The thesis describes commonly used measures of risk, such as volatility, Value at Risk (VaR) and Expected Shortfall (ES), and is tasked with creating models for measuring market risk. It is concerned with the risk over daily and over monthly horizons and shows the shortcomings of a square-root-of-time approach for converting VaR and ES between horizons. Parametric models, geometric Brownian motion (GBM) and GARCH process, and non-parametric models, historical simulation (HS) and some its possible improvements, are presented. The application of these mentioned models is demonstrated using real data. The accuracy of VaR models is proved through backtesting and the results are discussed. Part of this thesis is also a simulation study, which reveals the precision of VaR and ES estimates.

In the Thesis, linear stochastic differential equations in a Hilbert space driven by a cylindrical fractional Brownian motion with the Hurst parameter in the interval H < 1/2 are considered. Under the conditions on the range of the diffusion coefficient, existence of the mild solution is proved together with measurability and continuity. Existence of a limiting distribution is shown for exponentially stable semigroups. The theory is modified for the case of analytical semigroups. In this case, the conditions for the diffusion coefficient are weakened. The scope of the theory is illustrated on the Heath-Jarrow-Morton model, the wave equation, and the heat equation. 1

Title: Stochastic Models in Financial Mathematics Author: Bc. Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis looks into the problems of ordinary stochastic models used in financial mathematics, which are often influenced by unrealistic assumptions of Brownian motion. The thesis deals with and suggests more sophisticated alternatives to Brownian motion models. By applying the fractional Brownian motion we derive a modification of the Black-Scholes pricing formula for a mixed fractional Bro- wnian motion. We use Lévy processes to introduce subordinated stable process of Ornstein-Uhlenbeck type serving for modeling interest rates. We present the calibration procedures for these models along with a simulation study for estima- tion of Hurst parameter. To illustrate the practical use of the models introduced in the paper we have used real financial data and custom procedures program- med in the system Wolfram Mathematica. We have achieved almost 90% decline in the value of Kolmogorov-Smirnov statistics by the application of subordinated stable process of Ornstein-Uhlenbeck type for the historical values of the monthly PRIBOR (Prague Interbank Offered Rate) rates in...

[pt] O objetivo deste trabalho é determinar o melhor timing para a abertura de capital de empresas fechadas. Empresas fechadas detêm a opção de abertura de capital por tempo indeterminado deixando a decisão nas mãos do empreendedor ou do corpo administrativo. A aplicação da teoria de opções reais neste contexto tem como objetivo valorar a flexibilidade da administração em um ambiente dinâmico, no qual métodos tradicionais falham. O momento ideal para lançamento de ações no mercado primário torna-se viável a partir de certo estágio de vida da firma e, a partir deste momento, busca-se o momento mais oportuno para a captação. Sabendo que os preços das ações são variáveis estocásticas e seguem um Movimento Geométrico Browniano, a Simulação de Monte Carlo é capaz de gerar os preços de ações futuros para a tomada de decisão e valoração da opção de espera no tempo corrente. Neste cenário, o gestor tenderá a esperar choques positivos nos preços antes de lançar ações e, como consequência, o período ótimo de exercício da opção de abertura de capital deve ocorrer após aumentos anormais nos preços das ações. Seja o motivo principal do lançamento dar liquidez às ações ou financiar investimentos, o benefício da abertura deverá ser maior do que o valor da opção de espera. / [en] The purpose of this study is to identify the best timing for private firms to go public. Private firms possess an option to go public at the time their managers believe it would be a valuable strategy. In the same way, exchange traded companies also have the option of issuing more equities or even taking them over. In this sense, the theory of real options can be applied in order to value management flexibility in an uncertain and dynamic ambience in which traditional methods such as net present value fail to evaluate investments. For the management team, the more uncertain the market is, the more value it adds to the option to wait and postpone the issuance. Given that stock prices fluctuate randomly over time and follow the Geometric Brownian Motion, the Monte Carlo Simulation provides the forecasted stock prices to help the decision making process and also to value the timing option. Therefore, the entrepreneur is willing to wait for positive price shocks before taking the firm public. As a consequence of optimally exercising the timing option IPOs should occur after abnormal price increases. Any firm contemplating an IPO should factor the timing option into the decision. The main reason for the IPO can be either capital raising or the desire to increase the shares’ liquidity but, in all cases, the benefit of going public should be greater than the value of the timing option and issuing shares from private equity.

[pt] As vantagens de incluir a flexibilidade gerencial e a analogia às opções financeiras nos critérios de avaliação de projetos têm sido alvo de discussões teóricas no ramo das finanças. Diversos autores criticam os métodos de análise de investimentos utilizados correntemente, que têm como principal representante o fluxo de caixa descontado (FCD), apoiando-se na noção de que os gerentes ao tomarem decisões devem utilizar técnicas que reflitam as flexibilidades disponíveis. Nesse sentido, a presente dissertação tem por finalidade validar as vantagens sugeridas na utilização da teoria de Opções Reais através de um back-testing, que tem como objeto campos de petróleo com características representativas da indústria petrolífera brasileira. Estes testes serão realizados para o período de 1970 a 1990, sendo contemplada a incerteza econômica e excluindo-se as incertezas técnicas. O modelo desenvolvido em Excel e VBA (Visual Basic for Applications) para decisões de investimento considera as opções de espera de até dois anos e de escolha entre três intensidades de produção. O Movimento Geométrico Browniano foi assumido como o processo estocástico para representar a evolução dos preços reais do petróleo em dólares americanos ao longo do tempo, e sua volatilidade foi variada a título de análise de sensibilidade. Por fim, cabe ressaltar que os resultados obtidos não devem ser aceitos como definitivos, e sim como base de futuros trabalhos na linha de estudos empíricos para verificar e validar as vantagens teóricas das Opções Reais em relação aos demais critérios utilizados na prática. / [en] Financial researchers have discussed a lot about the theoretical advantages of including the managerial flexibility and the financial options analogy in projects valuation criteria. Plenty of authors criticize the currently used investment analysis methods, mainly represented by the discounted cash flow, supported by the notion that the managers should use techniques that better reflect the available flexibility to take their decisions. In this sense, the present dissertation has the objective of validating the suggested advantages of using the Real Options theory through a back-testing focused on oil fields with Brazilian oil industry representative characteristics. Those tests will be carried out for the 1970-1990 period, considering the economic uncertainty and excluding the technical uncertainties. The investment decisions model developed in Excel and VBA (Visual Basic for Applications) contemplates the options of waiting till two years and of choosing among three exploitation intensities. The Geometric Brownian Motion was assumed as the stochastic process to represent the real oil prices time evolution, and its volatility was varied to generate a sensibility analysis. Finally it is worthy to state that the results shall not be accepted as definitive, and just as a foundation to future studies on the empirical research line of verifying and validating the theoretical advantages of the Real Options with regard to others currently used criteria.

A literatura referente a testes de hipótese em modelos auto-regressivos que apresentam uma possível raiz unitária é bastante vasta e engloba pesquisas oriundas de diversas áreas. Nesta dissertação, inicialmente, buscou-se realizar uma revisão dos principais resultados existentes, oriundos tanto da visão clássica quanto da bayesiana de inferência. No que concerne ao ferramental clássico, o papel do movimento browniano foi apresentado de forma detalhada, buscando-se enfatizar a sua aplicabilidade na dedução de estatísticas assintóticas para a realização dos testes de hipótese relativos à presença de uma raíz unitária. Com relação à inferência bayesiana, foi inicialmente conduzido um exame detalhado do status corrente da literatura. A seguir, foi realizado um estudo comparativo em que se testa a hipótese de raiz unitária com base na probabilidade da densidade a posteriori do parâmetro do modelo, considerando as seguintes densidades a priori: Flat, Jeffreys, Normal e Beta. A inferência foi realizada com base no algoritmo Metropolis-Hastings, usando a técnica de simulação de Monte Carlo por Cadeias de Markov (MCMC). Poder, tamanho e confiança dos testes apresentados foram computados com o uso de séries simuladas. Finalmente, foi proposto um critério bayesiano de seleção de modelos, utilizando as mesmas distribuições a priori do teste de hipótese. Ambos os procedimentos foram ilustrados com aplicações empíricas à séries temporais macroeconômicas. / Testing for unit root hypothesis in non stationary autoregressive models has been a research topic disseminated along many academic areas. As a first step for approaching this issue, this dissertation includes an extensive review highlighting the main results provided by Classical and Bayesian inferences methods. Concerning Classical approach, the role of brownian motion is discussed in a very detailed way, clearly emphasizing its application for obtaining good asymptotic statistics when we are testing for the existence of a unit root in a time series. Alternatively, for Bayesian approach, a detailed discussion is also introduced in the main text. Then, exploring an empirical façade of this dissertation, we implemented a comparative study for testing unit root based on a posteriori model's parameter density probability, taking into account the following a priori densities: Flat, Jeffreys, Normal and Beta. The inference is based on the Metropolis-Hastings algorithm and on the Monte Carlo Markov Chains (MCMC) technique. Simulated time series are used for calculating size, power and confidence intervals for the developed unit root hypothesis test. Finally, we proposed a Bayesian criterion for selecting models based on the same a priori distributions used for developing the same hypothesis tests. Obviously, both procedures are empirically illustrated through application to macroeconomic time series.

Neste trabalho propomos um modelo de mercado através de uma discretização aleatória do movimento browniano proposta por Leão & Ohashi (2010). Com este modelo, dada uma função payoff, vamos desenvolver uma estratégia de hedging e uma metodologia para precificação de opções / In this work we propose a market model using a discretization scheme of the random Brownian motion proposed by Leão & Ohashi (2010). With this model, for any given payoff function, we develop a hedging strategy and a methodology to option pricing

La connaissance et la maîtrise de la présence d’un biofilm représentent aujourd’hui un challenge important. Dans le contexte d’étude des capacités de développement des biofilms, BioFilm Control fait figure de pionnier grâce à leur test nommé Biofilm Ring Test. Basé sur une sollicitation du biofilm via l’attraction, par un aimant, de microbilles magnétiques au centre du puits, le test évalue la présence de biofilm par l’absence de regroupement des billes au centre du puits à un instant donné. L’enjeu de ce travail est de décliner le BioFilm Ring Test® en un examen dynamique, non destructif et à l’échelle microscopique. Dans le biofilm, la matrice polymérique assure la cohésion entre cellules et confère une protection aux bactéries qui vivent au sein du biofilm. Les propriétés mécaniques de la matrice sont donc un indicateur de l’état local du biofilm. La recherche de ces paramètres permet de pouvoir prédire et contrôler la formation, l’accumulation et la dissémination de bactéries propageant les infections et/ou l’encrassement. Néanmoins, la détermination des propriétés mécaniques des biofilms nécessite des précautions et l’usage d’un vocabulaire homogénéisé et de méthodes unifiées au sein de la communauté. Pour cela, une première partie de ce travail de thèse consiste en la proposition d’un guide de bonnes pratiques quant à la caractérisation mécanique du matériau biofilm. Dans la deuxième partie de ce travail de thèse, une méthodologie pour le suivi de particules micrométriques au sein d’un matériau vivant est développée. Le recours à des techniques d’imagerie telle que la corrélation d’images numériques permet de remonter à la cinématique du mouvement de chacune des microbilles, qui servent de marqueurs au sein des images traitées, par une mesure sans contact. Cette méthode est ensuite appliquée à l’étude de la formation de biofilm. L’originalité de ce travail repose sur la caractérisation de l’évolution de la typologie du mouvement des microbilles métalliques lors de la formation des biofilms. Il s’agit là de discriminer des comportements de billes révélateurs de la genèse d’un biofilm. En tirant parti de l’observation du mouvement de microbilles inertes introduites dans le milieu bactérien, on détecte des changements de typologies de trajectoires qui semblent être reliés à l’activité de bactéries sessiles, adhésion ou formation de matériel extracellulaire. Les résultats montrent que les diverses étapes de la formation de biofilms sont caractérisées, ce qui permet notamment de discriminer la présence ou non d’antibiotiques mélangés avec les bactéries et d’apprécier leur efficacité. Dans une dernière partie, des recherches encore en phase de développement sont exposées. Elles s’intéressent au comportement du biofilm sous sollicitation volumique. Il s’agit dans ce cas d’observer le biofilm en champ lointain et de suivre le déplacement ou la déformation d’un marquage constitué d’un agglomérat de microbilles, plongées dans un champ magnétique. Ces premiers travaux pourront servir d’ébauche à des travaux futurs dans le but de caractériser quantitativement le matériau biofilm. / The control of biofilm formation constitutes an important challenge in many industrial and biomedical applications. In this context, BioFilm Control is a pioneer thanks to its test named BioFilm Ring Test. Based on the immobilisation of magnetic microbeads by adherent cells, the assay allows to detect the presence of biofilm at a given time. The aim of this phD project is to translate the BioFilm Ring Test® into a dynamic, non-destructive and microscopic examination of the biofilm state. Whithin the biofilm, the matrix provides a strong cohesion between cells and therefore increases their resistance against chemical or mechanical stress in comparison to their planktonic counterparts. The mechanical properties of the matrix are therefore an indicator of the local state of the biofilm. The search for these parameters makes it possible to predict and control the formation, accumulation and spread of bacteria that propagate infections and/or biofouling. Nevertheless, the determination of the mechanical properties of biofilms requires precautions and the use of an homogenized vocabulary and methods that are unified within the community. To this end, a first part of this thesis work consists in proposing a guide of good practices for the mechanical characterization of biofilm material. In the second part of this work, a methodology for tracking of micrometric particles within a living material is developed. The use of full field measurement method such as digital image correlation makes it possible to trace the kinematics of the motion of each particle, which is a probe of the local environment. This method is then applied to the study of the biofilm formation, by non-contact measurement. The originality of this work is based on the characterization of the change in the microbeads movement during the biofilm formation steps. The aim is to discriminate bead behaviours that reveal the genesis of a biofilm. By taking advantage of the observation of the movement of inert microbeads embedded into the bacterial environment, we detect changes of type of trajectories which seem to be correlated to the activity of sessiles bacteria, adhesion or formation of extracellular material. The results show that the various stages of the biofilm formation are characterized by a non-destructive test. Especially, It allows to appreciate the efficiency of an antibiotic. In the last part, research still in a development phase is presented. It concerns the behaviour of biofilm under mechanical solicitation. This involves observing the biofilm in the far field and following the displacement or deformation of a pattern consisting of an agglomerate of microbeads immersed in a magnetic field. This initial work can be used as a draft for future work to quantitatively characterize the biofilm material.

Motivated by a zero-intelligence approach, the aim of this thesis is to unify the microscopic (discrete price and volume), mesoscopic (discrete price and continuous volume) and macroscopic (continuous price and volume) frameworks of limit order books, with a view to providing a novel yet analytically tractable description of their behaviour in a high to ultra high-frequency setting. Starting with the canonical microscopic framework, the first part of the thesis examines the limiting behaviour of the order book process when order arrival and cancellation rates are sent to infinity and when volumes are considered to be of infinitesimal size. Mathematically speaking, this amounts to establishing the weak convergence of a discrete-space process to a mesoscopic diffusion limit. This step is initially carried out in a reduced-form context, in other words, by simply looking at the best bid and ask queues, before the procedure is extended to the whole book. This subsequently leads us to the second part of the thesis, which is devoted to the transition between mesoscopic and macroscopic models of limit order books, where the general idea is to send the tick size to zero, or equivalently, to consider infinitely many price levels. The macroscopic limit is then described in terms of reflected SPDEs which typically arise in stochastic interface models. Numerical applications are finally presented, notably via the simulation of the mesocopic and macroscopic limits, which can be used as market simulators for short-term price prediction or optimal execution strategies.

Dans le cadre de cette thèse, nous nous intéressons à ”l’exposant de Lyapu-nov” pour deux modèles en milieu aléatoire : la marche aléatoire en potentiel aléatoire, le mouvement brownien en potentiel poissonnien.Dans la première partie de la thèse (chapitre II), on étudie une marche aléatoire dans un potentiel aléatoire donné par une famille de variables aléa¬toires i.i.d. non-négatives. La continuité des exposants de Lyapunov par rap¬port à la loi du potentiel est démontrée dans le cas transient, c’est-à-dire en dimension d ≥ 3 ou en dimension 2 pour un potentiel borné inférieurement. On poursuit avec l’étude des exposants critiques : l’exposant de volume ξ et l’exposant de fluctuation X. On obtient l’une des inégalités suggérée par la conjecture de KPZ sous une condition de courbure de la forme asymptotique. Les exposants de Lyapunov jouent un rôle important dans cette étude.La deuxième partie de la thèse (chapitre III) est surtout consacrée à l’étude du brownien dans un potentiel aléatoire de longue portée. On débute cependant par un potentiel classique à portée finie. Sznitman (1987-1998) a étudié plusieurs aspects de ce modèle. Un premier résultat de cette partie est la continuité des exposants de Lyapunov par rapport au paramètre du pro¬cessus de Poisson. On étudie ensuite le modèle proposé par Lacoin (2012) qui est un modèle avec un potentiel à longue portée. Il a obtenu des estimations des exposants critiques sensiblement différentes de celles de Wüthrich (1998) pour le modèle de Sznitman. Dans cette thèse, on poursuit l’étude du modèle de Lacoin. On montre l’existence des exposants de Lyapunov, le théorème de la forme limite et une estimation de grandes déviations. / In this thesis, we are interested in Lyapunov exponent for two models in random media : random walk in random potential, Brownian motion in Poisson potential.In the first part (chapter II), we study a random walk in a random potential given by a family of i.i.d random non-negative variables. The continuity of Lyapunov exponents with respect to the law of potential is shown in the case transient, that is, in the dimension d ≥ 3 or in the dimension d = 2 for a lower bounded potential. Next, we consider the critical exponents : the exponent of volume ξ and the exponent of fluctuation X. We give an inequality suggested by the KPZ conjecture under a condition of asymptotic form. Lyapunov exponents play an important role in this work.The second part (chapter III) is mainly devoted to the study Brownian motion in a long-range random potential. However, we begin with a classical finite-range potential. Sznitman (1987-1998) investigated several aspects of this model. The first result of this part is the continuity of the Lyapunov exponents with respect to the parameter of the Poisson process. Then, we study the model proposed by Lacoin (2012) which is a long-range potential model. He obtained some estimations of critical exponents that are significantly different from those of Wüthrich (1998) for the model of Sznitman.In this thesis, we pursue the study of Lacoin model. We show the existence of Lyapunov exponents, the shape limit theorem and an estimation of large deviations