High Frequency Trading (HFT) algorithms are automated feedback systems interacting with markets to maximize returns on investments. These systems have the potential to read different resolutions of market information at any given time, where Level I information is the minimal information about an equity--essentially its price--and Level II information is the full order book at that time for that equity. This paper presents a study of using Recurrent Neural Network (RNN) models to predict the spread of the DOW Industrial 30 index traded on NASDAQ, using Level I and Level II data as inputs. The results show that Level II data does not significantly improve the prediction of spread when predicting less than 100 millisecond into the future, while it is increasingly informative for spread predictions further into the future. This suggests that HFT algorithms should not attempt to make use of Level II information, and instead reallocate that computation power for improved trading performance, while slower trading algorithms may very well benefit from processing the complete order book.

We study optimal order placement in a limit order book. By modelling the limit order book dynamics as a Markov chain, we can frame the purchase of a single share as a Markov Decision Process. Within the framework of the model, we can estimate optimal decision policies numerically. The trade rate is varied using a running cost control variable. The optimal policy is found to result in a lower cost of trading as a function of the trade rate compared to a market order only strategy. / Vi studerar optimal orderläggning i en limit orderbok. Genom att modellera dynamiken av inkommande ordrar som en Markov kedja så kan vi formula optimal orderläggning som en Markov Decision Process. Inom ramverket av modellen så kan vi skatta optimala strategier numeriskt. En löpande kostnad används som en kontrollvariabel för handelstakten av den optimala strategin. Vi finner att den optimala strategin resulterar i en lägre handelskostnad som funktion av deltagande jämfört med en marknadsorder strategi.

We introduce a financial model for limit order book with two main features: First, the limit orders and market orders for the given asset both appear and interact with each other. Second, the high frequency trading (HFT, for short) activities are allowed and described by the scaling limit of nearly-unstable multi-dimensional Hawkes processes with power law decay. The model eventually becomes a stochastic partial differential equation (SPDE, for short) with the diffusion coefficient determined by a Volterra integral equation governed by a Hawkes process, whose Hurst exponent is less than 1/2, which makes the volatility path of the stochastic PDE rougher than that driven by a Brownian motion. We have further established the well-posedness of such a system so that a foundation is laid down for further studies in this direction.

Neste trabalho, apresentamos um conjunto de fatos empíricos e propriedades estatística de negociações em alta frequência, e discutimos algumas questões gerais comuns a dados de alta frequência tais: como discretização, espaçamento temporal irregular, durações correlacionadas, periodicidade diária, correlações temporais e as propriedades estatísticas dos fluxos de ordens. Logo apresentamos dois modelos da literatura,estilizados para a dinâmica do limit order book. No primeiro modelo os fluxo de ordens é descrito por processos de Poisson independentes, propomos para ele uma forma alternativa da prova de ergodicidade basejada em funções de Lyapunov. O segundo modelo é um modelo reduzido que toma em consideração dinâmicas tipo difusão para os tamanhos do bid e ask, e se foca só nas ordens como melhores preços, e modela explicitamente as cotações do bid e ask na presença de liquidez oculta. E por ultimo, propomos um modelo alternativo para a dinâmica do preço e do spread no limit order book, estudamos o comportamento assintótico do modelo e estabelecemos condições de ergodicidade e transitoridade. Além disso, consideramos a uma família de cadeias de Markov definidos nas sequências de caracteres (strings, ou palavras) com infinito alfabeto e para alguns exemplos inspirados nos modelos de negociações em alta frequência, obtemos condições para ergodicidade, transitoriedade e recorrência nula, para a qual usamos as técnicas de construção de funções Lyapunov. / In this work, we present a set of empirical facts and statistical properties of negotiations at high frequency and discuss some general issues common to high-frequency data such: as discretization, irregular spacing, correlated durations, daily periodicity, temporal correlations and the statistical properties of flows orders. Soon we present two models stylized in the literature for the dynamic limit order book. In the first model the order flow described by separate Poisson processes and we propose it to an alternative form of test ergodicity based on Lyapunov function. The second model is a reduced model that takes into consideration diffusion-type dynamics for the sizes of the bid and ask, and focus only on orders as best price and model explicitly quotes the bid and ask in the presence of hidden liquidity. And finally, we propose an alternative model for the price dynamics and spread in the limit order book, we study the asymptotic behavior of the model and established conditions of ergodicity. Furthermore, we consider the a family of Markov chains defined on the sequences of characters (strings, or words) with infinite alphabet. For some examples inspired by the models of high frequency trading we obtain a conditions for ergodicity, transience and null-recurrence. In order to prove this we use the construction of Lyapunov functions techniques.

Neste trabalho, apresentamos um conjunto de fatos empíricos e propriedades estatística de negociações em alta frequência, e discutimos algumas questões gerais comuns a dados de alta frequência tais: como discretização, espaçamento temporal irregular, durações correlacionadas, periodicidade diária, correlações temporais e as propriedades estatísticas dos fluxos de ordens. Logo apresentamos dois modelos da literatura,estilizados para a dinâmica do limit order book. No primeiro modelo os fluxo de ordens é descrito por processos de Poisson independentes, propomos para ele uma forma alternativa da prova de ergodicidade basejada em funções de Lyapunov. O segundo modelo é um modelo reduzido que toma em consideração dinâmicas tipo difusão para os tamanhos do bid e ask, e se foca só nas ordens como melhores preços, e modela explicitamente as cotações do bid e ask na presença de liquidez oculta. E por ultimo, propomos um modelo alternativo para a dinâmica do preço e do spread no limit order book, estudamos o comportamento assintótico do modelo e estabelecemos condições de ergodicidade e transitoridade. Além disso, consideramos a uma família de cadeias de Markov definidos nas sequências de caracteres (strings, ou palavras) com infinito alfabeto e para alguns exemplos inspirados nos modelos de negociações em alta frequência, obtemos condições para ergodicidade, transitoriedade e recorrência nula, para a qual usamos as técnicas de construção de funções Lyapunov. / In this work, we present a set of empirical facts and statistical properties of negotiations at high frequency and discuss some general issues common to high-frequency data such: as discretization, irregular spacing, correlated durations, daily periodicity, temporal correlations and the statistical properties of flows orders. Soon we present two models stylized in the literature for the dynamic limit order book. In the first model the order flow described by separate Poisson processes and we propose it to an alternative form of test ergodicity based on Lyapunov function. The second model is a reduced model that takes into consideration diffusion-type dynamics for the sizes of the bid and ask, and focus only on orders as best price and model explicitly quotes the bid and ask in the presence of hidden liquidity. And finally, we propose an alternative model for the price dynamics and spread in the limit order book, we study the asymptotic behavior of the model and established conditions of ergodicity. Furthermore, we consider the a family of Markov chains defined on the sequences of characters (strings, or words) with infinite alphabet. For some examples inspired by the models of high frequency trading we obtain a conditions for ergodicity, transience and null-recurrence. In order to prove this we use the construction of Lyapunov functions techniques.

During the past 30 years, the proliferation of electronic trading has catalysed profound structural change in the global foreign exchange (FX) spot market. Today, more than 60% of the market's volume occurs via electronic trading platforms, which provide traders with round-the-clock market access from anywhere in the world. Such platforms offer several practical benefits that have encouraged market participation from a broad new class of financial institutions and have thereby spurred market growth. The most widely used electronic trading platforms in the FX spot market incorporate several features that differentiate them from those used in other financial markets. These features raise many important questions about order flow, market state, price formation, trader behaviour, and volatility. Despite the enormous trade volumes that such platforms facilitate, these questions have received almost no attention to date. In this thesis, we study a recent, high-quality data set from a large electronic trading platform in the FX spot market in order to investigate several aspects of trading via this mechanism. We calculate a wide range of statistics regarding order flow and market state, and we highlight how our findings contrast to those reported by empirical studies of electronic trading platforms in other markets. We study the autocorrelation properties of returns, absolute returns, and order flow, and we investigate the extent to which the market's organization impacts price formation. We also introduce a model designed to reproduce the most important properties of trading via such a platform. We derive several results regarding the model's temporal evolution, and we simulate the model to investigate how the interactions between individual traders influence volatility. We conclude that electronic trading platforms in the FX spot market retain many desirable features of centralized markets while providing traders with explicit control over their personal trading partnerships.

THE STIGLER-LUCKOCK MODEL FOR A LIMIT ORDER BOOK Abstract One of the types of modern-day markets are so-called order-driven markets whose core component is a database of all incoming buy and sell orders (order book). The main goal of this thesis is to extend the Stigler-Luckock model for order books to give a better insight into the price forming process and behaviour of the market participants themselves. The model introduced in this thesis focuses on a comparison of behaviour and various strategies of market makers who are sophisticated market participants profiting from extensive trading. The market is described using Markov chains, and the strategies are compared using Monte Carlo simulations and game theory. The results showed that market makers' orders should have small spread and large volumes. The final model compares two strategies in which market makers monitor their portfolio. In case of having more cash than asset (or vice versa), they shift prices of their orders to equalise the portfolio. The model recommends checking the market quite often, but acting conservatively, which means not changing prices that frequently and not jumping to conclusions just from a small imbalance in the portfolio.

On propose une perspective originale d'analyser les différents flux hautes fréquences d'information provenant des marchés financiers et fournit des modèles simples et intuitives qui reflètent étroitement la réalité. On observe les données empiriques et note certains faits stylisés et propose des modèles pour capturer ces faits. Dans le chapitre 1, on passe en revue les définitions et propriétés de base des marchés électroniques. En particulier, on revoit les travaux de microstructure et de modélisation du marché, et leurs relations à ce travail. On introduit la taille du "tick", qu'on utilise pour classifier les actifs et interpréter les différents résultats. Dans le chapitre 2, on montre empiriquement que l'impact d'une seule transaction dépend de la durée inter-transactions. En effet, lorsque le taux des échanges devient plus rapide, la variance des rendements des transactions augmente fortement et que ce comportement persiste à des échelles de temps plus grossières. On montre également que la valeur du spread augmente avec l'activité et on en déduit que les carnets d'ordres sont plus vide lorsque le taux des échanges est élevé. Dans le chapitre 3, on présente un modèle pour capturer le bruit de microstructure. Les prix des actifs sont représentés par la somme des rendements "tick" arrivant à des temps de Poisson aléatoires. Le modèle se compose d'une martingale diffusive contaminée par un bruit autocorrélé mais disparaissant aux échelles grossières. On est capable de capturer la signature de la variance et l'autocorrélation faible mais significative des rendements "tick". Dans le chapitre 4, on utilise les processus ponctuels de Hawkes pour modéliser l'arrivée aléatoire des transactions. On modélise la transformation échelle fine - échelle grossière des prix et en particulier l'effet sur les moments des rendements des prix. On propose une technique simple d'estimation non paramétrique de la structure de dépendance des processus de Hawkes dans le cas unidimensionnel et dans quelques cas particuliers multidimensionnels. On applique la méthode à des actifs de Future et trouve des noyaux de dépendance en loi de puissance.

Accurate forecasting of stock price movements is crucial for optimizing trade execution and mitigating risk in automated trading environments, especially when leveraging Limit Order Book (LOB) data. However, developing predictive models from LOB data presents substantial challenges due to its inherent complexities and high-frequency nature. In this thesis, the application of the General Compound Hawkes Process (GCHP) is explored to predict price volatility. Within this framework, a Hawkes process is employed to estimate the times of price changes, and a Markovian model is utilized to determine their amplitudes. The price volatility is obtained through both numerical and analytical methodologies. The performance of the GCHP is assessed on a publicly available dataset, including five distinct stocks. To enhance accuracy, the number of states in the Markov chain is gradually increased, and the advantages of incorporating a higher-order Markov chain for refined volatility estimation are demonstrated.

In this work, we propose a regressional framework, built on the work ”Deep Order Flow Imbalance: Extracting Alpha at Multiple Horizons from the Limit Order Book” by Kolm, et al. (2023), for predicting short term returns of odds on binary betting exchange markets. Using the framework, we apply five different deep learning models that leverage order book data from tennis betting exchanges during the calendar month of July 2023 with the purpose of examining the predictive capabilities of deep learning models in this setting. We train each model on either raw limit order book states or order flow. The models predict the returns of the best available odds returns on five different short term time horizons on the four order book sides, back and lay for each of the two players in a given tennis match. Applying windowing, for each vector prediction we use the 100 latest market messages consisting of 81 features (odds and volumes per the ten first levels in the order book and time delta between market messages) in the case of the raw limit order book state and 41 features (order book flow per the ten first levels in the order book and time delta between market messages) in the case of the order book flow. All code is written in Python and run on Google Colab, leveraging cloud computing, off-the-shelf models and popular libraries, TensorFlow and Keras, for data processing and pipelining, model implementation, training and testing. The models are evaluated relative to a benchmark in the form of a naive predictor based on the average odds returns on the training set. The models do not converge towards an optimal parameter composition duringtraining, indicating low predictive capabilities of the input data. Despite this, we generally find all models to outperform the benchmark on the lay order book sides and while some perform better than others, we see similar relative performance distributions within each model across horizon-order book side combinations. To enhance discussion and suggest the direction of future research we examine relationships between key game characteristics such asthe variation of odds returns and the accuracy of predictions on a given market.