Optimal High Frequency Trading in a Pro-Rata Microstructure with Predictive Information

We propose a framework to study optimal trading policies in a one-tick pro-rata limit order book, as typically arises in short-term interest rate futures contracts. The high-frequency trader has the choice to trade via market orders or limit orders, which are represented respectively by impulse controls and regular controls. We model and discuss the consequences of the two main features of this particular microstructure: first, the limit orders sent by the high frequency trader are only partially executed, and therefore she has no control on the executed quantity. For this purpose, cumulative executed volumes are modelled by compound Poisson processes. Second, the high frequency trader faces the overtrading risk, which is the risk of brutal variations in her inventory. The consequences of this risk are investigated in the context of optimal liquidation. The optimal trading problem is studied by stochastic control and dynamic programming methods, which lead to a characterization of the value function in terms of an integro quasi-variational inequality. We then provide the associated numerical resolution procedure, and convergence of this computational scheme is proved. Next, we examine several situations where we can on one hand simplify the numerical procedure by reducing the number of state variables, and on the other hand focus on specific cases of practical interest. We examine both a market making problem and a best execution problem in the case where the mid-price process is a martingale. We also detail a high frequency trading strategy in the case where a (predictive) directional information on the mid-price is available. Each of the resulting strategies are illustrated by numerical tests

Optimal High Frequency Trading with limit and market orders

We propose a framework for studying optimal market making policies in a limit order book (LOB). The bid-ask spread of the LOB is modelled by a Markov chain with finite values, multiple of the tick size, and subordinated by the Poisson process of the tick-time clock. We consider a small agent who continuously submits limit buy/sell orders and submits market orders at discrete dates. The objective of the market maker is to maximize her expected utility from revenue over a short term horizon by a tradeoff between limit and market orders, while controlling her inventory position. This is formulated as a mixed regime switching regular/ impulse control problem that we characterize in terms of quasi-variational system by dynamic programming methods. In the case of a mean-variance criterion with martingale reference price or when the asset price follows a Levy process and with exponential utility criterion, the dynamic programming system can be reduced to a system of simple equations involving only the inventory and spread variables. Calibration procedures are derived for estimating the transition matrix and intensity parameters for the spread and for Cox processes modelling the execution of limit orders. Several computational tests are performed both on simulated and real data, and illustrate the impact and profit when considering execution priority in limit orders and market ordersMarket making; limit order book; inventory risk; point process; stochastic control

Numerical methods for an optimal order execution problem

This paper deals with numerical solutions to an impulse control problem arising from optimal portfolio liquidation with bid-ask spread and market price impact penalizing speedy execution trades. The corresponding dynamic programming (DP) equation is a quasi-variational inequality (QVI) with solvency constraint satisfied by the value function in the sense of constrained viscosity solutions. By taking advantage of the lag variable tracking the time interval between trades, we can provide an explicit backward numerical scheme for the time discretization of the DPQVI. The convergence of this discrete-time scheme is shown by viscosity solutions arguments. An optimal quantization method is used for computing the (conditional) expectations arising in this scheme. Numerical results are presented by examining the behaviour of optimal liquidation strategies, and comparative performance analysis with respect to some benchmark execution strategies. We also illustrate our optimal liquidation algorithm on real data, and observe various interesting patterns of order execution strategies. Finally, we provide some numerical tests of sensitivity with respect to the bid/ask spread and market impact parameters

Optimal High Frequency Trading with limit and market orders

We propose a framework for studying optimal market making policies in a limit order book (LOB). The bid-ask spread of the LOB is modelled by a Markov chain with finite values, multiple of the tick size, and subordinated by the Poisson process of the tick-time clock. We consider a small agent who continuously submits limit buy/sell orders and submits market orders at discrete dates. The objective of the market maker is to maximize her expected utility from revenue over a short term horizon by a tradeoff between limit and market orders, while controlling her inventory position. This is formulated as a mixed regime switching regular/ impulse control problem that we characterize in terms of quasi-variational system by dynamic programming methods. In the case of a mean-variance criterion with martingale reference price or when the asset price follows a Levy process and with exponential utility criterion, the dynamic programming system can be reduced to a system of simple equations involving only the inventory and spread variables. Calibration procedures are derived for estimating the transition matrix and intensity parameters for the spread and for Cox processes modelling the execution of limit orders. Several computational tests are performed both on simulated and real data, and illustrate the impact and profit when considering execution priority in limit orders and market ordersComment: 22 page

CRH-GRHEn – Groupe de recherche sur l’histoire de l’environnement

Geneviève Massard-Guilbaud, directrice d’étudeMarc Élie, Frédéric Graber, Fabien Locher, Thomas Le Roux, chargés de recherche au CNRSMarie-Hélène Mandrillon Séminaire RUCHE « Nouvelles recherches en histoire environnementale » Le RUCHE a poursuivi pour la troisième année consécutive son séminaire d’histoire environnementale. Alors que les interventions des deux années précédentes étaient de nature historiographique, celles du cycle 2011-2012 ont permis la présentation de recherches en cours o..

Centre de recherches historiques – CRH

Hinnerk Bruhns, directeur de recherche au CNRSCatherine Colliot-Thélène, professeur à l’Université Rennes-IPatrice Duran, professeur à l’ENS Autour de Max Weber Le séminaire a abordé, dans différents domaines des sciences sociales, des questions relevant des œuvres de Max Weber et de ses contemporains, ainsi que de leur postérité au cours du XXe siècle. Deux séances ont été consacrées à des relectures de parties essentielles de l’œuvre de Max Weber, en rapport avec des moments clefs de sa bio..

Centre de recherches historiques – CRH

Hinnerk Bruhns, directeur de recherche au CNRSCatherine Colliot-Thélène, professeur à l’Université Rennes-IPatrice Duran, professeur à l’ENS Autour de Max Weber Le séminaire a abordé, dans différents domaines des sciences sociales, des questions relevant des œuvres de Max Weber et de ses contemporains, ainsi que de leur postérité au cours du XXe siècle. Deux séances ont été consacrées à des relectures de parties essentielles de l’œuvre de Max Weber, en rapport avec des moments clefs de sa bio..

CRH – Centre de recherches historiques

Hinnerk Bruhns, directeur de recherche au CNRSCatherine Colliot-Thélène, professeur à l’Université Rennes-IPatrice Duran, professeur à l’ENS-Cachan Autour de Max Weber Au cours de l’année ont été abordées, dans différents domaines des sciences sociales, des questions relevant des œuvres de Max Weber et de ses contemporains, ainsi que de leur postérité au cours du vingtième siècle. Le séminaire s’est ouvert le 16 novembre 2010 par une conférence de Monsieur Ulrich Bielefeld, directeur de l’uni..

Replication Fork Polarity Gradients Revealed by Megabase-Sized U-Shaped Replication Timing Domains in Human Cell Lines

In higher eukaryotes, replication program specification in different cell types remains to be fully understood. We show for seven human cell lines that about half of the genome is divided in domains that display a characteristic U-shaped replication timing profile with early initiation zones at borders and late replication at centers. Significant overlap is observed between U-domains of different cell lines and also with germline replication domains exhibiting a N-shaped nucleotide compositional skew. From the demonstration that the average fork polarity is directly reflected by both the compositional skew and the derivative of the replication timing profile, we argue that the fact that this derivative displays a N-shape in U-domains sustains the existence of large-scale gradients of replication fork polarity in somatic and germline cells. Analysis of chromatin interaction (Hi-C) and chromatin marker data reveals that U-domains correspond to high-order chromatin structural units. We discuss possible models for replication origin activation within U/N-domains. The compartmentalization of the genome into replication U/N-domains provides new insights on the organization of the replication program in the human genome

Contrôle optimal dans des carnets d'ordres limites

We propose a quantitative approach to some high frequency trading problematics. We are interested in several aspects of this field, from minimizing indirect trading costs to market making, and more generally in profit maximization strategies over a finite time horizon. We build an original framework that reflects specificities of high frequency trading, and especially the distinction between passive and active trading, thanks to mixed stochastic control methods. We carefully model high fequency market phenomena, and for each of them we propose calibration methods that are compatible with practical constraints of algorithmic trading.On propose un traitement quantitatif de différentes problématiques du trading haute fréquence. On s'intéresse à plusieurs aspects de cette pratique, allant de la minimisation des frais indirects de trading, jusqu'à la tenue de marché, et plus généralement des stratégies de maximisation du profit sur un horizon de temps fini. On établit un cadre de travail original qui permet de refléter les spécificités du trading haute fréquence, notamment la distinction entre le trading passif et le trading actif, à l'aide de méthodes de contrôle stochastique mixte. On porte un soin particulier à la modélisation des phénomènes de marché en haute fréquence, et on propose pour chacun des méthodes de calibration compatibles avec les contraintes pratiques du trading algorithmique