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: 5706

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

Finite Difference Methods in Financial Engineering

A Partial Differential Equation Approach

Author: Daniel J. Duffy

Publisher: John Wiley & Sons

ISBN: 1118856481

Category: Business & Economics

Page: 464

View: 4530

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.

Pricing Financial Instruments

The Finite Difference Method

Author: Domingo Tavella,Curt Randall

Publisher: Wiley

ISBN: 9780471197607

Category: Business & Economics

Page: 256

View: 1068

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

A Course in Derivative Securities

Introduction to Theory and Computation

Author: Kerry Back

Publisher: Springer Science & Business Media

ISBN: 3540279008

Category: Business & Economics

Page: 356

View: 7308

"Deals with pricing and hedging financial derivatives.... Computational methods are introduced and the text contains the Excel VBA routines corresponding to the formulas and procedures described in the book. This is valuable since computer simulation can help readers understand the theory....The book...succeeds in presenting intuitively advanced derivative modelling... it provides a useful bridge between introductory books and the more advanced literature." --MATHEMATICAL REVIEWS

Computational Methods in Finance

Author: Ali Hirsa

Publisher: CRC Press

ISBN: 1466576049

Category: Business & Economics

Page: 444

View: 3962

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 Methods in Finance with C++

Author: Maciej J. Capiński,Tomasz Zastawniak

Publisher: Cambridge University Press

ISBN: 1139536273

Category: Business & Economics

Page: N.A

View: 6174

Driven by concrete computational problems in quantitative finance, this book provides aspiring quant developers with the numerical techniques and programming skills they need. The authors start from scratch, so the reader does not need any previous experience of C++. Beginning with straightforward option pricing on binomial trees, the book gradually progresses towards more advanced topics, including nonlinear solvers, Monte Carlo techniques for path-dependent derivative securities, finite difference methods for partial differential equations, and American option pricing by solving a linear complementarity problem. Further material, including solutions to all exercises and C++ code, is available online. The book is ideal preparation for work as an entry-level quant programmer and it gives readers the confidence to progress to more advanced skill sets involving C++ design patterns as applied in finance.

Implementing Models in Quantitative Finance: Methods and Cases

Author: Gianluca Fusai,Andrea Roncoroni

Publisher: Springer Science & Business Media

ISBN: 9783540499596

Category: Business & Economics

Page: 607

View: 1203

This book puts numerical methods in action for the purpose of solving practical problems in quantitative finance. The first part develops a toolkit in numerical methods for finance. The second part proposes twenty self-contained cases covering model simulation, asset pricing and hedging, risk management, statistical estimation and model calibration. Each case develops a detailed solution to a concrete problem arising in applied financial management and guides the user towards a computer implementation. The appendices contain "crash courses" in VBA and Matlab programming languages.

High-Performance Computing in Finance

Problems, Methods, and Solutions

Author: M. A. H. Dempster,Juho Kanniainen,John Keane,Erik Vynckier

Publisher: CRC Press

ISBN: 1315354691

Category: Computers

Page: 614

View: 5357

High-Performance Computing (HPC) delivers higher computational performance to solve problems in science, engineering and finance. There are various HPC resources available for different needs, ranging from cloud computing– that can be used without much expertise and expense – to more tailored hardware, such as Field-Programmable Gate Arrays (FPGAs) or D-Wave’s quantum computer systems. High-Performance Computing in Finance is the first book that provides a state-of-the-art introduction to HPC for finance, capturing both academically and practically relevant problems.

Quantitative Modeling of Derivative Securities

From Theory To Practice

Author: Peter Laurence

Publisher: Routledge

ISBN: 1351420461

Category: Mathematics

Page: 336

View: 4358

Quantitative Modeling of Derivative Securities demonstrates how to take the basic ideas of arbitrage theory and apply them - in a very concrete way - to the design and analysis of financial products. Based primarily (but not exclusively) on the analysis of derivatives, the book emphasizes relative-value and hedging ideas applied to different financial instruments. Using a ""financial engineering approach,"" the theory is developed progressively, focusing on specific aspects of pricing and hedging and with problems that the technical analyst or trader has to consider in practice. More than just an introductory text, the reader who has mastered the contents of this one book will have breached the gap separating the novice from the technical and research literature.

The Mathematics of Derivatives Securities with Applications in MATLAB

Author: Mario Cerrato

Publisher: John Wiley & Sons

ISBN: 1119973414

Category: Business & Economics

Page: 248

View: 5355

Quantitative Finance is expanding rapidly. One of the aspects of the recent financial crisis is that, given the complexity of financial products, the demand for people with high numeracy skills is likely to grow and this means more recognition will be given to Quantitative Finance in existing and new course structures worldwide. Evidence has suggested that many holders of complex financial securities before the financial crisis did not have in-house experts or rely on a third-party in order to assess the risk exposure of their investments. Therefore, this experience shows the need for better understanding of risk associate with complex financial securities in the future. The Mathematics of Derivative Securities with Applications in MATLAB provides readers with an introduction to probability theory, stochastic calculus and stochastic processes, followed by discussion on the application of that knowledge to solve complex financial problems such as pricing and hedging exotic options, pricing American derivatives, pricing and hedging under stochastic volatility and an introduction to interest rates modelling. The book begins with an overview of MATLAB and the various components that will be used alongside it throughout the textbook. Following this, the first part of the book is an in depth introduction to Probability theory, Stochastic Processes and Ito Calculus and Ito Integral. This is essential to fully understand some of the mathematical concepts used in the following part of the book. The second part focuses on financial engineering and guides the reader through the fundamental theorem of asset pricing using the Black and Scholes Economy and Formula, Options Pricing through European and American style options, summaries of Exotic Options, Stochastic Volatility Models and Interest rate Modelling. Topics covered in this part are explained using MATLAB codes showing how the theoretical models are used practically. Authored from an academic’s perspective, the book discusses complex analytical issues and intricate financial instruments in a way that it is accessible to postgraduate students with or without a previous background in probability theory and finance. It is written to be the ideal primary reference book or a perfect companion to other related works. The book uses clear and detailed mathematical explanation accompanied by examples involving real case scenarios throughout and provides MATLAB codes for a variety of topics.

Option Prices as Probabilities

A New Look at Generalized Black-Scholes Formulae

Author: Christophe Profeta,Bernard Roynette,Marc Yor

Publisher: Springer Science & Business Media

ISBN: 9783642103957

Category: Mathematics

Page: 270

View: 6216

Discovered in the seventies, Black-Scholes formula continues to play a central role in Mathematical Finance. We recall this formula. Let (B ,t? 0; F ,t? 0, P) - t t note a standard Brownian motion with B = 0, (F ,t? 0) being its natural ?ltra- 0 t t tion. Let E := exp B? ,t? 0 denote the exponential martingale associated t t 2 to (B ,t? 0). This martingale, also called geometric Brownian motion, is a model t to describe the evolution of prices of a risky asset. Let, for every K? 0: + ? (t) :=E (K?E ) (0.1) K t and + C (t) :=E (E?K) (0.2) K t denote respectively the price of a European put, resp. of a European call, associated with this martingale. Let N be the cumulative distribution function of a reduced Gaussian variable: x 2 y 1 ? 2 ? N (x) := e dy. (0.3) 2? ?? The celebrated Black-Scholes formula gives an explicit expression of? (t) and K C (t) in terms ofN : K ? ? log(K) t log(K) t ? (t)= KN ? + ?N ? ? (0.4) K t 2 t 2 and ? ?

Computational Methods for Quantitative Finance

Finite Element Methods for Derivative Pricing

Author: Norbert Hilber,Oleg Reichmann,Christoph Schwab,Christoph Winter

Publisher: Springer Science & Business Media

ISBN: 3642354017

Category: Mathematics

Page: 299

View: 2081

Many mathematical assumptions on which classical derivative pricing methods are based have come under scrutiny in recent years. The present volume offers an introduction to deterministic algorithms for the fast and accurate pricing of derivative contracts in modern finance. This unified, non-Monte-Carlo computational pricing methodology is capable of handling rather general classes of stochastic market models with jumps, including, in particular, all currently used Lévy and stochastic volatility models. It allows us e.g. to quantify model risk in computed prices on plain vanilla, as well as on various types of exotic contracts. The algorithms are developed in classical Black-Scholes markets, and then extended to market models based on multiscale stochastic volatility, to Lévy, additive and certain classes of Feller processes. This book is intended for graduate students and researchers, as well as for practitioners in the fields of quantitative finance and applied and computational mathematics with a solid background in mathematics, statistics or economics.​

Pricing Financial Instruments

The Finite Difference Method

Author: Domingo Tavella,Curt Randall

Publisher: John Wiley & Sons

ISBN: 9780471197607

Category: Business & Economics

Page: 256

View: 6950

Pricing Financial Instruments Numerical methods for the solution of financial instrument pricing equations are fast becoming essential for practitioners of modern quantitative finance. Among the most promising of these new computational finance techniques is the finite difference method-yet, to date, no single resource has presented a quality, comprehensive overview of this revolutionary quantitative approach to risk management. Pricing Financial Instruments, researched and written by Domingo Tavella and Curt Randall, two of the chief proponents of the finite difference method, presents a logical framework for applying the method of finite difference to the pricing of financial derivatives. Detailing the algorithmic and numerical procedures that are the foundation of both modern mathematical finance and the creation of financial products-while purposely keeping mathematical complexity to a minimum-this long-awaited book demonstrates how the techniques described can be used to accurately price simple and complex derivative structures. From a summary of stochastic pricing processes and arbitrage pricing arguments, through the analysis of numerical schemes and the implications of discretization-and ending with case studies that are simple yet detailed enough to demonstrate the capabilities of the methodology-Pricing Financial Instruments explores areas that include: * Pricing equations and the relationship between European and American derivatives * Detailed analyses of different stability analysis approaches * Continuous and discrete sampling models for path dependent options * One-dimensional and multi-dimensional coordinate transformations * Numerical examples of barrier options, Asian options, forward swaps, and more With an emphasis on how numerical solutions work and how the approximations involved affect the accuracy of the solutions, Pricing Financial Instruments takes us through doors opened wide by Black, Scholes, and Merton-and the arbitrage pricing principles they introduced in the early 1970s-to provide a step-by-step outline for sensibly interpreting the output of standard numerical schemes. It covers the understanding and application of today's finite difference method, and takes the reader to the next level of pricing financial instruments and managing financial risk.

Mathematical Techniques in Finance

Tools for Incomplete Markets

Author: Ales Cerný

Publisher: Princeton University Press

ISBN: 1400831482

Category: Business & Economics

Page: 416

View: 4248

Originally published in 2003, Mathematical Techniques in Finance has become a standard textbook for master's-level finance courses containing a significant quantitative element while also being suitable for finance PhD students. This fully revised second edition continues to offer a carefully crafted blend of numerical applications and theoretical grounding in economics, finance, and mathematics, and provides plenty of opportunities for students to practice applied mathematics and cutting-edge finance. Ales Cerný mixes tools from calculus, linear algebra, probability theory, numerical mathematics, and programming to analyze in an accessible way some of the most intriguing problems in financial economics. The textbook is the perfect hands-on introduction to asset pricing, optimal portfolio selection, risk measurement, and investment evaluation. The new edition includes the most recent research in the area of incomplete markets and unhedgeable risks, adds a chapter on finite difference methods, and thoroughly updates all bibliographic references. Eighty figures, over seventy examples, twenty-five simple ready-to-run computer programs, and several spreadsheets enhance the learning experience. All computer codes have been rewritten using MATLAB and online supplementary materials have been completely updated. A standard textbook for graduate finance courses Introduction to asset pricing, portfolio selection, risk measurement, and investment evaluation Detailed examples and MATLAB codes integrated throughout the text Exercises and summaries of main points conclude each chapter

Topics in Numerical Methods for Finance

Author: Mark Cummins,Finbarr Murphy,John J.H. Miller

Publisher: Springer Science & Business Media

ISBN: 1461434335

Category: Mathematics

Page: 204

View: 1054

Presenting state-of-the-art methods in the area, the book begins with a presentation of weak discrete time approximations of jump-diffusion stochastic differential equations for derivatives pricing and risk measurement. Using a moving least squares reconstruction, a numerical approach is then developed that allows for the construction of arbitrage-free surfaces. Free boundary problems are considered next, with particular focus on stochastic impulse control problems that arise when the cost of control includes a fixed cost, common in financial applications. The text proceeds with the development of a fear index based on equity option surfaces, allowing for the measurement of overall fear levels in the market. The problem of American option pricing is considered next, applying simulation methods combined with regression techniques and discussing convergence properties. Changing focus to integral transform methods, a variety of option pricing problems are considered. The COS method is practically applied for the pricing of options under uncertain volatility, a method developed by the authors that relies on the dynamic programming principle and Fourier cosine series expansions. Efficient approximation methods are next developed for the application of the fast Fourier transform for option pricing under multifactor affine models with stochastic volatility and jumps. Following this, fast and accurate pricing techniques are showcased for the pricing of credit derivative contracts with discrete monitoring based on the Wiener-Hopf factorisation. With an energy theme, a recombining pentanomial lattice is developed for the pricing of gas swing contracts under regime switching dynamics. The book concludes with a linear and nonlinear review of the arbitrage-free parity theory for the CDS and bond markets.

The Econometrics of Financial Markets

Author: John Y. Campbell,Andrew W. Lo,A. Craig MacKinlay

Publisher: Princeton University Press

ISBN: 1400830214

Category: Business & Economics

Page: 632

View: 325

The past twenty years have seen an extraordinary growth in the use of quantitative methods in financial markets. Finance professionals now routinely use sophisticated statistical techniques in portfolio management, proprietary trading, risk management, financial consulting, and securities regulation. This graduate-level textbook is intended for PhD students, advanced MBA students, and industry professionals interested in the econometrics of financial modeling. The book covers the entire spectrum of empirical finance, including: the predictability of asset returns, tests of the Random Walk Hypothesis, the microstructure of securities markets, event analysis, the Capital Asset Pricing Model and the Arbitrage Pricing Theory, the term structure of interest rates, dynamic models of economic equilibrium, and nonlinear financial models such as ARCH, neural networks, statistical fractals, and chaos theory. Each chapter develops statistical techniques within the context of a particular financial application. This exciting new text contains a unique and accessible combination of theory and practice, bringing state-of-the-art statistical techniques to the forefront of financial applications. Each chapter also includes a discussion of recent empirical evidence, for example, the rejection of the Random Walk Hypothesis, as well as problems designed to help readers incorporate what they have read into their own applications.

Monte Carlo Methods in Financial Engineering

Author: Paul Glasserman

Publisher: Springer Science & Business Media

ISBN: 0387216170

Category: Mathematics

Page: 596

View: 3449

From the reviews: "Paul Glasserman has written an astonishingly good book that bridges financial engineering and the Monte Carlo method. The book will appeal to graduate students, researchers, and most of all, practicing financial engineers [...] So often, financial engineering texts are very theoretical. This book is not." --Glyn Holton, Contingency Analysis

Financial Derivatives

Pricing and Risk Management

Author: Robert W. Kolb,James A. Overdahl

Publisher: John Wiley & Sons

ISBN: 0470499109

Category: Business & Economics

Page: 600

View: 8383

Essential insights on the various aspects of financialderivatives If you want to understand derivatives without getting boggeddown by the mathematics surrounding their pricing and valuation,Financial Derivatives is the book for you. Through in-depthinsights gleaned from years of financial experience, Robert Kolband James Overdahl clearly explain what derivatives are and how youcan prudently use them within the context of your underlyingbusiness activities. Financial Derivatives introduces you to the wide range ofmarkets for financial derivatives. This invaluable guide offers abroad overview of the different types of derivatives-futures,options, swaps, and structured products-while focusing on theprinciples that determine market prices. This comprehensiveresource also provides a thorough introduction to financialderivatives and their importance to risk management in a corporatesetting. Filled with helpful tables and charts, FinancialDerivatives offers a wealth of knowledge on futures, options,swaps, financial engineering, and structured products. Discusses what derivatives are and how you can prudentlyimplement them within the context of your underlying businessactivities Provides thorough coverage of financial derivatives and theirrole in risk management Explores financial derivatives without getting bogged down bythe mathematics surrounding their pricing and valuation This informative guide will help you unlock the incrediblepotential of financial derivatives.