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O objetivo deste trabalho de dissertação foi estudar a importância da implementação do Cálculo Diferencial e Integral no ensino médio. Também houve o interesse em analisar a importância no método utilizado para essa implementação. Especificamente, nos preocupamos em tentar beneficiar os alunos do ensino médio de uma maneira geral, ou seja, os alunos que pretendem continuar seus estudos em nível superior em áreas exatas ou não. Motivados por isto, foi feito um estudo de caso em que se ministrou essa disciplina para dois grupos de alunos do ensino médio (grupos 1 e 2). Para um grupo (grupo 1) a disciplina foi apresentada de maneira mais intuitiva, com muitos exemplos (algébricos ou ligados ao nosso cotidiano) e não muito formal e para o outro grupo (grupo 2) a disciplina foi apresentada de forma mais tradicional e próxima ao abordado no ensino superior. Ao final do nosso estudo verificamos que os dois grupos foram beneficiados, pois tiveram a oportunidade de enriquecer seus conhecimentos através do estudo de uma disciplina muito importante nas áreas de ciências exatas e aplicadas. No entanto, como esperávamos, conseguimos atingir nosso objetivo de maneira mais satisfatória com o grupo 1, pois percebemos que este grupo sempre mostrou mais interesse e, consequentemente, mais facilidade em aprender e aplicar o conhecimento adquirido. Dessa forma concluímos que é possível inserir o Cálculo Diferencial e Integral no ensino médio e que métodos similares ao que utilizamos com o grupo 1 poderão contribuir para qualidade do ensino da matemática no ensino médio, já que oferecem para os alunos um conhecimento mais rico e abrangente da matemática. / The objective of this dissertation was to study the importance of implementing the Differential and Integral Calculus in high school. There has also been interest in analyzing the importance of the method used for its implementation. Specifically, we are concerned in trying to benefit high school students in general, that is, students who intend to continue their studies at a higher level in exact areas or not. Motivated by this, a case study in which they gave this discipline for two high school students groups was made (groups 1 and 2). For a group (group 1) the subject was presented in a more intuitive way, with many examples (algebraic or connected to our everyday) and not too formal and the other group (group 2) the discipline was presented in a more traditional and closely addressed to higher education. At the end of our study we found that both groups benefited, as had the opportunity to enrich their knowledge through the study of a very important discipline in the exact sciences and applied sciences. However, as we expected, we achieved our goal of more satisfactorily with the group 1 because we realize that this group has always shown more interest and, therefore, easier to learn and apply the knowledge gained. Thus we concluded that you can enter the Differential and Integral Calculus in high school and that similar methods to that used with the group 1 may contribute to quality of mathematics teaching in high school, as they provide for students a richer knowledge and embracing of mathematics.

Neste trabalho foi realizada a síntese e caracterização de um derivado de S,S-dioxidotioxantona, 7,8-benzotioxanton-9-ona-10,10-dióxido (TX-Np-SO2). A TX-Np-SO2 foi caracterizada estruturalmente por espectros eletrônico e vibracional (UV-vis e fluorescência, Infravermelho) e também por ressonância magnética nuclear (RMN) de 1H e 13C. As propriedades fotofísicas, espectros de emissão de fluorescência e fosforescência e o rendimento quântico de fluorescência da TX-Np-SO2 foram determinadas, assim como os tempos de vida de fluorescência (τF = 3,62 ns) e fosforescência (τPh = 800 ms). Os espectros de absorção triplete-triplete dos transientes da TX-Np-SO2, em soluções desoxigenadas de metanol, acetonitrila e metilciclohexano, foram determinados por Fotólise por Pulso de Laser, bem como as constantes de velocidade de supressão bimoleculares do derivado pelos supressores trans-estilbeno, trietilamina, p-dimetilamina benzoato de etila e 2-propanol. A fotopolimerização do metacrilato de metila (MMA) iniciada por TX-Np-SO2 foi estudada por dilatometria. A reação de polimerização do iniciador (TX-Np-SO2) na presença de amina e o monômero MMA foi estudada de modo a determinar a taxa de polimerização (Rp) a diferentes concentrações de trietilamina (TEA) e p-dimetilamina benzoato de etila (EDB). / The aim of this work was the synthesis and characterization of a S,S-dioxidethioxanthone derivative, 7,8-benzo-thioxanthen-9-one-10,10-dioxide (TX-Np-SO2). The TX-Np-SO2 was structurally characterized by its electronic and vibrational spectra (UV-vis and fluorescence, IR) and also by 1H and 13C RMN methods. The photophysical properties, fluorescence and phosphorescence emission spectra, and fluorescence quantum yield of TX-Np-SO2 were determined, as well as the fluorescence lifetime (τF = 3,62 ns) and the phosphorescence lifetime (τPh = 800 ms). The transient absorption spectra of TX-Np-SO2 in degassed solutions of methanol, acetonitrile and methylcyclohexane were determined by Laser Flash Photolysis, as well as the bimolecular triplet quenching rate constants for of the derivative by various compounds. The photopolymerization of methyl methacrylate (MMA) initiated by TX-Np-SO2 was studied by dilatometry. The polymerization reaction of the initiator (TX-Np-SO2) in the presence of amine and MMA monomer was studied in order to determine the polymerization rate (Rp) at different loadings of triethylamine (TEA) and ethyl-4-dimethylaminobenzoate (EDB).

O Cálculo Diferencial e Integral é o ramo da matemática no qual um dos objetivos é o estudo do movimento e da variação. Desenvolvido a partir da Álgebra e da Geometria, esse ramo dedica-se ao estudo de taxas de variação de grandezas, como a inclinação de uma reta tangente a uma curva em um dos seus pontos, e a acumulação de quantidades, como a área sob uma curva e o volume de um sólido. Por ser utilizado como uma ferramenta auxiliar em várias áreas das ciências pura e aplicada e, dada a sua aplicabilidade nas mais diversas áreas do conhecimento, é considerado um dos conteúdos mais importantes trabalhado nos cursos superiores. Todavia, no Ensino médio, quando aparece nos livros didáticos, vem embasado no ensino clássico de limites das funções. No presente trabalho, foi apresentada uma proposta que não utilizará o conceito de limite, mas outros conceitos que figuram, mais frequentemente no Ensino Médio, para se discutir uma possível abordagem do ensino de derivadas, aplicado às cônicas. / The differential and integral calculus is a Mathematical field which aims the study of movement and variation. Developed from Algebra and geometry, this field is applied to the study of magnitude variation taxes, like the inclination of a tangent line to a curve in one of its points, and the accumulation of quantities, like the area underneath a curve and the volume of a solid. Because its used as an aid tool in several fields of pure and applied sciences and, due to its applicability in many varied areas of knowledge, its considered one of the most important worked contents in higher education. However, in High School, when it comes in didactic material, its grounded in the classic limit of functions. In this present work, it was presented a proposal which will not use the concept of limit, but others concepts that appear, more often in High School, to discuss a possible approach of derivatives teaching, applied to conics.

This thesis studies two types of problems in financial derivatives pricing. The first type is the free boundary problem, which can be formulated as a partial differential equation (PDE) subject to a set of free boundary condition. Although the functional form of the free boundary condition is given explicitly, the location of the free boundary is unknown and can only be determined implicitly by imposing continuity conditions on the solution. Two specific problems are studied in details, namely the valuation of fixed-rate mortgages and CEV American options. The second type is the multi-dimensional problem, which involves multiple correlated stochastic variables and their governing PDE. One typical problem we focus on is the valuation of basket-spread options, whose underlying asset prices are driven by correlated geometric Brownian motions (GBMs). Analytic approximate solutions are derived for each of these three problems. / For each of the two free boundary problems, we propose a parametric moving boundary to approximate the unknown free boundary, so that the original problem transforms into a moving boundary problem which can be solved analytically. The governing parameter of the moving boundary is determined by imposing the first derivative continuity condition on the solution. The analytic form of the solution allows the price and the hedging parameters to be computed very efficiently. When compared against the benchmark finite-difference method, the computational time is significantly reduced without compromising the accuracy. The multi-stage scheme further allows the approximate results to systematically converge to the benchmark results as one recasts the moving boundary into a piecewise smooth continuous function. / For the multi-dimensional problem, we generalize the Kirk (1995) approximate two-asset spread option formula to the case of multi-asset basket-spread option. Since the final formula is in closed form, all the hedging parameters can also be derived in closed form. Numerical examples demonstrate that the pricing and hedging errors are in general less than 1% relative to the benchmark prices obtained by numerical integration or Monte Carlo simulation. By exploiting an explicit relationship between the option price and the underlying probability distribution, we further derive an approximate distribution function for the general basket-spread variable. It can be used to approximate the transition probability distribution of any linear combination of correlated GBMs. Finally, an implicit perturbation is applied to reduce the pricing errors by factors of up to 100. When compared against the existing methods, the basket-spread option formula coupled with the implicit perturbation turns out to be one of the most robust and accurate approximation methods. / 本論文為金融衍生產品定價的兩類問題作出了研究。第一類是自由邊界問題，它可以制定一個受制於自由邊界條件的偏微分方程式(PDE)，雖然當中自由邊界條件的函數形式是已知的，但自由邊界的位置是未知的，只能通過為實際解施加連續性條件作隱式確定。這裡為兩個具體問題進行了研究，分別是固定利率按揭合約(fixed-rate mortgages)定價和方差恆彈性模型的美式期權(CEV American options)定價。第二類是多維問題，它涉及到多個相關隨機變量及他們引申出的多維PDE。這裡為一個典型例子進行了研究，稱為籃子差異期權(basket-spread options)，其基礎資產價格由相關的幾何布朗運動驅動。我們為這三個問題提出了解析近似解。 / 對於上述的自由邊界問題，我們提出了一項參數移動邊界來近似模仿未知的自由邊界，使原來的自由邊界問題轉化為移動邊界問題，從而提出一種解析近似解。控制移動邊界的參數是通過滿足近似解的一階導數連續性條件來定。得到了解析近似解令當中的衍生產品定價和避險參數能有效快速地計算出，相比於有限差分法(finite-difference method)，精度保持了但計算時間顯著降低。再透過應用一個多階段方案，將移動邊界重鑄成一項分段光滑的連續函數，能有系統地將近似解的結果逼近有限差分法的結果。 / 對於上述的多維問題，我們從Kirk(1995)的二維差異期權(spread option)近似解定價公式推廣到多維的籃子差異期權。由於最終的定價公式是封閉形式，所有避險參數也從而得到封閉式近似解。從一些模擬例子顯示出，近似解的定價和避險參數，與通過數值積分法(numerical integration)或蒙地卡羅模擬法(Monte Carlo simulation)獲得的基準值比較，只有小於百分之一的誤差。此外，透過利用一種期權價格和相關基礎變量的概率分佈關係，我們進一步推論出一項籃子差異變量的近似解分佈函數，這可應用到任何多維幾何布朗運動的線性組合變量分佈。最後，我們提出一種隱式攝動方法，把定價誤差減少高達一百倍，跟現有的近似解定價方法相比，這是其中一種最健全和準確的籃子差異期權定價方法。 / Lau, Chun Sing = 自由邊界和多維的金融衍生產品定價問題 : 解析近似解 / 劉振聲. / Thesis Ph.D. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 174-186). / Abstracts also in Chinese. / Title from PDF title page (viewed on 12, September, 2016). / Lau, Chun Sing = Zi you bian jie he duo wei de jin rong yan sheng chan pin ding jia wen ti : jie xi jin si jie / Liu Zhensheng. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

An important problem in mathematical finance is to develop option pricing models that are able to capture implied volatility “smile” or “skew” commonly observed in financial markets. Many existing models are based on time-homogeneous Markov processes and they often have difficulty in calibrating implied volatilities across both strikes and maturities. In this dissertation, we develop two parsimonious and analytically tractable option pricing models to evaluate VIX options and crack spread options, respectively. Our modeling approach is based on additive subordination, which is a natural generalization of classical Bochner’s subordination. Probabilistically, additive subordination corresponds to a stochastic time change with respect to an independent additive subordinator. To model the VIX dynamics, we timechange a non-affine mean-reverting 3/2 diffusion with an independent additive subordinator to capture its empirical features, such as mean reversion and jumps, as well as upward-sloping implied volatility skew in VIX options. Moreover, we develop a parsimonious and analytically tractable two-factor model for crude oil and its refined product to evaluate crack spread option, where each factor is an additive subordinate Cox-Ingersoll-Ross process. This model captures key empirical features of individual commodities, such as mean-reversion and jumps, as well as of their spread. Analytical formulas for related options prices under each model are derived via an eigenfunction expansion approach. Empirical results show that our models have great flexibility in calibrating implied volatilities across strikes and maturities of each underlying with excellent performance. Our results suggest that additive subordination is a useful technique that allows one to construct a large family of jump-diffusions and/or pure jump processes with rich time- and state-dependent local characteristics, which are suited for parsimoniously reproducing empirical features with analytical tractability. / 金融數學中的一個重要問題是建立能夠捕獲金融市場普遍觀察到的隱含波動率微笑現象的期權定價模型。許多現存的模型基於時間齊次的馬爾可夫過程且這些模型一般難以同時校準具有各種執行價格和到期時間的隱含波動率。在此博士論文中，我們建立了兩個簡潔且易於分析的期權定價模型，分別用於定價VIX期權和裂變價差期權。我們的建模方法基於additivesubordination，該方法是經典的Bochner的Subordination方法的自然延伸。從概率論上講，additive subordination定義了一個關於additive subordinator的隨機時間變換。為了對VIX的動態變化建模，我們對一個具有非仿射均值回复的3/2擴散過程進行時間變換來捕獲VIX的相關性質，如均值回复和跳躍，以及VIX期權中的向上偏的隱含波動率曲線。進一步，我們對原油和其成品油創建了一個簡潔的且易於分析的俩因子模型來定價裂變價差期權，其中每一個因子都是一個additive subordinate Cox-Ingersoll-Ross過程。這個模型可以捕獲每個油品價格的關鍵屬性，如均值回复和跳躍，以及其他之間的價差。每個模型下的相應期權價格的解析公式通過特偵函數展開的方式求解得到。實證研究表明我們的模型具有較好的靈活性，在校準每個期權品種的隱含波動率曲面方面都具有非常好的表現。我們的研究結果也表明additive subordination是一個非常有用的方法。它可以用於創建一大類具有時間和狀態相依特偵的跳躍擴散或純跳過程，這些過程可用於簡便的建模一些實證特徵且便於分析。 / Li, Jing. / Thesis Ph.D. Chinese University of Hong Kong 2015. / Includes bibliographical references (leaves 129-142). / Abstracts also in Chinese. / Title from PDF title page (viewed on 13, September, 2016). / Detailed summary in vernacular field only.

Neste trabalho foi realizada a síntese e caracterização de um derivado de S,S-dioxidotioxantona, 7,8-benzotioxanton-9-ona-10,10-dióxido (TX-Np-SO2). A TX-Np-SO2 foi caracterizada estruturalmente por espectros eletrônico e vibracional (UV-vis e fluorescência, Infravermelho) e também por ressonância magnética nuclear (RMN) de 1H e 13C. As propriedades fotofísicas, espectros de emissão de fluorescência e fosforescência e o rendimento quântico de fluorescência da TX-Np-SO2 foram determinadas, assim como os tempos de vida de fluorescência (τF = 3,62 ns) e fosforescência (τPh = 800 ms). Os espectros de absorção triplete-triplete dos transientes da TX-Np-SO2, em soluções desoxigenadas de metanol, acetonitrila e metilciclohexano, foram determinados por Fotólise por Pulso de Laser, bem como as constantes de velocidade de supressão bimoleculares do derivado pelos supressores trans-estilbeno, trietilamina, p-dimetilamina benzoato de etila e 2-propanol. A fotopolimerização do metacrilato de metila (MMA) iniciada por TX-Np-SO2 foi estudada por dilatometria. A reação de polimerização do iniciador (TX-Np-SO2) na presença de amina e o monômero MMA foi estudada de modo a determinar a taxa de polimerização (Rp) a diferentes concentrações de trietilamina (TEA) e p-dimetilamina benzoato de etila (EDB). / The aim of this work was the synthesis and characterization of a S,S-dioxidethioxanthone derivative, 7,8-benzo-thioxanthen-9-one-10,10-dioxide (TX-Np-SO2). The TX-Np-SO2 was structurally characterized by its electronic and vibrational spectra (UV-vis and fluorescence, IR) and also by 1H and 13C RMN methods. The photophysical properties, fluorescence and phosphorescence emission spectra, and fluorescence quantum yield of TX-Np-SO2 were determined, as well as the fluorescence lifetime (τF = 3,62 ns) and the phosphorescence lifetime (τPh = 800 ms). The transient absorption spectra of TX-Np-SO2 in degassed solutions of methanol, acetonitrile and methylcyclohexane were determined by Laser Flash Photolysis, as well as the bimolecular triplet quenching rate constants for of the derivative by various compounds. The photopolymerization of methyl methacrylate (MMA) initiated by TX-Np-SO2 was studied by dilatometry. The polymerization reaction of the initiator (TX-Np-SO2) in the presence of amine and MMA monomer was studied in order to determine the polymerization rate (Rp) at different loadings of triethylamine (TEA) and ethyl-4-dimethylaminobenzoate (EDB).

Lai, Kwok Tung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 64-66). / Abstracts in English and Chinese. / Abstract --- p.i / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Literature Review --- p.9 / Chapter 3 --- Preliminary Analysis of Data --- p.17 / Chapter 3.1 --- Data --- p.17 / Chapter 3.2 --- Preliminary Analysis of Unspanned Stochastic Volatility --- p.20 / Chapter 4 --- A Jump-Diffusion Model for Federal Funds Target Rate --- p.23 / Chapter 4.1 --- Model Specification --- p.23 / Chapter 4.2 --- Estimation Result --- p.31 / Chapter 5 --- Pricing and Hedging Performance of Interest Rate Derivatives --- p.34 / Chapter 5.1 --- Pricing Performance of Interest Rate Cap --- p.34 / Chapter 5.2 --- Hedging Performance of Interest Rate Caplet --- p.38 / Chapter 5.3 --- Hedging Performance of Interest Rate Straddle --- p.42 / Chapter 6 --- Conclusion --- p.49 / Figures --- p.51 / Tables --- p.55 / Bibliography --- p.64

Computer Algebra Systems (CAS), a powerful mathematical software currently available on hand held calculators, is becoming increasingly available to assist secondary students learn school mathematics. This study investigates how two teachers taught introductory differential calculus to their Year 11 classes using multiple representations in a CAS-supported curriculim. This thesis aism to explore the impact of the teaching on students' understanding of the concept of derivative. Understanding the concept of derivative was gauged using an innovative Differentiation Competency Framework that was developed to describe understanding of the concept of derivative. It consist of eighteen competencies for formulation and interpretation of derivatives with, and without, translation between different representations. It clarifies the objectives of the curriculim, purpose for using particular CAS activities, and also guided the construction of individual test items on the Differentiation Competency Test that enabled individual and class learning about the concept of derivative to be identified. The framework also helped identify each teacher's privileging characteristics and facilitated analysis of the learning in relation to the teaching. This study found that using multiple representations was important in developing understanding of the concept of derivative but that the graphical and the symbolic representations were the most useful and important to emphasize and link. Analysis of the teaching actions showed that the teachers used CAS in ways that were consistent with their teaching approach and preferred use of representations and that a conceptual teaching method and student-centred style supported understanding of the concept of derivative. Teaching is directly linked to learning and each class developed a different understanding of the concept of derivative that related to the combined effect of their teacher's privileging characteristics: calculus content, teaching approach, and use of CAS. This study shows that if a CAS-supported curriculum is to be successfully implemented, it needs to acquire institutional status including a corresponding change in assessment to legitimize new teaching practices.

This thesis is a culmination of a research of a particular branch of Corporate Law, which has grown in several major parts of civilized jurisdictions. The thrust of the study was to evaluate the past, present and the future of a particular type of action known in Corporate Law under the umbrella of shareholder remedies - the 'Derivative Action' with emphasis to develop the law in one jurisdiction profiting from another. The research thus reveals how, when and where the so called action originated, the initial effects these actions had on the corporate world including shareholders, companies and related persons natural or juristic. Though much has been written by way of books, treatises and articles and several researches have dealt with the common topic shareholder remedies in its broad perspect, there is no separate study carried out on this topic in its global context with a comparative focus. This study has therefore given me the drive, initiative and courage to look at the conceptual view or the macro view of the so called 'Derivative Action' with of course special emphasis on the Australian and Sri Lankan jurisdictions in its micro aspects. This, I believe is the first time anyone has undertaken such a task. The study thus travels through distant roads of common law action to the statutory form of the action in the relevant jurisdictions and finds it driving with much purpose in jurisdictions such as Australia and Sri Lanka which are both in the transitional era from the common law action to the statutory action. The research is based on the collection of material namely, case law - Australian, Sri Lankan and international on the matters in issue, Legal treatises on the subject matter local and international, Law reform material - Australian, Sri Lankan and international on the topic, Bills and Statutes available on the topic in Australia, Sri Lanka and other countries. I have met resource personnel with regard to Law Reform in several jurisdictions on the matters in issue and visited the Australian Stock Exchange and the Colombo Stock Exchange. The research findings depend mainly on the electronic data available in addition to resources available at the University of Canberra, the Australian National University, Colombo Law Library, The Colombo Law Society Library and the Sri Lanka Supreme Court Judges' Chambers Library and the Sri Lanka Attorney General's Department Library. Visits to the McGill University in Montreal, Canada and the corporate law sector in New Zealand, including Universities and Law Offices in Christchurch and Auckland too has helped me considerably in the process. Review of the literature of the proposed statutory Derivative Action in Australia and the proposed statutory Derivative Action in Sri Lanka, are based mainly on; Enforcement of the duties of directors and officers of a company by means of a statutory derivative action (Report No. 12) Companies and Securities Law Review Committee. (November 1990.), Corporate Practices and the Rights of Shareholders (Report of the House of Representatives Standing Committee on Legal and Constitutional Affairs) Parliament of the Commonwealth of Australia. (November 1991.), Report on A Statutory Derivative Action Companies and Securities Advisory Committee. (July 1993.), Corporate Law Economic Reform Program (CLERP) Proposal Paper No 3 (1997), the CLERP draft legislative provisions (1998), Australian case law on the application of the common law Derivative Action, both in the High Court and in individual States and Australian articles on Derivative Action as a common law remedy and on the introduction of the statutory action. In the Sri Lankan context, the proposals in Sri Lanka for the statutory Derivative Action and the case law in Sri Lanka on the application of the common law remedy has been referred to. Other literature include, material available on the Canadian formula of Derivative Action, including Statutes, Rules, case law, articles and other relevant data, material available on the Derivative Actions in the United States, material available in New Zealand on Derivative Actions, material available in England on Derivative Actions, namely on the common law approach, case law, articles, Bills, Rules and other connected material, Statutes on Derivative Actions in other jurisdictions at present and Hong Kong proposals for a statutory Derivative Action, to name some. The aforesaid material and the review of the same have assisted the study as follows: -To place the past, present and the future of the common law Derivative Action. -Examine the objectives of the Derivative Action. -The operation of the common law aspects of the action. -The benefits of the statutory form of the action. -Experiences of other countries in the recent past on the subject. -The Australian reform process presently underway. -The best experiences in Australia with regard to case law. -To evaluate whether the remedy should be limited to fraud on the minority or whether it should be extended further even to negligence. -How best Sri Lanka could benefit from the Australian formula of the statutory form of the action. -To evaluate whether the proposed model of the statutory action in Sri Lanka is adequate in view of the Australian and other accepted formulae on the subject. -Whether the common law action should be expressly abolished in Sri Lanka. -Consider the possible introduction of the best methods to Sri Lanka. Finally, the research speaks for itself the need for a statutory Derivative Action for Sri Lanka in the future, to be an improvement on the Canadian, New Zealand and Australian models. The research findings, especially in its conclusions and recommendation in Chapter 8, will no doubt help to improve the proposed statutory Derivative Action in Sri Lanka in a small way.