This thesis proposes methods exploiting the vast informational content of limit order books (LOBs). The first part of this thesis discovers LOB inefficiencies that are sources of statistical arbitrage for high-frequency traders. Chapter 1 develops new theoretical relationships between cross-listed stocks, so their prices are arbitrage free. Price deviations are captured by a novel strategy that is then evaluated in a new backtesting environment enabling the study of latency and its importance for high-frequency traders. Chapter 2 empirically demonstrates the existence of lead-lag arbitrage at high-frequency. Lead-lag relationships have been well documented in the past, but no study has shown their true economic potential. An original econometric model is proposed to forecast returns on the lagging asset, and does so accurately out-of-sample, resulting in short-lived arbitrage opportunities. In both chapters, the discovered LOB inefficiencies are shown to be profitable, thus providing a better understanding of high-frequency traders’ activities. The second part of this thesis investigates anomalous patterns in LOBs. Chapter 3 studies the performance of machine learning methods in the detection of fraudulent orders. Because of the large amount of LOB data generated daily, trade frauds are challenging to catch, and very few cases are available to fit detection models. A novel unsupervised deep learning–based framework is proposed to discern abnormal LOB behavior in this difficult context. It is asset independent and can evolve alongside markets, providing better fraud detection capabilities to market regulators.

The main objective of this Master’s thesis is in the modelling of non-stationary time series data. While classical statistical models attempt to correct non- stationary data through differencing and de-trending, I attempt to create localized clusters of stationary time series data through the use of the self-organizing map algorithm. While numerous techniques have been developed that model time series using the self-organizing map, I attempt to build a mathematical framework that justifies its use in the forecasting of financial times series. Additionally, I compare existing forecasting methods using the SOM with those for which a framework has been developed and which have not been applied in a forecasting context. I then compare these methods with the well known ARIMA method of time series forecasting. The second objective of this thesis is to demonstrate the self-organizing map’s ability to cluster data vectors as it was originally developed as a neural network approach to clustering. Specifically I will demonstrate its clustering abilities on limit order book data and present various visualization methods of its output.

The aim of this thesis is to study fundamental problems in financial and insurance mathematics particularly the problem of measuring risk and its application within financial and insurance frameworks. The main contributions of this thesis can be classified in three main axes: the theory of risk measures, the problem of capital allocation and the theory of fluctuation. In Chapter 2, we design new coherent risk measures and study the associated capital allocation in the context of collective risk theory. We introduce the family of Cumulative Entropic Risk Measures. In Chapter 3, we study the optimal portfolio problem for the Entropic Value at Risk coherent risk measure for particular return models which are based on relevant cases of Jump-Diffusion models. In Chapter 4, we extending the notion of natural risk statistics to the multivariate setting. This non-trivial extension will endow us with multivariate data-based risk measures that are bound to have applications in finance and insurance. In Chapter 5, we introduce the concepts of drawdown and speed of depletion to the ruin theory literature and study them for the class of spectrally negative Lévy risk processes.

The Markov-switching GARCH model is the foundation of this thesis. This model offers rich dynamics to model financial data by allowing for a GARCH structure with time-varying parameters. This flexibility is unfortunately undermined by a path dependence problem which has prevented maximum likelihood estimation of this model since its introduction, almost 20 years ago. The first half of this thesis provides a solution to this problem by developing two original estimation approaches allowing us to calculate the maximum likelihood estimator of the Markov-switching GARCH model. The first method is based on both the Monte Carlo expectation-maximization algorithm and importance sampling, while the second consists of a generalization of previously proposed approximations of the model, known as collapsing procedures. This generalization establishes a novel relationship in the econometric literature between particle filtering and collapsing procedures. The discovery of this relationship is important because it provides the missing link needed to justify the validity of the collapsing approach for estimating the Markov-switching GARCH model. The second half of this thesis is motivated by the events of the financial crisis of the late 2000s during which numerous institutional failures occurred because risk exposures were inappropriately measured. Using 78 different econometric models, including many generalizations of the Markov-switching GARCH model, it is shown that model risk plays an important role in the measurement and management of long-term investment risk in the context of variable annuities. Although the finance literature has devoted a lot of research into the development of advanced models for improving pricing and hedging performance, the approaches for measuring dynamic hedging effectiveness have evolved little. This thesis offers a methodological contribution in this area by proposing a statistical framework, based on regression analysis, for measuring the effectiveness of dynamic hedges for long-term investment guarantees.

In this contribution, we introduce a new class of bivariate distributions of Marshall-Olkin type, called bivariate Erlang distributions. The Laplace transform, product moments and conditional densities are derived. Potential applications of bivariate Erlang distributions in life insurance and finance are considered. Further, our research project is devoted to the study of multivariate risk processes, which may be useful in analyzing ruin problems for insurance companies with a portfolio of dependent classes of business. We apply results from the theory of piecewise deterministic Markov processes in order to derive exponential martingales needed to establish computable upper bounds of the ruin probabilities, as their exact expressions are intractable.

This thesis consists mainly of three papers concerned with Markov additive processes, Lévy processes and applications on finance and insurance. The first chapter is an introduction to Markov additive processes (MAP) and a presentation of the ruin problem and basic topics of Mathematical Finance. The second chapter contains the paper "Lévy Systems and the Time Value of Ruin for Markov Additive Processes" written with Manuel Morales and that is published in the European Actuarial Journal. This paper studies the ruin problem for a Markov additive risk process. An expression of the expected discounted penalty function is obtained via identification of the Lévy systems. The third chapter contains the paper "On a Generalization of the Expected Discounted Penalty Function to Include Deficits at and Beyond Ruin" that is submitted for publication. This paper presents an extension of the expected discounted penalty function in a setting involving aggregate claims modelled by a subordinator, and Brownian perturbation. This extension involves a sequence of expected discounted functions of successive minima reached by a jump of the risk process after ruin. It has important applications in risk management and in particular, it is used to compute the expected discounted value of capital injection. Finally, the fourth chapter contains the paper "The Minimal Entropy Martingale Measure (MEMM) for a Markov-Modulated Exponential" written with Romuald Hérvé Momeya and that is published in the journal Asia Pacific Financial Market. It presents new results related to the incompleteness problem in a financial market, where the risky asset is driven by Markov additive exponential model. These results characterize the martingale measure satisfying the entropy criterion. This measure is used to compute the price of the option and the portfolio of hedging in an exponential Markov-modulated Lévy model.

This thesis focuses on the pricing and hedging problems of financial derivatives in a Markov-modulated exponential-Lévy model. Such model is built on a Markov additive process as much as the Black-Scholes model is based on Brownian motion. Since there exist many sources of randomness, we are dealing with an incomplete market and this makes inoperative techniques initiated by Black, Scholes and Merton in the context of a complete market. We show that, by using some results of the theory of Markov additive processes it is possible to provide solutions to the previous problems. In particular, we characterize the martingale measure which minimizes the relative entropy with respect to the physical probability measure. Also under some conditions, we derive explicitly the optimal portfolio which allows an agent to minimize the local quadratic risk associated. Furthermore, in a more practical perspective we characterize the price of a European type option as the unique viscosity solution of a system of nonlinear integro-di erential equations. This is a rst step towards the construction of e ective numerical schemes to approximate options price.

Nested Archimedean copulas recently gained interest since they generalize the well-known class of Archimedean copulas to allow for partial asymmetry. Sampling algorithms and strategies have been well investigated for nested Archimedean copulas. However, for likelihood based inference such as estimation or goodness-of-fit testing it is important to have the density. The present work fills this gap. After a short introduction on copula and nested Archimedean copulas, a general formula for the derivatives of the nodes and inner generators appearing in nested Archimedean copulas is developed. This leads to a tractable formula for the density of nested Archimedean copulas. Various examples including famous Archimedean families and transformations of such are given. Furthermore, a numerically efficient way to evaluate the log-density is presented.

The field of auto insurance is working by cycles with phases of profitability and other of non-profitability. In the phases of non-profitability, insurance companies generally have the reflex to increase the cost of premiums in an attempt to reduce losses. For cons, very large increases may have the effect of massive attrition of the customers. A too high attrition rate could have a negative effect on long-term profitability of the company. Proper management of rate increases thus appears crucial to an insurance company. This thesis aims to build a simulation tool to predict the content of the insurance portfolio held by an insurer based on the rate change proposed to each insured. A proce- dure using univariate Gaussian Processes regression is developed. This procedure offers a superior performance than the logistic regression model typically used to perform such tasks.

The aim of this thesis is to study several aspects of risk measures particularly in the context of financial applications. The primary framework that we use is that of coherent risk measures as defined in Artzner et al (1999). But this is not the only class of risk measures that we study here. We also investigate the concepts of natural risk statistics Kou et al (2006) and convex risk measure Follmer/ and Schied (2002). The main contributions of this Thesis can be classified in three main axes: Capital allocation, risk measurement and capital requirement and solvency. In chapter 2, we characterize risk measures with the Lebesgue property on bounded càdlàg processes. This allows to present two applications in risk assessment and capital allocation. In chapter 3, we extend the concept of natural risk statistics to the space of infinite sequences. This has been done in order to introduce a consistent way of constructing risk measures for data bases of any size. In chapter 4, we discuss the concept of Good Deals and how to deal with a situation where these pathological positions are present in the market. Finally, in chapter 5, we try to relate all three chapters by extending the definition of Good Deals to a larger set of risk measures that somehow includes the discussions in chapters 2 and 3.

During the last decade, the European Union has regulated emissions of Greenhouse Gases on its own territory. Consequently, a European Carbon Market (EU ETS) is currently emerging where CO2 emission certificates (EUA and CER) and derivatives are traded on Exchanges. The objectif of this research is to evaluate and compare different models to represent the emission certificates' price and to price derivatives of the carbon markets.

We discuss a simulation approach for the joint density function of the surplus prior to ruin and deficit at ruin for risk models driven by Lévy subordinators. This approach is inspired by the Ladder Height decomposition for the probability of ruin of such models. The Classical Risk Model driven by a Compound Poisson process is a particular case of this more generalized one. The Expected Discounted Penalty Function, also referred to as the Gerber-Shiu Function (GS Function), was introduced as a unifying approach to deal with different quantities related to the event of ruin. The probability of ruin and the joint density function of surplus prior to ruin and deficit at ruin are particular cases of this function. Expressions for those two quantities have been derived from the GS Function, but those are not easily evaluated nor handled as they are infinite series of convolutions with no analytical closed form. However they share a similar structure, thus allowing to use the Ladder Height decomposition of the Probability of Ruin as a guiding method to generate simulated values for this joint density function. We present an introduction to risk models driven by subordinators, and describe those models for which it is possible to process the simulation. To motivate this work, we also present an application for this distribution, in order to calculate different risk measures for those risk models. An brief introduction to the vast field of Risk Measures is conducted where we present selected measures calculated in this empirical study. This work contributes to better understanding the behavior of subordinators driven risk models, as it offers a numerical point of view, which is absent in the literature.

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