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Rating: 
Summary: Excellent Content - Sloppy Editing
Review: Reading this book produced three instances where previously encountered material was explained from a new and different point of view, the clouds parted, and I was bathed in the bright sunshine of understanding. Creation and valuation of hedging portfolios, reduced to their fundamentals, is now far less taxing to the memory. Explanations for the discretization of time and space now render previously mysterious numerical algorithms obvious in their intent. American options discused in a free boundry framework are far more intuitive than in the optimal stoping time approach. As much as I enjoyed the content, the occurence of numerous editing oversights was a considerable annoyance. This was even more surprising when I read the other customer review which indicated that one of the authors was an editor. Editors should be held to a higher standard and so I deducted an extra star!
Rating: 
Summary: A clear treatment, with well-chosen subjects
Review: 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.
Rating:
Summary: A clear treatment, with well-chosen subjects
Review: 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.
Rating:
Summary: A focussed book on an important subject
Review: 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.
Rating:
Summary: A specialised book for special Instruments
Review: 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.
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