Pricing Path-Independent Payoffs with Exotic Features in the Fractional Diffusion Model

We provide several practical formulas for pricing path-independent exotic instruments (log options and log contracts, digital options, gap options, power options with or without capped payoffs …) in the context of the fractional diffusion model. This model combines a tail parameter governed by the space fractional derivative, and a subordination parameter governed by the time-fractional derivative. The pricing formulas we derive take the form of quickly convergent series of powers of the moneyness and of the convexity adjustment; they are obtained thanks to a factorized formula in the Mellin space valid for arbitrary payoffs, and by means of residue theory. We also discuss other aspects of option pricing such as volatility modeling, and provide comparisons of our results with other financial models.

Duality and socle generators for residual intersections

We prove duality results for residual intersections that unify and complete results of van Straten, Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt.Suppose that I is an ideal of codimension g in a Gorenstein ring, and {J\subset I} is an ideal with {s=g+t} generators such that {K:=J:I} has codimension s. Let {{\overline{I}}} be the image of I in {{\overline{R}}:=R/K}.In the first part of the paper we prove, among other things, that under suitable hypotheses on I, the truncated Rees ring {{\overline{R}}\oplus{\overline{I}}\oplus\cdots\oplus{\overline{I}}{}^{t+1}} is a Gorenstein ring, and that the modules {{\overline{I}}{}^{u}} and {{\overline{I}}{}^{t+1-u}} are dual to one another via the multiplication pairing into {{{\overline{I}}{}^{t+1}}\cong{\omega_{\overline{R}}}}.In the second part of the paper we study the analogue of residue theory, and prove that, when {R/K} is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of {I^{t+1}/JI^{t}\cong{\omega_{R/K}}} is generated by a Jacobian determinant.

Engenharia Didática para a Teoria do Resíduo: Análises Preliminares, Análise a Priori e Descrição de Situações-Problema

O presente escrito aborda elementos de ordem teórico-conceitual, característicos de um design de investigação em Didática da Matemática, com o escopo de assinalar uma discussão e a possibilidade do ganho de conhecimentos didático-metodológicos acerca da Teoria do Resíduo. Dessa forma, tendo em vista extensa literatura, com influência da vertente de pesquisa francesa, descrevem-se com maior ênfase, as etapas de análise preliminar, análise a priori de uma Engenharia Didática - ED, com atenção especial dedicada à concepção e a modelização de duas situações-problema. Por intermédio de uma perspectiva afetada pela Teoria das Situações Didáticas – TSD, o trabalho aborda quatro etapas dialéticas, relativas ao planejamento teórico de uma experimentação, que promovem um debate científico envolvendo professor, alunos e um saber matemático específico, significado e mobilizado por intermédio da visualização (com o uso da tecnologia), e do entendimento tácito preliminar de determinadas propriedades fundamentais na Teoria dos Resíduos. Outrossim, o trabalho acentua o caráter epistemologicamente intrincado da referida teoria matemática, fato que justifica um olhar pormenorizado e cientificizado, proporcionado por uma proposta de ED.Palavras-chave: Engenharia Didática. Teoria do Resíduo. Análises Preliminares.AbstractThis writing deals with conceptual and theorical order elements, characteristic of a particular design of research in the Didactics of Mathematics, varygin between offering a discussion and the possibility of a gain of didactical and methodological knowledge of the Residue Theory. Thus, aiming at extensive literature, with the French research, the preliminary stages nominated preliminary and a priori analysis of a Didactical Engineering – DE are described with a greater emphasis, with especial attention dedicated to the design and modeling of two didactic situations. Thus through a perspective affected by the Theory of Didactic Situations – TDS, the present work addresses four dialectical steps, regarding the theorical planning related to experimentation, with the goal to promote a scientific debate involving the teacher, students and a specific mathematical knowledge, signified and mobilized through the visualization (with the using of technology) and a tacit understanding of certain fundamental properties in the Residue Theory. Furthermore, the work emphasizes the mathematical character of a epistemological complex theory, which justifies a careful and scientific look provided by the present ED proposal.Keywords: Didactic Engineering. Theory of Residue. Preliminary Analysis.