We construct a mathematical model of an order driven market where traders can submit limit orders and market orders to buy and sell securities. We adapt the notion of no free lunch of Harrison and Kreps and Jouini and Kallal to our setting and we prove a no-arbitrage theorem for the model of the order driven market. Furthermore, we compute signatures of order books of different financial markets. Signatures, i.e. the full sequence of definite iterated integrals of a path, are one of the fundamental elements of the theory of rough paths. The theory of rough paths provides a framework to describe the evolution of dynamical systems that are driven by rough signals, including rough paths based on Brownian motion and fractional Brownian motion (see the work of Lyons). We show how we can obtain the solution of a polynomial differential equation and its (truncated) signature from the signature of the driving signal and the initial value. We also present and analyse an ODE method for the numerical solution of rough differential equations. We derive error estimates and we prove that it achieves the same rate of convergence as the corresponding higher order Euler schemes studied by Davie and Friz and Victoir. At the same time, it enhances stability. The method has been implemented for the case of polynomial vector fields as part of the CoRoPa software package which is available at http://coropa.sourceforge.net. We describe both the algorithm and the implementation and we show by giving examples how it can be used to compute the pathwise solution of stochastic rough differential equations driven by Brownian rough paths and fractional Brownian rough paths.

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.

Today, better numerical approximations are required for multi-dimensional SDEs to improve on the poor performance of the standard Monte Carlo integration. With this aim in mind, the material in the thesis is divided into two main categories, stochastic calculus and mathematical finance. In the former, we introduce a new scheme or discrete time approximation based on an idea of Paul Malliavin where, for some conditions, a better strong convergence order is obtained than the standard Milstein scheme without the expensive simulation of the Lévy Area. We demonstrate when the conditions of the 2−Dimensional problem permit this and give an exact solution for the orthogonal transformation (θ Scheme or Orthogonal Milstein Scheme). Our applications are focused on continuous time diffusion models for the volatility and variance with their discrete time approximations (ARV). Two theorems that measure with confidence the order of strong and weak convergence of schemes without an exact solution or expectation of the system are formally proved and tested with numerical examples. In addition, some methods for simulating the double integrals or Lévy Area in the Milstein approximation are introduced. For mathematical finance, we review evidence of non-constant volatility and consider the implications for option pricing using stochastic volatility models. A general stochastic volatility model that represents most of the stochastic volatility models that are outlined in the literature is proposed. This was necessary in order to both study and understand the option price properties. The analytic closed-form solution for a European/Digital option for both the Square Root Model and the 3/2 Model are given. We present the Multilevel Monte Carlo path simulation method which is a powerful tool for pricing exotic options. An improved/updated version of the ML-MC algorithm using multi-schemes and a non-zero starting level is introduced. To link the contents of the thesis, we present a wide variety of pricing exotic option examples where considerable computational savings are demonstrated using the new θ Scheme and the improved Multischeme Multilevel Monte Carlo method (MSL-MC). The computational cost to achieve an accuracy of O(e) is reduced from O(e−3) to O(e−2) for some applications.

Thema der Dissertation ist zum einen die Quantifizierung und zum anderen die endogene Modellierung von Finanzrisiken. Die mathematische Analyse führt unter anderem auf Zusammenhänge finanzmathematischer Probleme mit der Theorie großer Abweichungen, der Choquet-Theorie, der Theorie interagierender Teilchensysteme und der Theorie dynamischer Systeme. Die ersten zwei Kapitel der Arbeit beleuchten die Bemessung von Finanzrisiken aus zwei unterschiedlichen Perspektiven. In Kapitel 1 analysieren wir die Berechnung von Risikomaßen mittels Monte Carlo Methoden. In Kapitel 2 wird die Rolle von Information und Zeit bei der Bewertung von Finanzrisiken untersucht. Die Modellierung von Finanzrisiken auf Märkten interagierender Akteure wird in den beiden letzten Kapiteln der Arbeit in zwei Fallstudien betrachtet. In der ersten Fallstudie in Kapitel 3 befassen wir uns dabei mit dem Zusammenhang von Kreditrisiken und Ansteckungsprozessen von Firmen, die mit ihren Geschäftspartnern interagieren. In der zweiten Fallstudie in Kapitel 4 beleuchten wir die Marktinteraktion von eingeschränkt rationalen Investoren in einem evolutionären Marktselektionsmodell. / In this thesis, we study monetary measures and endogenous models of financial risk. The mathematical analysis identifies connections between problems in mathematical finance on the one hand and large deviations, Choquet-theory, interacting particle systems, and dynamical systems on the other hand. The first part of the thesis considers two aspects of the quantification of financial risk. In the first chapter, we focus on the calculation of risk measurements by Monte Carlo simulation. In the second chapter, we investigate a particular class of dynamic risk measures. In the second part we analyze two models of financial risk in economies with interacting agents. First, we focus in the third chapter on credit contagion of firms which interact with each other in a network of business partners. Second, we investigate in the fourth chapter the market interaction of investors with bounded rationality in an evolutionary selection market model.

An assembled algorithm for pricing American options with absolute, discrete dividends using adaptive lattice methods is described. Considerations for hardware-conscious programming on both CPU and GPU platforms are discussed, to provide a foundation for the investigation of several approaches for deploying the program onto GPU architectures. The performance results of the approaches are compared to that of a central processing unit reference implementation, and to each other. In particular, an approach of designating subtrees to be calculated in parallel by allowing multiple calculation of overlapping elements is described. Among the examined methods, this attains the best performance results in a "realistic" region of calculation parameters. A fifteen- to thirty-fold improvement in performance over the CPU reference implementation is observed as the problem size grows sufficiently large.

Some recent works in the field of mathematical finance have brought new light on the importance of studying the regularity and the tail asymptotics of distributions for certain classes of diffusions with non-globally smooth coefficients. In this Ph.D. dissertation we deal with some issues in this framework. In a first part, we study the existence, smoothness and space asymptotics of densities for the solutions of stochastic differential equations assuming only local conditions on the coefficients of the equation. Our analysis is based on Malliavin calculus tools and on " tube estimates " for Ito processes, namely estimates for the probability that the trajectory of an Ito process remains close to a deterministic curve. We obtain significant estimates of densities and distribution functions in general classes of option pricing models, including generalisations of CIR and CEV processes and Local-Stochastic Volatility models. In the latter case, the estimates we derive have an impact on the moment explosion of the underlying price and, consequently, on the large-strike behaviour of the implied volatility. Parametric implied volatility modeling, in its turn, makes the object of the second part. In particular, we focus on J. Gatheral's SVI model, first proposing an effective quasi-explicit calibration procedure and displaying its performances on market data. Then, we analyse the capability of SVI to generate efficient approximations of symmetric smiles, building an explicit time-dependent parameterization. We provide and test the numerical application to the Heston model (without and with displacement), for which we generate semi-closed expressions of the smile

Made available in DSpace on 2016-06-02T20:29:21Z (GMT). No. of bitstreams: 1 5037.pdf: 6660249 bytes, checksum: fc8a2a80467bf7cf6a78c98c57c830f7 (MD5) Previous issue date: 2013-03-27 / Financiadora de Estudos e Projetos / This work intends to show the preparation and application of a Teaching Sequence on Mathematical Finance with two Middle School 3rd grade classes of a state school in São Paulo inland. The research methodology applied was Didactic Engineering. The study started by the research about the way Mathematical Finance was and is treated in Brazilian education, especially on the plans of São Paulo State, and about the knowledge of the two Middle School last grade classes on the subject. This work arose from the observations done, intending to relate the Mathematical Finance study with the mathematical concepts in the base, like the arithmetic and the geometric progressions and the increase factor, so avoiding the use of known by heart formulas, the most meaningless for the student. The proposed problems consist of current and fundamental applications for the student to analyze and make rational decisions in the product purchase options, as well as adherence to financing like that provided by the FIES (Student Financing Fund), highly publicized today. / O presente trabalho objetiva mostrar a elaboração e aplicação de uma Sequência Didática sobre a Matemática Financeira em duas turmas de alunos da 3ª série do Ensino Médio de uma escola estadual do interior de São Paulo. A metodologia de pesquisa aplicada foi a Engenharia Didática. Iniciou-se o estudo com a pesquisa de como a matemática financeira foi e é abordada na educação brasileira, principalmente nas propostas do estado de São Paulo e sobre o conhecimento dos alunos das duas turmas concluintes do ensino médio em relação ao tema. Das constatações efetuadas surgiu este trabalho, com o intuito de associar o estudo da matemática financeira com os conceitos matemáticos que a embasam, como as Progressões Aritméticas e Geométricas e o fator de aumento, evitando, assim, o uso de fórmulas decoradas e muitas vezes sem sentido para o aluno. Os problemas propostos consistem em aplicações atuais e fundamentais para que o aluno possa analisar e tomar decisões racionais nas opções de compra de produtos, bem como na adesão a financiamentos como os proporcionados pelo FIES (Fundo de Financiamento Estudantil), muito divulgados atualmente.

Submitted by Luciana Sebin (lusebin@ufscar.br) on 2016-09-28T13:56:50Z No. of bitstreams: 1 DissDGG.pdf: 3285381 bytes, checksum: e32f895e81dfab9212b1b022551720aa (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-10T14:51:11Z (GMT) No. of bitstreams: 1 DissDGG.pdf: 3285381 bytes, checksum: e32f895e81dfab9212b1b022551720aa (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-10T14:51:17Z (GMT) No. of bitstreams: 1 DissDGG.pdf: 3285381 bytes, checksum: e32f895e81dfab9212b1b022551720aa (MD5) / Made available in DSpace on 2016-10-10T14:51:23Z (GMT). No. of bitstreams: 1 DissDGG.pdf: 3285381 bytes, checksum: e32f895e81dfab9212b1b022551720aa (MD5) Previous issue date: 2016-03-19 / Não recebi financiamento / This work deals with developing, structuring and implementation of a didactic sequence taught Financial Mathematics to high school students through reproduction environments able to provide them the opportunity to be protagonists in decisionmaking. Articulated with the French methodology of mathematics education known as Didactic Engineering, this pedagogical proposal has been designed with the aim of contributing to the youth actively participate in family financial planning. The first activity of the sequence focused on the inclusion of the student body in financial contexts in which situations were simulated conducive to learning and incorporating fundamental contents of Financial Mathematics. The second, favored the application of knowledge acquired, in scenarios where the Investment Analysis is essential for the provision of financial success. The results show that there was a significant learning capable of conferring the full potential students to apply the concepts covered in real situations, helping them to develop a new perspective on this important branch of mathematics. / Esse trabalho tratou da elaboração, estruturação e aplicação de uma sequência didática que ensinou Matemática Financeira a alunos do ensino médio por meio da reprodução de ambientes capazes de proporcionar aos mesmos a oportunidade de serem protagonistas na tomada de decisões. Articulada com a metodologia francesa de educação matemática conhecida como Engenharia Didática, essa proposta pedagógica foi concebida com o propósito de contribuir para que o jovem participe ativamente do planejamento financeiro familiar. A primeira atividade da sequência teve como foco a inserção do corpo discente em contextos financeiros onde foram simuladas situações propícias ao aprendizado e incorporação de conteúdos fundamentais da Matemática Financeira. Já a segunda, favoreceu a aplicação do conhecimento adquirido, em cenários onde a Análise de Investimentos é imprescindível para o provimento de êxito financeiro. Os resultados obtidos atestam que houve aprendizado significativo capaz de conferir aos alunos capacidade plena de aplicar os conceitos abordados em situações reais, contribuindo para que eles desenvolvam um novo olhar sobre esse importante ramo da matemática.

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-11-04T12:26:04Z No. of bitstreams: 1 arquivototal.pdf: 1817584 bytes, checksum: 1fc2338fcfbcb84a97b2aa64265456fa (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-11-04T14:32:14Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 1817584 bytes, checksum: 1fc2338fcfbcb84a97b2aa64265456fa (MD5) / Made available in DSpace on 2015-11-04T14:32:14Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1817584 bytes, checksum: 1fc2338fcfbcb84a97b2aa64265456fa (MD5) Previous issue date: 2013-03-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study the main topics about Mathematical Finance, seeking for immediate connection with our reality. We study some subjects discussed in High School in order to provide basic tools for decision making in our daily lives. We also studied an electronic tool that helps us to solve several problems that have extensive calculations. / Neste trabalho, estudamos os principais tópicos da Matemática Financeira, buscando sempre fazer ligação imediata com eventos de nossa realidade. Passamos por alguns assuntos não abordados no Ensino Médio com intuito de fornecer ferramentas básicas para a tomada de decisão em nosso cotidiano. Estudamos também uma ferramenta eletrônica que nos auxilia a resolver diversos problemas que possuem extensos cálculos.

We develop expansion methods as a new computational approach towards high-dimensional partial differential equations (PDEs), particularly of such type as arising in the valuation of financial derivatives. The proposed methods are extended from [41] and use principal component analysis (PCA) of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. They enable calculation of highly accurate approximate solutions with computational complexity polynomial in the number of dimensions for PDEs with a low number of dominant principal components. For the case of PDEs with constant coefficients, we show existence of expansion solutions and prove theoretical error bounds. We give a precise characterisation of when our methods can be applied and construct specific examples of a first and second order version. We provide numerical results showing that the empirically observed convergence speeds are in agreement with the theoretical predictions. For the case of PDEs with varying coefficients, we give a heuristic motivation using the Parametrix approach and empirically test the methods' accuracy for a range of variable parameter stock models. We demonstrate the applicability of our expansion methods to real-world securities pricing problems by considering path-dependent and early-exercise options in the LIBOR market model. Using the example of Bermudan swaptions and Ratchet floors, which are considered difficult benchmark problems, we give a careful analysis of the numerical accuracy and computational complexity. We are able to demonstrate that for problems with medium to high dimensionality, around 60-100, and moderate time horizons, the presented PDE methods deliver results comparable in accuracy to benchmark state-of-the-art Monte Carlo methods in similar or (significantly) faster run time.