Price dynamics in a Markovian limit order market

We propose and study a simple stochastic model for the dynamics of a limit order book, in which arrivals of market order, limit orders and order cancellations are described in terms of a Markovian queueing system. Through its analytical tractability, the model allows to obtain analytical expressions for various quantities of interest such as the distribution of the duration between price changes, the distribution and autocorrelation of price changes, and the probability of an upward move in the price, {\it conditional} on the state of the order book. We study the diffusion limit of the price process and express the volatility of price changes in terms of parameters describing the arrival rates of buy and sell orders and cancelations. These analytical results provide some insight into the relation between order flow and price dynamics in order-driven markets.Comment: 18 pages, 5 figure

Price Dynamics in a Markovian Limit Order Market

We propose and study a simple stochastic model for the dynamics of a limit order book, in which arrivals of market order, limit orders and order cancellations are described in terms of a Markovian queueing system. Through its analytical tractability, the model allows to obtain analytical expressions for various quantities of interest such as the distribution of the duration between price changes, the distribution and autocorrelation of price changes, and the probability of an upward move in the price, conditional on the state of the order book. We study the diffusion limit of the price process and express the volatility of price changes in terms of parameters describing the arrival rates of buy and sell orders and cancelations. These analytical results provide some insight into the relation between order flow and price dynamics in order-driven markets.limit order book, market microstructure, queueing, diffusion limit, high-frequency data, liquidity, duration analysis, point process

Dynamique de carnets d'ordres boursiers : modèles stochastiques et théorèmes limites

This thesis proposes a mathematical framework for the modeling the intraday dynamics of prices and order ow in limit order markets: electronic markets where participants buy and sell a nancial contract by submitting market orders and limit orders at high frequency to a centralized limit order book. We propose a stochastic model of a limit order book as a queueing system representing the dynamics of the queues of buy/sell limit orders at the best available (bid/ask) price levels and argue that the main features of price dynamics in limit order markets may be understood in this framework. We study in detail the relation between the statistical properties of the price and the dynamics of the point process describing the arrival and execution of orders, rst in a Markovian setting (Chapter 2) then, using asymptotic methods, in a more general setting of a stationary point process in the heavy tra c limit, where orders arrive very frequently, as in most liquid stock markets (Chapters 3 and 4). Chapter 2 studies a Markovian model for limit order book dynamics, in which arrivals of market order, limit orders and order cancelations are described in terms of a Poisson point process. The state of the order book is then described as a time-changed random walk in the positive quadrant regenerated at each hitting time of the boundary. This model allows to obtain analytical expressions for the distribution of the duration between price changes, the distribution and autocorrelation of price changes, and the probability of an upward move in the price, conditional on the state of the order book, by mapping them into quantities related to hitting times of a random walk in Z2 + killed at the boundary. We study the di usion limit of the price process and express the volatility of price changes in terms of parameters describing the arrival rates of buy and sell orders and cancelations. These analytical results provide some insight into the relation between order ow and price dynamics in order-driven markets. Chapter 3 studies a more general queueing model in which order arrivals and order sizes are described by a stationary point process, allowing for a wide range of distributional assumptions and temporal dependence structures in the order ow. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, in a liquid market where buy and sell orders are submitted at high frequency, the intraday dynamics of the limit order book may be approximated by a Markovian jump-di usion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the underlying order ow. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book. Both quantities are expressed in terms of the solution of elliptic equation in the positive orthant, for which solutions are given in important special cases. These results apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, self-exciting point processes and models of the ACD-GARCH family. Chapter 4 is a more detailed study of price dynamics in a limit order market where market orders, limit orders and order cancelations occur with high frequency according to a stationary marked point process. We rst study the discrete, high-frequency dynamics of the price and derive analytical relations between the statistical properties of intraday price changes -distribution of increments, mean reversion and autocorr elation- and properties of the process describing the order ow and depth of the order book. We then study the behavior of the price process at lower frequencies under various heavyviii tel-00738647, version 1 - 4 Oct 2012 CONTENTS ix tra c limits { uid limits and di usion limits{ and derive in each case the price trend and intraday volatility in terms of the arrival rates of buy and sell orders and cancelations and the variance of order sizes. These analytical formulae provide insights into the link between price volatility on one hand and high-frequency order ow and liquidity on the other hand and are shown to be in good agreement with high-frequency data for US stocks.Cette thèse propose un cadre mathématique pour la modélisation de la dynamique du prix et du flux d'ordres dans un marché électronique ou' les participants achètent et vendent un produit financier en soumettant des ordres limites et des ordres de marche à haute fréquence à un carnet d'ordres centralisé. Nous proposons un modèle stochastique de carnet d'ordres en tant que système de files d'attente représentant la totalité des ordres d'achat et de vente au meilleur niveau de prix (bid/ask) et nous montrons que les principales caractéristiques de la dynamique du prix dans un tel marche peuvent être comprises dans ce cadre. Nous étudions en détail la relation entre les principales propriétés du prix et la dynamique du processus ponctuel décrivant l'arrivée et l'exécution des ordres, d'abord dans un cadre Markovien (Chapitre 2) puis, en utilisant des méthodes asymptotiques, dans le cadre plus général d'un processus ponctuel stationnaire dans sa limite heavy traffic, pour lequel les ordres arrivent fréquemment, comme c'est le cas pour la plupart des marches liquides (Chapitres 3 et 4). Le Chapitre 2 étudie un modèle Markovien de dynamique de carnet d'ordres, dans lequel l'arrivée d'ordres de marche, d'ordres limites et d'annulations est d'écrite à l'aide d'un processus de Poisson ponctuel. L'état du carnet d'ordres est d'écrit par une marche aléatoire changée de temps dans le quadrant positif et régénérée à chaque fois qu'elle atteint le bord. Ce modèle permet d'obtenir des expressions analytiques pour la distribution des durées entre changements de prix, la distribution et les autocorrelations des changements de prix, ainsi que la probabilité que le prix augmente, conditionnellement à l'état du carnet d'ordres. Nous étudions la limite de diffusion du prix et exprimons la volatilité des changements de prix à l'aide de paramètres décrivant l'intensité des ordres d'achat, de vente et d'annulations. Ces résultats analytiques permettent de mieux comprendre le lien entre volatilité du prix et flux d'ordres. Le Chapitre 3 étudie un modèle plus général de carnet d'ordres pour lequel les arrivées d'ordres et les tailles d'ordres proviennent d'un processus ponctuel stationnaire très général. Nous obtenons un théorème central limite fonctionnel pour la dynamique jointe des files d'attente des ordres de vente et d'achat, et prouvons que, pour un marche liquide, dans lequel les ordres d'achat et de vente arrivent à haute fréquence, la dynamique du carnet d'ordres peut être approximée par un processus à sauts Markovien diffusant dans l'orthant et dont les caractéristiques peuvent être exprimées à l'aide de propriétés statistiques du flux d'ordres sous-jacent. Ce résultat permet d'obtenir des approximations analytiques pour plusieurs quantities d'intérêt telles que la probabilité que le prix augmente ou la distribution de la durée avant le prochain changement de prix, conditionnellement à l'état du carnet d'ordres. Ces quantités sont exprimées en tant que solutions d'équations elliptiques, pour lesquelles nous donnons des solutions explicites dans certains cas importants. Ces résultats s'appliquent à une classe importante de modèles stochastiques, incluant les mod'eles bas'es sur les processus de Poisson, les processus auto-excitants ou la famille de processus ACD-GARCH. Le Chapitre 4 est une étude plus détaillée de la dynamique du prix dans un marche où les ordres de marche, les ordres limites et les annulations arrivent à haute fréquence. Nous étudions d'abord la dynamique discrète du prix à l'échelle de la seconde et nous obtenons des relations analytiques entre les propriétés statistiques des changements de prix dans une journée -distribution des incréments du prix, retour à la moyenne et autocorrelations- et des propriétés du processus décrivant le flux d'ordres et la profondeur du carnet d'ordres. Ensuite nous étudions le comportement du prix à des fréquences vi CONTENTS vii plus faibles pour plusieurs régimes asymptotiques -limites fluides et diffusives- et nous obtenons pour chaque cas la tendance du prix et sa volatilité en fonction des intensités d'arrivées d'ordres d'achat, de vente et d'annulations ainsi que la variance des tailles d'ordres. Ces formules permettent de mieux comprendre le lien entre volatilité du prix d'un côté et le flux d'ordres, décrivant la liquidité, d'un autre cote. Nous montrons que ces résultats sont en accord avec la réalité des marches liquides

Order book dynamics in liquid markets: limit theorems and diffusion approximations

Revision 2012We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the limit order book may be approximated by a Markovian jump-diffusion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the order flow and only depend on rate of arrival of orders and the covariance structure of order sizes. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book. Our results allow for a wide range of distributional assumptions and temporal dependence in the order flow and apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, self-exciting point processes and models of the ACD-GARCH family

Dynamique de carnets d'ordres boursiers (modeles stochastiques et theoremes limites)

This thesis proposes a mathematical framework for the modeling the intraday dynamics of prices and order flow in it limit order markets: electronic markets where participants buy and sell a financial contract by submitting market orders and limit orders at high frequency to a centralized it limit order book. We propose a stochastic model of a limit order book as a queueing system representing the dynamics of the queues of buysell limit orders at the best available (bid/ask) price levels and argue that the main features of price dynamics in limit order markets may be understood in this framework. We study in detail the relation between the statistical properties of the price and the dynamics of the point process describing the arrival and execution of orders, first in a Markovian setting (Chapter ref chapter.markov) then, using asymptotic methods, in a more general setting of a stationary point process in the it heavy traffic limit, where orders arrive very frequently, as in most liquid stock markets (Chapters ref chapter.heavytraffic and \ref chapter.price).Cette thèse propose un cadre mathématique pour la modélisation de la dynamique du prix et du flux d'ordre dans un marché électronique ou les participants achetent et vendent un produit financier soumettant des ordres limités et des ordres de marché à haute frequence à un textit{carnet d'ordres} centralisé. Nous proposons un modèle stochastique de carnet d'ordres en tant que système de files d'attente représentant la totalit\'e des ordres d'achat et de vente au meilleur niveau de prix (bid/ask) et nous affirmons que les principales caract\'eristiques de la dynamique du prix dans un tel marché peuvent etre comprises dans ce cadre. Nous étudions en détail la relation entre les principales propriétés du prix et la dynamique du processus ponctuel décrivant l'arrivée et l'exécution des ordres, d'abord dans un cadre Markovien (Chapitre \ref{chapter.markov}) puis, en utilisant des m\'ethodes asymptotiques, dans le cadre plus général d'un processus ponctuel stationnaire dans sa limite textit{heavy traffic}, pour lequel les ordres arrivent fréquemment, comme c'est le cas pour la plupart des marchés liquides (Chapitres \ref{chapter.heavytraffic} et ref{chapter.price}).PARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF

Price dynamics in a Markovian limit order market.

We propose and study a simple stochastic model for the dynamics of a limit order book, in which arrivals of market order, limit orders and order cancellations are described in terms of a Markovian queueing system. Through its analytical tractability, the model allows to obtain analytical expressions for various quantities of interest such as the distribution of the duration between price changes, the distribution and autocorrelation of price changes, and the probability of an upward move in the price, conditional on the state of the order book. We study the diffusion limit of the price process and express the volatility of price changes in terms of parameters describing the arrival rates of buy and sell orders and cancelations. These analytical results provide some insight into the relation between order flow and price dynamics in order-driven markets

Order book dynamics in liquid markets: limit theorems and diffusion approximations

We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the limit order book may be approximated by a Markovian jump-diffusion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the order flow and only depend on rate of arrival of orders and the covariance structure of order sizes. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book. Our results allow for a wide range of distributional assumptions and temporal dependence in the order flow and apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, self-exciting point processes and models of the ACD-GARCH family.limit order book ; queueing systems ; heavy traffic limit ; functional central limit theorem ; diffusion approximation ; high-frequency data ; market microstructure ; point process

Price dynamics in a Markovian limit order market

We propose and study a simple stochastic model for the dynamics of a limit order book, in which arrivals of market order, limit orders and order cancellations are described in terms of a Markovian queueing system. Through its analytical tractability, the model allows to obtain analytical expressions for various quantities of interest such as the distribution of the duration between price changes, the distribution and autocorrelation of price changes, and the probability of an upward move in the price, {\it conditional} on the state of the order book. We study the diffusion limit of the price process and express the volatility of price changes in terms of parameters describing the arrival rates of buy and sell orders and cancelations. These analytical results provide some insight into the relation between order flow and price dynamics in order-driven markets.

Order book dynamics in liquid markets: limit theorems and diffusion approximations

We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the limit order book may be approximated by a Markovian jump-diffusion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the underlying order flow. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book. Our results allow for a wide range of distributional assumptions and temporal dependence in the order flow and apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, self-exciting point processes and models of the ACD-GARCH family.