du bid, c’est à dire αC = n|b(n) etβC = b(n) si l’état du carnet est n (εC = 1)

— et d’avoir une annulation partielle, c’est à dire αC 6= n|βC ou βC 6= b(n) sil’état du carnet est n (εC = 0)

Nous obtenons pour l’estimateur du paramètre de la variable aléatoire εB, pεB =0.981(IC = 7.5 × 10−4) et pour εC , pεC = 0.991(IC = 5 × 10−4). Nous réalisonsdes tests d’adéquation du khi-deux. Pour εB, εC , αC , nous ne pouvons pas rejeterl’hypothèse nulle (égalité entre les données et la loi de Bernoulli avec les paramètresestimés).

Pour estimer αS dans le cas dépendant (voir le paragraphe 2.3.3), nous utilisonsla moyenne temporelle empirique de n|b(n)−l, c’est à dire limn→∞

∑nk=0 n|b(N)−l

k . Si αS

est uniforme alors αS est uniforme dans 1, . . . , n|b(n)−l. Il n’y a pas de paramètreà estimer. En revanche, si αS suit une loi binomiale alors αS suit une loi bino-miale de paramètres (n|b(n)−l, p). Nous estimons p par une méthode de maximum devraisemblance avec la moyenne temporelle empirique de n|b(n)−l.

Pour estimer αC dans le cas dépendant (voir le paragraphe 2.3.5), nous utilisonsla moyenne temporelle empirique de ∆βCn−1, c’est à dire limn→∞

∑nk=0(∆βC

knk−1).

Si αC est uniforme alors αC est uniforme dans 1, . . . , ∆βCn − 1. Il n’y a pas deparamètre à estimer. En revanche, si αC suit une loi binomiale alors αS suit uneloi binomiale de paramètres (∆βCn − 1, p). Nous estimons p par une méthode demaximum de vraisemblance avec la moyenne temporelle empirique de ∆βCn− 1.

69


4.3 Calibration par rapport au problème de ré-currence

D’après la section 3.2, nous exhibons une condition sur les paramètres pourconnaitre la nature de la chaine de Markov N. Une condition suffisante est donnéepar :

Soit x ∈ N , D = Ex[αb] + Ex[βb] et pour x|1 assez grand, si les paramètresvérifient cette inégalité :

pb(x)D ≤ pc(x)Ex[αc],

alors N est récurrente. Nous cherchons si les données du marché donnent une chainerécurrente ou non.

Nous testons l’inégalité avec les paramètres et cherchons s’il existe un entier Npour caractériser la notion d’assez grand pour x|1. Afin d’étudier cette inégalité, nousreprésentons sur une figure, pour chaque endroit de l’espace N , si la condition estvérifiée. Nous utilisons les caractéristiques principales de N , c’est à dire le prix dubid b(x) et le volume total x|1 pour représenter graphiquement des éléments de N .b(x) et x|1 suffisent car ce sont les caractéristiques misent en jeu pour la questionde récurrence. Dans les données, nous n’avons pas accès àx|1, nous choisissons deconsidérer x|b(x)−p avec p=9. Sur chaque figure, nous représentons en abscisse levolume total (x|b(x)−p − x|b(x)−p

min)/120 et en ordonnée, le prix du bid b(x) − bmin

avec x|b(x)−pmin la plus petite valeur de x|b(x)−p dans l’échantillon.

Nous étudions les deux cas pour αC : indépendant et dépendant de l’état courantdu carnet. Nous utilisons les lois et les paramètres obtenues de la section 4.2. Pour lesvariables βC− et βB−, nous utilisons par défaut la loi log-normale avec les paramètresestimés dans le tableau 4.4. Pour les autres variables, il n’y a pas d’ambiguïté voirrésultat des tests d’adéquations 4.2.

αC indépendant de l’état du carnet Lorsque αC est indépendant de l’état ducarnet, l’inégalité ne dépend pas de x.

Dans le cas où αC , βC suivent une loi log-normale de paramètres données dansle tableau4.4 alors la condition de récurrence est vérifiée nulle part (Figure 4.6).

αC dépendant de l’état du carnet Nous considérons plusieurs cas : dans le casoù αC suit une loi uniforme, nous retombons dans le cadre précédent. L’inégalité estvrai pour tout x ∈ N . La représentation est la mime que 4.6.

Dans le cas où αC suit une loi binomiale, nous avons un cadre intéressant. Nousavons un ensemble de positions qui vérifie la condition et un autre ensemble qui nela vérifie pas. Les deux ensembles sont disjoints et sont séparés par une frontière. Lafrontière est différente selon le choix de βC (Figure 2a, 4.7a et 4.7b)

70

4.3. Calibration par rapport au problème de récurrence

Figure 4.6 – Représentation des positions qui vérifient la condition de récurrence :les parties noires sont les positions non atteintes par les données du marché et lapartie grise correspond aux positions vérifiant la condition. En abscisse, nous avonsle volume total (x|b(x)−p− x|b(x)−p

min)/120 et en ordonnée, le prix du bid b(x)−bmin.αC suit une loi log-normale et βC suit une loi log-normale de paramètres donnéesdans le tableau 4.4

(a) βC suit une loi géométrique (b) βC suit une loi log-normale

Figure 4.7 – Représentation des positions qui vérifient la condition de récurrence :les parties noires sont les positions non atteintes par les données du marché, lapartie blanche correspond aux positions ne vérifiant pas la condition, la partie grisecorrespond aux positions vérifiant la condition. En abscisse, nous avons le volumetotal (x|b(x)−p − x|b(x)−p


71


Figure 4.8 – Représentation de l’espace N des positions les plus visitées. Plus leszones sont blanches, plus elles sont visitées. En abscisse, nous avons le volume total(x|b(x)−p − x|b(x)−p


La frontière entre les deux zones caractérise la notion de "assez grand" dans lacondition de récurrence. La zone vérifiant la condition dans le cas de la loi géomé-trique est plus large (Figure 4.7a). La partie grise correspond à la zone où le driftest négatif, donc revient vers le "centre" de l’espace N . Nous traçons une représen-tation de l’espace N pour caractériser le "centre" de l’espace N pour illustrer. Le"centre" de l’espace N a une forme de croissant correspondant à la partie proche dela frontière.

Conclusion Le cas αC dépendant de l’état et suivant une loi binomiale semble pluscorrespondre à une certaine réalité notamment en décrivant le caractère récurrenten séparant en deux zones l’espace N . Nous allons examiner de manière empirique,les inconvénients et les avantages du modèle simulé par rapport aux faits stylisésdans la section 4.4 et par rapport aux critères de liquidité de marché dans la section4.5.

4.4 Faits stylisés

La section précédente nous indique l’importance de la dépendance de αC parrapport à l’état. Nous considérons ce cas là. Nous utilisons les lois et les paramètresobtenues de la section 4.2. Pour les variables βC− et βB−, nous utilisons par défautla loi log-normale avec les paramètres estimés dans le tableau 4.4. Pour les autresvariables, il n’y a pas d’ambiguïté (voir les résultats des tests d’adéquations 4.2).

72

4.4. Faits stylisés

(a) dt = 1 (b) dt = 10

(c) dt = 100 (d) dt = 1000

Figure 4.9 – Distribution des incréments de prix pour différents ∆t

Nous nous référons à la sous section 1.2.1 du chapitre 1 pour une introduction surles faits stylisés.

Afin d’étudier la distribution des incréments de prix, nous étudions la distribu-tion empirique des incréments de prix. Pour définir la distribution empirique desincréments de prix, nous simulons une trajectoire de N et nous prenons la série desRn(Nn, ∆t) := (b(Nn+∆t)− b(Nn)) avec ∆t ∈ N∗ une échelle de temps.

Distribution des incréments de prix Nous traçons l’histogramme de la sériedes (Rn) pour différentes valeurs de ∆t avec ∆t = 1 (figure 4.9a), ∆t = 10 (figure4.9b), ∆t = 100 (figure 4.9c),∆t = 1000 (figure 4.9d). Nous calculons aussi la sériedes Rn pour les données de marché.

Pour les petites valeurs de ∆t, la distribution empirique simulée correspond à ladistribution des données du marché. Cependant plus ∆t est grand, moins la distri-bution concorde avec les données du marché. La masse de la distribution simuléedes Rn est concentrée au centre ce qui signifie que les données simulées produisentmoins de variance pour les incréments. La non concordance est dû à un problème dechoix de fenêtre de ∆t. Nous observons aussi une asymétrie au niveau de la distri-bution simulée qui est dû à la sensibilité du choix des lois de βC−, βB−, βB+ et desparamètres à estimer.

Autocorrélation des incréments de prix Nous calculons l’auto-corrélation em-pirique des incréments de prix à l’aide de la série des (Rn). Pour tracer l’autocorré-

73


(a) Fonction d’autocorrélation des incré-ments de prix

(b) Fonction d’autocorrélation de la valeurabsolue des incréments de prix

(c) Fonction d’autocorrélation du carré desincréments de prix

Figure 4.10 – Représentation de différentes fonctions d’autocorrélation pour dt = 1

logramme, nous calculons pour τ ∈ 0, K :

C(τ,∆t) =∑nRn(∆t)Rn+τ (∆t)−

∑nRn(∆t)2

V (∆t)

avec V (∆t) la variance de la série des (Rn(∆t)) Nous obtenons pour τ = 0, C(τ,∆t) =1. Puis nous traçons donc la fonction d’auto-corrélation sur la figure 4.10a avec pourabscisse la valeur de τ . Nous n’avons pas de valeurs positives significatives pourl’auto-corrélation des incréments de prix du bid, ce qui suit l’intuition évoqué dansle paragraphe 1.2.1.1.

Regroupement de la volatilité Nous calculons l’autocorrélation empirique ducarré des incréments de prix à l’aide de la série des (Rn). Pour tracer l’autocorrélo-gramme, on calcule pour τ ∈ 0, K :

C2(τ,∆t) =∑nR

2n(∆t)R2

n+τ (∆t)−∑nR

2n(∆t)R2

n(∆t)V2(∆t)

avec V2(∆t) la variance de la série des (R2n(dt)) et

Cabs(τ,∆t) =∑n |Rn(∆t)||Rn+τ (∆t)| −

∑n |Rn(dt)|2

Vabs(∆t)

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4.5. Critères de liquidité de marché

avec Vabs(∆t) la variance de la série des (|Rn(∆t)|) On a pour τ = 0, C2(τ,∆t) = 1.Nous n’avons pas de valeurs positives significatives pour l’auto-corrélation du carrésur la figure 4.10c et de la valeur absolu sur la figure 4.10b des incréments de prixdu bid.

4.5 Critères de liquidité de marché

Nous utilisons les lois et les paramètres obtenues de la section 4.2. Pour lesvariables βC− et βB−, nous utilisons par défaut la loi log-normale avec les paramètresestimés dans le tableau 4.4. Pour les autres variables, il n’y a pas d’ambiguïté (voirles résultats des tests d’adéquations 4.2). Nous nous référons à la sous section 1.2.1du chapitre 1 pour une introduction sur les faits stylisés.

Profil moyen Nous définissons le profil moyen comme la moyenne temporelle desvolumes à chaque niveau de prix. Nous prenons comme référence le prix du bid :

〈∆b(X)−pX〉 = limn→∞

n∑k=0

∆b(Xk)−pXk.

Pour calculer cette quantité, nous simulons une trajectoire de la chaine de MarkovN et nous gardons en mémoire ∆b(Xk)−pN pour tout p niveau de prix et tout k ∈ N∗

le nombre de pas de temps pris pour la simulation. Dans le cas de chaine de Markovrécurrente positive, la valeur empirique converge bien vers le profile moyen théorique.Nous retrouvons bien le début de la forme du profil moyen avec une courbe croissantejusqu’à un maximum puis une décroissance. Sur les figures 4.11a et 4.11b, nous avonsreprésenté en abscisse, les niveaux de prix en prenant comme référence le prix dubid, b(Xk) − p, c’est à dire pour p = 0 nous avons le prix du bid et en ordonnée,nous représentons la moyenne temporelle du volume dans le carnet. Pour p = 0,nous avons Dans le cas où αS dépend de l’état du carnet (Figure 4.11b), la courbeempirique s’accorde plus avec le profil moyen calculé par les données du marchéen comparaison avec le profil moyen empirique avec αS indépendant de l’état ducarnet (Figure 4.11a). Cependant, la queue des profils ne décroit pas ce qui est dûau manque de données sur le carnet sur les volumes 10 niveaux en dessous du prixdu bid. Pour avoir une meilleur concordance, Bouchaud [15] prend une distributionempirique de placement de prix à priori. Ils ne prennent pas de loi théorique, n’estimepas de paramètres et ne calibre pas par rapport aux données. Abergel [2] modéliseet calibre à chaque niveau de prix un processus de Poisson.

Dans notre cas, nous avons modélisé le prix de placement d’ordre limite et leprix de placement d’annulation par une loi que nous calibrons avec les données demarché. Nous perdons de l’information en calibrant de cette manière mais nous

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(a) αS indépendant de l’état du carnet (b) αS dépendant de l’état du carnet

Figure 4.11 – Représentation du profil moyen

avons réduit le nombre de paramètres à estimer. De plus, nous obtenons une bonneconcordance du profil moyen pour les valeurs proches du prix du bid.

Impact de prix Nous définissons l’impact de prix, dans notre cas :

Iind(∆t, q) := E[R(Nt, ∆t)|δt = S , qt = q]

Nous n’étudions dans notre cas, l’impact individuel de prix, c’est à dire l’espérancedu changement de prix après ∆t provoquer par un ordre marché de taille q. Dansnotre cas, le prix considéré sera le prix du bid. Pour calculer cette quantité, noussimulons une trajectoire de la chaine de Markov N et gardons en mémoire l’ensembledes ordres marché, leur taille. à ∆t fixé, nous récupérons pour chaque ordre marchél’incrément de prix R(Nt, ∆t) := b(Nt+∆t) − b(Nt). Nous avons un problème dereprésentation dû à la distribution des tailles des ordres marchés qui possèdent unequeue épaisse. Pour avoir une bonne représentation, nous prenons les quantiles de ladistribution et nous calculons pour chaque quantile la moyenne du changement deprix. Nous obtenons dans le cas où αS indépendant de l’état du carnet les courbessuivantes représentés sur les figures pour ∆t = 100 4.12a et pour ∆t = 1000 4.12b.La courbe empirique simulée ne concorde pas avec la courbe des données de marché.

Dans le cas où αS dépendant de l’état du carnet, nous représentons l’impact deprix sur les figures pour ∆t = 100 4.12c et pour ∆t = 1000 4.12d. Nous réussissons àavoir une meilleure concordance entre la courbe simulée et la courbe des données demarché. Cela peut s’expliquer notamment par la sélection de liquidité dû aux traders.Les traders choisissent d’envoyer des ordres de marché de grande taille lorsque laliquidité est élevée. En faisant dépendre αS de l’état courant, nous caractérisonsce phénomène qui caractérise la concavité de la courbe de l’impact de prix. Nousretrouvons bien la concavité dans le cas αS dépendant.

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4.5. Critères de liquidité de marché

(a) αS indépendant de l’état du carnet avec∆t = 100

(b) αS indépendant de l’état du carnet avec∆t = 1000

(c) αS dépendant de l’état du carnet avec∆t = 100

(d) αS dépendant de l’état du carnet avec∆t = 1000

Figure 4.12 – Représentation de l’impact de prix

Conclusion

Nous avons donné une procédure pour traiter les données. Nous expliquons,l’intérêt de la dépendance de αC par rapport à l’état courant n dans le cadre duproblème de récurrence. αC ne doit pas suivre une loi uniforme et une loi binomialepermet d’avoir une chaine récurrente. Le choix de βC− n’est pas important dansce problème. Nous n’obtenons pas de résultats satisfaisants pour les faits stylisésnotamment pour la distribution des incréments de prix. Mais nous arrivons à unrésultat intéressant pour les critères de liquidité et nous montrons l’intérêt de ladépendance de la variable aléatoire de αS par rapport à l’état courant n.

Un problème plus général de calibration consisterait à calibrer selon les faitsstylisés et les critères de liquidité.

Soit (wi)i∈1,...,5 > 0 fixé, notons l’ensemble Θcalib les probabilités dans Θrec quiminimisent :

w1d(RM(p,∆t), RQ(p,∆t)) + w2d(CM(τ,∆t)− CQ(τ,∆t)

w3d(CM2 (τ,∆t), CQ

2 (τ,∆t))

w4d(MM(p)−MQ(p)) + w5d(IM(dt, q), IQ(dt, q))

avec d(XM , XQ) une distance entre la métrique donnée XM par les données et la

77


métrique donnée par le modèle.Alors le problème de calibration consisterait à trouver les paramètres dans l’en-

semble des probabilités Q dans Θcalib. Le calcul explicite des métriques, dépendantdes paramètres du modèle, est nécessaire dans ce problème de calibration.

78

Chapitre 5

Optimal liquidation in a limitorder book

Contents5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . 80

5.1.1 Model settings . . . . . . . . . . . . . . . . . . . . . . . . 80

5.1.1.1 Order book mechanism . . . . . . . . . . . . . . 80

5.1.2 The control problem . . . . . . . . . . . . . . . . . . . . . 82

5.1.2.1 Control strategies . . . . . . . . . . . . . . . . . 82

5.1.2.2 State space and state process . . . . . . . . . . . 82

5.1.2.3 Admissible strategies . . . . . . . . . . . . . . . 83

5.1.2.4 The gain function . . . . . . . . . . . . . . . . . 83

5.1.2.5 The value function . . . . . . . . . . . . . . . . . 84

5.2 Analytical properties . . . . . . . . . . . . . . . . . . . . . 84

5.2.1 Dynamic programming principle . . . . . . . . . . . . . . 84

5.2.2 One step model . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.2.1 Algorithm . . . . . . . . . . . . . . . . . . . . . 87

5.3 Characterization of the optimal strategy for one stepmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.2 Computations of JS , JB, JC functions . . . . . . . . . . . 91

Introduction

This chapter deals with the optimal liquidation problem in a one-sided limitorder book framework, we presented in 2.

In section 1.4, we describe the general optimal liquidation problem. In this fra-mework, the (bid) price is derived through the depth, the state of the limit orderbook by the operator bid b. As our knowledge, this problem is not studied in theliterature. Abergel et al [1] study a related problem in the market making contextwith a limit order book modeling.

79

Partie I, Chapitre 5 – Optimal liquidation in a limit order book

In the case of limit order book framework, short-time scaled is relevant, perma-nent cost can be omitted. We choose the discrete time framework, so the strategy isthe choice of the quantities x0, . . . , xK . In the introduction of chapter 2, we showedthe interest of the depth representation with the transaction gain function. As oppo-sed to the optimal liquidation problem in a one-sided order book shape frameworkas Predoiu et al [56], we don’t assume a reference price (semi-martingale).

In the section 5.1, we explain the optimal liquidation problem by defining thestrategies, the gain function and the value function. In the section 5.2, we give thedynamic programming principle, we use for solving the one step model. The onestep model is the optimal control with two instants. In order to give some intuition,we give an algorithm for the one step model and draw the strategy and the costfunction with respect to the remaining quantity to sell. With the conjecture, wecharacterize the optimal strategy in the case of one step model in the section 5.3.

5.1 Problem Formulation

Let (Ω,F ,P) be a probability space equipped with a discrete filtration F =(Fk)k∈0,...,T where T ∈ N∗ is a finite horizon. We assume that F0 contains all theP-null sets of F. We consider an order-driven market, in which there is a strategictrader who has to sell X ≥ 0 shares before T . As we shall restrict her strategies tomarket orders, she will acts only on a one-sided Bid limit order book through sellmarket orders. Other investors may submit sell market orders, buy limit orders ormay cancel limit orders.

5.1.1 Model settings

5.1.1.1 Order book mechanism

We assume that the dynamic of the limit order book will follow the same modelas the one described and studied in the previous parts. We recall some notations.

1) N is the set of non negative and non increasing integer sequences n = (n|1, n|2, . . .)such that there exists an i ≥ 0 such that n|j = 0 from any j ≥ i.

The depth of the LOB n is an element of N . Notice that n1 represents thetotal volume of limit buy orders in the LOB.

2) The sell operator TS : N × N→ N is defined by :

TS(n, a) = (n− a)+ = ((n|1 − a)+, (n|2 − a)+, . . .)

It represents the order book after the arrival of sell order with size a.

80

5.1. Problem Formulation

3) The buy operator TB : N × N× N∗ → N is defined by :

TB(n, a, b) :=

n|i + a i ≤ b

n|i otherwise

It represents the order book after an arrival of buy order with a buy size of aand a buy price level of b.

4) The cancel operator TC : N × N× N∗ → N is defined by :

TC(n, a, b) :=

n|i − a ∨∆bn = ni + (−∆bn ∧ −a) i ≤ b

n|i otherwise

It represents the order book after an arrival of cancel order at a price level band with size a.

With these operators, the dynamic of the limit order book may be described.The dynamic of the limit order book is the sequence of event arrival or thecomposition of corresponding events operators.

5) The Bid Price b : N → N∗ is defined by

b(n) = maxi ∈ N∗/n|i > 0

orb(n) =

+∞∑i=1

11n|i>0

After an event arrival, we can compute the new Bid price. Its dynamic is anobvious consequence of the limit order book dynamic.We define the next Bid Price as the new Bid price induced by sell orderarrival bS : N × N→ N∗ by

bS(n, a) := maxi ∈ N∗/n|i > a = b(TS(n, a))

6) We define the operator ∆ : N × N→ N such that

∆pn := ∆(n, p) = n|p − n|p+1, for n ∈ N

∆pn represents the volume of limit buy orders at the level price p.

Randomness flow :We denote by(δk)k∈0,...,T the sequence of random variables corresponding to the events type.(αSk )k∈0,...,T the sequence of random variables corresponding to the size of sell or-ders.

81


(αBk )k∈0,...,T, (βBk )k∈0,...,T the sequence of random variables corresponding to thesize and the price level of buy orders.(αCk )k∈0,...,T, (βCk )k∈0,...,T the sequence of random variables corresponding to thesize and the price level of cancel orders.

5.1.2 The control problem

5.1.2.1 Control strategies

For the strategic trader, a control strategy Q = (Qk)k∈0,...,T is a predictable andnon decreasing process where, for all k ∈ 0, . . . , T, Qk is a random variable valuedin N and Q0 = 0.We denote by ∆Qk := Qk+1 −Qk the quantity of assets sold at time k, whereas Qk

is the number of shares sold by the strategic trader up to time k−.

When the strategic trader knows the depth n, she may interact instantaneouslywith sell orders. The controlled dynamics of the LOB is given by the followingequation :

N|ik+1 = N|ik+ − 11δk=S(αSk ∧ N|ik+) (5.1)


kN|ik+) for all (i, k) ∈ N∗ × 0, ..., T − 1


Bk ,

where we have defined the process representing the LOB just after the strategictrader sells :

N|ik+ = (N|ik −∆Qk)+

We also consider that the strategic trader has the obligation to respect the liqui-dation constraint. The number of shares sold by the strategic trader has to be asclose as possible to X at the horizon time T . However, the liquidity could fall andthe strategic trader could not respect her inventory constraint. For k ∈ 1, T, weimpose that an admissible strategy Q should satisfy :

Q0 = 00 ≤ ∆Qk ≤ N1

k−1

∆QT = N|1T ∧ (X −QT )( terminal liquidation contraint)

5.1.2.2 State space and state process

The strategic trader sell orders induces the same mechanism as small investorssell orders. If, at time k, the state of depth is Nk = n ∈ N , and the strategic trader

82

5.1. Problem Formulation

decides to sell ck. The new state induced by the strategic trader is TS(n, ck). We canthen define the state processes and their dynamics.

First we introduce the state space :

Y := 0, . . . , T × N × 0, . . . , X

We then define the state process as follows : Y := (τ,N, Q) where τ represents thetime and is such that τk = k for any k ∈ 0, . . . , T, N is the LOB process associatedto the strategy Q and its dynamic is given by equation (5.1).

N|ik+1 = N|ik+ − 11δk=S(αSk ∧ N|ik+) (5.2)


kN|ik+) for all (i, k) ∈ N∗ × 0, ..., T − 1


Bk ,

where we have defined N|ik+ = (N|ik −∆Qk)+. For (k, n) ∈ 0, . . . , T×N , we denoteby Nk,n,Q the solution of equation (5.1) such that Nk,n,Q

k = n.

5.1.2.3 Admissible strategies

For (k, n, x) ∈ Y , we are now able to define the set of admissible strategies by

A(k, n, x) := Q = (Qj)j∈k,...,T+1 : Q is a F− predictable and non decreasing process s.t.

Qk = x, ∀j ∈ k, . . . , T − 1, 0 ≤ ∆Qj ≤ (Nk,n,Q)|1j (5.3)

and ∆QT = (Nk,n,Q)|1T ∧ (X −QT )

Notice that, for (k, n, x) ∈ 0, . . . , T × N × [0, . . . , X], A(k, n, x) is finite.

5.1.2.4 The gain function

Let k ∈ 0, . . . , T and n ∈ N be the state of the LOB at time k. When thestrategic trader submits a sell order of size q ≥ 0 at time k, the strategic trader sellsq ∧ n|b(n) at price b(n).If q < b(n), the gain is qb(n). The strategic trader does not have direct impact onthe Bid price in this case.If b(n)− bS(n, q) = j > 0, the strategic trader sells at price b(n)− p with p rangingfrom 0 to j. For each p ranging from 0 to j − 1, she sells ∆b(n)−pn and for p = z, shesells q − n|b(n)−z.Consequently, we are now able to define the gain function g for a sell order of sizeq in an order book n by :

83


g(n, q) :=∞∑i=1

i(q ∧ n|i − n|i+1)+

Notice that

g(n, q) :=∞∑i=1

i(n|i − n|i+1)−∞∑i=1

i(TS(n, q)|i − TS(n, q)|i+1)

An integration by parts formula gives ∑∞i=1 i(n|i − n|i+1) = ∑∞i=1 n|i.

Therefore, if we define s(n) := ∑∞k=0 n|i, we may rewrite g in the following way :

g(n, q) = s(n)− s(TS(n, q))

5.1.2.5 The value function

The strategic trader aims at maximizing the expected value of the total gainobtained at the final time T. We consider the following value function v defined onY by,

v(k, n, x) := supQ∈A(k,n,x)

JQ(k, n, x),

where we have set

JQ(k, n, x) := Ek,n,q[ T−1∑j=k

g(Nk,n,Qj , ∆Qs) + g

(Nk,n,QT , ∆QT

) ]

Recall that ∆QT = (X −QT ) ∧(Nk,n,QT

)|1.

Since A(k, n, x) is finite, the supremum is reached and there exists an optimal stra-tegy. Hence we have

v(k, n, x) := maxQ∈A(k,n,x)

JQ(k, n, x)

Notice that we have the following boundary conditions :

v(T, n, x) = g(n, x ∧ n|1) and v(k, n, X) = 0

5.2 Analytical properties

5.2.1 Dynamic programming principle

Theorem 5.2.1 (Dynamic programming principle).Let (k, n, x) ∈ Y. For any stopping time τ taking values in k+ 1, . . . , T, we have :

84

5.2. Analytical properties


Et,n,x[τ−1∑j=k

g(Nk,n,Qj , ∆Qj) + v(τ,Nk,n,Q

τ , Qτ )]

Proof : Let (k, n, x) ∈ Y , from the definition of the value function, it follows that :


Ek,n,q[T∑j=k

g(Nk,n,Qj , ∆Qj)]

For any stopping time τ taking values in k+ 1, . . . , T, we split the sum inside theexpectation and get :


EPk,n,q[

τ−1∑j=k

g(Nk,n,Qj , ∆Qj) +

T∑j=τ

g(Nk,n,Qj , ∆Qj)]

We consider an optimal strategy Q∗ ∈ A(τ,Nk,n,Q, Qτ ), and conclude.

Notice that if we apply the dynamic programming principle for τ := k + 1 then, for(k, n, x) ∈ Y , we have :


g(n, ∆Qk) + v(k + 1,Nk,n,Qk+1 , Qk+1)]

5.2.2 One step model

In this section, we study the one period optimal control problem. We considerthat the strategic trader has x remaining quantity to sell in two transactions, oneat the beginning of the period and one at the end of the period. We illustrate thepossible events during this period in the figure 5.1.

Figure 5.1 is a representation of controlled path in a one step model. At timeK-1, the strategic trader owns x shares to sell and the market is at the state n.The strategic trader choose to sell q ∈ 0, . . . , x. The new state of the market isTS(n, q) and the remaining share is x − q. We emphasize that the trader can reactjust after looking the state n. The market reacts and evolves according the model.At time K, the strategic trader owns x− q shares and sell x− q in order to respectthe liquidation constraint.In order to give some intuition, we compute the value function and an optimalstrategy for different LOB profiles of LOB such as a block order shape as in Predoiu[56] or such as randomly generated LOB profiles.

By using the dynamic programming principle and for k = K − 1, we obtain :

v(K − 1, n, x) = maxq∈[0,...,x∧n|1]

g(n, q) + E[g(Nk+1, x− q)|Nk = TS(n, q)]

85


nv(K − 1, n, x)

TS(n, q)

g(n, q)

TS(n, 0) = n

TS(n, x)

TS(n, q)

v(K,TS(n, q), x− q)

TS(TS(n, q), a)

TC (TS(n, q), a, b)

TB(TS(n, q), a, b)

TS(n, x)

g(n, x)

TS(TS(TS(n, q), a), x− q)

TS(TC (TS(n, q), a, b), x− q)

TS(TB(TS(n, q), a, b), x− q)

k = T − 1 k = T− k = T k = T+

Figure 5.1 – Representation of controlled path in a one step model

86


And we define J

J(K − 1, n, x, q) := g(n, q) + E[g(Nk+1, x− q)|Nk = TS(n, q)]

5.2.2.1 Algorithm

We start with initializing the LOB n, the number of shares to liquidate x, thelaw parameters ppp := (pppE,pppS,pppPC ,pppV C ,pppPB,pppV B) for the events arrivals. In or-der to optimize the algorithm, we compute the probability vectors Pγ of γ =δ, αS, αC , βC , αB, βB defined by

Pγ := (P[γ = i])i∈supportγ

We now describe the algorithm for computing the value function and finding anoptimal strategy for the one period model.

— For each iteration q ∈ [0, . . . , x], we follow the following steps :

— Step 1 : Compute the new depth m := TS(n, q), the Bid Price of the newdepth bS := bS(n, q)

— Step 2 : Compute the gain J0 := g(n, q) and the gain Jx := g(m, x− q)

— Step 3 : Compute JS, JC , JB :

JS :=∑

a∈suppαSg(TS(m, a), x− q)P[αS = i]

JC :=∑

a,b∈suppαC×suppβCg(TC(m, a, bS − βC), x− q)P[αC = a]P[βC = b]

JB :=∑

a,b∈suppαB×suppβBg(TB(m, a, bS−βB+u), x−q)P[αB = a]P[βB = b]

— Step 4 : Compute J [q] :

J [q] := J0 + P[δ = N ]Jx + P[δ = S]JS + P[δ = C]JC + P[δ = B]JB

— Step 5 : Compute q∗ and v :

q∗ := argmaxJ [q]

v := maxJ [q]

We plot in figure 5.2 a representation of different depth profiles. The horizontalaxis represent the price scale and the vertical axis represent the cumulative volume.All curves are represented by piece-wise linear curve. The green curve correspond to

87


Data:— the depth n— x : remaining share (integer)— E : events (vector size 4)

for k ∈ J0, xK doCompute newdepth = executionMarket(depth, k);Compute J0 = gain(depth, k);Compute JN = gain(newdepth, x− k);for i ∈ J1, nsK do

ComputeJS = JS + prsellsize[i]× gain(executionMarket(newdepth, i), x− k)

endfor i, j ∈ J1, nbK× J1,mbK do

Compute JB = JB + prbuysize[i]× prbuyprice[j]×gain(executionLimit(newdepth, i, bestPrice(newdepth)− j +buyprice[2]), x− k) )

endfor i, j ∈ J1, ncK× J1,mcK do

Compute JC = JC + prcancelsize[i]× prcancelprice[j]×gain(executionCancel(newdepth, i, bestPrice(newdepth)− j), x− k)

endCompute J [k] = J0 + JN + JS + JB + JC

endResult: J : (vector size x+1)

Algorithm 2: Algorithm for computing the value function and finding an optimalstrategy for the one period model

Figure 5.2 – Depth profiles

88


(a) Recurence (b) Transience

Figure 5.3 – Strategy for linear n

the block shape order book. The block shape order is define by n := (n|k)k∈N∗ suchas :

n|k = 10[(475− k)+ − (450− k)+

]and for k ∈ 450, . . . , 475 ∆kn = 10 (5.4)

The orange and the blue curves correspond to randomly generated order book.We generate the values∆kn for k ∈ 450, . . . , 475 and∆kn = 0 outside 450, . . . , 475.For each k, we generate a realization from uniform law with parameters (a =0, b = 20). For each k, we generate a realization from binomial law with parameters(n = 20, p = 0.5).

We represent the optimal share q/10 at time K − 1 against the remaining sharex to liquidate in the case of the block shape order book in the case of recurrenceparameters in figure 5.3a and transient parameters in figure 5.3b. We refer to thechapter 4 for the discussion of parameters which lead to the recurrence property orthe transience property. In the recurrent case, the optimal share q is proportional tox until q = n|1. in the transient case, the optimal share q is proportional to x fromq = 0 to q = q0 and from q = q1 to q = n|1. Between q0 and q1, the optimal shareq = n|b(n) − 1.

In the figure 5.4, we compute the function J . At the x-axis, we represent theremaining share x to liquidate in the case of the block shape order book and inthe other axis, the function J . In figures 5.4a, 5.4b, 5.4c and 5.4d, we notice thesame shape for J . For all x and parameters, J is a sawtooth function. We conjecturethat J is increasing on each interval q = n|k+1 ∧ x, . . . , n|k − 1 ∧ x for all k ∈b(n) + 1, . . . , 1.

89


(a) Recurrence J for x = 50 (b) Transience J for x = 50

(c) Recurrence J for x = 250 (d) Transience J for x = 250

Figure 5.4 – Computation of J for x = 50, 250

5.3 Characterization of the optimal strategy forone step model

5.3.1 Main result

We want to simplify the expression of J . By noticing that

g(TS(n, q), x− q) = g(n, x)− g(n, q) (5.5)

and by computing the drift for the function g with m = TS(n, q), we have

Lg(m, x− q) = E[g(Nk+1, x− q)− g(Nk, x− q)|Nk = m]. (5.6)

Thus,

J(K − 1, n, x, q) = g(n, q) + E[g(Nk+1, x− q)|Nk = TS(n, q)] (5.7)

= g(n, q) + g(TS(n, q), x− q) + Lg(m, x− q)

= g(n, x) + Lg(m, x− q)

We define Lg(m, x− q) := JS(n, x, q) + JB(n, x, q) + JC(n, x, q) with

— JS(n, x, q) := pSE[g(TS(TS(n, q), αS), x− q)− g(TS(n, q), x− q)]

— JB(n, x, q) := pBE[g(TB(TS(n, q), αB, βB), x− q)− g(TS(n, q), x− q)]

— JC(n, x, q) := pCE[g(TC(TS(n, q), αC , βC), x− q)− g(TS(n, q), x− q)]

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5.3. Characterization of the optimal strategy for one step model

By following the conjecture,

Definition 5.3.1. We define Ab the set of sell size which leads to the same next bidprice :

For b ∈ [0, b(n)],Ab(n, x) := c ∈ [0, x], bS(n, c) = b

The main result of this part is the following :

Theorem 5.3.1. For all n ∈ N , for all q ∈ Ab with bS(n, q), the function J(K −1, n, x, .) is non decreasing in q. Then, for all b ∈ [0, b(n)], for all q ∈ Ab,

arg maxq∈Ab

J(K − 1, n, x, q) = nB − 1

This theorem means that in this model, the strategic trader only need to know thenext bid price to define his sell size at time K − 1. For proving this theorem, weneed to compute the functions JS, JB, JC . In order to compute these functions, weneed to understand compositions of transition operators such as TSTS, TCTS andTBTS. Since for all n ∈ N , (a, b) ∈ N we have

TS(TS(n, a), b) = TS(n, a+ b), (5.8)

the function JS is the easiest to compute.

5.3.2 Computations of JS, JB, JC functions

In order to compute the functions JS, JB, JC , we need to define some notationssuch as

— Dα the support of the random variable α,

— Fα(y) := ∑a∈Dα 11a≤yP[α = a] the cumulative distribution of α,

— Dα(y) := ∑a∈Dα a11a≤yP[α = a],

— Hα(y) := yFα(y)−Dα(y)

— Fα(y) := 1− Fα(y)

— Dα(y) = α−Dα(y)

— Hα(y) := yFα(y)− Dα(y)

— H is positive and non decreasing and H is non decreasing

We define for a function q → f(q) with q ∈ N, the increment operator for f ∆qf

such as ∆qf(q) := f(q + 1)− f(q).We give a lemma for the decomposition of g.

Lemma 5.3.1. For n ∈ N , x ∈ N, we have

g(n, x) = xbS(n, x) + s(n, bS(n, x))

91


.

We use these lemma by separating the price part and the volume part. Weemphasize the importance of the dependence of bS(n, x). For b ∈ N fixed and for allx ∈ Ab, we notice the same shape for g as the J function.

— For n ∈ N fixed, the function b→ s(n, b) is decreasing in N.

— For n ∈ N fixed, the function x→ bS(n, x) is decreasing in N.

Then, For n ∈ N fixed, the function x→ s(n, bS(n, x)) is increasing in N.

Lemma 5.3.2 (Computation of JS). For all n ∈ N , for all q ∈ 0, . . . , x, we have

JS(n, x, q) := pSE[g(TS(TS(n, q), αS), x− q)− g(TS(n, q), x− q)] = A(n, x)−A(n, q)

with A(n, q) := ∑i≤bS(n,q)

[HαS(n|i − 1− q) + E[αS]

]. Moreover, for b ∈ [0, b(n)], for

all q ∈ Ab, JS(n, x, q) is non decreasing in q.

Proof : Firstly, we want to compute E[g(TS(n, αS), x)− g(n, x)]. Then, we replacen := TS(n, q) and x := x− q for computing the term JS.

Since we have 5.3.1, we focus on the two functions bS and s. By definition of thefunction bS, for n ∈ N , we have bS(n, αS) = ∑

i≥1 11αS≤n|i−1 and bS(TS(n, αS), x) =bS(n, αS + x). We have :

E[bS(TS(n, αS), x)− bS(n, x)] = −bS(n,x)∑i=1

FαS(n|i − 1− x) (5.9)

By definition of the function s, we have :

s(TS(n, a), bS(TS(n, a), x)) =∑

i>bS(n,a+x)(n|i − a)11i≤bS(n,a) (5.10)

= s(n, bS(n, a+ x))− s(n, bS(n, a))

+ a(bS(n, a+ x)− bS(n, a))

We compute the expectation of s(n, bS(n, αS + x)) and the αSbS(n, αS + x). Sincethe set i ∈ N, i > bS(n, x+ αS) is equivalent to the set i ∈ N, αS ≥ n|i − x and

92


for i > bS(n, x),FαS(n|i − x− 1) = 1, then

E[s(n, bS(n, x+ αS))] = E[∑

i>bS(n,x+αS)n|i] (5.11)

= E[∑i≥1

11αS≥n|i−xn|i]

=∑i≥1

n|iFαS(n|i − x− 1)

=∑

i≤bS(n,x)n|iFαS(n|i − x− 1) +

∑i>bS(n,x)

n|iFαS(n|i − x− 1)

=∑

i≤bS(n,x)n|iFαS(n|i − x− 1) + s(n, bS(n, x))

We compute

E[αSbS(n, x+ αS)] =∑

i≤bS(n,x)DαS(n|i − 1− x) (5.12)

=∑

i≤bS(n,x)E[αS]− DαS(n|i − 1− x)

Then for the function s, by using the equations 5.11 and 5.12 we obtain :

E[s(TS(n, αS) , bS(TS(n, a), x))− s(n, bS(n, x))] (5.13)

=∑

i≤bS(n,x)

[n|iFαS(n|i − 1− x)− DαS(n|i − 1− x) + E[αS]

]−

∑i≤b(n)

[n|iFαS(n|i − 1)− DαS(n|i − 1) + E[αS]

]

Then, we conclude for g with equations 5.9 and 5.13 :

E[g(TS(n, αS), x)− g(n, x)] =∑

i≤bS(n,x)

[HαS(n|i − 1− x) + FαS(n|i − 1− x) + E[αS]

]−

∑i≤b(n)

[HαS(n|i − 1) + FαS(n|i − 1) + E[αS]

](5.14)

By setting n := TS(n, q) and x := x− q in equation 5.14, we compute JS,

JS(n, x, q) = E[g(TS(TS(n, q), αS), x− q)− g(TS(n, q), x− q)] (5.15)

=∑

i≤bS(n,x)

[HαS(n|i − 1− x) + FαS(n|i − 1− x) + E[αS]

]−

∑i≤bS(n,q)

[HαS(n|i − 1− q) + FαS(n|i − 1− q) + E[αS]

]

93


We compute for for b ∈ N and q, q + 1 ∈ Ab(n, x),

∆qJS(n, x, q) =

∑i≤bS(n,q)

FαS(n|i − q − 1) ≥ 0 (5.16)

Then JS is increasing in q for all q ∈ Ab(n, x). The lemma is proved.

If we define JC(n, x, q) := pcE[g(TC(TS(n, q), αC , βC), x− q)−g(TS(n, q), x− q)],

Lemma 5.3.3 (Computation of JC). for all n ∈ N , for all q ∈ 0, . . . , x, we have

JC(n, x, q)pc

= Ac(n, x, bS(n, q)) + Q[βC− = 0]Bc(n, x, bS(n, q), q)

Moreover, for b ∈ [0, b(n)], for all q ∈ Ab, JC(n, x, q) is non decreasing in q.

Proof : If there is a cancel event arrival and the cancel price level is lower than thebS(n, x), the new bid price doesn’t change. It means that, for b(n) − b < bS(n, x),bS(TC(n, a, b(n)− b), x)− bS(n, x) = 0.

Otherwise, we have for yb := b(n)− b,

bS(TC(n, a, yb), x)− bS(n, x) =(bS(n, x+ a)− bS(n, x)

)11b(n)−bS(n,x)≥b11a≤∆ybn(5.17)

We have αC(∆βcn) = 11εc=1∆βcn + 11εc=0ξc∆βcn and βC(n) = b(n) − βC−. By

using equation 5.17, we compute,

E[bS(TC(n, αC , βC), x)− bS(n, x)] = −∑

i≤bS(n,x)E[11βC−≤∆bS(n,0,x)F

αC(∆βcn)(n|i − 1− x)]

(5.18)

Since the expectation of bS is computed, we analyze s

s(m, bS(m, x))− s(n, bS(n, x)) =(s(n, bS(n, x+ a)) + a(bS(n, x+ a)− yb)

)11yb≥bS(n,x)11a≤∆ybn

(5.19)

E[(s(n, bS(n, x+ αC(∆βcn)))− s(n, bS(n, x))] =∑

i≤bS(n,x)n|iE[11βC−≤∆bS(n,0,x)F

αC(∆βcn)(n|i − 1− x)]

(5.20)

94


E[αC(∆βcn)(bS(n, x+ αC(∆βcn))− b(n) + βC−)] = E[11βC−≤∆bS(n,0,x)αC(∆βcn)(βC− − b(n))]

+∑

i≤bS(n,x)E[11βC−≤∆bS(n,0,x)(DαC(∆βcn)(n|i − 1− x)]

(5.21)

We conclude then for g :

E[g(TC(n, αC , βC), x) − g(n, x)] =∑

i≤bS(n,x)(n|i − x)E[11βC−≤∆bS(n,0,x)F

αC(∆βcn)(n|i − 1− x)]

+∑

i≤bS(n,x)E[11βC−≤∆bS(n,0,x)D

αC(∆βcn)(n|i − 1− x)]

+ E[11βC−≤∆bS(n,0,x)αC(∆βcn)(βC− − b(n))] (5.22)

We compute for n := TS(n, q) and x := x−q and we recall that∆bS(n,q)−βC−TS(n, q) =∆bS(n,q)−βC−n11βC−>0 + (nbS(n,q) − q)11βC−=0

JC(n, x, q)pC

=∑

i≤bS(n,x)(n|i − x)E[11βC−≤∆bS(n,q,x)F

αC(∆βcTS(n,q))(n|i − 1− x)]

+∑

i≤bS(n,x)E[11βC−≤∆bS(n,q,x)D

αC(∆βcTS(n,q))(n|i − 1− x)]

+ E[11βC−≤∆bS(n,q,x)αC(∆βcTS(n, q))(βC− − bS(n, q))]

= Ac(n, x, bS(n, q)) +Bc(n, x, bS(n, q), q)Q[βC− = 0] (5.23)

with

Ac(n, x, bS(n, q)) =∑

i≤bS(n,x)(n|i − x)E[11βC−>011βC−≤∆bS(n,q,x)F

αC(∆βcn)(n|i − 1− x)]

+∑

i≤bS(n,x)E[11βC−>011βC−≤∆bS(n,q,x)D

αC(∆βcn)(n|i − 1− x)]

+ E[11βC−>011βC−≤∆bS(n,q,x)αC(∆βcn)(βC− − bS(n, q))] (5.24)

and

Bc(n, x, bS(n, q), q) =∑

i≤bS(n,x)(n|i − x)FαC(nbS(n,q)−q)(n|i − 1− x)]

+∑

i≤bS(n,x)DαC(nbS(n,q)−q)(n|i − 1− x)]

− E[αC(nbS(n,q) − q)(bS(n, q))] (5.25)

It is easy to notice for b ∈ N fixed and q, q+ 1 ∈ Arb(n, x), ∆qAc(n, x, bS(n, q)) = 0

We assume that E[αC(N)/N ] = m > 1 with m constant in N. This assumptionis true in the case of the specification of αC 2.3.5 when αC follows a uniform lawwith parameters (1, ∆bn) or αC follow a binomial law with parameters (∆bn, p).

95


We compute ∆qBc(n, x, bS(n, q), q) and if we define ∆fαCq (j) := fα

C

n|bS(n,q)−q−1(j) −fα

C

n|bS(n,q)−q(j)

∆qBc(n, x, bS(n, q), q) =

∑i≤bS(n,x)

n|i−x−1∑j=1

j∆fαC

q (j) +∑

i≤bS(n,x)(n|i − x)∆FαC

q (n|i − x− 1)

+ mbS(n, q) (5.26)

For example, if the random variable αC is state dependent (we refer to to thespecification of αC 2.3.5 or the section for parameter estimation 4.2) and follows auniform law with parameters (1, ∆bn), we have

fαC

q (j) = 1n|b − q and ∆fαCq (j) = 1

(n|b − q − 1)(n|b − q) > 0

FαC

q (j) = 1− j

n|b − q and ∆FαC

q (j) = −j(n|b − q − 1)(n|b − q) < 0

Then, we compute ∆qBc,

∆qBc(n, x, bS(n, q), q) = −1

2∑

i≤bS(n,x)

(n|i − x− 1)(n|i − x)(n|b − q − 1)(n|b − q) +mbS(n, q) > 0

For q ∈ Ab(n), ∆qBc > 0. The lemma is proved.

Lemma 5.3.4 (Computation of JB). For all n ∈ N , for all q ∈ 0, . . . , x, we have

JB(n, x, q)pB

= E[g(TB(TS(n, q), αB, βB), x− q)− g(TS(n, q), x− q)] = AB(n, x, x− q)

Moreover JB(n, x, q) is non decreasing in q.

Proof : If there is a buy event arrival and the buy price level yb is lower than thebS(n, x), the new bid price doesn’t change. Since the volume x of a sell event andthe volume a of a buy event are opposite sign, we need to distinguish the two case.When the volume a is greater than x (absolute value), the new bid price is the buyprice level. When the volume x is greater than a, the new bid price is the maximumbetween the buy price level and the new bid price after the difference of x and a.Then :

bS(TB(n, a, yb), x)−bS(n, x) =

yb − bS(n, x) if a > x and bS(n, x) ≤ yb

bS(n, x− a)− bS(n, x) if a ≤ x and bS(n, x− a) > yb

yb − bS(n, x) if a ≤ x and bS(n, x− a) ≤ yb

96


We have yb = b(n)− 11εB=0βB− + 11εB=1β

B+.

We recall that I(n, x) = b(n)− bS(n, x). By definition, βB+ is positive and βB−

is non negative.

bS(TB(n, a, yb), x)−bS(n, x) =

[βB+ + I(n, x)]11εB=1

[−βB− + I(n, x)]11a>x11βB−≤I(n,x)11εB=0

[bS(n, x− a)− bS(n, x)]11a≤x11βB−>I(n,x−a)11εB=0

[−βB− + I(n, x)]11a≤x1111βB−≤I(n,x−a)11εB=0

Then for the expectation of bS with βB := yb and bS(n, x − αB) − bS(n, x) =I(n, x)− I(n, x− αB), we have

E[(bS(TB(n, αB, βB), x)− bS(n, x))] = E[11εB=1(βB+ + I(n, x))] (5.27)

+ E[11εB=0(−βB− + I(n, x))]

− E[11εB=011αB>xHβB−(I(n, x))]

− E[11εB=011αB≤xHβB−(I(n, x− αB))]

With m := TB(n, a, yb), we define ∆s(m, bS(m, x)) := s(m, bS(m, x))− s(n, bS(n, x)).Thus,

∆s(m, bS(m, x)) =

s(n, yb)− s(n, bS(n, x)) if a > x and bS(n, x) ≤ ybs(n, bS(n, x− a))− s(n, bS(n, x)) + a(yb − bS(n, x− a)) if a ≤ x and bS(n, x− a) > yb

s(n, yb)− s(n, bS(n, x)) if a ≤ x and bS(n, x− a) ≤ yb

We have yb = b(n)− 11εB=0βB− + 11εB=1β

B+ and s(n, b) = 0 for b ≥ b(n)

∆s(m, bS(m, x)) =

[−s(n, bS(n, x))]11εB=1

[s(n, b(n)− βB−)− s(n, bS(n, x))]11a>x11I(n,x)≥βB−11εB=0

[s(n, bS(n, x− a))− s(n, bS(n, x)) + a(−βB− + I(n, x− a))]11a≤x11I(n,x−a)<βB−11εB=0

[s(n, b(n)− βB−)− s(n, bS(n, x))]11a≤x11I(n,x−a)≥βB−11εB=0

Then for the s :

E[∆s(m, bS(m, x))] = −E[11εB=1s(n, bS(n, x))] (5.28)

+ E[11εB=011αB>x11I(n,x)≥βB−s(n, b(n)− βB−)− s(n, bS(n, x))]

+ E[11εB=011αB≤xαBHβB−(I(n, x− αB))]

+ E[11εB=011αB≤x11I(n,x−αB)<βB−s(n, bS(n, x− αB))− s(n, bS(n, x))]

+ E[11εB=011αB≤x11I(n,x−αB)≥βB−s(n, b(n)− βB−)− s(n, bS(n, x))]

97



+ E[11εB=011αB>x11I(n,x)≥βB−s(n, b(n)− βB−)− s(n, bS(n, x))]



+ E[11εB=011αB≤x11I(n,x−αB)≥βB−s(n, b(n)− βB−)− s(n, bS(n, x))]

We focus on the term 11I(n,x)≥βB−s(n, b(n)− βB−) :

E[11I(n,x)≥βB−s(n, b(n)− βB−)] =I(n,x)∑b=0

b(n)∑i>b(n)−b

n|iP[βB− = b] (5.30)

=∑

i>bS(n,x)n|i(FαB(I(n, x))− FαB(b(n)− i))

and then

E[11I(n,x−αB)≥βB−s(n, b(n)− βB−)] =∑

i>bS(n,x−αB)n|i(FαB(I(n, x− αB))− FαB(b(n)− i))

(5.31)




− E[11εB=011αB>x∑

i>bS(n,x)n|iF βB−(b(n)− i)]

− E[11εB=011αB≤x∑

i>bS(n,x−αB)n|iF βB−(b(n)− i)]



− E[11εB=011αB≤x∑

i>bS(n,x)n|iF βB−(I(n, x− αB)]

− E[11εB=011αB>x∑


+ E[11εB=011αB≤x∑

i>bS(n,x−αB)n|i(F βB−(I(n, x− αB))− F βB−(b(n)− i))]

98


E[∆s(m, bS(m, x))] = E[11εB=011αB≤xαBHβB−(I(n, x− αB))] (5.34)

− E[11εB=1s(n, bS(n, x))]− E[11εB=0s(n, bS(n, x))]

+ E[11εB=011αB>x∑


+ E[11εB=011αB≤xbS(n,x−αB)∑i>bS(n,x)

n|iF βB−(I(n, x− αB)]

+ E[11εB=011αB≤x∑

i>bS(n,x−αB)n|iF βB−(b(n)− i))]

E[(bS(TB(n, αB, βB), x)− bS(n, x))] = E[11εB=1(βB+ + I(n, x))]

+ E[11εB=0(−βB− + I(n, x))]

− E[11εB=011αB>xHβB−(I(n, x))]

− E[11εB=011αB≤xHβB−(I(n, x− αB))]

(5.35)

We obtain for g :

E[(g(TB(n, αB, βB), x) − g(n, x))] = E[11εB=1x(βB+ + I(n, x))− s(n, bS(n, x))]

+ E[11αB≤x11εB=0x(−βB− + I(n, x))− s(n, bS(n, x))]

− FαB(x)xHβB−(I(n, x))E[11εB=0]

+ E[11αB≤x11εB=0(αB − x)HβB−(I(n, x− αB))]

+ E[11εB=011αB>x∑


+ E[11εB=011αB≤xbS(n,x−αB)∑i>bS(n,x)

n|iF βB−(I(n, x− αB)]

+ E[11εB=011αB≤x∑

i>bS(n,x−αB)n|iF βB−(b(n)− i))]

We compute for n := TS(n, q) and x := x−q. We denote∆bS(n, q, x) := bS(n, q)−bS(n, x). We have

s(TS(n, q), bS(TS(n, q), x− q)) = s(n, bS(n, x))− s(n, bS(n, q))− q∆bS(n, q, x)

99


JB(n, x, q)pB

= E[(q − x)[11εB=0βB− − 11εB=1β

B+] + x∆bS(n, q, x) + s(n, bS(n, x))− s(n, bS(n, q))]

− FαB(x− q)(x− q)HβB−(∆bS(n, q, x))E[11εB=0]

+ E[11αB≤x−q11εB=0(αB − x+ q)HβB−(∆bS(n, q, x− αB))]

+ E[11εB=011αB>x−qbS(n,q)∑i>bS(n,x)

(n|i − q)F βB−(bS(n, q)− i)]

+ E[11εB=011αB≤x−qbS(n,x−αB)∑i>bS(n,x)

(n|i − q)F βB−(∆bS(n, q, x− αB)]

+ E[11εB=011αB≤x−qbS(n,q)∑

i>bS(n,x−αB)(n|i − q)F βB−(bS(n, q)− i))]

We compute ∆qJB(n, x, q) for q ∈ Ab in the same way as ∆qJ

C(n, x, q). We find∆qJ

B(n, x, q) > 0. The lemma is proved.

Conclusion

The main result of this chapter is the strategy is reduced to choose the priceimpact instead of the quantity, in the case of the one step model. In order to find thisresult, we just use the dynamical programming principle and the different relationsbetween the operators of N .

100

Deuxième partie

Optimal execution in a one-sidedorder book with stochastic volume

process

101

Chapitre 6

Optimal execution in a one-sidedlimit order book with stochastic

volume process

Contents6.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Analytical properties of the value function . . . . . . . . 108

6.2.1 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2.2 Dynamic programming principle . . . . . . . . . . . . . . 116

6.2.3 Heuristic HJB . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3 Viscosity Characterization of the value function . . . . . 117

6.4 Examples and numerical results . . . . . . . . . . . . . . . 125

6.4.1 Linear price impact . . . . . . . . . . . . . . . . . . . . . 125

6.4.2 Deterministic resilience . . . . . . . . . . . . . . . . . . . 126

6.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . 128

6.4.3.1 Parameters . . . . . . . . . . . . . . . . . . . . . 128

6.4.3.2 Modified Block Order Book . . . . . . . . . . . . 130

6.4.3.3 Discrete Order Book . . . . . . . . . . . . . . . . 131

This part is independent of the first part of the thesis.

Introduction

This chapter deals with the optimal liquidation problem in a one-sided limitorder shape framework, we presented in chapter 2.

In section 1.4, we describe the general optimal liquidation problem and we referto an introduction of the limit order book shape model in subsection 1.4.2. In ourmodel, we are in a continuous-time framework, the strategy can be decomposed witha jump part, and a singular part and a continuous part. It leads to a singular controlproblem. We refer to the Pham’s book [55] for general stochastic control theory forcontinuous diffusion. Since we allow to a general diffusion with a jump part, we usethe Oksendal and Sulem’s book [54].

103

Partie II, Chapitre 6 – Optimal execution in a one-sided limit order book with stochasticvolume process

We explain the optimal liquidation problem by defining the strategies, the gainfunction and the value function in the section 6.1. In the section 6.2, we give thedynamic programming principle and the Hamilton- Jacobi-Bellman equations forthe value function. We characterize the value function as the unique continuousviscosity solution of the Hamilton-Jacobi-Bellman equations in the section 6.3. Wegive specific examples of our model and present numerical results obtained by anumerical approach in the section 6.4.

6.1 Model

The cost of a financial transaction depends on the two following quantities : theliquidity in the market, which is summarized through the limit order book and thevolume of the trade, this is the quantity of shares to be traded. In the usual setup ofmathematical finance, liquidity is assumed to be infinite and the mean price paid forthe transaction does not depend on the volume. In this case, the quantity of sharestraded does not impact the shape of the limit order book. In contrast, when largetraders participate in the market this assumption no longer holds. To liquidate largepositions or to perform big purchases, large traders incur costs that depend on thevolume of the transaction and the current shape of the limit order book. A largetrader who wants to liquidate or purchase a large quantity of shares over a fixedperiod of time would like to split the big trade into smaller trades, and benefit fromthe execution timing to minimize the costs. This problem is known as the optimalorder execution trade.

To model the mean price of a transaction there are two aspects to take intoaccount :

— The quantity of shares to be traded (∆Xt)

— The liquidity in the market (ft), modeled through the Limit Order Book.Liquidity in a LOB has three components : Depth, Resilience, and Tightness.Since we work on a one-sided LOB we disregard this last aspect and will modelLOB only through its density. We will, have

Smeant = F (∆Xt, ft)

where ∆Xt is the quantity to trade and ft : Ω 7→ F(R+) is the density ofthe order book. In general, to model ft is complicated because it involves infinitedimensional processes. We take the following approach : assume that there is a fixform f from the start. Then we model

ft(ω)(a) = f(a+ Yt(ω))

104

6.1. Model

where Yt models the volume up to time t, not only including the investor but alsoother agents in the market.

We work on a filtered probability space (Ω,F , (Ft)t∈[0,T ],P) satisfying the stan-dard conditions of right-continuity and completeness. We assume that this filteredprobability space supports a standard P-Brownian motionW andM a random Pois-son measure on R+×R with mean measure γtdtm(dz) where γ : [0, T ]→ (0, γ] andm is a probability measure on R.

A financial agent wants to buy X shares of an illiquid asset over the time interval[0, T ]. Without loss of generality we will assume that all quantities are discounted.

Let (At)t≥0 be the reference price of the assets, which we assume to be a conti-nuous P-martingale. In our model, we assume that, in the absence of trading, thenumber of available shares at time t in the price interval [At, At + x) is F (x). F is anon-decreasing and left-continuous function associated to an infinite measure µ on[0,+∞) in the following way :

F (x) := µ([0, x)), for all x ≥ 0. (6.1)

Assumption 6.1.1. We will make some assumption on the function F

i. There exists b > 0, α > 0 and a > 0 such that

F (x) ≥ bxα, for all x ≥ a. (6.2)

ii. There exists K > 0 and q > 0 such that, for any x, x′ ≥ 0,

if | F (x)− F (x′) |< q then | x− x′ |≤ K. (6.3)

Examples :

F

µ

a b x

y

Figure 6.1 – Modified Block Order Book

105


F

µ

x

y

Figure 6.2 – Discrete Order Book

Investor’s strategies. The strategies of the agent are given by non-decreasingright-continuous adapted processes (Xt)0≤t≤T with XT = X. We assume that X0− =0 and denote by ∆Xt = Xt −Xt− the jump at time t.

The dynamics of the volume effect process (Yt). We assume that the strategyof our investor has impact on the price. When the financial agent follows strategy X,we assume that at time t, the ask price is no longer the reference price but is givenby At + Dt where Dt := ψ(Yt), with Yt representing the dynamics of the volumeeffect process

dYt = dXt − h(Yt−)dt+ σ(Yt−)dWt +∫RYt−q(Yt− , z)M(dt, dz);Y0− = y. (6.4)

and the left-continuous function ψ given by follows

ψ(y) := supa ≥ 0|F (a) < y, for y > 0 and ψ(0) := 0. (6.5)

At

Yt

At + ψ(Yt) Price

Qua

ntity

106

6.1. Model

Assumption 6.1.2. We assume that h, σ, q are such that,

i) For all n ∈ N and y ≥ 0, we have

supt∈[0,T ]; X∈A

E[(Y y,X

t )n]< +∞, (6.6)

where A is the set of non-decreasing right-continuous adapted processes (Xt)0≤t≤T

with XT = X and X0− = 0.

ii) There exists β > 0 such that

limy→+∞

σ(y)2yβ−2 < +∞ (6.7)

We denoteYt− := Yt− +∆MYt

where ∆MYt is the jump of the measure M at time t.

Remarks :

1.) The volume effect process Y incorporates both the impact from the investor’strading through the term dXt and from other agents in the market.

2.) When the investor follows the strategy (Xt)0≤t≤T , for any given a ≥ 0, thenumber of assets available between prices At and At+a becomes (F (a)−Yt−)+.

3.) Equation (6.4) specifies how the order book is affected by the transactions ofthe trader. Suppose that there is a large transaction at time t with the investorbuying ∆Xt shares, then by equation (6.4), Yt = Yt− + ∆Xt. Right after thistransaction the ask price jumps from At + ψ(Yt−) to At + ψ(Yt− +∆Xt).

Strategy cost. We now can write the cost of the strategy X = (Xt)0≤t≤T as

C(X) :=∫ T

0(At + Dt−)dXc

t +∑

0≤t≤T[At∆Xt + (Φ(Yt)− Φ(Yt−))],

=∫ T

0ψ(Yt−)dXc

t +∑

0≤t≤T(Φ(Yt)− Φ(Yt−)) +

∫ T

0AtdXt.

where Xc denotes the continuous part of X, Dt− := ψ(Yt−) and Φ(y) ,∫ ψ(y)

0 ξdF (ξ).Notice that it follows from the monotonicity and the left-continuity of F , that forall y > 0 we have F (ψ(y)) = y and then

Φ(y) =∫ y

0ψ(ζ) dζ, y > 0. (6.8)

107


Notice that the integration by parts formula implies that

E[∫ T

0AtdXt

]= E

[ATX − A0X0− +

∫ T

0Xt−dAs

].

Under suitable conditions on A, since X is nonnegative and bounded by X andX0− = 0, this can be simplified to obtain

E[∫ T

0AtdXt

]= XA0.

The expected cost for the investor following strategy X becomes

E

∫ T

0ψ(Yt−)dXc

t +∑

0≤t≤T(Φ(Yt)− Φ(Yt−)) + XA0

. (6.9)

Control Problem We analyze (6.9) as a control problem in which the controls aregiven by the strategies X and Y is a state variable whose dynamics are given by(6.4). As usual we introduce a time variable t and make the problem dynamic byintroducing the following value function for 0 ≤ t < T and x ∈ [0, X] and y ≥ 0.

v(t, x, y) = infX∈A(t,x)

E

∫ T

tψ(Y t,y,X

s− )dXcs +

∑t≤s≤T


s− )) , (6.10)

where Y t,y,Xs for t ≤ s ≤ T denotes the solution of (6.4) with Y t,y,X

t− = y and the setof admissible controls A(t, x) is given by

A(t, x) , X : X ; Xt− = x; XT = X. (6.11)

The value function at terminal time T is given by

v(T, x, y) = Φ(y +X − x)− Φ(y). (6.12)

Notice thatv(t,X, y) = 0. (6.13)

6.2 Analytical properties of the value function

Let (t, x, y) ∈ S := [0, T ) × [0, X] × [0,+∞) be the state variable at time t.If the investor immediately buys X − x, the associated cost would be equal to

108

6.2. Analytical properties of the value function

Φ(y +X − x)− Φ(y) therefore, we have

v(t, x, y) ≤ Φ(y +X − x)− Φ(y) on [0, T )× [0, X]× [0,+∞).

6.2.1 Continuity

In this section we will denote for any t, y ≥ 0 and X ∈ A(t, x)

C(t, x, y,X) :=∫ T

tψ(Y t,y,X

u− )dXcu +

∑t≤u≤T

(Φ(Y t,y,Xu )− Φ(Y t,y,X

u− )). (6.14)

To prove continuity of the value function we need the following lemmas.

Lemma 6.2.1. For any non-decreasing right-continuous adapted process X and anyinitial time t, the flow y → Y t,y,X

. is continuous.

Proof : See Theorems V.37 and V.38 of [58]

Lemma 6.2.2. For any t ∈ [0, T ] and y ≥ 0, the value function x → v(t, x, y) isdecreasing.

Proof : Let t ∈ [0, T ] and y ≥ 0. Suppose that 0 ≤ x < x′ ≤ X. Let ε > 0 bearbitrary and assume that X ∈ A(t, x) satisfies

v(t, x, y) + ε ≥ E[C(t, x, y,X)].

Letτ , infs ≥ t : Xs − x ≥ X − x′.

We define the strategy X ′ ∈ A(t, x′) as follows. X ′s = x′ + (Xs − x) for s < τ andXs = X for s > τ . We have that

C(t, x′, y,X ′) ≤ C(t, x, y,X).

Taking expectations on both sides we deduce that

v(t, x′, y) ≤ E[C(t, x′, y,X ′)] ≤ E[C(t, x, y,X)] ≤ v(t, x, y) + ε.

Since ε > 0 was arbitrary the conclusion follows.

Lemma 6.2.3. Let 0 ≤ s ≤ t ≤ T , x ∈ [0, X], y ≥ 0 and X ∈ A(t, x). For anyrandom variable ξ such that for any η > 0, P

(ξ >√η)≤ √η and ξ admits moments

of any order, we have

E[Φ(Y s,y,Xt + ξ

)− Φ

(Y s,y,Xt

) ]≤ ρy(η). (6.15)

109


where ρy is a continuous function defined on R+ such that limζ→0 ρy(ζ) = 0.

Proof : Let ξ be a random variable such that for any η > 0, P(ξ >√η)≤ √η and

ξ admits moments of any order. Let η > 0, we have

∆ := E[Φ(Y s,y,Xt + ξ

)− Φ

(Y s,y,Xt

) ]= E

[ ∫ ξ

0ψ(Y s,y,Xt + ζ

)d ζ]

≤ E[ ∫ √η

0ψ(Y s,y,Xt + ζ

)d ζ11ξ≤√η

]+E

[ ∫ ξ

0ψ(Y s,y,Xt + ζ

)d ζ11ξ>√η

]. (6.16)

We will find an upper bound for the first term of inequality (6.16). First, we noticethat assumption (6.2) implies that we have

For all y ≥ F (a), ψ(y) ≤ b−1αy

1α (6.17)

As ψ is non decreasing, we have

∆1 := E[ ∫ √η

0ψ(Y s,y,Xt + ζ

)d ζ11ξ≤√η

]

≤ E[ ∫ √η

0

(F (a) + ζ

b

) 1α

11Y s,y,Xt ≤F (a)d ζ]

+E[ ∫ √η

0

(Y s,y,Xt + ζ

b

) 1α

11Y s,y,Xt >F (a)d ζ]

= α

b1α (1 + α)

E[

(F (a))α+1α 11Y s,y,Xt ≤F (a)

((1 +

√η

F (a))α+1α − 1

) ]α

b1α (1 + α)

E[ (Y s,y,Xt

)α+1α 11Y s,y,Xt >F (a)

((1 +

√η

Y s,y,Xt

)α+1α − 1

) ]

≤ Cy

((1 +

√η

F (a))α+1α − 1

),

where we have set

Cy := α

b1α (1 + α)

(F (a)

α+1α + E

[ (Y s,y,Xt

)α+1α

]).

Now we turn to evaluating the second term of inequality (6.2.3). As ψ is non de-creasing, we deduce from inequality (6.17) and Cauchy-Schwartz inequality that

110


∆2 := E[ ∫ ξ

0ψ(Y s,y,Xt + ζ

)d ζ11ξ>√η

]

≤ E[ξ

(Y s,y,Xt + ξ

b

) 1α

11ξ>√η]

≤

E[ξ2(Y s,y,Xt + ξ

b

) 2α ]

P (ξ > √η)

12

As ξ admits moments of any order and P(ξ >√η)≤ √η, we conclude the proof

thanks to assumption (6.6).

Theorem 6.2.1. The value function v defined in (6.10) is a continuous functionon [0, T ]× [0, X]× [0,∞).

Proof : We shall study separately the continuity in each of the variables.(1.) Continuity in x uniformly with respect to t :We fix t ∈ [0, T ], y ≥ 0 and 0 ≤ x′ < x ≤ X. Let ε > 0, there exists X ∈ A(t, x)which satisfies

v(t, x, y) + ε ≥ E[C(t, x, y,X)].

By adding a jump of size x − x′ at time T to X we obtain X ∈ A(t, x′). We havethat

C(t, x′, y, X) = C(t, x, y,X) + Φ(Y t,y,XT + (x− x′))− Φ(Y t,y,X

T ).

Taking expectations on both sides of this equation we deduce that

E[C(t, x′, y, X)] ≤ v(t, x, y) + ε+ E[(Φ(Y t,y,XT + (x− x′))− Φ(Y t,y,X

T ))].

On the other hand since, by Lemma 6.2.2, the value function is decreasing in x, weconclude that

v(t, x, y) ≤ v(t, x′, y) ≤ v(t, x, y) + ε+ E[(Φ(Y t,y,XT + (x− x′))− Φ(Y t,y,X

T ))].

We deduce from Lemma (6.2.3) that

0 ≤ v(t, x′, y)− v(t, x, y) ≤ ε+ ρy(x− x′).

Therefore, we have obtained the continuity of the value function v in x uniformly int.

(2.) Continuity in y uniformly with respect to t and x :Fix y ≥ 0, y′ ≥ 0 and ε > 0. Let t ∈ [0, T ] and x ∈ [0, X]. There exists X ∈ A(t, x)

111


which satisfies

v(t, x, y) + ε ≥ E[C(t, x, y,X)].

Let X ∈ A(t, x) such that

dXcs = 0 and ∆Xs = ∆Xs for all t ≤ s < T. (6.18)

We define the following stopping time :

τ := infs ≥ t : Y t,y′,Xs ≤ Y t,y,X

s . (6.19)

We define the following random variable :

ξτ :=∫ τ

tdXc

u −[Y t,y,Xτ − Y t,y′,X

τ

], (6.20)

Notice that when y′ ≤ y, τ = t and ξτ ≤ 0. Now we construct the following strategyX belonging to A(t, x) :

dXs :=

dXs, for t ≤ s < τ(∆Xs + (Y t,y,X

s − Y t,y′,Xs )

)∧(X − Xτ−

), for s = τ

dZs, for τ < s ≤ T,

where Z is the non decreasing process such that Zτ = 0,

on ξτ ≥ 0, dZs :=

dXs, for τ < s < T,

∆Xs + ξτ , for s = T

and,

on ξτ < 0, dZs =

dXs, for τ < s < θ,

X −(Xτ +Xθ− −Xτ

)for s = θ < T

0, for θ < s ≤ T,

where θ is defined by

θ := infu ≥ τ : Xτ +Xu −Xτ ≥ X.

We then obtain the following inequality

v(t, x, y′)− v(t, x, y) ≤ ε+ E[C(t, x, y′, X)− C(t, x, y,X)]

= ε+R1 +R2 +R3 +R4 +R5,

112


where we have set

R1 = E∫ T

tψ(Y t,y′,X

u− )dXcu −

∫ T

tψ(Y t,y,Xu− )

)dXc

u ≤ 0

R2 = E∑

t≤u<τ

[Φ(Y t,y′,X

u )− Φ(Y t,y′,Xu− )

]−[Φ(Y t,y,X

u )− Φ(Y t,y,Xu− )

]

R3 = E[Φ(Y t,y′,X

τ )− Φ(Y t,y′,Xτ− )

]−[Φ(Y t,y,X

τ )− Φ(Y t,y,Xτ− )

]

R4 = E1lξτ≥0

[Φ(Y t,y′,X

T + ξτ )− [Φ(Y t,y,XT )

]

R5 ≤ E1lξτ<0

[Φ(Y t,y′,X

θ− +∆Xθ)− Φ(Y t,y,Xθ− +∆Xθ)

]≤ 0.

We begin with finding an upper bound for R2. As we have

∑t≤u<τ

∆Xu ≤ X,

there exists δ > 0 such that

∑t≤u<τ ; ∆Xu<δ

∆Xu < ε.

We split the sum in R2, between a sum of big jumps of X and small jumps of X.More precisely we can write that R2 = S1 + S2 where

S1 := E∑

t≤u<τ ; ∆Xu≥δ

[Φ(Y t,y′,X

u )− Φ(Y t,y′,Xu− )

]−[Φ(Y t,y,X

u )− Φ(Y t,y,Xu− )

]

S2 := E∑

t≤u<τ ; ∆Xu<δ

[Φ(Y t,y′,X

u )− Φ(Y t,y′,Xu− )

]−[Φ(Y t,y,X

u )− Φ(Y t,y,Xu− )

]

First, notice that Φ is non decreasing and that Y t,y,Xu− ≤ Y t,y′,X

u− ≤ Y t,y′,Xu− for u < τ ,

therefore we have

S1 ≤ E∑

t≤u<τ ; ∆Xu≥δΦ(Y t,y′,X

u− +∆Xu)− Φ(Y t,y,Xu− +∆Xu).

We denote by ui ∈ [t, τ) the random times at which ∆Xu ≥ δ. We have a finitenumber of ui, lower than dXδ e.For u ∈ [t, τ), we set γu := Y t,y′,X

u− − Y t,y,Xu− and notice that γu ≥ 0 a.s.Hence, we

113


obtain

S1 ≤ E[ ∑t≤u<τ ; ∆Xu≥δ

∫ γu

0ψ(Y t,y,Xu− + ζ

)dζ]

≤dXδe∑

i=1E[ ∫ γui

0ψ(Y t,y,X

u−i+ ζ

)dζ]

Moreover, for any u ∈ [t, τ), it follows from Itô’s formula that,

E[γu] = y′ − y + E[ ∫ τ

0h(Y t,y,X

u− )− h(Y t,y′,Xu− ) du

]≤ y′ − y.

Thanks to Lemma (6.2.3), we can then assert that

S1 ≤⌈X

δ

⌉ρy(y′ − y). (6.21)

Once again, it follows from the inequality Y t,y′,Xu− ≤ Y t,y′,X

u− for any u < τ , that

S2 ≤ E[ ∑t≤u<τ ; ∆Xu<δ

∫ ∆Xu

0ψ(Y t,y′,Xu− + ζ

)− ψ

(Y t,y,Xu− + ζ

)dζ]

From the uniform continuity of the flow and the assumption (6.3), we know that forε small enough, we have

S2 ≤ E[ ∑t≤u<τ ; ∆Xu<δ

K∆Xu

]≤ Kε.

We obviously have R3 ≤ 0 because Φ is non decreasing and that Y t,y,Xτ− ≤ Y t,y′,X

τ− ≤Y t,y′,Xτ− for u < τ .

We turn to finding an upper bound for R4. Notice that we have R4 = 0 when y′ ≤ y

therefore we assume here that y′ > y. From Itô’s formula we deduce that there existsa martingale M such that :

Y t,y,Xτ − Y t,y′,X

τ = y − y′ +∫ τ

tdXc

u +∫ τ

th(Y t,y′,X

u− )− h(Y t,y,Xu− ) du+Mτ ,

thus, as h is non increasing, we have E[ξτ

]≤ y′ − y. We may apply Lemma (6.2.3)

114


to get

R4 = E[ ∫ ξ+

τ

0ψ(Y t,y′,X

T + ζ) dζ]

≤ ρy′(y′ − y)

We conclude this proof by recalling that ζ → ρζ(η) is continuous for any η ≥ 0.

(3.) Continuity in t : Let x ∈ [0, X], y ≥ 0 and 0 ≤ s < t. We first observe thatgiven a strategy X ∈ A(t, x) as in (6.11), we can construct a strategy X ∈ A(s, x)by not doing anything on [s, t) and then, we follow the strategy X = X ′, wheredX ′u = δXu on [t, T ], up to time τ where the stopping time τ is defined by

τ := infu ≥ t : Y s,y,Xu− ≤ y.

then, we set

dXu =

(∆Xτ + (y − Y s,y,X

τ− ))∧ (X − x), for u = τ

dXu, for τ < u < θ

X − Xu− , for u = θ

0, for θ < u ≤ T,

where we have setθ := infu ≥ t : Xu− +∆Xu ≥ X.

Considering an ε-optimal strategyX ∈ A(t, x) and following the lines of the previousstep of the proof, we can show that

v(s, x, y)− v(t, x, y) ≤ C(s, x, y, X)− C(t, x, y,X) + ε

≤ E[C(t, x, Y s,y,X

t− , X)− C(t, x, y,X)]

+ ε

≤ f(ε) where f is s.t. lim0f = 0

Now, we consider a strategy X ∈ A(s, x) such that v(s, x, y) ≥ C(s, x, y,X)− ε. Wedefine the strategy X ∈ A(t, x) by

dXu =

(∆Xt + (Y s,y,X

t− − y)+)∧ (X − x), for u = t

dXu, for t < u < θ

X − Xu− , for u = θ and θ ≤ T

∆XT +Xt −Xs− − (Y s,y,Xt− − y)+(= X − XT−), for u = T and θ = +∞,

115


where we have set θ = 0 on ∆Xt + (Y s,y,Xt− − y)+ ≥ X − x and

θ := infu ≥ t : Xu− +∆Xu ≥ X on ∆Xt + (Y s,y,Xt− − y)+ < X − x,

with the convention inf ∅ = +∞. Following again the lines of the previous step ofthe proof, we can show that

v(s, x, y)− v(t, x, y) ≥ C(s, x, y,X)− C(t, x, y, X)− ε

≥ E[C(t,Xt− , Y

s,y,Xt− , X)− C(t, x, y, X)

]+ ε

≥ g(ε) where f is s.t. lim0g = 0

(3.) Conclusion :We can combine the previous considerations to get join continuitybecause the arguments are uniform in the other variables.

6.2.2 Dynamic programming principle

We may state the standard Dynamic programming principle associated to ourcontrol problemDynamic Programming Principle (DPP)Let (t, x, y) ∈ [0, T )× [0, X]× R+. For any stopping time τ taking values in (t, T ),we have


E[ ∫ τ

tψ(Y t,y,X

s− )dXcs +

∑t≤s≤τ


s− )) (6.22)

+ v(τ,Xτ , Yt,y,Xτ )

].

6.2.3 Heuristic HJB

If the value function is smooth, we would have by Itô’s lemma that for r > t

v(r,Xr−, Et,y,Xr− ) = v(t, x, y) +

∫ r

t

(∂v

∂t+ Lπ

)(u,Xu− , Y

t,y,Xu− ) du

+∑t≤u<r

∆v(u,Xu, Yt,y,Xu ) + local martingale,

where

Lπv , π

(∂v

∂x+ ∂v

∂y

)+ 1

2∂2v

∂y2σ2−h∂v

∂y+ γt

∫R

(v(t, x, y − q(y, z))− v(t, x, y))m(dz);

anddXc

u = πu du.

116

6.3. Viscosity Characterization of the value function

We should have then that

0 = infX∈A(t,x)

E[ ∫ r

t

(∂v

∂t+ Lπ

)(u,Xu− , Y

t,y,Xu− ) du (6.23)

+∑

t≤u<τ(Φ(Et,y,X

u )− Φ(Et,y,Xu− ) +∆v(u,Xu, Y

t,y,Xu ))

].

whereLπv , Lπ + ψπ.

Along an optimal strategy X∗ it should be the case that at all times

Φ(Et,y,X∗

u )− Φ(Et,y,X∗

u− ) +∆v(u,X∗u, Y t,y,X∗

u ) = 0.

Taking τ = t + h with h > 0 in (6.23), dividing by h and taking the limit as hgoes to 0, we conjecture that the value function should satisfy

0 ≤ ∂v

∂t+ inf

π≥0Lπv,

as long as the infimum is finite and equal to

L , L0, (6.24)

which occurs iff∂v

∂x+ ∂v

∂y+ ψ ≥ 0.

These considerations would lead us to propose the following HJB quasi-variationalinequality for v

max(−∂v∂t− Lv,−∂v

∂x− ∂v

∂y− ψ

)= 0. (6.25)

where L is given by (6.24).

6.3 Viscosity Characterization of the value func-tion

Theorem 6.3.1. The value function v is the unique continuous viscosity solutionon S to the variational inequality :

max(−∂v∂t− Lv, −∂v

∂x− ∂v

∂y− ψ

)= 0, (6.26)

117


satisfying the following growth condition :

0 ≤ v(t, x, y) ≤ Φ(y +X − x)− Φ(y) on [0, T )× [0, X]× [0,+∞), (6.27)

and with boundary data v(t,X, y) = 0 and v(T, x, y) = Φ(y +X − x)− Φ(y),

Proof : We divide the proof in three steps : we first show that v is a subsolution ofequation 6.26 then that v is a supersolution of equation 6.26 and finally we establisha comparison theorem which will lead to the unicity of the solution of equation 6.26.Proof of subsolution property.Consider any z0 := (t0, x0, y0) ∈ S and ϕ ∈ C2(S) s.t. v − ϕ ≤ 0 on S andv(z0) = ϕ(z0). We define ε > 0 such that t0 + ε < T . Let us consider an admissiblecontrol X ∈ A(t0, x0) where we decide to buy 0 ≤ η < X − x0 assets at time t0 andthen to do nothing until the time t0 + ε. From the dynamic programming principle(DPP), we have

ϕ(z0) = v(z0) ≤ Φ(y0 + η)− Φ(y0) + E[v(t0 + ε, x0 + η, Y t0,y0,X

t0+ε )]

≤ Φ(y0 + η)− Φ(y0) + E[ϕ(t0 + ε, x0 + η, Y t0,y0,X

t0+ε )]. (6.28)

Applying Itô’s formula to the process ϕ(t, x0 +η, Y t0,y0,Xt ) between t0 and t0 +ε, and

taking the expectation, we obtain

E[ϕ(t0 + ε, x0 + η, Y t0,y0,X

t0+ε )]

= ϕ(t0, x0 + η, y0 + η)

+E[∫ t0+ε

t0(∂ϕ∂t

+ Lϕ)(t, x0 + η, Y t0,y0,Xt− )dt

].(6.29)

Combining relations (6.28) and (6.29), we have

E[∫ t0+ε

t0(∂ϕ∂t

+ Lϕ)(t, x0 + η, Y t0,y0,Xt− )dt

]≥ ϕ(t0, x0, y0)− ϕ(t0, x0 + η, y0 + η)

− [Φ(y0 + η)− Φ(y0)] (6.30)

? Take first η = 0. By dividing the above inequality by ε and letting ε going to0, we conclude that

[∂ϕ

∂t+ Lϕ

](z0) ≥ 0. (6.31)

? Take now η > 0 in (6.30). By sending ε to 0, we get

0 ≤ ϕ(t0, x0 + η, y0 + η)− ϕ(t0, x0, y0) + Φ(y0 + η)− Φ(y0).

118


It follows from equation (6.8) that

0 ≤∫ η

0

[∂ϕ

∂x+ ∂ϕ

∂y

](t0, x0 + s, y0 + s) ds+

∫ ψ(y0+η)

ψ(y0)ξ dF (ξ)

=∫ η

0

[∂ϕ

∂x+ ∂ϕ

∂y+ ψ

](t0, x0 + s, y0 + s) ds

Dividing by η and letting η → 0, we obtain

0 ≤[∂ϕ

∂x+ ∂ϕ

∂y+ ψ

](z0). (6.32)

This proves the required subsolution property

max(−∂ϕ∂t− Lϕ, −∂ϕ

∂x− ∂ϕ

∂y− ψ,

)(z0) ≤ 0, (6.33)

Proof of the supersolution property.

We prove the supersolution property by contradiction. Suppose that the claim is nottrue. Therefore, there exists z0 ∈ S, a C2 function ϕ with (ϕ−v)(z0) = 0 and ϕ ≤ v

on S, and η > 0 such that

−∂ϕ∂t

(z)− Lϕ(z0) < −η and −[∂ϕ

∂x+ ∂ϕ

∂y+ ψ

](z0) < −η (6.34)

From the regularity of ϕ, we deduce that there exists ε > 0 such that t0 + ε < T

and, for all z ∈ Bε(z0), where Bε(z0) := ((t0 − ε)+, t0 + ε) × ((x0 − ε)+, (x0 + ε) ∧X)× ((y0 − ε)+, y0 + ε) ⊂ S, we have

−∂ϕ∂t

(z)− Lϕ(z) < −η, (6.35)

−[∂ϕ

∂x+ ∂ϕ

∂y+ ψ

](z) < −η, (6.36)

Notice that inequality (6.36) derives from the monotony and the left-continuity ofψ. For any admissible control X ∈ A(t0, x0), consider the exit time τε = inft ≥t0, (t,Xt, Y

t0,y0,Xt ) /∈ Bε(z0). We notice that τε < T . Applying Itô’s formula to the

119


process ϕ(t,Xt, Yt0,y0,Xt ) between t0 and τ−ε , we obtain

E[ϕ(τ−ε , Xτ−ε

, Y t0,y0,X

τ−ε)]

= ϕ(z0) + E[∫ τε

t0

[∂ϕ

∂t+ Lϕ

](t,Xt−, Y

t0,y0,Xt− )dt

]

+E[∫ τε

0

[∂ϕ

∂x+ ∂ϕ

∂y

](t,Xt− , Y

t0,y0,Xt− ) dXc

t

]

+E

∑t0≤t<τε

[ϕ(t,Xt, Yt0,y0,Xt )− ϕ(t,Xt− , Y

t0,y0,Xt− )]

.(6.37)

From (6.36), and noting that∆Xt := Xt−Xt− = Y t0,y0,Xt −Y t0,y0,X

t− for all t0 ≤ t < τε,we have

ϕ(t,Xt, Yt0,y0,Xt )− ϕ(t,Xt− , Y

t0,y0,Xt− ) =

∫ ∆Xt

0

[∂ϕ

∂x+ ∂ϕ

∂y

](t,Xt− + ζ, Y t0,y0,X

t− + ζ) dζ

≥ η∆Xt −∫ ∆Xt

0ψ(Y t0,y0,X

t− + ζ) dζ

= η∆Xt −[Φ(Y t0,y0,X

t )− Φ(Y t0,y0,Xt− )

]. (6.38)

Hence, plugging last inequality, (6.35) and (6.36) in (6.37), we obtain

E[ϕ(τ−ε , Xτ−ε

, Y t0,y0,X

τ−ε)]≥ ϕ(z0) + ηE

[τε − t0 +

∫ τ−ε

t0dXt

]−E

[ ∫ τε

t0ψ(Y t0,y0,X

t− )dXct

]−E

[ ∑t0≤t<τε

[Φ(Y t0,y0,Xt )− Φ(Y t0,y0,X

t− )]]

(6.39)

Therefore, we obtain

v(z0) = ϕ(z0)

≤ E

∫ τε

t0ψ(Y t0,y0,X

t− )dXct +

∑t0≤t<τε

[Φ(Y t0,y0,Xt )− Φ(Y t0,y0,X

t− )] + ϕ(τ−ε , Xτ−ε, Y t0,y0,X

τ−ε)

−η(E[∫ τ−ε

t0dt+

∫ τ−ε

t0dXt

])(6.40)

On the set Y t0,y0,X

τ−ε∈ [y0− ε, y0 + ε], we have Zz0,X

τ−ε:= (τε, Y t0,y0,X

τ−ε, Xτ−ε

) ∈ Bε(z0),then Zz0,X

τε belongs to the boundary δBε(z0) or is out of Bε(z0). Hence, there existsa random variable γ ∈ [0, 1] such that, on Y t0,y0,X

τ−ε∈ [y0 − ε, y0 + ε], we have

Z(γ)τε :=

(τε, Y

t0,y0,X

τ−ε+ γ∆Xτε , Xτ−ε

+ γ∆Xτε

)∈ δBε(z0).

120


On the one hand, it follows from the DPP, that, on Y t0,y0,X

τ−ε∈ [y0− ε, y0 + ε], we

havev(Z(γ

τε ) ≤ v(Zz0,Xτε ) + Φ(Y t0,y0,X

τε )− Φ(Y t0,y0,X

τ−ε+ γ∆Xτε).

On the other hand, on Y t0,y0,X

τ−ε∈ [y0 − ε, y0 + ε], we have

ϕ(Zz0,X

τ−ε) = ϕ(Z(γ)

τε )−∫ γ∆Xτε

0

[∂ϕ

∂x+ ∂ϕ

∂y

](t,Xt− + ζ, Y t0,y0,X

t− + ζ) dζ

≤ ϕ(Z(γ)τε )− γη∆Xτε +

∫ γ∆Xτε

0ψ(Y t0,y0,X

t− + ζ) dζ

≤ v(Z(γ)τε )− γη∆Xτε + Φ(Y t0,y0,X

τ−ε+ γ∆Xτε)− Φ(Y t0,y0,X

τ−ε)

≤ v(Zz0,Xτε )− γη∆Xτε + Φ(Y t0,y0,X

τε )− Φ(Y t0,y0,X

τ−ε)

(6.41)

Plugging equation (6.41) in (6.40), we obtain

v(z0) ≤ E

∫ τε

t0ψ(Y t0,y0,X

t− )dXct +

∑t0≤t≤τε

[Φ(Y t0,y0,Xt )− Φ(Y t0,y0,X

t− )] + v(Zz0,Xτε )

−η

(E[∫ τ−ε

t0dt+

∫ τ−ε

t0dXt + γ∆Xτε1lY t0,y0,X

τ−ε∈[y0−ε,y0+ε]

])(6.42)

From the dynamic programming principle (DPP), it follows that (6.40) implies that

0 ≤ − infX

E[τε − t0 +

∫ τ−ε

t0dXt + γ∆Xτε1lY t0,y0,X

τ−ε∈[y0−ε,y0+ε]

].

Therefore there exists a sequence of admissible control (Xn)n≥0 such that

E[τnε − t0 +Xn

τn−ε− x0 + γ∆Xn

τnε1lY t0,y0,Xn

τn−ε∈[y0−ε,y0+ε]

]≤ 1n,

where τnε = inft ≥ t0, (t,Xnt , Y

t0,y0,Xn

t ) /∈ Bε(z0). As τnε − t0 ≥ 0, Xnτn−ε− x0 ≥ 0

and γ∆Xnτnε∈ [0, ε], on the one hand, we have

1n≥ εP

(y0 − ε < Y t0,y0,Xn

τn−ε< y0 + ε− γ∆Xn

τnε

)≥ εP

(y0 − ε < Y t0,y0,Xn

τn−ε< y0

),

and, on the other hand, we have

limn→+∞

τnε = t0 and limn→+∞

Xnτn−ε

= x0.

Therefore, limn→+∞ Yt0,y0,Xn

τn−ε= y0 and P

(y0 − ε < Y t0,y0,Xn

τn−ε< y0

)tends to 1 when

121


n goes to +∞. It leads to a contradiction, then we obtain the required viscositysupersolution property :

max(−∂ϕ∂t− Lϕ, −∂ϕ

∂x− ∂ϕ

∂y− ψ

)≥ 0, (6.43)

Lemma 6.3.1 (Comparison Principle). If v is a continuous viscosity subsolution of(6.26) and w is a continuous viscosity supersolution of (6.26), such that v and wsatisfy the growth condition 6.27 and

v(t,X, y) ≤ w(t,X, y) and v(T, x, y) ≤ w(T, x, y),

then v ≤ w.

Proof of the comparison principle.

Construction of a strict subsolution. Let α > 0 such that

limy→+∞

σ(y)2yα−2 < +∞ and limy→+∞

Φ(y,X − x)− Φ(y)yα

= 0.

On [0, T ]× [0, X]× R+, we set

ϕ(t, x, y) = −e−ct ((−a1x+ a2)yα − b1x+ b2) ,

where a1, a2, b1, b2 and c are positive constants such that a2 > a1X and b2 > b1X.We set A(x) = −a1x+ a2 > 0 on [0, X]. On the one hand, we have

∂ϕ

∂t+ Lϕ = e−ct

(c (A(x)yα − b1x+ b2)− A(x)σ2

2 α(α− 1)σ(y)2yα−2)

+e−ct(

+αh(y)A(x)yα−1 + γtA(x)∫Ryα − (y − q(y, z))α dz

)≥ A(x)e−ct

(cyα − σ2

2 α(α− 1)σ(y)2yα−2)

+ ce−cT (−b1X + b2)

≥ ce−cT (−b1X + b2).

The last inequality comes from the linear growth assumption on σ and is true for cbig enough. On the other hand, we have

∂ϕ

∂x+ ∂ϕ

∂y= e−ct

(a1y

α + b1 − αA(x)yα−1)≥ ε,

for b1 big enough compared to a1 and a2.Therefore, ϕ is a continuous function defined on [0, T ] × [0, X] × R+ such that for

122


all m > 0

limy→+∞

1mϕ(t, x, y) + Φ(y,X − x)− Φ(y) = −∞, ∀(t, x) ∈ [0, T ]× [0, X].

and vm := v + 1mϕ is a strict subsolution of equation (6.26) in the sense that

max(−∂vm∂t− Lvm, −

∂vm∂x− ∂vm

∂y− ψ

)≤ − ε

m< 0,

Toward Ishii’s lemma :Let m ≥ 1. We need to show that % := sup(t,x) vm − w ≤ 0, with vm := v + 1

mϕ.

Suppose on the contrary that % > 0. Since,

limy→∞

vm − w = −∞, limx→X

vm − w ≤ 0 and limt→T

vm − w ≤ 0, (6.44)

it is clear that this supremum is attained at some point (t0, x0, y0) ∈ [0, T )× [0, X)×R+ i.e. % = vm(t0, x0, y0)− w(t0, x0, y0), with 0 ≤ t0 < T , 0 ≤ x0 < X and 0 ≤ y0.

Let y ≥ y0, for k ≥ 1, define Φi(t, x, y, z) = vm(t, x, y)−w(t, x, z)−φk(y, z), whereφk(y, z) := k/2 | y − z |2. Let %k = sup[0,T ]×[0,X]×[0,y]2 Φk(t, x, y, z), which is attainedat some point (tk, xk, yk, zk) ∈ [0, T ] × [0, X] × [0, y]2. By taking a subsequence,we can also assume that there exists a point (t0, x0, y0, z0) to which (tk, xk, yk, zk)converges as k →∞. enough, we can then assume that tk < T, and xk > 0.

In order to show that limk xk = limk yk = x0, consider the following inequality :

Φk(t0, x0, y0, y0) ≤ Φk(tk, xk, yk, zk).

In particular, we have

k

2 | yk − zk |2 ≤ −vm(t0, x0, y0) + w(t0, x0, y0) + vm(tk, xk, yk)− w(tk, xk, zk).

As vm and w are continuous on the compact set [0, T ]× [0, X]× [0, y], there existsC > 0 such that

| yk − zk |2 ≤C

k.

Letting k go to +∞, we find y0 = z0. Finally, we show that %k tends to % when kgoes to +∞. To do so, note that % ≤ %k since Φk(tk, xk, yk, zk) ≥ Φk(t0, x0, y0, y0) =vm(t0, x0, y0)− w(t0, x0, y0) = %. Moreover, we have

%k ≤ vm(tk, xk, yk)− w(tk, xk, zk)−i

2 | yk − zk |2≤ vm(tk, xk, yk)− w(tk, xk, yk)

which converges to vm(t0, x0, y0)− w(t0, x0, y0) ≤ % since vm and w are continuous.

123


Since this limit is less or equal to %, we conclude that %k → % and k2 | yk − zk |

2→ 0when k →∞. Moreover, we have vm(t0, x0, y0)− w(t0, x0, y0) = %.

Ishii’s lemma :We can now apply Theorem 3.2 of [25] at the point (tk, xk, yk, zk). There exist Mand M ′ ∈ R such that −k− || A || 0

0 −k− || A ||

≤ M 0

0 M ′

≤ A+ 1kA2 (6.45)

with A = D2φk(yk, zk) = k −k−k k

, where φk(y, z) := k/2 | y − z |2 and

,from the relation between the notion of superjets and our definition of viscositysupersolutions, we have

ε/m ≤ min(K[∂φk

∂y(yk, zk),M ](yk) + γtkI[vm](tk, xk, yk);

∂φk∂y

(yk, zk) + ψ(yk))

(6.46)

0 ≥ min(K[−∂φk

∂z(yk, zk),−M ′](zk) + γtkI[w](tk, xk, zk); −

∂φk∂z

(yk, zk) + ψ(zk))

(6.47)

in which we have set

K[p,M ](y) = σ2(y)2 M−h(y)p and I[g](t, x, y) =

∫R(g(t, x, y−q(y, ζ))−g(t, x, y))m(dζ).

From the second inequality, we have two cases :

i. −∂φk∂z

(yk, zk) + ψ(zk) ≤ 0.

ii. K[−∂φk∂z,−M ′](yk, zk) + γtkI[w](tk, xk, zk) ≤ 0.

Subtracting inequalities 6.46 and 6.47, we obtain, in the first case

ε

m≤ ∂φk

∂y(yk, zk) + ∂φk

∂z(yk, zk) + ψ(yk)− ψ(zk)

≤ ψ(yk)− ψ(zk).

We get a contradiction by letting k going to +∞

124

6.4. Examples and numerical results

In the second case, we have

ε

m≤ K[∂φk

∂y(yk, zk),M ](yk) + γtkI[vm](tk, xk, yk)

−(K[−∂φk

∂z(yk, zk),−M ′](zk) + γtkI[w](tk, xk, zk)

)

≤ 12(Mσ(yk)2 +M ′σ(zk)2)− k(h(yk)− h(zk))(yk − zk)

+γtk(I[vm](tk, xk, yk)− I[w](tk, xk, zk)

)From the Lipschitz continuity of h, the continuity of γ and vm and w, we obtain

ε

m≤ 1

2 limk→∞

⟨ M 00 M ′

(σ(yk), σ(zk)), (σ(yk), σ(zk))⟩

+γt0(I[vm](t0, x0, y0)− I[w](t0, x0, z0)

)≤ 1

2 limk→∞

⟨(A+ 1

kA2)

(σ(yk), σ(zk)), (σ(yk), σ(zk))⟩

+γt0I[Φ](t0, x0, y0),

≤ 12 limk→∞

⟨(A+ 1

kA2)

(σ(yk), σ(zk)), (σ(yk), σ(zk))⟩

= limk→∞

32k(σ(yk)− σ(zk))2,

= 0.

The last equality follows from the Lipschitz continuity of σ and the property :

limk→+∞

k(yk − zk)2 = 0.

6.4 Examples and numerical results

6.4.1 Linear price impact

Suppose that ψ(y) = y and h(y) = ρy. Observe that Φ(y) = y2

2 . In this case, weare under the framework of Alfonsi and Blanc [4].Suppose further that the process

Nt =∫ t

0σ2(Ys) dWs

is a martingale, where Y is the solution of (6.4) with X ≡ 0 and M ≡ 0. Theargument in the proof of Theorem 2.1 of Alfonsi and Blanc [4] shows that in thiscase there are no Price Manipulation Strategies and the control problem is equivalent

125


to the liquidation problem of Obizhaeva and Wang [53]. Indeed, we can verify thatin this case the function

v(t, x, y) = (y + (X − x))2

2 + ρ(T − t) −y2

2 = 22 + ρ(T − t)Φ(y + (X − x))− Φ(y) (6.48)

solves the variational inequality (6.25) and satisfies the terminal condition (6.12).Notice that the solution does not depend on σ(y). Observe that there is no NoTrading region because

∂v

∂x+ ∂v

∂y+ ψ = 0.

In addition∂v

∂t+ Lv = 0

if and only if σ(y) ≡ 0 and

(X − x) = (1 + ρ(T − t))y. (6.49)

Hence only if σ(y) = 0 there is absolutely continuous trading on the linear fron-tier (6.49). Otherwise, the optimal trading strategy should be represented by a”local time” on the frontier.

6.4.2 Deterministic resilience

Let’s assume that σ = q = 0. In particular the volume effect process Y is of finitevariation. Let’s further assume that 0 ≤ t < T , X ∈ A(t, x) and Yt− = y. Then onecan show that

Φ(YT )− Φ(y) =∫ T

tψ(Ys)(dXc

s − h(Ys)ds) +∑

t≤s≤TΦ(Ys)− Φ(Ys−).

Hence,


E[Φ(Et,y,X

T ) +∫ T

tg(h(Et,y,X

s ))ds]− Φ(y),

whereg(y) = yψ(h−1(y)).

If one further assumes that g is a convex function, by Jensen’s inequality

∫ T

tg(h(Et,y,X

s ))ds ≥ (T − t)g(

1T − t

∫ T

th(Y t,y,X

s ) ds).

126


But we also know that

Y t,y,XT = y + (X − x)−

∫ T

th(Y t,y,X

s ) ds.

Hence we have that

v(t, x, y) ≥ infX∈A(t,x)

E[Gt,x,y(Y t,y,X

T )]]− Φ(y),

whereGt,x,y(e) = Φ(e) + (T − t)g

(X − x+ y − e

T − t

).

Assume that e∗t,x,y minimizes Gt,x,y and

e∗t,x,y ∈ Yt,y,XT : X ∈ A(t, x) is of type A. (6.50)

Type A strategies are those that jump only at times t and T and such that dXs =dXc

s = h(Yt)ds on (t, T ). Then we would have that

v(t, x, y) = G(e∗t,x,y)− Φ(y).

If furthermore e∗t,x,y is smooth in the variables t, x, y and Gt,x,y is smooth, so thatG′(e∗t,x,y) = 0, then

∂v

∂t+ Lv = ∂v

∂t− h∂v

∂y= hψ;

and,∂v

∂x+ ∂v

∂y+ ψ = −ψ + ψ = 0.

Assumption (6.50) is not always satisfied. For instance, suppose that we have a blockshape for the order book with F (x) = ψ(x) = h(x) = x. In this case Φ(x) = x2

2 andg(x) = x2. One can show that

Y t,y,XT : X ∈ A(t, x) is of type A = X − x+ y − (y + ξ)(T − t) : ξ ≥ 0.

Also in this casee∗t,x,y = 2(X − x+ y)

T − t+ 2 .

We conclude that (6.50) holds if and only if

2(X − x+ y)T − t+ 2 ≤ X − x+ y − y(T − t);

if and only ifX − x+ y − ((T − t) + 2)y ≥ 0;

127


if and only ifX − x− (1 + (T − t))y ≥ 0.

If this condition is not satisfied then one can show that among the type A strategiesthe best one, does not jump at time t. For this strategy the corresponding cost is

(X − x+ y(1− (T − t)))2

2 + ((T − t)− 12)y2.

Now consider a strategy of the following form : Do not buy any shares on [t, t+ δ) ⊂[t, T ]. After time t+ δ follow a type A strategy with jump at time t+ δ equal to 0.The cost of this strategy is

(X − x+ ye−δ(1− (T − t− δ)))2

2 + (T − t− δ)(y2e−2δ) + y2(1− e−2δ)− 12y

2.

It can be shown that for certain choices of the parameters x,X, y, t, T , e.g. T − t =2, X −x = 1, y = 1, the last expression as a function of δ decreases around 0. Henceδ = 0 is not optimal and a strategy as the one proposed above for some δ > 0outperforms any type A strategy. Therefore in this case type A strategies do notlonger contain optimal strategies even under the condition of convexity for g.

6.4.3 Numerical Results

In this section, we present some numerical results obtained by a numerical ap-proximation of (6.26). We have implemented a finite difference method for singularcontrol problem as studied in [42]. Since the scheme has monotonicity, consistencyand stability properties, it converges to the viscosity solution of (6.26) (see [11] and[10]).

6.4.3.1 Parameters

Time parameters :

— Time step : ∆t=0.002 hour=7.2 seconds

— Horizon time T : T = 1 hour

Space parameters :

— Space step : ∆x = ∆y = 0.01

— Asset position X : X = 1

— Bound of Y Y :Y = 3

Diffusion and jumps parameters

— Resilience function h(y) = y

— Volatility function σ2(y) = 0.25y(Y − y)

128


— Jump sizes q(y, z) = yez, mean measure : γt = 1

129


6.4.3.2 Modified Block Order Book

Φ(y) =

12y

2 if y ≤ a

12 ((y + b− a)2 + a2 − b2) else

with a = 30, b = 70.

(a) t=0 (b) t=250 (c) t=490

Figure 6.3 – Modified Block Order Book

In the previous and the following illustration, we represent the buying and thewaiting regions respectively in yellow and in blue. On the x-axis we have the quantityof already bought and the present value of Y is the second coordinate.When the order book follows the so-called modified block shape, there is a gapbetween two prices a and b. Therefore, as soon as the present price is greater thana, there is a jump in the cost of purchasing new shares and the agent has better towait that new limit sell order arrive and fill the gap. Obviously, close to maturity,the agent cannot wait any longer and this is illustrated in figures (6.3b) and (6.3c).

130


6.4.3.3 Discrete Order Book

Φ(y) = ∑∞k=0 k(y − 1

2k −12)1l(k,k+1] = byc(y − 1

2byc −12)

(a) t=0 (b) t=250 (c) t=450

Figure 6.4 – Discrete Order Book

For a discrete order book, there are gaps with sizes of one or several ticks betweentwo consecutive prices. Therefore, as soon as the agent has consumed the liquidityat a given price, there is a jump in the cost of purchasing new shares and the agentmay prefer to wait that new limit sell orders arrive and fill the gap. The closer tomaturity, she will be, the less she should wait. This extends the ideas of the exampleof modified block order books.

Conclusion

In this chapter, we solved the optimal execution problem for the limit orderbook shape model, with general order book shape and resilience function form. Weused the Dynamic Programming Principle and viscosity solution to find a variatio-nal inequality that the value function satisfies. We approach numerically the valuefunction and find for a fixed time, that the strategy is completely linked to the gapof the order book shape.

131

Conclusion

In the first part of the thesis, in chapter 2, we designed a limit order book modelwith one side of the limit order book, the bid side. We relaxed the constant size eventand allowed size randomness. It gave a flexible model for the state dependence forsize random variables. We represented the state of the limit order book by the depthn. With this representation, the gain function is easier to use for optimal liquidationproblem. We constructed a theory around the space of depthN by defining operatorsof N . Through the Markovian framework, we described the transition probabilitiesand the Markov chain generator. We specified the space of the random variables δ,αS, αB, αC , βB, βC by using the empirical studies in the literature on the limit orderbook. We explained the concept of homogeneity and gave an important lemma 2.3.1based on the strong law of large number theorem for martingale.

In chapter 3, we found that our model is enough flexible for letting the MarkovChain to be recurrent or transient. For the transient case, the introduction of thescope process is an original way to express the fact that all limit orders are not nottoo far to the bid price. With the fact that the volume process is bounded, the totalvolume H doesn’t explode to infinity which is the financial reality. We proved in allcases, the existence of parameters and we derived in particular cases the set of theparameters which lead to the properties. Since the scope is not available in financialdata, we need to find another characteristics for bounding the volume distribution.Instead of using the scope b(n)−d(n), we can define another scope as b(n)−bS(n, a)with an appropriate a ∈ N.

In chapter 4, we explain the interest of the current state n dependence of therandom variable αC for the recurrence problem. The random variable αC shouldnot follow a uniform law and binomial law lead to a recurrent Markov chain. Weconclude an interesting result for the liquidity criteria by showing the interest of thethe current state n dependence of the random variable αS. A more general problemof calibration would be to calibrate according to the stylized facts and the liquiditycriteria as metrics. Then the calibration problem would be to find the parameters inthe set of probabilities Q ∈ Θcalib. The explicit calculation of the metrics, dependingon the model parameters, is necessary in this calibration problem.

In chapter 5, the main result is the strategy is reduced to choose the price impactinstead of the quantity, in the case of the one step model. In order to find this result,we just use the dynamical programming principle and the different relations betweenthe operators of N . Further research needs to be done for solving completely the

133

optimal liquidation problem.In the second part of the thesis, in chapter 6 we solved the optimal liquidation

problem for the limit order book shape model, with general order book shape andresilience function form. We used the Dynamic Programming Principle and visco-sity solution to find a variational inequality that the value function satisfies. Weapproach numerically the value function and find for a fixed time, that the strategyis completely linked to the gaps of the order book shape. Further research needs tobe done for the calibration of the model. If we assume that the order book shape µis the mean average profile, we need to determine the reference price. We can useDelattre et al [27] for computing the reference price (At) in the financial data. Simul-taneously, we need to compute the volume effect (Y data

t ) in the financial data. Sincethe diffusion Y model is a jump diffusion, we need to distinguish the continuous andthe jump part. In the case of Level 1 financial data, we only have the volume diffe-rence ∆dataY between two consecutive events at the ask (∆dataY > 0 for limit orderarrival and ∆dataY < 0 for canceling or market order arrival). There is no ambiguityfor the volume effect change, we can assume for the calibration ∆dataY = ∆modelY .In the case of Level 2 financial data, we only have the volume change for 10 pricelevels. In a naive consideration, the volume effect evolve for any event in the 10 pricelevels. It means that a volume change at the bid has the same impact as a volumechange at six ticks lower to the bid. In this case, we can add another structures suchas ∆dataψ(Y ) and ∆dataφ(Y ) for taking account the event level price.

134

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Titre : Modelisation de carnet d’ordres et gestion de risque de liquidite

Mots cles : Carnet d’ordres, Microstructure des marches financiers, Chaıne de Markov denombrable,Donnees a haute frequence, Controle stochastique, Strategie optimale de liquidation

Resume : Cette these porte sur l’etude demodelisation stochastique de carnet d’ordres, et dedeux problemes de controle stochastique dans uncontexte de risque de liquidite et d’impact sur le prixdes actifs. La these est constituee de deux parties dis-tinctes.Dans la premiere partie, nous traitons, sous differentsaspects, un modele markovien de carnet d’ordres.En particulier, dans le chapitre 2, nous introdui-sons un modele de representation par profondeurcumulee. Nous considerons differents types d’ar-rivees d’evenements avec une dependance de l’etatcourant. Le chapitre 3 traite le probleme de sta-bilite du modele a travers une approche semi-martingale pour la classification d’une chaıne deMarkov denombrable. Nous donnons, pour chaqueprobleme de classification, une calibration du modelea partir des faits empiriques comme le profil moyende la densite du carnet d’ordres. Le chapitre 4 estconsacre a l’estimation et a la calibration de notre

modele a partir des flux de donnees du marche. Ainsi,nous comparons notre modele et les donnees aumoyen des faits stylises et des criteres de liquidite.Nous donnons une calibration concrete aux differentsproblemes de classification. Puis, dans le chapitre 5,nous traitons le probleme de liquidation optimale dansle cadre du modele de representation par profondeurcumulee.Dans la deuxieme partie, dans le chapitre 6, nousproposons une modelisation d’un probleme de liqui-dation optimale d’un investisseur avec une resiliencestochastique. Nous nous ramenons a un probleme decontrole stochastique singulier. Nous montrons que lafonction valeur associee est l’unique solution de vis-cosite d’une equation d’Hamilton-Jacobi-Bellman. Deplus, nous utilisons une methode numerique iterativepour calculer la strategie optimale. La convergence dece schema numerique est obtenue via des criteres demonotonicite, de stabilite et de consistance.

Title : Limit order book modeling and liquidity risk management

Keywords : Limit order book, Financial market microstructure, High frequency data, Denumerable MarkovChain, Stochastic control, Liquidation optimal strategy

Abstract : This thesis deals with the study of stochas-tic modeling of limit order book and two stochasticcontrol problems under liquidity risk and price impact.The thesis is made of two distinct parts.In the first part, we investigate Markovian limit or-der book model under different aspects. In particu-lar, in chapter 2, we introduce a model of cumula-tive depth representation. We consider different arri-val events with dependencies on current state. Chap-ter 3 handles the model stability problem through asemi-martingale approach for the denumerable Mar-kov Chain classification. We give for each problem amodel calibration from empirical facts such as meanaverage profile of limit order book density. Chapter 4is dedicated to model estimation and calibration bymeans of market data flow. Thus, we compare our

model to market data through stylized facts and mar-ket liquidity criteria. We give a concrete calibration todifferent stability problems. Finally, in chapter 5, wehandle an optimal liquidation problem in the cumula-tive depth representation model framework.We study, in the second part, in chapter 6, an optimalliquidation problem of an investor under stochastic re-silience. This problem may be formulated as a sto-chastic singular control problem. We show that the as-sociated value function is the unique viscosity solutionof an Hamilton-Jacobi-Bellman equation. We suggestan iterative numerical method to compute the optimalstrategy. The numerical scheme convergence is ob-tained through the monotonicity, stability and consis-tency criteria.

Universite Paris-SaclayEspace Technologique / Immeuble DiscoveryRoute de l’Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

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