Narrow Results

With the fragmentation of electronic markets, exchanges are now competing in order to attract trading activity on their platform. Consequently, they developed several regulatory tools to control liquidity provision/consumption on their liquidity pool. In this paper, we study the problem of an exchange using incentives in order to increase market liquidity. We model the limit order book as the solution of a stochastic partial differential equation (SPDE) as in [R. Cont and M. S. Müller, 2021, A stochastic partial differential equation model for limit order book dynamics, SIAM Journal on Financial Mathematics 12(2), 744–787]. The incentives proposed to the market participants are functions of the time and the distance of their limit order to the mid-price. We formulate the control problem of the exchange who wishes to modify the shape of the order book by increasing the volume at specific limits. Due to the particular nature of the SPDE control problem, we are able to characterize the solution with a classic Feynman–Kac representation theorem. Moreover, when studying the asymptotic behavior of the solution, a specific penalty function enables the exchange to obtain closed-form incentives at each limit of the order book. We study numerically the form of the incentives and their impact on the shape of the order book, and analyze the sensitivity of the incentives to the market parameters.

Based on the order flow data of a stock and its warrant, the immediate price impacts of market orders are estimated by two competitive models, the power-law model (PL model) and the logarithmic model (LG model). We find that the PL model is overwhelmingly superior to the LG model, regarding the robustness of the estimated parameters and the accuracy of out-of-sample forecasting. We also find that the price impacts of ask and bid orders are consistent with each other for filled trades, since significant positive correlations are observed between the model parameters of both types of orders. Our findings may provide valuable insights for optimal trade execution.

Motivated by the desire to bridge the gap between the microscopic description of price formation (agent-based modeling) and the stochastic differential equations approach used classically to describe price evolution at macroscopic time scales, we present a mathematical study of the order book as a multidimensional continuous-time Markov chain and derive several mathematical results in the case of independent Poissonian arrival times. In particular, we show that the cancellation structure is an important factor ensuring the existence of a stationary distribution and the exponential convergence towards it. We also prove, by means of the functional central limit theorem (FCLT), that the rescaled-centered price process converges to a Brownian motion. We illustrate the analysis with numerical simulation and comparison against market data.

We consider an optimal trading problem over a finite period of time during which an investor has access to both a standard exchange and a dark pool. We take the exchange to be an order-driven market and propose a continuous-time setup for the best bid price and the market spread, both modeled by Lévy processes. Effects on the best bid price arising from the arrival of limit buy orders at more favorable prices, the incoming market sell orders potentially walking the book, and deriving from the cancellations of limit sell orders at the best ask price are incorporated in the proposed price dynamics. A permanent impact that occurs when ‘lit’ pool trades cannot be avoided is built in, and an instantaneous impact that models the slippage, to which all lit exchange trades are subject, is also considered. We assume that the trading price in the dark pool is the mid-price and that no fees are due for posting orders. We allow for partial trade executions in the dark pool, and we find the optimal trading strategy in both venues. Since the mid-price is taken from the exchange, the dynamics of the limit order book also affects the optimal allocation of shares in the dark pool. We propose a general objective function and we show that, subject to suitable technical conditions, the value function can be characterized by the unique continuous viscosity solution to the associated partial integro-differential equation. We present two explicit examples of the price and the spread models, derive the associated optimal trading strategy numerically. We discuss the various degrees of the agent's risk aversion and further show that roundtrips are not necessarily beneficial.

The paper considers a general semi-Markov model for limit order books with two states that incorporates price changes that are not fixed to one tick. Furthermore, we introduce an even more general case of the semi-Markov model for limit order books that incorporates an arbitrary number of states for the price changes. For both cases, the justifications, diffusion limits, implementations and numerical results are presented for different limit order book data: Apple, Amazon, Google, Microsoft, Intel on 21 June 2012 and Cisco, Facebook, Intel, Liberty Global, Liberty Interactive, Microsoft, Vodafone from 3 November 2014 to 7 November 2014.

We develop a Markovian model that deals with the volume offered at the best quote of an electronic order book. The volume of the first limit is a stochastic process whose paths are periodically interrupted and reset to a new value, either by a new limit order submitted inside the spread or by a market order that removes the first limit. Using applied probability results on killing and resurrecting Markov processes, we derive the stationary distribution of the volume offered at the best quote. All proposed models are empirically fitted and compared, stressing the importance of the proposed mechanisms.

In this paper, we consider the pricing problem of European options and spread options for the Hawkes-based model in the limit order book (LOB). We introduce a variant of Hawkes process and consider its limit theorems, namely the exponential multivariate general compound Hawkes process (EMGCHP). We also consider a special case of one-dimensional EMGCHP and its limit theorems. Option pricing with one-dimensional EMGCHP in LOB and numerical examples are presented. We also discuss implied volatility and implied order flow. It reveals the relationship between stock volatility and the order flow in the LOB system. In this way, the Hawkes-based model can provide more market forecast information than the classical Black–Scholes model. Margrabe’s spread options valuations with two one-dimensional and one two-dimensional Hawkes-based models for two assets are presented.

This study examines the impact of increasing pre-trade transparency using intraday data from the Taiwanese stock market, which has recently experienced gradually increasing transparency. The analytical results indicate the disclosed quotes are more informative than the accompanied depths, and the orders of institutional traders are more informative than those of individual traders. Additionally, the best quotes of unexecuted orders for individual traders always contain more information than the average quotes from Steps 2 to 5, whereas this does not apply for institutional investors. The feature is more obvious for the sub-samples with high and medium turnover rate, but not for the sub-samples with low turnover rate.

A one-sided limit order book is modeled as a noncooperative game for several players. An external buyer asks for an amount $X>0$ of a given asset. This amount will be bought at the lowest available price, as long as the price does not exceed an upper bound $\overline{P}$. One or more sellers offer various quantities of the asset at different prices, competing to fulfill the incoming order. The size $X$ of the order and the maximum acceptable price $\overline{P}$ are not a priori known, and thus regarded as random variables. In this setting, we prove that a unique Nash equilibrium exists, where each seller optimally prices his assets in order to maximize his own expected profit. Furthermore, a dynamics is introduced, assuming that each player gradually adjusts his pricing strategy in reply to the strategies adopted by all other players. In the case of (i) infinitely many small players or (ii) two large players with one dominating the other, we show that the pricing strategies asymptotically converge to the Nash equilibrium.

We propose an optimization framework for market-making in a limit order book, based on the theory of stochastic approximation. The idea is to take advantage of the iterative nature of the process of updating bid and ask quotes in order to make the algorithm optimize its strategy on a trial-and-error basis (i.e., online learning) using a variation of the stochastic gradient-descent algorithm. An advantage of this approach is that the exploration of the system by the algorithm is performed in run-time, so explicit specifications of price dynamics are not necessary, as is the case in the stochastic-control approach [(Gueant et al., 2013, Dealing with the Inventory Risk: A Solution to the Market Making Problem, Mathematics and Financial Economics 7(4), 477–507)]. For price/liquidity modeling, we consider a discrete-time variant of the Avellaneda–Stoikov model [(Avellaneda, M. and S. Stoikov, 2008, Liquidation in Limit Order Books with Controlled Intensity, Mathematical Finance 24(4), 627–650)] similar to its developent in the paper of Laruelle et al. [(Laruelle et al., 2013, Optimal Posting Price of Limit Orders: Learning by trading, Mathematics and Financial Economics 7(3), 359–403)] in the context of optimal liquidation tactics. Our aim is to set the ground for more advanced reinforcement learning techniques and to argue that the rationale of our method is generic enough to be extended to other classes of trading problems besides market-making.

We study the one-sided limit order book corresponding to limit sell orders and model it as a measure-valued process. Limit orders arrive to the book according to a Poisson process and are placed on the book according to a distribution which varies depending on the current best price. Market orders to buy periodically arrive to the book according to a second, independent Poisson process and remove from the book the order corresponding to the current best price. We consider the above described limit order book in a high frequency regime in which the rate of incoming limit and market orders is large and traders place their limit sell orders close to the current best price. Our first set of results provide weak limits for the unscaled price process and the properly scaled measure-valued limit order book process in the high frequency regime. In particular, we characterize the limiting measure-valued limit order book process as the solution to a measure-valued stochastic differential equation. We then provide an analysis of both the transient and long-run behavior of the limiting limit order book process.

We investigate TRTH tick-by-tick data on three exchanges (Paris, London and Frankfurt) and on a five-year span. A simple algorithm helps the synchronization of the trades and quotes data, enhancing the basic procedure. The analysis of the performance of this algorithm turns out to be a forensic tool assessing the quality of the database: significant technical changes affecting the exchanges are tracked through the data. Moreover, the choices made when reconstructing order flows have consequences on the quantitative models that are calibrated afterwards on such data. Finally, this order flow reconstruction provides a refined look at the Lee–Ready procedure and its optimal lags. Findings are in line with both financial reasoning and the analysis of an illustrative Poisson model.

This paper is split in three parts: first, we use labeled trade data to exhibit how market participants’ decisions depend on liquidity imbalance; then, we develop a stochastic control framework where agents monitor limit orders, by exploiting liquidity imbalance, to reduce adverse selection. For limit orders, we need optimal strategies essentially to find a balance between fast execution and avoiding adverse selection: if the price has chances to go down, the probability to be filled is high, but it is better to wait a little more to get a better price. In a third part, we show how the added value of exploiting liquidity imbalance is eroded by latency: being able to predict future liquidity consuming flows is of less use if you do not have enough time to cancel and reinsert your limit orders. There is thus a rationale for market makers to be as fast as possible to reduce adverse selection. Latency costs of our limit order driven strategy can be measured numerically.

To authors’ knowledge, this paper is the first to make the connection between empirical evidences, a stochastic framework for limit orders including adverse selection, and the cost of latency. Our work is a first step to shed light on the role played by latency and adverse selection in optimal limit order placement.

We study the multi-level order-flow imbalance (MLOFI), which is a vector quantity that measures the net flow of buy and sell orders at different price levels in a limit order book (LOB). Using a recent, high-quality data set for six liquid stocks on Nasdaq, we fit a simple, linear relationship between MLOFI and the contemporaneous change in mid-price. For all six stocks that we study, we find that the out-of-sample goodness-of-fit of the relationship improves with each additional price level that we include in the MLOFI vector. Our results underline how order-flow activity deep into the LOB can influence the price-formation process.

We show that the excessive use of hidden orders causes artificial price pressures and abnormal asset returns. Using a simple game-theoretical setting, we demonstrate that this effect naturally arises from mis-coordination in trading schedules between traders, when suppliers of liquidity do not sufficiently disclose their trade intentions. As a result, hidden liquidity can increase trading costs and induce excess price fluctuations unrelated to information. Using NASDAQ order book data, we find strong empirical support and illustrate that hidden liquidity is higher if bid–ask spreads are smaller and relative tick sizes are higher.

The modeling of the limit order book is directly related to the assumptions on the behavior of real market participants. This paper is two-fold. We first present empirical findings that lay the ground for two improvements to these models. The first one is concerned with market participants by adding the additional dimension of informed market makers, whereas the second, and maybe more original one, addresses the race in the book between informed traders and informed market makers leading to different shapes of the order book. Namely, we build an agent-based model for the order book with four types of market participants: informed trader, noise trader, informed market makers and noise market makers. We build our model based on the Glosten–Milgrom approach and the most recent Huang–Rosenbaum–Saliba approach. We introduce a parameter capturing the race between informed liquidity traders and suppliers after a new information on the fundamental value of the asset. We then derive the whole “static” limit order book and its characteristics, namely, the bid-ask spread and volumes available at each level price- from the interactions between the agents and compare it with the pre-existing model. We then discuss the case where noise traders have an impact on the fundamental value of the asset and extend the model to take into account many kinds of informed market makers.

In this paper, we develop a new form of simulation model for limit order books based on heterogeneous trading agents, whose motivations are liquidity driven. These agents are abstractions of real market participants, expressed in a stochastic model framework. We develop an efficient way to perform statistical calibration of the model parameters on Level 2 limit order book data from Chi-X, based on a combination of indirect inference and multi-objective optimization. We then demonstrate how such a modeling framework can be of use in testing exchange regulations, as well as informing brokerage decisions and other trading based scenarios.

In the present paper, we study the optimal execution problem under stochastic price recovery based on limit order book dynamics. We model price recovery after execution of a large order by accelerating the arrival of the refilling order, which is defined as a Cox process whose intensity increases by the degree of the market impact. We include not only the market order, but also the limit order in our strategy in a restricted fashion. We formulate the problem as a combined stochastic control problem over a finite time horizon. The corresponding Hamilton–Jacobi–Bellman quasi-variational inequality is solved numerically. The optimal strategy obtained consists of three components: (i) the initial large trade; (ii) the unscheduled small trades during the period; (iii) the terminal large trade. The size and timing of the trade is governed by the tolerance for market impact depending on the state at each time step, and hence the strategy behaves dynamically. We also provide competitive results due to inclusion of the limit order, even though a limit order is allowed under conservative evaluation of the execution price.

The ability to postpone one's execution in the market without penalty in search of a better price is an important strategic advantage in high-frequency trading. To elucidate competition between traders one has to formulate to a quantitative theory of formation of the execution price from market expectations and quotes. Equilibrium theory was provided in 2005 by Foucault, Kadan and Kandel. I derive an asymptotic distribution of the bids/offers as a function of the ratio of patient and impatient traders using the dynamic version of the Foucault, Kadan and Kandel limit order book (LOB) model. Our version of the LOB model allows stylized, but sufficiently realistic representation of the trading markets. In particular, dynamic LOB allows simulation of the distribution of execution times and spreads from high-frequency quotes. Significant analytic progress is made toward framing of short-term trading as competition for immediacy of execution between traders under imperfect information. The results are qualitatively compared with empirical volume-at-price distribution of highly liquid stocks.

This paper investigates fundamental stochastic attributes of the random structures of the volume profiles of the limit order book. We find statistical evidence that heavy-tailed sub-exponential volume profiles occur on the limit order book and these features are best captured via the generalized Pareto distribution MLE method. In futures exchanges, the heavy tail features are not asset class dependent and occur on ultra or mid-range high frequency. Volume forecasting models should account for heavy tails, time varying parameters and long memory. In application, utilizing the generalized Pareto distribution to model volume profiles allows one to avoid over-estimating the round trip cost of trading.