Download or read book Fourth Order Accurate Implicit Finite Difference Method for Evaluating American Options written by International Business Machines Corporation. Research Division and published by . This book was released on 1998 with total page 21 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We present a numerical method for valuing vanilla American options on a single asset that is fourth order accurate in the log of the asset price, and second order accurate in time. The method overcomes the standard difficulty encountered in developing high order accurate finite difference schemes for valuing American options, that is the lack of smoothness in the option price at the critical boundary. To do this we make special corrections to the right hand side of the difference equations near the boundary so they retain their level of accuracy. These corrections are easily evaluated using estimates of the boundary location and jump in the gamma that occurs there, such as those developed by Carr. The method can also be used for evaluating American options depending on more than one asset whenever estimates of the location of the critical boundary are available. Furthermore, the provable error estimates we obtain also allow development of extrapolation techniques. Results of numerical experiments comparing our method with more standard finite difference methods are provided."

Download or read book Tools for Computational Finance written by Rüdiger U. Seydel and published by Springer Science & Business Media. This book was released on 2012-03-09 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: The disciplines of financial engineering and numerical computation differ greatly, however computational methods are used in a number of ways across the field of finance. It is the aim of this book to explain how such methods work in financial engineering; specifically the use of numerical methods as tools for computational finance. By concentrating on the field of option pricing, a core task of financial engineering and risk analysis, this book explores a wide range of computational tools in a coherent and focused manner and will be of use to the entire field of computational finance. Starting with an introductory chapter that presents the financial and stochastic background, the remainder of the book goes on to detail computational methods using both stochastic and deterministic approaches. Now in its fifth edition, Tools for Computational Finance has been significantly revised and contains: A new chapter on incomplete markets which links to new appendices on Viscosity solutions and the Dupire equation; Several new parts throughout the book such as that on the calculation of sensitivities (Sect. 3.7) and the introduction of penalty methods and their application to a two-factor model (Sect. 6.7) Additional material in the field of analytical methods including Kim’s integral representation and its computation Guidelines for comparing algorithms and judging their efficiency An extended chapter on finite elements that now includes a discussion of two-asset options Additional exercises, figures and references Written from the perspective of an applied mathematician, methods are introduced as tools within the book for immediate and straightforward application. A ‘learning by calculating’ approach is adopted throughout this book enabling readers to explore several areas of the financial world. Interdisciplinary in nature, this book will appeal to advanced undergraduate students in mathematics, engineering and other scientific disciplines as well as professionals in financial engineering.

Download or read book Tools for Computational Finance written by Rüdiger Seydel and published by Springer Science & Business Media. This book was released on 2004 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt: This edition contains more material. The largest addition is a new section on jump processes (Section 1.9). The derivation of a related partial integro differential equation is included in Appendix A3. More material is devoted to Monte Carlo simulation. An algorithm for the standard workhorse of in verting the normal distribution is added to Appendix A7. New figures and more exercises are intended to improve the clarity at some places. Several further references give hints on more advanced material and on important developments. Many small changes are hoped to improve the readability of this book. Further I have made an effort to correct misprints and errors that I knew about. A new domain is being prepared to serve the needs of the computational finance community, and to provide complementary material to this book. The address of the domain is www.compfin.de The domain is under construction; it replaces the website address www . mi. uni koeln.de/numerik/compfin/. Suggestions and remarks both on this book and on the domain are most welcome.

Download or read book Frontiers of High Performance Computing and Networking written by Geyong Min and published by Springer Science & Business Media. This book was released on 2006-11-22 with total page 1176 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed joint proceedings of ten international workshops held in conjunction with the 4th International Symposium on Parallel and Distributed Processing and Applications, ISPA 2006, held in Sorrento, Italy in December 2006. It contains 116 papers that contribute to enlarging the spectrum of the more general topics treated in the ISPA 2006 main conference.

Download or read book The Numerical Solution of the American Option Pricing Problem written by Carl Chiarella and published by World Scientific. This book was released on 2014-10-14 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt: The early exercise opportunity of an American option makes it challenging to price and an array of approaches have been proposed in the vast literature on this topic. In The Numerical Solution of the American Option Pricing Problem, Carl Chiarella, Boda Kang and Gunter Meyer focus on two numerical approaches that have proved useful for finding all prices, hedge ratios and early exercise boundaries of an American option. One is a finite difference approach which is based on the numerical solution of the partial differential equations with the free boundary problem arising in American option pricing, including the method of lines, the component wise splitting and the finite difference with PSOR. The other approach is the integral transform approach which includes Fourier or Fourier Cosine transforms. Written in a concise and systematic manner, Chiarella, Kang and Meyer explain and demonstrate the advantages and limitations of each of them based on their and their co-workers'' experiences with these approaches over the years. Contents: Introduction; The Merton and Heston Model for a Call; American Call Options under Jump-Diffusion Processes; American Option Prices under Stochastic Volatility and Jump-Diffusion Dynamics OCo The Transform Approach; Representation and Numerical Approximation of American Option Prices under Heston; Fourier Cosine Expansion Approach; A Numerical Approach to Pricing American Call Options under SVJD; Conclusion; Bibliography; Index; About the Authors. Readership: Post-graduates/ Researchers in finance and applied mathematics with interest in numerical methods for American option pricing; mathematicians/physicists doing applied research in option pricing. Key Features: Complete discussion of different numerical methods for American options; Able to handle stochastic volatility and/or jump diffusion dynamics; Able to produce hedge ratios efficiently and accurately"

Download or read book Computational Sciences Modelling Computing and Soft Computing written by Ashish Awasthi and published by Springer Nature. This book was released on 2021-07-27 with total page 271 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes revised and selected papers of the First International Conference on Computational Sciences - Modelling, Computing and Soft Computing, held in Kozhikode, Kerala, India, in September 2020. The 15 full papers and 6 short papers presented were thoroughly reviewed and selected from the 150 submissions. They are organized in the topical secions on computing; soft computing; general computing; modelling.

Download or read book Proceedings of the Ninth International Conference on Mathematics and Computing written by Debasis Giri and published by Springer Nature. This book was released on 2023-07-31 with total page 433 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book features selected papers from the 9th International Conference on Mathematics and Computing (ICMC 2023), organized at BITS Pilani K. K. Birla Goa Campus, India, during 6–8 January 2023. It covers recent advances in the field of mathematics, statistics, and scientific computing. The book presents innovative work by leading academics, researchers, and experts from industry in mathematics, statistics, cryptography, network security, cybersecurity, machine learning, data analytics, and blockchain technology in computer science and information technology.

Download or read book The Journal of Computational Finance written by and published by . This book was released on 2009 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Adaptive Finite Difference Methods for Valuing American Options written by Duy M. Dang and published by . This book was released on 2007 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: We develop space-time adaptive methods for valuing American options with strong emphasis on American put options. We examine the application of adaptive techniques to the Black-Scholes partial differential equation problem associated with an American put option in the context of non-uniform second-order finite differences. At certain timesteps, we obtain a redistribution of the spatial points based on a monitor function that attempts to equidistribute the error. The proposed finite difference discretization on non-uniform grids and redistribution of the spatial points lead to linear complementarity problems with M-matrices. The Projected Successive Over-relaxation and a penalty method are considered to handle the free boundaries. We study and compare the accuracy and efficiency of the considered methods. A complete proof of convergence and uniqueness of the projected SOR method under certain conditions is also presented.

Download or read book Valuing American Put Options Using Gaussian Quadrature written by Michael A. Sullivan and published by . This book was released on 1999 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper develops an efficient and accurate method for numerical evaluation of the integral equation which defines the American put option value function. Numerical integration using Gaussian quadrature and function approximation using Chebyshev polynomials are combined to evaluate recursive expectations and produce an approximation of the option value function in two dimensions, across stock prices and over time to maturity. A set of such solutions results in a multidimensional approximation that is extremely accurate and very quick to compute. The method is an effective alternative to finite difference methods, the binomial model, and various analytic approximations.

Download or read book Advances in Finance and Stochastics written by Klaus Sandmann and published by Springer Science & Business Media. This book was released on 2013-04-18 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: In many areas of finance and stochastics, significant advances have been made since this field of research was opened by Black, Scholes and Merton in 1973. This volume contains a collection of original articles by a number of highly distinguished authors, on research topics that are currently in the focus of interest of both academics and practitioners.

Download or read book Abstracts of Papers Presented to the American Mathematical Society written by American Mathematical Society and published by . This book was released on 2007 with total page 754 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Numerical Solution of Differential Equations written by Zhilin Li and published by Cambridge University Press. This book was released on 2017-11-30 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: A practical and concise guide to finite difference and finite element methods. Well-tested MATLAB® codes are available online.

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 Numerical Solutions for American Options on Assets with Stochastic Volatilities written by Jinliang Li and published by . This book was released on 2001 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: ABSTRACT: This dissertation discusses American options on assets with stochastic volatilities. First, it gives a proof of the solution uniqueness of the 2-D PDE to evaluate options for both general two-factor model and the model used in this dissertation. Second, it formulates the two factor American option as a 2-D PDE free boundary problem. Third, because the solution of this 2-D PDE free boundary problem is not a very smooth function and the free boundary changes rapidly near maturity, most of the numerical methods could fail to find a reasonable solution or the numerical solution has a large truncation error. Instead of solving this 2-D PDE directly, the difference between the solution of the original 2-D PDE free boundary problem and the solution of a 1-D parabolic equation with the same final condition is calculated. The difference function is very smooth in the entire region. We can solve this new 2-D PDE free boundary problem more accurately and more efficiently. Fourth, this paper uses a coordinate transformation to map the moving boundary to a fixed boundary and applies the Singularity Separating Method (SSM) technique to separate the free boundary and find the exact location of the free boundary (the optimal exercise price). This will be very useful for arbitrage activities. Fifth, it develops numerical methods to solve the new free boundary problem and focuses on the high order implicit finite difference method. It provides several methods to solve the nonlinear system. Sixth, It discovers the put-call symmetry relation between American options in the two factor stochastic volatility model. Seventh, it uses the extrapolation technique to improve the approximation accuracy of the numerical solution. Chapter 5 gives several numerical examples.

Download or read book Finite Difference Methods in Financial Engineering written by Daniel J. Duffy and published by John Wiley & Sons. This book was released on 2013-10-28 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: The world of quantitative finance (QF) is one of the fastest growing areas of research and its practical applications to derivatives pricing problem. Since the discovery of the famous Black-Scholes equation in the 1970's we have seen a surge in the number of models for a wide range of products such as plain and exotic options, interest rate derivatives, real options and many others. Gone are the days when it was possible to price these derivatives analytically. For most problems we must resort to some kind of approximate method. In this book we employ partial differential equations (PDE) to describe a range of one-factor and multi-factor derivatives products such as plain European and American options, multi-asset options, Asian options, interest rate options and real options. PDE techniques allow us to create a framework for modeling complex and interesting derivatives products. Having defined the PDE problem we then approximate it using the Finite Difference Method (FDM). This method has been used for many application areas such as fluid dynamics, heat transfer, semiconductor simulation and astrophysics, to name just a few. In this book we apply the same techniques to pricing real-life derivative products. We use both traditional (or well-known) methods as well as a number of advanced schemes that are making their way into the QF literature: Crank-Nicolson, exponentially fitted and higher-order schemes for one-factor and multi-factor options Early exercise features and approximation using front-fixing, penalty and variational methods Modelling stochastic volatility models using Splitting methods Critique of ADI and Crank-Nicolson schemes; when they work and when they don't work Modelling jumps using Partial Integro Differential Equations (PIDE) Free and moving boundary value problems in QF Included with the book is a CD containing information on how to set up FDM algorithms, how to map these algorithms to C++ as well as several working programs for one-factor and two-factor models. We also provide source code so that you can customize the applications to suit your own needs.

Download or read book Finite Difference Computing with PDEs written by Hans Petter Langtangen and published by Springer. This book was released on 2017-06-21 with total page 522 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is open access under a CC BY 4.0 license. This easy-to-read book introduces the basics of solving partial differential equations by means of finite difference methods. Unlike many of the traditional academic works on the topic, this book was written for practitioners. Accordingly, it especially addresses: the construction of finite difference schemes, formulation and implementation of algorithms, verification of implementations, analyses of physical behavior as implied by the numerical solutions, and how to apply the methods and software to solve problems in the fields of physics and biology.