Pricing Financial Instruments: The Finite Difference Method [Hardcover]

Domingo Tavella (Author), Curt Randall (Author)

Book Description

ISBN-10: 0471197602 | ISBN-13: 978-0471197607 | Publication Date: April 15, 2000 | Edition: 1

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 be-tween 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.

Praise for Pricing Financial Instruments

"Pricing Financial Instruments is the first broad and accessible treatment of finite difference methods for pricing derivative securities. The authors have taken great care to clearly explain both the origins of the pricing problems in a financial setting, as well as many practical aspects of their numerical methods. The book covers a wide variety of applications, such as American options and credit derivatives. Both financial analysts and academic asset-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 a simple yet powerful tool for numerically solving the evolutionary PDEs that arise in modern mathematical finance. This book should finally dispel the widely held notion that these methods are somehow difficult or abstract. I highly recommend it to anyone interested in the implementation of these methods in the financial arena."-Peter Carr, Principal Bank of America Securities

"A very comprehensive treatment of the application of finite difference techniques to derivatives finance. Practitioners will find the many extensive examples very valuable and students will appreciate the rigorous attention paid to the many subtleties of finite difference techniques."-Francis Longstaff, Professor The Anderson School at UCLA

"The finite difference approach is central to the numerical pricing of financial securities. This book gives a clear and succinct introduction to this important subject. Highly recommended."-Mark Broadie, Associate Professor School of Business, Columbia University

For updates on new and bestselling Wiley Finance books: wiley.com/wbns

Editorial Reviews

From the Inside Flap

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.

From the Back Cover

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 be-tween 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.

Praise for Pricing Financial Instruments

"Pricing Financial Instruments is the first broad and accessible treatment of finite difference methods for pricing derivative securities. The authors have taken great care to clearly explain both the origins of the pricing problems in a financial setting, as well as many practical aspects of their numerical methods. The book covers a wide variety of applications, such as American options and credit derivatives. Both financial analysts and academic asset-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 a simple yet powerful tool for numerically solving the evolutionary PDEs that arise in modern mathematical finance. This book should finally dispel the widely held notion that these methods are somehow difficult or abstract. I highly recommend it to anyone interested in the implementation of these methods in the financial arena."-Peter Carr, Principal Bank of America Securities

"A very comprehensive treatment of the application of finite difference techniques to derivatives finance. Practitioners will find the many extensive examples very valuable and students will appreciate the rigorous attention paid to the many subtleties of finite difference techniques."-Francis Longstaff, Professor The Anderson School at UCLA

"The finite difference approach is central to the numerical pricing of financial securities. This book gives a clear and succinct introduction to this important subject. Highly recommended."-Mark Broadie, Associate Professor School of Business, Columbia University

For updates on new and bestselling Wiley Finance books: wiley.com/wbns

See all Editorial Reviews

Product Details

• Hardcover: 237 pages
• Publisher: Wiley; 1 edition (April 15, 2000)
• Language: English
• ISBN-10: 0471197602
• ISBN-13: 978-0471197607
• Product Dimensions: 9.3 x 6.3 x 0.9 inches
• Shipping Weight: 1.1 pounds (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #1,118,542 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

62 of 66 people found the following review helpful:

5.0 out of 5 stars A focussed book on an important subject, May 9, 2000

By 

Peter Carr, PhD. (New York, NY) - See all my reviews

This review is from: Pricing Financial Instruments: The Finite Difference Method (Hardcover)

The pricing theory due to Black Scholes and Merton is widely recognized as one of the most significant contributions of economics to practice. There are now many good introductory books surveying the vast literature on the subject. So what is needed is a book showing how to implement various models in practice. The book by Tavella and Randall is the first in what I hope is a series of such books. The authors are well known in computational circles: Tavella is the editor of the highly regarded Journal of Computational Finance and Randall is a Principal at SciComp, a leading developer of software synthesis technology for the finance industry. The authors focus on finite differences, which is an important generalization of the common tree approach, and thus is one of the most widely used numerical techniques in finance.

The book's first two chapters introduce the mathematics of financial derivatives in an intuitively appealing manner. For example, measure changes are introduced as a powerful tool without strong demands on the reader in terms of background. Also, linear complementarity is used in the context of valuing American options, again in an intuitive fashion. The third chapter introduces finite differences in the context of the familiar parabolic PDE governing derivative security values. It is in this and the remaining chapters that much useful material can be found. For example, the cell averaging technique in chapter 4 is a very useful device for dampening the error introduced by slope discontinuities which commonly occur in financial problems. The authors also give the first textbook treatment of the important class of pure jump models such as the variance gamma model, which are growing in popularity. The chapter on coordinate transformations gives the finite difference version of what some finance people term the adaptive mesh method.

In summary, this book is a must-read for anyone seriously interested in implementing derivative security pricing models. I hope the authors plan to follow up with a more advanced version which can cover such interesting topics as multi-grid, hopskotch, operator splitting, and the like.

19 of 19 people found the following review helpful:

5.0 out of 5 stars A clear treatment, with well-chosen subjects, January 15, 2001

By 

Brian Boonstra (Chicago, IL USA) - See all my reviews
(REAL NAME)   

This review is from: Pricing Financial Instruments: The Finite Difference Method (Hardcover)

Tavella and Randall have produced a compact, yet complete treatment of finite difference techniques in finance. I met Curt Randall in 1996 when his SciFinance software was in its infancy (though there is currently no connection between us). This software automatically generates C code to solve PDE's. That is an order of magnitude -- maybe two orders -- harder than just writing the code by hand. I inferred that Dr. Randall has a unique understanding of finite-difference methods for solving PDE's. For these reasons, I was very interested to see his book. For a more general treatment finite difference schemes, see Gordon Smith's 1986 book.

The mathematical motivations for all the techniques presented are given, with no wasted exposition. I liked the lucid analyses of stability, which many books in finance gloss over. I also liked the mention and partial analysis of a large set of solvers of sparse linear systems. having not followed the literature on jump processes in recent years, I was quite happy to see their treatment as well.

This book is all of what it claims to be, and no more. I do not recommend it as a textbook, or as a reference for those not already somewhat familiar with the subject, either from the mathematics side or the finance side. You will not get an explanation of what an eigenvalue or fourier transform is. The Lax Equivalence Theorem is cited, but not motivated or proven. No mention is given of when it might make more sense to use, say, a Monte Carlo scheme to find an option price. You won't find much economics in the book. But you will find a clear, correct, and useful analysis of more or less all aspects of finite difference schemes as they relate to solving contingent claims pricing problems.

8 of 8 people found the following review helpful:

3.0 out of 5 stars A specialised book for special Instruments, May 9, 2004

By A Customer

This review is from: Pricing Financial Instruments: The Finite Difference Method (Hardcover)

This book approximates the solution of one-factor and multi-factor PDEs that describe derivatives such as barrier options, convertible bonds, Asian options and credit derivatives.
Standard finite difference schemes are used. In particular, 3-point centred difference schemes approximate the derivatives in the S directions while Crank-Nicolson (averaging) is used to approximate the t derivative. Stability and convergence of the schemes are proved using the Lax Equivalence theoerem. Special attention is paid to resolving the, by now well known problems associated with the Crank Nicolson method. The workarounds are choosing smaller meshes near discontinuous boundaries, coordinate transformations and choosing the right sampling points.
The book is a good attempt (in my opinion) to show how to apply FDM in financial engineering applications. It is probably most useful for those who have already experience of FDM. It is NOT an introductory book.
Some of the criticisms are (this is why I give it a 3 star):

1. The von Neumann stability analyis technique is only applicable to constant coefficient, linear PDE. It is outdated, better methods being the maximum principle and viscosity solutions.

2. The discrete set of equations need to be solved by rather esoteric matrix solvers bacause the authors discretise a PDE in all directions. Using ADI or operator splitting instead lets us solve one-dimensional problem with Tridiagonal LU Decomposition.

3. A lot of detail on meshes has unfortunately been left out.

4. Using Crank Nicolson only aggrevates the problems in FDM schemes. There ARE better methods out there.

5. TYPOS!! for example, equation (4.13) on page 122. The S term is missing.

On the other hand, this book is aimed at real-life problems. However, extra detail needs to be added in my opinion in order to make it more accessible to a wider audience.