Analytical and numerical methods for some problems related to financial option pricing

Doutoramento em Matemática Aplicada à Economia e à GestãoWe investigate second-order PDE problems arising in financial option pricing. Firstly, we consider a non-linear problem arising when transaction costs are included and investigate the existence, uniqueness, and localisation of the solution by using topological methods. Secondly, we study the discretisation of a generalised version of the linear Black-Scholes PDE, where both the asset appreciation rate and the volatility are taken time and space-dependent and growing in the space variable, under the strong assumption that the PDE is nondegenerate. The PDE solvability is considered in the framework of the variational approach and the discretisation is obtained by using basic finite-difference methods. For the time variable, we consider a general evolution equation problem the PDE problem can be cast into, and discretise it by using both the implicit and the explicit schemes. We obtain a convergence result under a smoothness assumption weaker than the usual Holder continuity. Furthermore, we investigate two main types of operator specification. For the spatial variable, we consider the discretised version of the PDE problem in discrete weighted Sobolev spaces and obtain a convergence result stronger than we could find in the literature. Finally, the rate of convergence for the approximation in space and time is computed.Neste estudo, investigamos EDPs de segunda ordem que surgem no apregamento de opções financeiras. Em primeiro lugar, consideramos urn problema não-linear que surge quando sao incluídos custos de transação e investigamos a existência, unicidade e localização da solução, usando métodos topológicos. Em segundo lugar, estudamos a discretização de urn versão generalizada da EDP de Black-Scholes linear, em que a taxa de apreciação do activo e a volatilidade sao dependentes do tempo e do espaço e crescentes na variavel espacial, assumindo que a EDP e não degenerada. A solvabilidade da EDP e considerada numa abordagem variacional e a discretização e obtida usando métodos basicos de diferenças finitas. Para a variável temporal, consideramos uma equação de evolução geral, de que a EDP e caso particular, que discretizamos usando os esquemas implicito e explicito. Obtemos urn resultado de convergência sob uma hipótese de regularidade mais fraca que a usual continuidade de Holder. Adicionalmente, investigamos dois tipos fundamentais de especificação do operador. Para a varia vel espacial, consideramos a versão discreta da EDP em espaços de Sobolev discretos e obtemos urn resultado de convergência mais forte do que pudemos encontrar na literatura. Finalmente, obtemos a taxa de convergência para a aproximação no espaço e no tempo

A fully nonlinear problem arising in financial modelling

We state existence and localisation results for a fully nonlinear boundary value problem using the upper and lower solutions method. With this study we aim to contribute to a better understanding of some analytical features of a problem arising in financial modelling related to the introduction of transaction costs in the classical Black-Scholes model. Our result concerns stationary solutions which become interesting in finance when the time does not play a relevant role such as, for instance, in perpetual options.info:eu-repo/semantics/publishedVersio

A fully nonlinear problem arising in financial modelling

We state existence and localisation results for a fully nonlinear boundary value problem using the upper and lower solutions method. With this study we aim to contribute to a better understanding of some analytical features of a problem arising in financial modelling related to the introduction of transaction costs in the classical Black-Scholes model. Our result concerns stationary solutions which become interesting in finance when the time does not play a relevant role such as, for instance, in perpetual options.info:eu-repo/semantics/publishedVersio

Discretisation of abstract linear evolution equations of parabolic type

We investigate the discretisation of the linear parabolic equation du/dt = A(t)u + f(t) in abstract spaces, making use of both the implicit and the explicit finite-difference schemes. The stability of the explicit scheme is obtained, and the schemes’ rates of convergence are estimated. Additionally, we study the special cases where A and f are approximated by integral averages and also by weighted arithmetic averages.info:eu-repo/semantics/publishedVersio

Aprendizaje cooperativo para promover el pensamiento creativo y la creatividad matemática en la Educación Superior

Creativity is an indispensable competence for the future of higher education students. Creative thinking skills play an important role in modern society. Creativity is acknowledged as a crucial aspect of business, research and development or arts. This study presents the assessment results of an intervention using cooperative learning and the conventional teaching method in order to promote creative thinking in higher education. The design used was quasi-experimental pretest and posttest intervention using the Creative Intelligence test (CREA), with experimental and control groups of higher education students of the Communication and Multimedia course in a Linear Algebra class. The participants were 50 students from a Portuguese public university. Results showed that students who participated in the intervention scored significantly higher in the creativity test at the end of the intervention and indicated that creative thinking, and divergent thinking abilities in particular, can be enhanced through the kind of intervention that was proposed in the studyLa creatividad es una competencia indispensable para el futuro de los estudiantes de educación superior. Las habilidades de pensamiento creativo juegan un importante papel en la sociedad moderna. La creatividad es reconocida como un aspecto fundamental en el campo de los negocios, la investigación y desarrollo y las artes. Este estudio presenta los resultados de la evaluación de una intervención que utiliza el aprendizaje cooperativo y el método de enseñanza convencional para promover el pensamiento creativo en la educación superior. El diseño utilizado fue cuasi-experimental con la aplicación del test de Inteligencia Creativa (CREA) antes y después de la intervención, con grupos experimentales y de control de estudiantes de educación superior del curso de Comunicación y Multimedia en una clase de Álgebra Lineal. Los participantes del estudio fueron 50 estudiantes de una universidad pública portuguesa. Los resultados mostraron que los estudiantes que participaron en la intervención obtuvieron puntuaciones significativamente más altas en el test de creatividad al final de la intervención e indicaron que el pensamiento creativo, y las habilidades de pensamiento divergentes en particular, pueden mejorarse a través del tipo de intervención que se propuso en el estudi

LET’S DO IT AGAIN: A SECOND QUALITATIVE ANALYSIS OF PORTUGUESE TEACHERS' ATTITUDES TOWARDS STATISTICS

We continue to believe that teachers' attitudes towards statistics can have a significant effect on their own statistical training, their way of teaching statistics, and on the future attitudes of their students. The influence of attitudes in teaching statistics in different contexts was previously studied in several researches, including some of the authors. This work is part of a broader study of Portuguese of attitudes towards statistics of Portuguese in-service teachers. Here we used a qualitative content analysis of survey responses from Portuguese 2nd cycle in-service teachers (students aged 10 or 11), focusing on 9 open-ended items from the EAEE – Escala de Actitudes hacia la Estadistica de Estrada. The EAEE survey was presented to Portuguese in-service 2nd cycle teachers and all of them were asked to complete the 9 open-ended explanations, and 132 (54% of 246) chose to do this for at least one of those 9 items, but only 41(17% of 246; 31% of 132) did it to all 9 items. The 246 in-service teachers had a mean attitude total score of 97 (SD 9.5), a median of 97, higher than the indifference/neutrality 75 (mid-point of EAEE). Now we present the qualitative content analysis of the EAEE survey open-ended answers from Portuguese in-service second cycle teachers. Among teachers’ explanations for positive attitudes in the items considered, is their need for and interest in teaching statistics, and for negative attitudes they emphasize mistrust of statistics presented on television and their manipulation. These analyses allowed us to investigate teachers’ attitudes towards statistics in order to devise a way to improve them

Impact of COVID-19 on undergraduate business students: a longitudinal study on academic motivation, engagement and attachment to university

Purpose – This study aims to explore whether the COVID-19-related circumstances hindered these academic-related variables. Design Methodology Approach – The authors surveyed two groups of undergraduate business students (42% male) who completed the questionnaires at the beginning and at the end of the semester. One group of students attended only face-to-face classes in the 2018/2019 academic year (n = 126) and the other group transitioned to online classes because of the COVID-19 outbreak in the 2019/2020 academic year (n = 99). Findings – The findings show no statistically significant group differences between the pre- and post-test in students’ intrinsic and extrinsic motivation, feelings of attachment to the university and engagement dimensions of absorption and vigour. Nevertheless, a moderate negative effect was found in the dedication engagement dimension. Practical Implications – The authors discuss the main results in terms of some practices that may contribute towards attenuating the effects of future emerging pandemics in the higher education setting. Originality Value – The COVID-19 pandemic imposed a rapid transition to online instruction in education institutions worldwide. However, it remains unclear to date how students’ engagement, motivation and attachment to the university were negatively affected by the first COVID-19 outbreak.info:eu-repo/semantics/acceptedVersio

Oral Communivation and M-LEARNING in the primary school English Classroom: photography and video as a resource

The process of design, implementation and evaluation of the project ‘Films & Photographs: An M-learning Experience in Primary Education’ is presented in this article through the action research methodology. The project’s main objective was to foster oral communication in the English language (as an additional language) using mobile devices in primary education through photography and video. Two primary school teachers of English, two university lecturers and an in-service teacher advisor from the Teachers’ Training and Resource Centre participated in the project. The pupils, between ten and twelve years old, belonged to two different schools in the Principality of Asturias, Spain and the project was developed during the school year 2015-2016. The techniques used to collect data were participant observation, semi structured-interviews and a questionnaire. The activities were focused on the use of mobile devices (smartphones and tablets) to produce and edit videos and photographs. The results show students and teachers’ positive perception about m-learning since they consider it has directly influenced the motivation to carry out the proposed tasks. All this has an effect on the improvement of the communicative, technological and artistic competences in the primary education context

Critical thinking test for elementary and secondary students

The importance of critical thinking in education is supported by numerous researches for decades. In the absence of a critical thinking test (CTT) designed to be used in Portugal, the focus of this paper is the presentation of a critical thinking test for Portuguese adolescents. We present the exploratory validation study of this instrument, administered to 128 students, aged 12 to 19, attending elementary and secondary schools of low and high socioeconomic status. The construct validity of the nine-item instrument was estimated using an Exploratory Factorial Analysis (AFE), to evaluate the multidimensionality of the CTT. There were three factors: creation, argumentation and explanation, explaining 70.07% of the total variance .We found significant differences between grades and schools. The results suggest that the development of critical thinking skills is related to the grade and students’ socioeconomic status