The modeling of the limit order book is directly related to the assumptions on the behavior of real market participants. This paper is twofold. 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.
We propose a limit order book (LOB) model with dynamics that account for both the impact of the most recent order and the shape of the LOB. We present an empirical analysis showing that the type of the last order significantly alters the submission rate of immediate future orders, even after accounting for the state of the LOB. To model these effects jointly we introduce a discrete Markov chain model. Then on these improved LOB dynamics, we find the policy for optimal order choice and placement in the share purchasing problem by framing it as a Markov decision process. The optimal policy deriv…
Limit order books (LOBs) match buyers and sellers in more than half of the world's financial markets. This survey highlights the insights that have emerged from the wealth of empirical and theoretical studies of LOBs. We examine the findings reported by statistical analyses of historical LOB data and discuss how several LOB models provide insight into certain aspects of the mechanism. We also illustrate that many such models poorly resemble real LOBs and that several well-established empirical facts have yet to be reproduced satisfactorily. Finally, we identify several key unresolved questions…
We develop a new market-making model, from the ground up, which is tailored towards high-frequency trading under a limit order book (LOB), based on the well-known classification of order types in market microstructure. Our flexible framework allows arbitrary order volume, price jump, and bid-ask spread distributions as well as the use of market orders. It also honors the consistency of price movements upon arrivals of different order types. For example, it is apparent that prices should never go down on buy market orders. In addition, it respects the price-time priority of LOB. In contrast to …
We showcase how Quantile Regression (QR) can be applied to forecast financial returns using Limit Order Books (LOBs), the canonical data source of high-frequency financial time-series. We develop a deep learning architecture that simultaneously models the return quantiles for both buy and sell positions. We test our model over millions of LOB updates across multiple different instruments on the London Stock Exchange. Our results suggest that the proposed network not only delivers excellent performance but also provides improved prediction robustness by combining quantile estimates.
We have analyzed the statistical probabilities of limit-order book (LOB) shape through building the book using the ultra-high-frequency data from 23 liquid stocks traded on the Shenzhen Stock Exchange in 2003. We find that the averaged LOB shape has a maximum away from the same best price for both buy and sell LOBs. The LOB shape function has nice exponential form in the right tail. The buy LOB is found to be abnormally thicker for the price levels close to the same best although there are much more sell orders on the book. We also find that the LOB shape functions for both buy and sell sides …
We investigate the behavior of limit order books on the meso-scale motivated by order execution scheduling algorithms. To do so we carry out empirical analysis of the order flows from market and limit order submissions, aggregated from tick-by-tick data via volume-based bucketing, as well as various LOB depth and shape metrics. We document a nonlinear relationship between trade imbalance and price change, which however can be converted into a linear link by considering a weighted average of market and limit order flows. We also document a hockey-stick dependence between trade imbalance and one…
We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modeling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e., of the volume of the LOB, or the exponential recovery of the bid-ask spread. W…
In order-driven markets, limit-order book (LOB) resiliency is an important microscopic indicator of market quality when the order book is hit by a liquidity shock and plays an essential role in the design of optimal submission strategies of large orders. However, the evolutionary behavior of LOB resilience around liquidity shocks is not well understood empirically. Using order flow data sets of Chinese stocks, we quantify and compare the LOB dynamics characterized by the bid-ask spread, the LOB depth and the order intensity surrounding effective market orders with different aggressiveness. We …
The limit order book (LOB) depicts the fine-grained demand and supply relationship for financial assets and is widely used in market microstructure studies. Nevertheless, the availability and high cost of LOB data restrict its wider application. The LOB recreation model (LOBRM) was recently proposed to bridge this gap by synthesizing the LOB from trades and quotes (TAQ) data. However, in the original LOBRM study, there were two limitations: (1) experiments were conducted on a relatively small dataset containing only one day of LOB data; and (2) the training and testing were performed in a non-…
In this work, we present a continuous-time large-population game for modeling market microstructure betweentwo consecutive trades. The proposed modeling framework is inspired by our previous work [23]. In this framework, the Limit Order Book (LOB) arises as an outcome of an equilibrium between multiple agents who have different beliefs about the future demand for the asset. The agents' beliefs may change according to the information they observe, triggering changes in their behavior. We present an example illustrating how the proposed models can be used to quantify the consequences of changes …
LOB-Bench: Benchmarking Generative AI for Finance -- an Application to Limit Order Book Data
Oxford-Man Institute of Quantitative Finance, University of Oxford（牛津大学量化金融研究所）；Department of Computer Science, University of Oxford（牛津大学计算机科学系）；Department of Statistics, University of Oxford（牛津大学统计学系）；Foerster Lab for AI Research, University of Oxford（福尔斯特人工智能研究实验室）；J.P. Morgan AI Research（摩根大通人工智能研究）
This paper focuses on an extension of the Limit Order Book (LOB) model with general shape introduced by Alfonsi, Fruth and Schied. Here, the additional feature allows a time-varying LOB depth. We solve the optimal execution problem in this framework for both discrete and continuous time strategies. This gives in particular sufficient conditions to exclude Price Manipulations in the sense of Huberman and Stanzl or Transaction-Triggered Price Manipulations (see Alfonsi, Schied and Slynko). These conditions give interesting qualitative insights on how market makers may create or not price manipul…
In an order-driven financial market, the price of a financial asset is discovered through the interaction of orders - requests to buy or sell at a particular price - that are posted to the public limit order book (LOB). Therefore, LOB data is extremely valuable for modelling market dynamics. However, LOB data is not freely accessible, which poses a challenge to market participants and researchers wishing to exploit this information. Fortunately, trades and quotes (TAQ) data - orders arriving at the top of the LOB, and trades executing in the market - are more readily available. In this paper, …
We derive a continuous time model for the joint evolution of the mid price and the bid-ask spread from a multiscale analysis of the whole limit order book (LOB) dynamics. We model the LOB as a multiclass queueing system and perform our asymptotic analysis using stylized features observed empirically. We argue that in the asymptotic regime supported by empirical observations the mid price and bid-ask-spread can be described using only certain parameters of the book (not the whole book itself). Our limit process is characterized by reflecting behavior and state-dependent jumps. Our analysis allo…
We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges in distribution to a fully coupled SDE-SPDE system when the order arrival rates tend to infinity and the impact of an individual order arrival on the book as well as the tick size tends to zero. The SDE describes the bid/ask price dynamics whil…
This paper studies a limit order book (LOB) model, in which the order dynamics depend on both, the current best available prices and the current volume density functions. For the joint dynamics of the best bid price, the best ask price, and the standing volume densities on both sides of the LOB we derive a weak law of large numbers, which states that the LOB model converges to a continuous-time limit when the size of an individual order as well as the tick size tend to zero and the order arrival rate tends to infinity. In the scaling limit the two volume densities follow each a non-linear PDE …
This work proposes DeepFolio, a new model for deep portfolio management based on data from limit order books (LOB). DeepFolio solves problems found in the state-of-the-art for LOB data to predict price movements. Our evaluation consists of two scenarios using a large dataset of millions of time series. The improvements deliver superior results both in cases of abundant as well as scarce data. The experiments show that DeepFolio outperforms the state-of-the-art on the benchmark FI-2010 LOB. Further, we use DeepFolio for optimal portfolio allocation of crypto-assets with rebalancing. For this pu…
Modern financial exchanges use an electronic limit order book (LOB) to store bid and ask orders for a specific financial asset. As the most fine-grained information depicting the demand and supply of an asset, LOB data is essential in understanding market dynamics. Therefore, realistic LOB simulations offer a valuable methodology for explaining empirical properties of markets. Mainstream simulation models include agent-based models (ABMs) and stochastic models (SMs). However, ABMs tend not to be grounded on real historical data, while SMs tend not to enable dynamic agent-interaction. To overco…
We introduce a methodology to visualize the limit order book (LOB) using a particle physics lens. Open-source data-analysis tool ROOT, developed by CERN, is used to reconstruct and visualize futures markets. Message-based data is used, rather than snapshots, as it offers numerous visualization advantages. The visualization method can include multiple variables and markets simultaneously and is not necessarily time dependent. Stakeholders can use it to visualize high-velocity data to gain a better understanding of markets or effectively monitor markets. In addition, the method is easily adjusta…
We present results demonstrating that an appropriately configured deep learning neural network (DLNN) can automatically learn to be a high-performing algorithmic trading system, operating purely from training-data inputs generated by passive observation of an existing successful trader T. That is, we can point our black-box DLNN system at trader T and successfully have it learn from T's trading activity, such that it trades at least as well as T. Our system, called DeepTrader, takes inputs derived from Level-2 market data, i.e. the market's Limit Order Book (LOB) or Ladder for a tradeable asse…
We use a recent, high-quality data set from Nasdaq to perform an empirical analysis of order flow in a limit order book (LOB) before and after the arrival of a market order. For each of the stocks that we study, we identify a sequence of distinct phases across which the net flow of orders differs considerably. We note some of our results are consist with the widely reported phenomenon of stimulated refill, but that others are not. We therefore propose alternative mechanical and strategic motivations for the behaviour that we observe. Based on our findings, we argue that strategic liquidity pro…
We propose a microstructural modeling framework for studying optimal market making policies in a FIFO (first in first out) limit order book (LOB). In this context, the limit orders, market orders, and cancel orders arrivals in the LOB are modeled as Cox point processes with intensities that only depend on the state of the LOB. These are high-dimensional models which are realistic from a micro-structure point of view and have been recently developed in the literature. In this context, we consider a market maker who stands ready to buy and sell stock on a regular and continuous basis at a public…
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 6 liquid stocks on Nasdaq, we fit a simple, linear relationship between MLOFI and the contemporaneous change in mid-price. For all 6 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 in…
Limit Order Book Simulation and Trade Evaluation with $K$-Nearest-Neighbor Resampling
University of Oxford（牛津大学）；Deutsche Bank（德意志银行）；Imperial College London（伦敦帝国学院）；Oxford-Man Institute of Quantitative Finance, University of Oxford（牛津-曼量化金融研究所（牛津大学））
Price gap, defined as the logarithmic price difference between the first two occupied price levels on the same side of a limit order book (LOB), is a key determinant of market depth, which is one of the dimensions of liquidity. However, the properties of price gaps have not been thoroughly studied due to the less availability of ultrahigh frequency data. In the paper, we rebuild the LOB dynamics based on the order flow data of 26 A-share stocks traded on the Shenzhen Stock Exchange in 2003. Three key empirical statistical properties of price gaps are investigated. We find that the distribution…
We propose a microscopic model to describe the dynamics of the fundamental events in the limit order book (LOB): order arrivals and cancellations. It is based on an operator algebra for individual orders and describes their effect on the LOB. The model inputs are arrival and cancellation rate distributions that emerge from individual behavior of traders, and we show how prices and liquidity arise from the LOB dynamics. In a simulation study we illustrate how the model works and highlight its sensitivity with respect to assumptions regarding the collective behavior of market participants. Empir…
Generative AI for End-to-End Limit Order Book Modelling: A Token-Level Autoregressive Generative Model of Message Flow Using a Deep State Space Network
University of Oxford（牛津大学）；Oxford-Man Institute of Quantitative Finance（牛津-曼量化金融研究院）；Alan Turing Institute（艾伦·图灵研究所）；Man Group（英仕曼集团）；Foerster Lab for AI Research（福斯特人工智能研究实验室）
Oxford-Man Institute of Quantitative Finance, University of Oxford（牛津大学牛津曼量化金融研究院）
Market by order (MBO) data - a detailed feed of individual trade instructions for a given stock on an exchange - is arguably one of the most granular sources of microstructure information. While limit order books (LOBs) are implicitly derived from it, MBO data is largely neglected by current academic literature which focuses primarily on LOB modelling. In this paper, we demonstrate the utility of MBO data for forecasting high-frequency price movements, providing an orthogonal source of information to LOB snapshots and expanding the universe of alpha discovery. We provide the first predictive a…
A quasi-centralized limit order book (QCLOB) is a limit order book (LOB) in which financial institutions can only access the trading opportunities offered by counterparties with whom they possess sufficient bilateral credit. We perform an empirical analysis of a recent, high-quality data set from a large electronic trading platform that utilizes QCLOBs to facilitate trade. We find many significant differences between our results and those widely reported for other LOBs. We also uncover a remarkable empirical universality: although the distributions describing order flow and market state vary c…
We propose a new model for the level I of a Limit Order Book (LOB), which incorporates the information about the standing orders at the opposite side of the book after each price change and the arrivals of new orders within the spread. Our main result gives a diffusion approximation for the mid-price process. To illustrate the applicability of the considered framework, we also propose a feasible method to compute several quantities of interest}, such as the distribution of the time span between price changes and the probability of consecutive price increments conditioned on the current state o…
University of Toronto（多伦多大学）；Department of Computer Science（计算机科学系）
With the proliferation of algorithmic high-frequency trading in financial markets, the Limit Order Book has generated increased research interest. Research is still at an early stage and there is much we do not understand about the dynamics of Limit Order Books. In this paper, we employ a machine learning approach to investigate Limit Order Book features and their potential to predict short term price movements. This is an initial broad-based investigation that results in some novel observations about LOB dynamics and identifies several promising directions for further research. Furthermore, w…
A point process model for order flows in limit order books is proposed, in which the conditional intensity is the product of a Hawkes component and a state-dependent factor. In the LOB context, state observations may include the observed imbalance or the observed spread. Full technical details for the computationally-efficient estimation of such a process are provided, using either direct likelihood maximization or EM-type estimation. Applications include models for bid and ask market orders, or for upwards and downwards price movements. Empirical results on multiple stocks traded in Euronext …
An ability to postpone one's execution without penalty provides 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. This theory was provided in 2005 by Foucault, Kadan and Kandel. I derive 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 dynamic Limit Order Book (LOB) model. Dynamic version of the LOB model allows st…
Optimal order execution is widely studied by industry practitioners and academic researchers because it determines the profitability of investment decisions and high-level trading strategies, particularly those involving large volumes of orders. However, complex and unknown market dynamics pose significant challenges for the development and validation of optimal execution strategies. In this paper, we propose a model-free approach by training Reinforcement Learning (RL) agents in a realistic market simulation environment with multiple agents. First, we configure a multi-agent historical order …
We investigate whether the bid/ask queue imbalance in a limit order book (LOB) provides significant predictive power for the direction of the next mid-price movement. We consider this question both in the context of a simple binary classifier, which seeks to predict the direction of the next mid-price movement, and a probabilistic classifier, which seeks to predict the probability that the next mid-price movement will be upwards. To implement these classifiers, we fit logistic regressions between the queue imbalance and the direction of the subsequent mid-price movement for each of 10 liquid s…
Detecting Multilevel Manipulation from Limit Order Book via Cascaded Contrastive Representation Learning
Guangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation, Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China（广东省脑启发智能计算重点实验室，计算机科学与工程系，南方科技大学，深圳518055，中国）；Department of Statistics and Data Science, Southern University of Science and Technology, Shenzhen 518055, China（统计与数据科学系，南方科技大学，深圳518055，中国）
This article presents a Hawkes process model with Markovian baseline intensities for high-frequency order book data modeling. We classify intraday order book trading events into a range of categories based on their order types and the price changes after their arrivals. To capture the stimulating effects between multiple types of order book events, we use the multivariate Hawkes process to model the self- and mutually-exciting event arrivals. We also integrate a Markovian baseline intensity into the event arrival dynamic, by including the impacts of order book liquidity state and time factor t…
Latency (i.e., time delay) in electronic markets affects the efficacy of liquidity taking strategies. During the time liquidity takers process information and send marketable limit orders (MLOs) to the exchange, the limit order book (LOB) might undergo updates, so there is no guarantee that MLOs are filled. We develop a latency-optimal trading strategy that improves the marksmanship of liquidity takers. The interaction between the LOB and MLOs is modelled as a marked point process. Each MLO specifies a price limit so the order can receive worse prices and quantities than those the liquidity ta…
University College London（伦敦大学学院）；UCL Centre for Blockchain Technologies（伦敦大学学院区块链技术中心）；London School of Economics（伦敦政治经济学院）；Systemic Risk Centre（系统风险中心）
The present work addresses theoretical and practical questions in the domain of Deep Learning for High Frequency Trading. State-of-the-art models such as Random models, Logistic Regressions, LSTMs, LSTMs equipped with an Attention mask, CNN-LSTMs and MLPs are reviewed and compared on the same tasks, feature space and dataset, and then clustered according to pairwise similarity and performance metrics. The underlying dimensions of the modeling techniques are hence investigated to understand whether these are intrinsic to the Limit Order Book's dynamics. We observe that the Multilayer Perceptron…
We report successful results from using deep learning neural networks (DLNNs) to learn, purely by observation, the behavior of profitable traders in an electronic market closely modelled on the limit-order-book (LOB) market mechanisms that are commonly found in the real-world global financial markets for equities (stocks & shares), currencies, bonds, commodities, and derivatives. Successful real human traders, and advanced automated algorithmic trading systems, learn from experience and adapt over time as market conditions change; our DLNN learns to copy this adaptive trading behavior. A novel…
Oxford-Man Institute of Quantitative Finance（牛津曼量化金融研究所）；University of Oxford（牛津大学）
We design multi-horizon forecasting models for limit order book (LOB) data by using deep learning techniques. Unlike standard structures where a single prediction is made, we adopt encoder-decoder models with sequence-to-sequence and Attention mechanisms to generate a forecasting path. Our methods achieve comparable performance to state-of-art algorithms at short prediction horizons. Importantly, they outperform when generating predictions over long horizons by leveraging the multi-horizon setup. Given that encoder-decoder models rely on recurrent neural layers, they generally suffer from slow…
