Review of modern numerical methods for a simple vanilla option pricing problem

Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better

Fuzzy Decision Making and Soft Computing Applications

This Special Issue collects original research articles discussing cutting-edge work as well as perspectives on future directions in the whole range of theoretical and practical aspects in these research areas: i) Theory of fuzzy systems and soft computing; ii) Learning procedures; iii) Decision-making applications employing fuzzy logic and soft computing

Selected aspects of complex, hypercomplex and fuzzy neural networks

This short report reviews the current state of the research and methodology on theoretical and practical aspects of Artificial Neural Networks (ANN). It was prepared to gather state-of-the-art knowledge needed to construct complex, hypercomplex and fuzzy neural networks. The report reflects the individual interests of the authors and, by now means, cannot be treated as a comprehensive review of the ANN discipline. Considering the fast development of this field, it is currently impossible to do a detailed review of a considerable number of pages. The report is an outcome of the Project 'The Strategic Research Partnership for the mathematical aspects of complex, hypercomplex and fuzzy neural networks' meeting at the University of Warmia and Mazury in Olsztyn, Poland, organized in September 2022.Comment: 46 page

Numerical Solution of Fuzzy Arbitrary Order Predator-Prey Equations

This paper seeks to investigate the numerical solution of fuzzy arbitrary order predator-prey equations using the Homotopy Perturbation Method (HPM). Fuzziness in the initial conditions is taken to mean convex normalised fuzzy sets viz. triangular fuzzy number. Comparisons are made between crisp solution given by others and fuzzy solution in special cases. The results obtained are depicted in plots and tables to demonstrate the efficacy and powerfulness of the methodology

New Trends in Differential and Difference Equations and Applications

This is a reprint of articles from the Special Issue published online in the open-access journal Axioms (ISSN 2075-1680) from 2018 to 2019 (available at https://www.mdpi.com/journal/axioms/special issues/differential difference equations)

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

Fuzzy transform applied in initial value problem with uncertain initial condition

Orientador: Laécio Carvalho de BarrosDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Este trabalho trata da formulação do problema de Cauchy com condição inicial fuzzy e apresenta um método de resolução para este problema. Enquanto o problema de Cauchy clássico tem como solução uma função, a solução do problema com condição inicial fuzzy é uma relação fuzzy. Para obtê-la, o problema é dividido em três problemas clássicos (determinísticos) em que as soluções são obtidas a partir das teorias de transformada fuzzy (F-transformada) e partição fuzzy. A transformada fuzzy é uma ferramenta que age sobre uma função no sentido de condensar informações a respeito da mesma num dado intervalo. Nesse sentido, a partição fuzzy divide o intervalo em que o problema de valor inicial é considerado, em subintervalos para posterior aplicação da transformada. Com isso, as soluções dos problemas clássicos são encontradas em forma de n-uplas de números reais que, combinadas com a partição do intervalo considerado, produzirão um sistema de equações relacionais fuzzy. A solução deste sistema será a relação fuzzy que é solução do problema de Cauchy com condição inicial fuzzy. Devido a dificuldade de encontrar uma solução analítica, são apresentadas aproximações de soluções do sistema. A metodologia é ilustrada através de modelos biomatemáticos de crescimento populacionalAbstract: This work studies the formulation of Cauchy problem with fuzzy initial condition and shows a method of resolution for this problem. While classical Cauchy problem has a function as solution, the solution of the problem with fuzzy initial condition is a fuzzy relation. For obtaining this relation, the problem is separated at three classical problems (deterministic) whose solutions are obtained from fuzzy transform (F-transform) and fuzzy partition theories. The fuzzy transform is a tool that acts on a function compressing its information on a given interval. In this sense, fuzzy partition separates the interval whose initial value problem is considered, at subintervals for subsequent application of the transform. In this way, the solutions of classical problems are found as n-uples of real numbers that, combined with the partition of the considered interval, will produce a system of fuzzy relation equations. Due to the difficulty of finding an analytical solution, it will be presented approximations of system solutions. The methodology is illustrated with biomathematical models of population growthMestradoMatematica AplicadaMestre em Matemática AplicadaCAPE