Non-standard Transformations Research Articles

Empirical analysis of rough and classical stochastic volatility models to the SPX and VIX markets

We conduct an empirical analysis of rough and classical stochastic volatility models to the SPX and VIX options markets. Our analysis focusses primarily on calibration quality and is split in two parts. In part one, we perform a historical calibration to SPX options over the years 2004–2019 of a selection of models that include the one-factor rough Bergomi and rough Heston models. In part two, we consider three calibration dates with low, typical, and high volatility, examine a wide selection of models, and calibrate to both SPX options as well as jointly to SPX and VIX options. The key results are as follows: The rough Bergomi and rough Heston models fail to create a term structure of smile effect that is sufficiently pronounced for SPX options. Moreover, we discover that short-expiry SPX smiles generally are more symmetric than long-expiry smiles, a feature we neither find that these models can reproduce. We propose an alternative volatility model driven by two Ornstein-Uhlenbeck processes that uses a non-standard transformation function. Calibrating it to SPX options, we obtain almost perfect fits, and calibrating it jointly to SPX and VIX options, we obtain very decent fits. This suggests, contrary to what one might be led to believe based on much of the existing literature, that the joint SPX-VIX calibration problem is largely solvable with classical two-factor volatility, all without roughness and jumps.

Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation

We consider five-point differential-difference equations. Our aim is to find integrable modifications of the Ito-Narita-Bogoyavlensky equation related to it by non-invertible discrete transformations. We enumerate all modifications associated to transformations of the first, second and third orders. As far as we know, such a classification problem is solved for the first time in the discrete case. We analyze transformations obtained to specify their nature. A number of new integrable five-point equations and new transformations have been found. Moreover, we have derived one new completely discrete equation. There are a few non-standard transformations which are of the Miura type or are linearizable in a non-standard way. We have also proved that the orders of possible transformations are restricted by the number five in this problem.

ФЛУКТУАЦІЙНІ ЯВИЩА В СУЧАСНИХ АНГЛОМОВНИХ ЛІНГВО-МЕНТАЛЬНИХ СТРУКТУРАХ

Стаття присвячена розгляду явищ флуктуації, що актуалізуються як на рівні конституентів (лінгво-ментальних одиниць), так і на рівні механізмів (метакогнітивних моделей). У роботі явище флуктуації детально розглядається на рівні таких англомовних концептуальних структур як морфологізовані метаконцепти та лексикалізовані концепти. Також увагу зосереджено на лінгвокогнітивному дослідженні процесів нестандартних трансформацій телескопізмів й абревіатур та їх подальшої реалізації у процесах концептуальної деривації, за дії якої утворюються похідні. Авторська теорія моделювання телескопних інновацій дає можливість дослідження рідкісних випадків деривації полікомпонентного характеру. В роботі розкривається думка про те, що принципи організації концептосистеми англійської мови перебувають у стані перманентної секвенції з умовами розвитку англомовного суспільства. Значною мірою саме дивергентність антропоцентричних процесів корелює особливості розвитку концептосистеми англійської мови, що змінюються в калейдоскопі часопросторових картин. Наведені в статті інноваційні моделі вказують, що внутрішньосистемні флуктуаційні явища абсолютно не знижують показник урівноваженості лінгво-ментального простору. Напроти, за їх допомогою відбувається перерозподіл складових у межах структури системи та досягається здатність до її реорганізації. У ролі ілюстративного матеріалу теоретичних викладок даної роботи подаються англомовні інновації. (The theme of the article is linked with the phenomena of fluctuations, actualized both at the level of constituents (lingvo-mental units), and at the level of mechanisms (metacognitive models). In the work, the phenomenon of fluctuations is considered in detail at the level of such English-language conceptual structures as morphological metaconcepts and lexical concepts. Also the research deals with the lingvo-cognitive study of the processes of non-standard transformations of blendings and abbreviations and their further implementation in the processes of conceptual derivation, which determines creation of derivatives. Under the author’s theory of modeling of blended innovations makes it possible to study the rare cases of polycomponent nature of blending. The paper reveals the idea that the principles of organizing of English conceptosystem are in a state of permanent sequencing with the conditions of development of the English-speaking society. To a large extent, the very divergence of anthropocentric processes correlates the peculiarities of the development of the conceptual system of the English language, which change in the kaleidoscope of chronotopic pictures. The innovative models presented in this article indicate that intra-system fluctuation phenomena absolutely do not reduce the balance of the lingvo-mental setting. On the contrary, with their help, there is a redistribution of components within the structure of the system and is achieving the ability to its reorganization. English lexical innovations are presented in the role of illustrative material of theoretical definitions of this work.)

Formal Indicators and Scientific Concepts

A principal theme of hermeneutical phenomenology of science has been to analyze the status of theoretical entities. In Ginev’s ambitious analysis of contemporary trends in hermeneutic phenomenology of science, for instance, one of the two “hermeneutic circles” he describes involves the constitution of objects of inquiry, as “mathematized entities” associated with data-models, which has a formal side in theory and an empirical side in experimentation. The question then arises of the relation between the two sides; the danger, he puts it, is a theoretical essentialism which is implied when the mathematical projection is conceived as operationalized by experiment. Ginev’s proposal to avoid this involves the concept of “inscription.” This paper proposes another approach, covariant realism, which draws from Heidegger’s notion of “formal indication” and which makes explicit the temporality of theoretical objects in the flow of the research process. Heidegger developed his notion as an integral part of his hermeneutics of facticity; its motivation was the need to develop a discourse adequate for pre-theoretical experience. While it may seem strange to apply this idea so far out of context, it seems poised to address certain long-standing problems in the philosophy of science, those of incommensurability and scientific theory change. Formal indication characterizes phenomena that are understood to be provisionally grasped, already interpreted, and anticipated as able to show themselves differently in different contexts. The value of this admittedly nonstandard transformation of Heidegger’s concept suggests deeper possibilities for continentally-inspired approaches to understanding science practice than have hitherto been explored.

Approximation Schemes for Packing with Item Fragmentation

We consider two variants of the classical bin packing problem in which items may be fragmented. This can potentially reduce the total number of bins needed for packing the instance. However, since fragmentation incurs overhead, we attempt to avoid it as much as possible. In bin packing with size increasing fragmentation (BP-SIF), fragmenting an item increases the input size (due to a header/footer of fixed size that is added to each fragment). In bin packing with size preserving fragmentation (BP-SPF), there is a bound on the total number of fragmented items. These two variants of bin packing capture many practical scenarios, including message transmission in community TV networks, VLSI circuit design and preemptive scheduling on parallel machines with setup times/setup costs. While both BP-SPF and BP-SIF do not belong to the class of problems that admit a polynomial time approximation scheme (PTAS), we show in this paper that both problems admit a dual PTAS and an asymptotic PTAS. We also develop for each of the problems a dual asymptotic fully polynomial time approximation scheme (AFPTAS). Our AFPTASs are based on a non-standard transformation of the mixed packing and covering linear program formulations of our problems into pure covering programs, which enables to efficiently solve these programs.

On the synthesis of fast algorithms for signal processing

Nonstandard transformations over matrices allow the derivation of different, optimal algorithms in digital signal processing.

Confinement of quarks and gluons in gauge quantum field theory

A SU(3)-gauge quantum field theory with a quark triplet, an antiquark triplet, and a self-conjugate gluon octet as basic fields is investigated. In virtue of a nontrivial coupling between the representation of the translation group and the SU(3)-color charge of the basic fields it is proved: (i) The basic quark, antiquark, and gluon fields are confined. (ii) Every state vector of the physical Hilbert space is a SU(3)-color singlet state. (iii) Poincaré invariance holds in the physical Hilbert space. These results rest on a nonstandard transformation law of the basic fields with respect to translations. Its meaning within the frame of classical Lagrangian field theory for chromodynamics is investigated.