Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency at which they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll [17] and formalized mathematically by Avellaneda and Stoikov [3]. The market is modeled using a reference price S t following a Brownian motion with standard deviation σ, arrival rates of buy or sell liquidity-consuming orders depend on the distance to the reference price S t and a market maker maximizes the expected utility of its P&L over a finite time horizon. We show that the Hamilton-Jacobi-Bellman equations associated to the stochastic optimal control problem can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also shed light on the asymptotic behavior of the optimal quotes and propose closed-form approxi-7 Our results would be mutatis mutandis the same if we added a penalization term −b(|q T |) for the shares remaining at time T . The rationale underlying this point is that price risk prevents the trader from having important exposure to the stock. Hence, qt should naturally mean-revert around 0.

In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader facing a "background noise" (or "mean field"). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price.In this paper the trader faces the uncertainty of fair price changes too but not only. He also has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically.Our MFG formulation of this problem belongs to the class of "extended MFG", we hence provide generic results to address these "MFG of controls", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of "heterogenous preferences" (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can "learn" it day after day, observing others' behaviors.

Through the analysis of a dataset of ultra high frequency order book updates, we introduce a model which accommodates the empirical properties of the full order book together with the stylized facts of lower frequency financial data. To do so, we split the time interval of interest into periods in which a well chosen reference price, typically the midprice, remains constant. Within these periods, we view the limit order book as a Markov queuing system. Indeed, we assume that the intensities of the order flows only depend on the current state of the order book. We establish the limiting behavior of this model and estimate its parameters from market data. Then, in order to design a relevant model for the whole period of interest, we use a stochastic mechanism that allows to switch from one period of constant reference price to another. Beyond enabling to reproduce accurately the behavior of market data, we show that our framework can be very useful for practitioners, notably as a market simulator or as a tool for the transaction cost analysis of complex trading algorithms.

This paper addresses portfolio liquidation using a new angle. Instead of focusing only on the scheduling aspect like Almgren and Chriss in [2], or only on the liquidity-consuming orders like Obizhaeva and Wang in [35], we link the optimal trade-schedule to the price of the limit orders that have to be sent to the limit order book to optimally liquidate a portfolio. Most practitioners address these two issues separately: they compute an optimal trading curve and they then send orders to the markets to try to follow it. The results obtained in this paper can be interpreted and used in two ways: (i) we solve simultaneously the two problems and provide a strategy to liquidate a portfolio over a few hours, (ii) we provide a tactic to follow a trading curve over slices of a few minutes. As far as the model is concerned, the interactions of limit orders with the market are modeled via a point process pegged to a diffusive "fair price". A Hamilton-Jacobi-Bellman equation is used to solve the control problem involving both non-execution risk and price risk. Backtests are carried out to exemplify the use of our results, both on long periods of time (for the entire liquidation process) and on slices of 5 minutes (to follow a given trading curve).

In this paper, we use a database of around 400,000 metaorders issued by investors and electronically traded on European markets in 2010 in order to study market impact at different scales.At the intraday scale we confirm a square root temporary impact in the daily participation, and we shed light on a duration factor in 1/T γ with γ 0.25. Including this factor in the fits reinforces the square root shape of impact. We observe a power-law for the transient impact with an exponent between 0.5 (for long metaorders) and 0.8 (for shorter ones). Moreover we show that the market does not anticipate the size of the meta-orders. The intraday decay seems to exhibit two regimes (though hard to identify precisely): a "slow" regime right after the execution of the meta-order followed by a faster one. At the daily time scale, we show price moves after a metaorder can be split between realizations of expected returns that have triggered the investing decision and an idiosynchratic impact that slowly decays to zero.Moreover we propose a class of toy models based on Hawkes processes (the Hawkes Impact Models, HIM) to illustrate our reasoning. We show how the Impulsive-HIM model, despite its simplicity, embeds appealing features like transience and decay of impact. The latter is parametrized by a parameter C having a macroscopic interpretation: the ratio of contrarian reaction (i.e. impact decay) and of the "herding" reaction (i.e. impact amplification).

The classical literature on optimal liquidation, rooted in Almgren–Chriss models, tackles the optimal liquidation problem using a trade‐off between market impact and price risk. It answers the general question of optimal scheduling but the very question of the actual way to proceed with liquidation is rarely dealt with. Our model, which incorporates both price risk and nonexecution risk, is an attempt to tackle this question using limit orders. The very general framework we propose to model liquidation with limit orders generalizes existing ones in two ways. We consider a risk‐averse agent, whereas the model of Bayraktar and Ludkovski only tackles the case of a risk‐neutral one. We consider very general functional forms for the execution process intensity, whereas Guéant, Lehalle and Fernandez‐Tapia are restricted to exponential intensity. Eventually, we link the execution cost function of Almgren–Chriss models to the intensity function in our model, providing then a way to see Almgren–Chriss models as a limit of ours.

Optimal trading is a recent field of research which was initiated by Almgren, Chriss, Bertsimas and Lo in the late 90's. Its main application is slicing large trading orders, in the interest of minimizing trading costs and potential perturbations of price dynamics due to liquidity shocks. The initial optimization frameworks were based on mean-variance minimization for the trading costs. In the past 15 years, finer modelling of price dynamics, more realistic control variables and different cost functionals were developed. The inclusion of signals (i.e. short term predictors of price dynamics) in optimal trading is a recent development and it is also the subject of this work.We incorporate a Markovian signal in the optimal trading framework which was initially proposed by Gatheral, Schied, and Slynko [21] and provide results on the existence and uniqueness of an optimal trading strategy. Moreover, we derive an explicit singular optimal strategy for the special case of an Ornstein-Uhlenbeck signal and an exponentially decaying transient market impact. The combination of a mean-reverting signal along with a market impact decay is of special interest, since they affect the short term price variations in opposite directions.Later, we show that in the asymptotic limit were the transient market impact becomes instantaneous, the optimal strategy becomes continuous. This result is compatible with the optimal trading framework which was proposed by Cartea and Jaimungal [10].In order to support our models, we analyse nine months of tick by tick data on 13 European stocks from the NASDAQ OMX exchange. We show that * http://eyaln13.wixsite.com/eyal-neuman arXiv:1704.00847v3 [q-fin.TR] 4 Jun 2018 orderbook imbalance is a predictor of the future price move and it has some mean-reverting properties. From this data we show that market participants, especially high frequency traders, use this signal in their trading strategies.

In this paper, we use a database of around 400,000 metaorders issued by investors and electronically traded on European markets in 2010 in order to study market impact at different scales.At the intraday scale we confirm a square root temporary impact in the daily participation, and we shed light on a duration factor in 1/T γ with γ 0.25. Including this factor in the fits reinforces the square root shape of impact. We observe a power-law for the transient impact with an exponent between 0.5 (for long metaorders) and 0.8 (for shorter ones). Moreover we show that the market does not anticipate the size of the meta-orders. The intraday decay seems to exhibit two regimes (though hard to identify precisely): a "slow" regime right after the execution of the meta-order followed by a faster one. At the daily time scale, we show price moves after a metaorder can be split between realizations of expected returns that have triggered the investing decision and an idiosynchratic impact that slowly decays to zero.Moreover we propose a class of toy models based on Hawkes processes (the Hawkes Impact Models, HIM) to illustrate our reasoning. We show how the Impulsive-HIM model, despite its simplicity, embeds appealing features like transience and decay of impact. The latter is parametrized by a parameter C having a macroscopic interpretation: the ratio of contrarian reaction (i.e. impact decay) and of the "herding" reaction (i.e. impact amplification). 1 A metaorder refers to a large investor buy or sell order which is executed as a succession of smaller orders.