Includes abstract. / Includes bibliographic references (leaves 34-35). / Engle and Granger’s (1987) co-integrating framework provides a useful method of analyzing the dynamics of non-stationary data in both the short and long run. However, despite its popularity in various areas of research, the application of co-integration to financial data has been limited. This paper provides an example of the application of co-integration in a pairs trading strategy to identify mean reverting spreads. The strategy is implemented with an algorithmic trading setup that models the spread in a state-space form...

Includes bibliographical references. / We focus on the implementation details for Lévy processes and their extension to stochastic volatility models for pricing European vanilla options and exotic options. We calibrated five models to European options on the S&P500 and used the calibrated models to price a cliquet option using Monte Carlo simulation. We provide the algorithms required to value the options when using Lévy processes. We found that these models were able to closely reproduce the market option prices for many strikes and maturities. We also found that the models we studied produced different prices for the cliquet option even though all the models produced the same prices for vanilla options. This highlighted a feature of model uncertainty when valuing a cliquet option. Further research is required to develop tools to understand and manage this model uncertainty. We make a recommendation on how to proceed with this research by studying the cliquet option’s sensitivity to the model parameters.

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In this thesis, we study the representation of dynamic risk measures based on backward stochastic differential equations (BSDEs) and ergodic-BSDEs, and capital allocation. We consider the equations driven by the Brownian motion and the compensated Poisson process. We obtain four results. Firstly, we consider the representation of dynamic risk measures defined under BSDE, with generators that have quadratic-exponential growth in the control variables. Under this setting, the dynamic capital allocation of the risk measure is obtained via the differentiability of BSDEs with jumps. In this case, we introduce the Malliavin directional derivative that generalises the classical Gˆateaux-derivative. Using the capital allocation results and the full allocation property of the Aumann-Shapley, we obtain the representation of the dynamic convex and coherent risk measures. The results are illustrated for the dynamic entropic risk and static coherent risk measures. Secondly, we consider the representation of dynamic convex risk measure based on the ergodic-BSDEs in the diffusion framework. The maturityindependent risk measure is defined as the first component to the solution of a BSDE whose generator depends on the second component of the solution to the ergodic-BSDE. Using the differentiability results of BSDEs, we determine the capital allocation. Furthermore, we give an example in the form of the forward entropic risk measure and the capital allocation. Thirdly, we investigate the representation of capital allocation for dynamic risk measures based on BSVIEs from Kromer and Overbeck 2017 and extend it to risk measures based on BSVIEs with jumps. The extension of dynamic risk measure based on BSVIEs with jumps is studied by Agram 2019. In our case, we study capital allocation for dynamic risk measures based on BSVIEs with jumps. In particular, we determine the capital allocation of the dynamic risk measures based on BSVIEs with jumps. Finally, we study the representation for a forward entropic risk measure using ergodic BSDEs under the jump-diffusion framework. In this case, we notice that when the ergodic BSDE includes jump term the forward entropic risk measure does not satisfy the translation property. / Thesis (PhD (Mathematical Sciences))--University of Pretoria, 2021. / The University of Pretoria, Department of Mathematics and Applied Mathematics. / The University Capacity Development Programme National Collaborative Project (UCDP) South Africa. / Mathematics and Applied Mathematics / PhD (Mathematical Sciences) / Unrestricted

This thesis studies binomial and trinomial lattice approximations in Black-Scholes type option pricing models. Also, it covers the basics of these models, derivations of model parameters by several methods under different kinds of distributions. Furthermore, the convergence of the binomial model to normal distribution, Geometric Brownian Motion and Black-Scholes model is discussed. Finally, the connections and interrelations between discrete random variables under the Lattice approach and continuous random variables under models which follow Geometric Brownian Motion are discussed, compared and contrasted.

Since the 2008 crisis, the hedging instruments have gained popularity with financial institutions. This is the case of the total return swap that is used today by major institutions like Goldman Sachs or J.P. Morgan. Murex is a software provider for financial institutions. The company already had a total return swap product, the RTRS (for Risky Total Return Swap), but with the growing demand Murex decided to develop a new product, the BRS (Bond Return Swap). So now they have two bond total return swaps. This master thesis aims to analyze total return swap and highlight the improvement of the BRS. After a theoretical analysis of the total return swap, a test campaign is realized. For different types of bond and different configurations of total return swap, formulas are derived to be compared to the returned values. The results given by the RTRS are good on basic bonds. If the bond is more complex, for instance a bond with credit risk or an amortized bond, the values returned by the RTRS are not reliable if not wrong. On the other hand, the BRS performs well in every situation and positions itself as the best total return swap proposed by Murex. / Sedan finanskrisen 2008 har hedginginstrument blivit allt viktigare för finansinstitut. Detta är fallet med det så kallade Total Return Swaps (TRS) som används idag av stora institutioner såsom Goldman Sachs och J.P. Morgan. Murex är en mjukvaruleverantör som redan hade en TRS produkt, den RTRS (Risky Total Return Swap) . Men med den växande efterfrågan beslutade fretaget att utveckla en ny produkt, den så kallade BRS (Bond Return Swap). Så nu har de två TRS:ar. Denna uppsats syftar till att analysera Total Return Swap och belysa de förbättringar som tillförs av BRS. Efter en teoretisk genomgång av TRS realiseras en serie tester. För olika typer av obligationer och olika konstellationer av TRS härleds formler och jämförs deras värden. Resultaten från RTRS verkar vara bra på basobligationer. Om obligationen är mer komplex, till exempel en obligation med kreditrisk eller en amorterad obligation, är RTRS returnerade värden inte tillförlitliga om inte fel. Å andra sidan presterar BRS bra i alla situationer och positionerar sig som det bästa Total Return Swap som föreslagits av Murex.

We report our investigation of how news stories influence the behaviour of tradable financial assets, in particular, equities. We consider the established methods of turning news events into a quantifiable measure and explore the models which connect these measures to financial decision making and risk control. The study of our thesis is built around two practical, as well as, research problems which are determining trading strategies and quantifying trading risk. We have constructed a new measure which takes into consideration (i) the volume of news and (ii) the decaying effect of news sentiment. In this way we derive the impact of aggregated news events for a given asset; we have defined this as the impact score. We also characterise the behaviour of assets using three parameters, which are return, volatility and liquidity, and construct predictive models which incorporate impact scores. The derivation of the impact measure and the characterisation of asset behaviour by introducing liquidity are two innovations reported in this thesis and are claimed to be contributions to knowledge. The impact of news on asset behaviour is explored using two sets of predictive models: the univariate models and the multivariate models. In our univariate predictive models, a universe of 53 assets were considered in order to justify the relationship of news and assets across 9 different sectors. For the multivariate case, we have selected 5 stocks from the financial sector only as this is relevant for the purpose of constructing trading strategies. We have analysed the celebrated Black-Litterman model (1991) and constructed our Bayesian multivariate predictive models such that we can incorporate domain expertise to improve the predictions. Not only does this suggest one of the best ways to choose priors in Bayesian inference for financial models using news sentiment, but it also allows the use of current and synchronised data with market information. This is also a novel aspect of our work and a further contribution to knowledge.

Orientador: Marcus Vinicius Maltempi / Banca: Rosana Giaretta Sguerra Miskulin / Banca: Celina Aparecida Almeida Pereira Abar / Resumo: Nesta dissertação abordamos questões concernentes à formação inicial de professores de Matemática, a partir da incorporação das Tecnologias de Informação e Comunicação (TIC) e de idéias relacionadas à necessidade de uma Educação Financeira para todos. Neste sentido, nosso objetivo foi investigar as contribuições que surgem à prática pedagógica dos licenciandos em Matemática quando constroem e aplicam WebQuests sob o contexto da Matemática Financeira. Para tanto, um curso de Extensão sobre construção de WebQuests foi oferecido aos alunos do Curso de Licenciatura em Matemática da UNESP de Rio Claro - SP, os quais, posteriormente, utilizaram as WebQuests construídas como material didático durante o Estágio Supervisionado (prática docente) que realizaram junto a uma escola pública de Rio Claro, SP. O curso de Extensão, assim como toda a investigação aqui apresentada, foi desenvolvido tendo por base a teoria de aprendizagem construcionista. Além disso, nos embasamos em trabalhos que tratam da formação inicial docente, em especial, do professor de Matemática, e também, da importância da Educação Financeira na vida das pessoas. Baseados no estudo realizado, cremos que a partir do momento que os licenciandos vivenciarem uma formação inicial diferenciada, que privilegie também o uso das TIC na Educação, além dos conteúdos específicos, estaremos formando professores mais propensos a usarem as TIC em sua prática docente, de forma a propor a seus alunos situações nas quais eles terão que criar, discutir e refletir sobre suas ações. / Abstract: In this dissertation we accosted concerning questions to the inicial formation of the Mathematics teachers, starting from the incorporation of theTechnologies of Information and Communication (TIC) and of ideas related to the need of a Financial Education for all. In this sense, our objective was to investigate the contributions that appear to the pedagogic practice of the graduates in Mathematics when they build and they apply WebQuests about the context of the Financial Mathematics. For so much, a course of Extension about construction of WebQuests was offered to the students of the Degree Course in Mathematics of UNESP, Rio Claro - SP, the ones which, later, they used WebQuests built as didactic material during the Supervised Apprenticeship (teaching practice) that accomplished a public school of Rio Claro, SP. The course of Extension, as well as all the investigation here presented, it was developed having for base the theory of learning construcionista. Besides, ourselves based in works that treat of the teaching initial formation, especially, of the Mathematics teachers, and also, of the importance of the Financial Education in the people's life. Based on the accomplished study, we believe that starting from the moment that the graduates live a differentiated initial formation, that it also privileges the use of TIC in the Education, besides the specific contents, we will be forming teachers more prone to use TIC in his educational practice, in way to propose to their students situations in which they will have to create, to discuss and to contemplate about their actions. / Mestre

Financial mathematics is a branch of mathematics that assesses the risk and value of various financial instruments. Banks, companies, and other institutions mitigate their risk through financial instruments known as derivatives,that derive their value from some underlying asset. The equations that arise from pricing and modeling can be very complex, leading to the necessity of numerical methods. This project studied the use of certain numerical methods for the pricing of a particular type of option called an Asian option. Asian options can provide favorable risk profiles because the payout is determined based on the average value over a time period, rather than the final value. The price of an Asian option is governed by a partial differential equation in three variables: stock price, average price over the current time interval, and time. The solution method was first to discretize the partial differential equation into a system of ordinary differential equations. Next, the ODE system was integrated using a stiff-ODE solver available in MATLAB. Enhancements to this solution method include specifying the sparsity pattern, implementing an iterative linear solver (GMRES) in place of MATLAB's built-in direct linear solver, and using preconditioning to improve the solution characteristics of that solver.

Most realistic financial derivatives models are too complex to allow explicit analytic solutions. The computational techniques used to implement those models fall into two broad categories: finite difference methods for the solution of partial differential equations (PDEs) and Monte Carlo simulation. Accordingly, the course consists of two sections. The first half of the course focuses on finite difference methods. The following topics are discussed; Parabolic PDEs, Black-Scholes PDE for European and American options; binomial and trinomial trees; explicit, implicit and Crank- Nicholson finite difference methods; far boundary conditions, convergence, stability, variance bias; early exercise and free boundary conditions; parabolic PDEs arising from fixed income derivatives; implied trees for exotic derivatives, adapted trees for interest rate derivatives. The second half of the course focuses on Monte Carlo. The following topics are discussed; Random number generation and testing; evaluation of expected payoff by Monte Carlo simulation; variance reduction techniques�antithetic variables, importance sampling, martingale control variables; stratification, low-discrepancy sequences and quasi-Monte Carlo methods; efficient evaluation of sensitivity measures; methods suitable for multifactor and term-structure dependent models. Computational Methods of Financial Mathematics is taught by Marcel Blais, a professor at Worcester Polytechnic Institute.