This thesis presents a novel approach to optimize execution strategies in the Foreign Exchange Spot Market, focusing on the application of data-driven methodologies and stochastic modeling.
It begins by proposing a new measure to evaluate the limit order book volume imbalance, which considers multiple price levels and their weighted impact on market predictions.
Then the research employs a combination of convolutional neural networks and long short-term memory models to analyze the limit order book data and its imbalances, i.e. the previously introduced limit order book volume imbalance and the order flow imbalance, which takes into consideration their time evolution. This methodology allows for the exploration of price movements and the identification of optimal execution strategies by predicting future mid-price movements, which are embedded in the drift term of the stochastic model. The findings indicate that the developed models outperform traditional strategies by adapting more effectively to market dynamics.
Further, this work develops a backtesting environment that simulates market conditions to empirically validate the effectiveness of the proposed strategies against historical data. The results demonstrate a superior performance in terms of profitability and cost-efficiency compared to traditional models that do not utilize the drift term from predictive analytics in their stochastic model of the mid-price.
Future work may extend these methodologies to other financial markets and explore prediction models that include the influence of multiple assets.

This thesis examines the approximation capabilities of path signatures within rough path spaces, focusing on both local and global universality. To this end, we provide a self-contained introduction to Rough Path theory, highlighting the interplay between additive and multiplicative functionals. This leads to the renowned Lyons' Extension theorem and the definition of rough path spaces. We also re-examine the concept of universality from a kernel-theoretic perspective, culminating in the classical universal approximation result for signatures over a compact subset of paths. To broaden the scope beyond compact domains, we introduce the framework of weighted spaces and elaborate on the notion of global universality. Specifically, we formally define globally universal kernels and prove sufficient conditions for their existence. The associated reproducing kernel Hilbert space is shown to approximate a wide range of functions over the entire domain, which may be non-(locally) compact. In particular, we apply these theoretical tools to rough path spaces, thereby cementing the global universality of signatures.

In this thesis, we aim to improve the application of deep reinforcement learning in portfo- lio optimization. Reinforcement learning has in recent years been applied to a wide range of problems, from games to control systems in the physical world and also to finance. While reinforcement learning has shown success in simulated environments (e.g. matching or exceeding human performance in games), its adoption in practical applications (non- simulated environments) has lagged. Dulac-Arnold et al. [2019] suggest this is caused by a discrepancy in the experimental set-up in research and the conditions in practice. Specifically, they present a list of challenges that make the application of reinforcement learning in real-world settings more difficult. One of these challenges is non-stationary environments, which is common in financial environments. It is a challenge since, given an observed state, the optimal action may not always be the same as it may change over time due to non-stationarity. Therefore, more specifically, the goal of this thesis is to overcome the challenge of non-stationarity in the application of reinforcement learning to portfolio optimization. In this thesis, we use reinforced deep Markov models (RDMM) introduced by Ferreira [2020] (applied to an optimal execution problem and later used by Cartea et al. [2021] for statistical arbitrage on simulated price movements of an FX triplet) for its data efficiency and ability to handle complex environments. RDMM involve a partially observ- able Markov decision process (POMDP) which is also the setting used by Xie et al. [2021] to model non-stationarity in reinforcement learning. We extend RDMM to incorporate non-stationarity, using the framework suggested by Xie et al. [2021], and apply it to port- folio optimization. Our implementation is sample efficient which allows for quick learning, by doing this we attempt to improve on another challenge of reinforcement learning — i.e. sample-inefficiency [Dulac-Arnold et al., 2019]. Moreover, our implementation can handle continuous state and action spaces.
We compare the performance of our algorithms to classical portfolio optimization tech- niques such as Mean-Variance (MV) and Equal Risk Contribution (ERC), and to popular reinforcement learning techniques such as Deep Deterministic Policy Gradient (DDPG) and Soft Actor-Critic (SAC). We observe our implementation has higher sample-efficiency compared DDPG and SAC, and higher cumulative returns on the test set compared to MV, ERC, DDPG, and SAC.

Portfolio optimization, a fundamental area of study in financial engineering,
plays a crucial role in creating efficient portfolios. In this thesis, we consider
a robust multi-period Mean-Variance portfolio optimization framework and
apply it to real-world market data. The approach we look at incorporates a
time-consistent structure that considers the variance of each period, aiming
to minimize their sum, while ensuring that the expected return for each period
exceeds a predefined threshold. Additionally, we introduce proportional
transaction costs to simulate real-world market conditions. To account for uncertainties
and increase robustness, we employ a distribution uncertainty set
within a Wasserstein ball around the empirical distribution of historical data.
This enables us to select worst-case portfolio scenarios for deriving robust optimal
solutions. We aim to compare this method with other existing portfolio
optimization methods, which we describe in depth in our analysis, to assess
its effectiveness.
To achieve the aforementioned research objectives, we conducted an extensive
review of portfolio optimization literature, exploring both Mean-Variance
and Mean-CVaR portfolio optimization problems. Our research also included
robust approaches on portfolio optimization including different distribution
and parameter uncertainty sets. We proceeded to construct a comprehensive
set of numerical experiments, evaluating portfolio optimization methods performance
on real market data. Moreover, we included the S&P500 index to
compare them against market performance. In these experiments, we randomly
selected stock sets and evaluation period to work on, in order to ensure
an unbiased assessment of the methods.
To evaluate the performance of each method, we used the Sharpe ratio of
the realized portfolio returns. Our key findings indicate that, in most cases,
at least one of the portfolio optimization models outperformed S&P500, suggesting
that portfolio optimization problems perform relatively well in the
real world. Furthermore, single-period models demonstrated better performance
compared to multi-period having higher Sharpe ratio most of the times.
Notably, robust optimization models exhibited superior performance compared
to nominal models, underlying the significance of accounting for uncertainty.
The implications of our research are twofold. Firstly, portfolio optimization
problems, especially in the single-period context, demonstrated
promising performance and should be embraced by financial practitioners
seeking optimal risk-return investment strategies. Secondly, we recommend
the preference for robust approaches over traditional models, as they offer improved
flexibility to market uncertainties and potentially mitigate downside
risks.

With the ever-increasing need to reduce the use of fossil fuels, Tesla is accelerating the world's transition to sustainable energy. This means replacing all internal combustion vehicles with electric ones over time. The growing number of Tesla vehicles on the road poses interesting scaling challenges for all departments especially for the Service Engineering team. To help prioritize issues and reduce service costs, frequencies, and duration, a way to generate an overview of all the costs separated by types of repair is needed. This thesis aims to automatically generate such an overview by borrowing techniques used in Natural Language Processing. In particular, the LDA and GSDMM algorithms for topic generation are tested. Additionally, a novel method based on the cross frequencies of items is presented. Methods are also presented to compute the novelty of every topic and a new metric, here called the growing score, is introduced as a mean to monitor the frequencies of the topics over time. The methods are applied to service records data from Tesla and the results are analyzed in detail. The results are also enhanced by computing additional information regarding costs for all parts of the service visit and also the configurations of the cars. It is found that the novel approach for the topic generation and the novelty and growing scores produce useful information that can be used to optimize and avoid the need for service procedures to reducing total cost of ownership.

Financial markets continue to see an increase in the share of trades executed by algorithmic trading systems. A key component of an efficient algorithmic trading system is its ability to accurately estimate the probability an order will be executed: the fill probability. This thesis aims to determine whether the dynamics of the order book can be captured accurately by a stochastic model, which subsequently leads to the following research question: Can we accurately compute fill probabilities of limit orders, conditional on the state of the order book? Our research builds upon the stochastic model proposed by Cont, Stoikov and Talreja, which assumes orders and cancellations have unit size and arrive according to independent Poisson processes [12]. There are two main advantages of the model proposed by Cont et al., which include: • Straightforward parameter estimation based on observable order book data, • The possibility to compute various probabilities of interest semi-analytically. These probabilities include the fill probability of limit orders and the direction of the next mid-price move. In this model, the number of orders at each price level in the order book is modelled as a birth-death process, in which births represent an increase in the number of orders and conversely, deaths represent a decrease. The first-passage time is the time it takes for such a process to reach a certain state for the first time. Since the order arrivals follow Poisson processes, we can find an expression for the Laplace transforms of the density functions of the first-passage times. By numerically inverting these Laplace transforms we can calculate the probabilities of interest. The parameters of the Poisson processes that determine the arrivals of orders are calibrated using observable order book data of the Euro-US Dollar currency pair of the foreign exchange (FX) spot market. We extend the model of Cont et al. by integrating spread-dependent arrival rates of orders and cancellations, meaning that the arrival rates change for different sizes of the spread. This extension is based on the order book data, which clearly shows a difference in arrival rates for various spread sizes. Additionally, we provide an expression to compute fill probabilities of orders posted not only at the best bid price, but also for one price level below. This expression can be generalised to obtain an expression for the fill probabilities of orders posted even deeper in the order book. Finally, by employing this stochastic model, we also show that this model has the potential to be applied to asset classes other than equities, which in our case is the FX market. To evaluate this model, we compare the computed probabilities with the empirical probabilities based on the FX data. Our findings indicate that the model has the ability to effectively capture the dynamics at each price level and the short-term movements of the mid-price. We are able to show that the fill probability for orders posted at the best quote can be estimated with reasonable accuracy when the order has the highest priority of execution. For orders with lower priority, the model is not able to capture all the dynamics, resulting in most cases in an overestimation of the fill probability. Additionally, we evaluate the method for computing the fill probability at one price level lower than the best quote. We also compare the results of this method to empirical probabilities. However, it is challenging to evaluate the performance of the model due to limited data availability and the small magnitude of the computed probabilities, which are typically less than 1.0%, resulting in large relative errors. Overall, this research contributes to better understanding order book dynamics and provides insights on the precision of the computation of fill probabilities via an extension of [12]. Further improvements and refinement of the model, such as a more realistic order flow or allowing multiple order sizes, could lead to more accurate estimations and a better comprehension of the market dynamics and fill probabilities.

In this paper, we consider optimal pairs trading strategies in terms of static optimality and dynamic optimality under mean–variance criterion. The spread of the entity pairs is assumed to be mean-reverting and follows an Ornstein–Uhlenbeck process. A constrained optimal control problem is considered, and the Lagrange multiplier technique is adopted to transform the primal problem into a family of linear-quadratic optimal control problems that can be solved by the classical dynamic programming principle. Both solutions for static and dynamic optimal pairs trading problems are derived and discussed. We show that the “static and dynamic optimality” is a viable approach to the time-inconsistent control problem. Furthermore, numerical experiments are presented to demonstrate the performance of the optimal pairs trading strategies.

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