Download or read book High Order Methods for Parabolic Equations in Multiple Space Dimensions for Option Pricing Problems written by Christian Hendricks and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book High Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions written by Bertram Düring and published by . This book was released on 2014 with total page 25 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present a high-order compact finite difference approach for a rather general class of parabolic partial differential equations with time and space dependent coefficients as well as with mixed second-order derivative terms in n spatial dimensions. Problems of this type arise frequently in computational fluid dynamics and computational finance. We derive general conditions on the coefficients which allow us to obtain a high-order compact scheme which is fourth-order accurate in space and second-order accurate in time. Moreover, we perform a thorough von Neumann stability analysis of the Cauchy problem in two and three spatial dimensions for vanishing mixed derivative terms, and also give partial results for the general case. The results suggest unconditional stability of the scheme. As an application example we consider the pricing of European Power Put Options in the multidimensional Black-Scholes model for two and three underlying assets. Due to the low regularity of typical initial conditions we employ the smoothing operators of Kreiss et al. to ensure high-order convergence of the approximations of the smoothed problem to the true solution.
Download or read book Novel Methods in Computational Finance written by Matthias Ehrhardt and published by Springer. This book was released on 2017-09-19 with total page 599 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses the state-of-the-art and open problems in computational finance. It presents a collection of research outcomes and reviews of the work from the STRIKE project, an FP7 Marie Curie Initial Training Network (ITN) project in which academic partners trained early-stage researchers in close cooperation with a broader range of associated partners, including from the private sector. The aim of the project was to arrive at a deeper understanding of complex (mostly nonlinear) financial models and to develop effective and robust numerical schemes for solving linear and nonlinear problems arising from the mathematical theory of pricing financial derivatives and related financial products. This was accomplished by means of financial modelling, mathematical analysis and numerical simulations, optimal control techniques and validation of models. In recent years the computational complexity of mathematical models employed in financial mathematics has witnessed tremendous growth. Advanced numerical techniques are now essential to the majority of present-day applications in the financial industry. Special attention is devoted to a uniform methodology for both testing the latest achievements and simultaneously educating young PhD students. Most of the mathematical codes are linked into a novel computational finance toolbox, which is provided in MATLAB and PYTHON with an open access license. The book offers a valuable guide for researchers in computational finance and related areas, e.g. energy markets, with an interest in industrial mathematics.
Download or read book Mathematical Modeling And Methods Of Option Pricing written by Lishang Jiang and published by World Scientific Publishing Company. This book was released on 2005-07-18 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the unique perspective of partial differential equations (PDE), this self-contained book presents a systematic, advanced introduction to the Black-Scholes-Merton's option pricing theory.A unified approach is used to model various types of option pricing as PDE problems, to derive pricing formulas as their solutions, and to design efficient algorithms from the numerical calculation of PDEs. In particular, the qualitative and quantitative analysis of American option pricing is treated based on free boundary problems, and the implied volatility as an inverse problem is solved in the optimal control framework of parabolic equations.
Download or read book Blow up for Higher Order Parabolic Hyperbolic Dispersion and Schrodinger Equations written by Victor A. Galaktionov and published by CRC Press. This book was released on 2014-09-22 with total page 565 pages. Available in PDF, EPUB and Kindle. Book excerpt: Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrodinger Equations shows how four types of higher-order nonlinear evolution partial differential equations (PDEs) have many commonalities through their special quasilinear degenerate representations. The authors present a unified approach to deal with these quasilinear PDEs.The book
Download or read book Advances in High Performance Computing written by Ivan Dimov and published by Springer Nature. This book was released on 2020-08-07 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: Every day we need to solve large problems for which supercomputers are needed. High performance computing (HPC) is a paradigm that allows to efficiently implement large-scale computational tasks on powerful supercomputers unthinkable without optimization. We try to minimize our effort and to maximize the achieved profit. Many challenging real world problems arising in engineering, economics, medicine and other areas can be formulated as large-scale computational tasks. The volume is a comprehensive collection of extended contributions from the High performance computing conference held in Borovets, Bulgaria, September 2019. This book presents recent advances in high performance computing. The topics of interest included into this volume are: HP software tools, Parallel Algorithms and Scalability, HPC in Big Data analytics, Modelling, Simulation & Optimization in a Data Rich Environment, Advanced numerical methods for HPC, Hybrid parallel or distributed algorithms. The volume is focused on important large-scale applications like Environmental and Climate Modeling, Computational Chemistry and Heuristic Algorithms.
Download or read book Efficient Parallel Solution Methods for High Dimensional Option Pricing Problems written by Peter Schober and published by . This book was released on 2017 with total page 21 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many problem classes in Finance lead to high-dimensional partial differential equations (PDEs), which need to be solved efficiently. Currently, several methods exist to either circumnavigate the curse of dimensionality or use parallel High Performance Computing to calculate solutions despite it. In Schröder et al. (2013b) the authors present a special class of decomposition techniques to decompose a high-dimensional PDE into a linear combination of independent, low-dimensional PDEs, which can be solved in parallel. We combine this decomposition with the combination technique introduced Griebel et al., 1992 to circumnavigate the curse of dimensionality for these low-dimensional PDEs using sparse grids. The combination technique also allows for a straightforward parallelization of the so-called component grids that are used to construct the solution in the sparse grid space. Therefore, we introduce a two-level parallelization technique, which facilitates the solution of the whole set of low-dimensional PDEs in parallel. For each of these PDEs, we employ the combination technique and compute the solution on the component grids again in parallel.The presented parallelization approach significantly reduces the overall runtime of solution routines for decomposed high-dimensional PDEs. We show strong scalability of our approach, even for problems of very high dimensionality, using basket options on the DAX 30 and the S&P 500 as numerical examples.
Download or read book Space Time Methods written by Ulrich Langer and published by Walter de Gruyter GmbH & Co KG. This book was released on 2019-09-23 with total page 261 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume provides an introduction to modern space-time discretization methods such as finite and boundary elements and isogeometric analysis for time-dependent initial-boundary value problems of parabolic and hyperbolic type. Particular focus is given on stable formulations, error estimates, adaptivity in space and time, efficient solution algorithms, parallelization of the solution pipeline, and applications in science and engineering.
Download or read book Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations written by Beatrice Riviere and published by SIAM. This book was released on 2008-12-18 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focuses on three primal DG methods, covering both theory and computation, and providing the basic tools for analysis.
Download or read book Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives written by National Aeronautics and Space Adm Nasa and published by . This book was released on 2018-09-27 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we review the existing and develop new continuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L(exp 2) stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh. Yan, Jue and Shu, Chi-Wang and Bushnell, Dennis M. (Technical Monitor) Langley Research Center NASA/CR-2002-211959, NAS 1.26:211959, ICASE-2002-42...
Download or read book Superlinear Parabolic Problems written by Pavol Quittner and published by Springer Science & Business Media. This book was released on 2007-12-16 with total page 593 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the qualitative study of solutions of superlinear elliptic and parabolic partial differential equations and systems. This class of problems contains, in particular, a number of reaction-diffusion systems which arise in various mathematical models, especially in chemistry, physics and biology. The book is self-contained and up-to-date, taking special care on the didactical preparation of the material. It is devoted to problems that are intensively studied but have not been treated thus far in depth in the book literature.
Download or read book Parabolic Equations with Irregular Data and Related Issues written by Claude Le Bris and published by Walter de Gruyter GmbH & Co KG. This book was released on 2019-06-17 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book studies the existence and uniqueness of solutions to parabolic-type equations with irregular coefficients and/or initial conditions. It elaborates on the DiPerna-Lions theory of renormalized solutions to linear transport equations and related equations, and also examines the connection between the results on the partial differential equation and the well-posedness of the underlying stochastic/ordinary differential equation.
Download or read book An Exponential Function Approach To Parabolic Equations written by Chin-yuan Lin and published by World Scientific. This book was released on 2014-08-08 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is on initial-boundary value problems for parabolic partial differential equations of second order. It rewrites the problems as abstract Cauchy problems or evolution equations, and then solves them by the technique of elementary difference equations. Because of this, the volume assumes less background and provides an easy approach for readers to understand.
Download or read book Finite Difference Methods for Ordinary and Partial Differential Equations written by Randall J. LeVeque and published by SIAM. This book was released on 2007-01-01 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.
Download or read book The Operator Compact Implicit Method for Parabolic Equations written by Melvyn Ciment and published by . This book was released on 1977 with total page 59 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper attempts to trace out the broad characteristics of a class of higher order finite difference schemes which are applicable to the solution of parabolic partial differential equations associated with viscous fluid flow problems. The basic method developed here uses the approach of the compact implicit techniques applied to the full spatial operator. The resulting spatial approximation, referred to here as the operator compact implicit method can be implemented with a variety of temporal integration schemes. In particular, a simple factorization technique is employed to resolve higher space dimension problems in terms of simple tridiagonal systems. The operator compact implicit method is compared to standard techniques and to some of the newer compact implicit methods. Stability characteristics, computational efficiency and the results of numerical experiments are discussed.
Download or read book The Numerical Solution of the American Option Pricing Problem written by Carl Chiarella and published by World Scientific Publishing Company Incorporated. This book was released on 2014 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: The early exercise opportunity of an American option makes it challenging to price. The Numerical Solution of the American Option Pricing Problem focuses on three numerical methods that have proved useful for the numerical solution of the partial differential equations with free boundary problem arising in American option pricing, namely the method of lines, the sparse grid approach and the integral transform approach. It clearly explains and demonstrates the advantages and limitations of each of them using several examples.
Download or read book High Order Difference Methods for Parabolic Equations written by Daniel Alan Matuska and published by . This book was released on 1971 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt:
