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

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

Analytical Solutions of the Black–Scholes Pricing Model for European Option Valuation via a Projected Differential Transformation Method

In this paper, a proposed computational method referred to as Projected Differential Transformation Method (PDTM) resulting from the modification of the classical Differential Transformation Method (DTM) is applied, for the first time, to the Black–Scholes Equation for European Option Valuation. The results obtained converge faster to their associated exact solution form; these easily computed results represent the analytical values of the associated European call options, and the same algorithm can be followed for European put options. It is shown that PDTM is more efficient, reliable and better than the classical DTM and other semi-analytical methods since less computational work is involved. Hence, it is strongly recommended for both linear and nonlinear stochastic differential equations (SDEs) encountered in financial mathematics

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

Epistemic Uncertainty Quantification in Scientific Models

In the field of uncertainty quantification (UQ), epistemic uncertainty often refers to the kind of uncertainty whose complete probabilistic description is not available, largely due to our lack of knowledge about the uncertainty. Quantification of the impacts of epistemic uncertainty is naturally difficult, because most of the existing stochastic tools rely on the specification of the probability distributions and thus do not readily apply to epistemic uncertainty. And there have been few studies and methods to deal with epistemic uncertainty. A recent work can be found in [J. Jakeman, M. Eldred, D. Xiu, Numerical approach for quantification of epistemic uncertainty, J. Comput. Phys. 229 (2010) 4648-4663], where a framework for numerical treatment of epistemic uncertainty was proposed. In this paper, firstly, we present a new method, similar to that of Jakeman et al. but significantly extending its capabilities. Most notably, the new method (1) does not require the encapsulation problem to be in a bounded domain such as a hypercube; (2) does not require the solution of the encapsulation problem to converge point-wise. In the current formulation, the encapsulation problem could reside in an unbounded domain, and more importantly, its numerical approximation could be sought in Lp norm. These features thus make the new approach more flexible and amicable to practical implementation. Both the mathematical framework and numerical analysis are presented to demonstrate the effectiveness of the new approach. And then, we apply this methods to work with one of the more restrictive uncertainty models, i.e., the fuzzy logic, where the p-distance, the weighted expected value and variance are defined to assess the accuracy of the solutions. At last, we give a brief introduction to our future work, which is epistemic uncertainty quantification using evidence theory

Tracking uncertainty in a spatially explicit susceptible-infected epidemic model

In this paper we conceive an interval-valued continuous cellular automaton for describing the spatio-temporal dynamics of an epidemic, in which the magnitude of the initial outbreak and/or the epidemic properties are only imprecisely known. In contrast to well-established approaches that rely on probability distributions for keeping track of the uncertainty in spatio-temporal models, we resort to an interval representation of uncertainty. Such an approach lowers the amount of computing power that is needed to run model simulations, and reduces the need for data that are indispensable for constructing the probability distributions upon which other paradigms are based

Lp-calculus approach to the random autonomous linear differential equation with discrete delay

[EN] In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay , with initial condition x(t)=g(t), -t0. The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. Using Lp-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an Lp-solution too. An analysis of Lp-convergence when the delay tends to 0 is also performed in detail.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). 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