# Finite Difference Methods in Financial Engineering

A Partial Differential Equation Approach

Author: Daniel J. Duffy

Publisher: John Wiley & Sons

ISBN: 1118856481

Page: 464

View: 6374

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.

# Finite Difference Methods in Financial Engineering

A Partial Differential Equation Approach

Author: Daniel J. Duffy

Publisher: Wiley

ISBN: 9780470858820

Page: 442

View: 5877

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.

# Finite Difference Methods in Financial Engineering

A Partial Differential Equation Approach

Author: Daniel J. Duffy

Publisher: John Wiley & Sons

ISBN: 0470858834

Page: 440

View: 7830

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.

# Derivative Securities and Difference Methods

Author: You-lan Zhu,Xiaonan Wu,I-Liang Chern,Zhi-zhong Sun

Publisher: Springer Science & Business Media

ISBN: 1461473063

Category: Mathematics

Page: 647

View: 4980

This book is mainly devoted to finite difference numerical methods for solving partial differential equations (PDEs) models of pricing a wide variety of financial derivative securities. With this objective, the book is divided into two main parts. In the first part, after an introduction concerning the basics on derivative securities, the authors explain how to establish the adequate PDE boundary value problems for different sets of derivative products (vanilla and exotic options, and interest rate derivatives). For many option problems, the analytic solutions are also derived with details. The second part is devoted to explaining and analyzing the application of finite differences techniques to the financial models stated in the first part of the book. For this, the authors recall some basics on finite difference methods, initial boundary value problems, and (having in view financial products with early exercise feature) linear complementarity and free boundary problems. In each chapter, the techniques related to these mathematical and numerical subjects are applied to a wide variety of financial products. This is a textbook for graduate students following a mathematical finance program as well as a valuable reference for those researchers working in numerical methods in financial derivatives. For this new edition, the book has been updated throughout with many new problems added. More details about numerical methods for some options, for example, Asian options with discrete sampling, are provided and the proof of solution-uniqueness of derivative security problems and the complete stability analysis of numerical methods for two-dimensional problems are added. Review of first edition: “...the book is highly well designed and structured as a textbook for graduate students following a mathematical finance program, which includes Black-Scholes dynamic hedging methodology to price financial derivatives. Also, it is a very valuable reference for those researchers working in numerical methods in financial derivatives, either with a more financial or mathematical background." -- MATHEMATICAL REVIEWS

# Numerical Solution of Partial Differential Equations

An Introduction

Author: K. W. Morton,D. F. Mayers

Publisher: Cambridge University Press

ISBN: 1139443208

Category: Mathematics

Page: N.A

View: 5352

This is the 2005 second edition of a highly successful and well-respected textbook on the numerical techniques used to solve partial differential equations arising from mathematical models in science, engineering and other fields. The authors maintain an emphasis on finite difference methods for simple but representative examples of parabolic, hyperbolic and elliptic equations from the first edition. However this is augmented by new sections on finite volume methods, modified equation analysis, symplectic integration schemes, convection-diffusion problems, multigrid, and conjugate gradient methods; and several sections, including that on the energy method of analysis, have been extensively rewritten to reflect modern developments. Already an excellent choice for students and teachers in mathematics, engineering and computer science departments, the revised text includes more latest theoretical and industrial developments.

# Monte Carlo Frameworks

Building Customisable High-performance C++ Applications

Author: Daniel J. Duffy,Joerg Kienitz

Publisher: John Wiley & Sons

ISBN: 0470684062

Page: 775

View: 4700

This is one of the first books that describe all the steps that are needed in order to analyze, design and implement Monte Carlo applications. It discusses the financial theory as well as the mathematical and numerical background that is needed to write flexible and efficient C++ code using state-of-the art design and system patterns, object-oriented and generic programming models in combination with standard libraries and tools. Includes a CD containing the source code for all examples. It is strongly advised that you experiment with the code by compiling it and extending it to suit your needs. Support is offered via a user forum on www.datasimfinancial.com where you can post queries and communicate with other purchasers of the book. This book is for those professionals who design and develop models in computational finance. This book assumes that you have a working knowledge of C ++.

# Introductory Finite Difference Methods for PDEs

Author: N.A

Publisher: Bookboon

ISBN: 8776816427

Category:

Page: N.A

View: 1288

# A First Course in the Numerical Analysis of Differential Equations

Author: A. Iserles

Publisher: Cambridge University Press

ISBN: 0521734908

Category: Mathematics

Page: 459

View: 8753

lead the reader to a theoretical understanding of the subject without neglecting its practical aspects. The outcome is a textbook that is mathematically honest and rigorous and provides its target audience with a wide range of skills in both ordinary and partial differential equations." --Book Jacket.

# Finite Difference Computing with PDEs

A Modern Software Approach

Author: Hans Petter Langtangen,Svein Linge

Publisher: Springer

ISBN: 3319554565

Category: Computers

Page: 507

View: 5502

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.

# Pricing Financial Instruments

The Finite Difference Method

Author: Domingo Tavella,Curt Randall

Publisher: Wiley

ISBN: 9780471197607

Page: 256

View: 1959

Numerical methods for the solution of financial instrument pricingequations are fast becoming essential for practitioners of modernquantitative finance. Among the most promising of these newcomputational finance techniques is the finite differencemethod-yet, to date, no single resource has presented a quality,comprehensive overview of this revolutionary quantitative approachto risk management. Pricing Financial Instruments, researched and written by DomingoTavella and Curt Randall, two of the chief proponents of the finitedifference method, presents a logical framework for applying themethod of finite difference to the pricing of financialderivatives. Detailing the algorithmic and numerical proceduresthat are the foundation of both modern mathematical finance and thecreation of financial products-while purposely keeping mathematicalcomplexity to a minimum-this long-awaited book demonstrates how thetechniques described can be used to accurately price simple andcomplex derivative structures. From a summary of stochastic pricing processes and arbitragepricing arguments, through the analysis of numerical schemes andthe implications of discretization-and ending with case studiesthat are simple yet detailed enough to demonstrate the capabilitiesof the methodology- Pricing Financial Instruments explores areasthat include: * Pricing equations and the relationship be-tween European andAmerican derivatives * Detailed analyses of different stability analysisapproaches * Continuous and discrete sampling models for path dependentoptions * One-dimensional and multi-dimensional coordinatetransformations * Numerical examples of barrier options, Asian options, forwardswaps, and more With an emphasis on how numerical solutions work and how theapproximations involved affect the accuracy of the solutions,Pricing Financial Instruments takes us through doors opened wide byBlack, Scholes, and Merton-and the arbitrage pricing principlesthey introduced in the early 1970s-to provide a step-by-stepoutline for sensibly interpreting the output of standard numericalschemes. It covers the understanding and application of today'sfinite difference method, and takes the reader to the next level ofpricing financial instruments and managing financial risk. Praise for Pricing Financial Instruments "Pricing Financial Instruments is the first broad and accessibletreatment of finite difference methods for pricing derivativesecurities. The authors have taken great care to clearly explainboth the origins of the pricing problems in a financial setting, aswell as many practical aspects of their numerical methods. The bookcovers a wide variety of applications, such as American options andcredit derivatives. Both financial analysts and academicasset-pricing specialists will want to own a copy."-Darrell Duffie,Professor of Finance Stanford University "In my experience, finite difference methods have proven to be asimple yet powerful tool for numerically solving the evolutionaryPDEs that arise in modern mathematical finance. This book shouldfinally dispel the widely held notion that these methods aresomehow difficult or abstract. I highly recommend it to anyoneinterested in the implementation of these methods in the financialarena."-Peter Carr, Principal Bank of America Securities "A very comprehensive treatment of the application of finitedifference techniques to derivatives finance. Practitioners willfind the many extensive examples very valuable and students willappreciate the rigorous attention paid to the many subtleties offinite difference techniques."-Francis Longstaff, Professor TheAnderson School at UCLA "The finite difference approach is central to the numerical pricingof financial securities. This book gives a clear and succinctintroduction to this important subject. Highly recommended."-MarkBroadie, Associate Professor School of Business, ColumbiaUniversity For updates on new and bestselling Wiley Finance books:wiley.com/wbns

# Computational Partial Differential Equations Using MATLAB

Author: Jichun Li,Yi-Tung Chen

Publisher: CRC Press

ISBN: 9781420089059

Category: Mathematics

Page: 378

View: 1346

This textbook introduces several major numerical methods for solving various partial differential equations (PDEs) in science and engineering, including elliptic, parabolic, and hyperbolic equations. It covers traditional techniques that include the classic finite difference method and the finite element method as well as state-of-the-art numerical methods, such as the high-order compact difference method and the radial basis function meshless method. Helps Students Better Understand Numerical Methods through Use of MATLAB® The authors uniquely emphasize both theoretical numerical analysis and practical implementation of the algorithms in MATLAB, making the book useful for students in computational science and engineering. They provide students with simple, clear implementations instead of sophisticated usages of MATLAB functions. All the Material Needed for a Numerical Analysis Course Based on the authors’ own courses, the text only requires some knowledge of computer programming, advanced calculus, and difference equations. It includes practical examples, exercises, references, and problems, along with a solutions manual for qualifying instructors. Students can download MATLAB code from www.crcpress.com, enabling them to easily modify or improve the codes to solve their own problems.

# Numerical Methods in Finance and Economics

A MATLAB-Based Introduction

Author: Paolo Brandimarte

Publisher: John Wiley & Sons

ISBN: 1118625579

Category: Mathematics

Page: 696

View: 977

A state-of-the-art introduction to the powerful mathematical and statistical tools used in the field of finance The use of mathematical models and numerical techniques is a practice employed by a growing number of applied mathematicians working on applications in finance. Reflecting this development, Numerical Methods in Finance and Economics: A MATLAB?-Based Introduction, Second Edition bridges the gap between financial theory and computational practice while showing readers how to utilize MATLAB?--the powerful numerical computing environment--for financial applications. The author provides an essential foundation in finance and numerical analysis in addition to background material for students from both engineering and economics perspectives. A wide range of topics is covered, including standard numerical analysis methods, Monte Carlo methods to simulate systems affected by significant uncertainty, and optimization methods to find an optimal set of decisions. Among this book's most outstanding features is the integration of MATLAB?, which helps students and practitioners solve relevant problems in finance, such as portfolio management and derivatives pricing. This tutorial is useful in connecting theory with practice in the application of classical numerical methods and advanced methods, while illustrating underlying algorithmic concepts in concrete terms. Newly featured in the Second Edition: * In-depth treatment of Monte Carlo methods with due attention paid to variance reduction strategies * New appendix on AMPL in order to better illustrate the optimization models in Chapters 11 and 12 * New chapter on binomial and trinomial lattices * Additional treatment of partial differential equations with two space dimensions * Expanded treatment within the chapter on financial theory to provide a more thorough background for engineers not familiar with finance * New coverage of advanced optimization methods and applications later in the text Numerical Methods in Finance and Economics: A MATLAB?-Based Introduction, Second Edition presents basic treatments and more specialized literature, and it also uses algebraic languages, such as AMPL, to connect the pencil-and-paper statement of an optimization model with its solution by a software library. Offering computational practice in both financial engineering and economics fields, this book equips practitioners with the necessary techniques to measure and manage risk.

# Finite Difference Methods for Ordinary and Partial Differential Equations

Author: Randall J. LeVeque

Publisher: SIAM

ISBN: 9780898717839

Category: Differential equations

Page: 339

View: 7358

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.

# Numerical Methods and Optimization in Finance

Author: Manfred Gilli,Dietmar Maringer,Enrico Schumann

ISBN: 0123756634

Category: Mathematics

Page: 600

View: 3550

This book describes computational finance tools. It covers fundamental numerical analysis and computational techniques, such as option pricing, and gives special attention to simulation and optimization. Many chapters are organized as case studies around portfolio insurance and risk estimation problems. In particular, several chapters explain optimization heuristics and how to use them for portfolio selection and in calibration of estimation and option pricing models. Such practical examples allow readers to learn the steps for solving specific problems and apply these steps to others. At the same time, the applications are relevant enough to make the book a useful reference. Matlab and R sample code is provided in the text and can be downloaded from the book's website. Shows ways to build and implement tools that help test ideas Focuses on the application of heuristics; standard methods receive limited attention Presents as separate chapters problems from portfolio optimization, estimation of econometric models, and calibration of option pricing models

# Numerical Methods in Finance with C++

Author: Maciej J. Capiński,Tomasz Zastawniak

Publisher: Cambridge University Press

ISBN: 0521177162

Page: 175

View: 2058

Provides aspiring quant developers with the numerical techniques and programming skills needed in quantitative finance. No programming background required.

# Introduction to C++ for Financial Engineers

An Object-Oriented Approach

Author: Daniel J. Duffy

Publisher: John Wiley & Sons

ISBN: 1118856465

Page: 440

View: 609

This book introduces the reader to the C++ programming language and how to use it to write applications in quantitative finance (QF) and related areas. No previous knowledge of C or C++ is required -- experience with VBA, Matlab or other programming language is sufficient. The book adopts an incremental approach; starting from basic principles then moving on to advanced complex techniques and then to real-life applications in financial engineering. There are five major parts in the book: C++ fundamentals and object-oriented thinking in QF Advanced object-oriented features such as inheritance and polymorphism Template programming and the Standard Template Library (STL) An introduction to GOF design patterns and their applications in QF Applications The kinds of applications include binomial and trinomial methods, Monte Carlo simulation, advanced trees, partial differential equations and finite difference methods. This book includes a companion website with all source code and many useful C++ classes that you can use in your own applications. Examples, test cases and applications are directly relevant to QF. This book is the perfect companion to Daniel J. Duffy’s book Financial Instrument Pricing using C++ (Wiley 2004, 0470855096 / 9780470021620)

# Financial Instrument Pricing Using C++

Author: Daniel J. Duffy

Publisher: John Wiley & Sons

ISBN: 1119170494

Page: 1168

View: 4819

# Finite Difference Computing with Exponential Decay Models

Author: Hans Petter Langtangen

Publisher: Springer

ISBN: 3319294393

Category: Computers

Page: 200

View: 955

This text provides a very simple, initial introduction to the complete scientific computing pipeline: models, discretization, algorithms, programming, verification, and visualization. The pedagogical strategy is to use one case study – an ordinary differential equation describing exponential decay processes – to illustrate fundamental concepts in mathematics and computer science. The book is easy to read and only requires a command of one-variable calculus and some very basic knowledge about computer programming. Contrary to similar texts on numerical methods and programming, this text has a much stronger focus on implementation and teaches testing and software engineering in particular.

# Computational Methods in Finance

Author: Ali Hirsa

Publisher: CRC Press

ISBN: 1439829578

Page: 444

View: 2867

As today’s financial products have become more complex, quantitative analysts, financial engineers, and others in the financial industry now require robust techniques for numerical analysis. Covering advanced quantitative techniques, Computational Methods in Finance explains how to solve complex functional equations through numerical methods. The first part of the book describes pricing methods for numerous derivatives under a variety of models. The book reviews common processes for modeling assets in different markets. It then examines many computational approaches for pricing derivatives. These include transform techniques, such as the fast Fourier transform, the fractional fast Fourier transform, the Fourier-cosine method, and saddlepoint method; the finite difference method for solving PDEs in the diffusion framework and PIDEs in the pure jump framework; and Monte Carlo simulation. The next part focuses on essential steps in real-world derivative pricing. The author discusses how to calibrate model parameters so that model prices are compatible with market prices. He also covers various filtering techniques and their implementations and gives examples of filtering and parameter estimation. Developed from the author’s courses at Columbia University and the Courant Institute of New York University, this self-contained text is designed for graduate students in financial engineering and mathematical finance as well as practitioners in the financial industry. It will help readers accurately price a vast array of derivatives.

# Numerical Solution of Partial Differential Equations in Science and Engineering

Author: Leon Lapidus,George F. Pinder

Publisher: Wiley-Interscience

ISBN: N.A

Category: Mathematics

Page: 677

View: 2137

"This book was written to provide a text for graduate and undergraduate students who took our courses in numerical methods. It incorporates the essential elements of all the numerical methods currently used extensively in the solution of partial differential equations encountered regularly in science and engineering. Because our courses were typically populated by students from varied backgrounds and with diverse interests, we attempted to eliminate jargon or nomenclature that would render the work unintelligible to any student. Moreover, in response to student needs, we incorporated not only classical (and not so classical) finite-difference methods but also finite-element, collocation, and boundary-element procedures. After an introduction to the various numerical schemes, each equation type--parabolic, elliptic, and hyperbolic--is allocated a separate chapter. Within each of these chapters the material is presented by numerical method. Thus one can read the book either by equation-type or numerical approach."--Preface, page [v].