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Nonlinear Optimization with Financial Applications

Kluwer Academic Publishers
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In the past few years the growth of the field of mathematical finance has led to the publication of many new books. This is but one addition to the — now long — list of mathematical finance books. There are many books in mathematical finance that contain chapters on nonlinear optimization and how it could be applied to financial problems. To say the least, that's usually not enough. So here is a whole book focused on achieving that goal. Nonlinear Optimization with Financial Applications is oriented towards the application of nonlinear optimization methods to financial problems such as portfolio optimization.

Just glancing over the table of contents we can note the large number of topics covered. This is mainly a book on numerical methods for nonlinear optimization. The application to portfolio theory is covered at the level required by a practitioner using the material in real world applications. For students the book is really accessible: it provides a base for intuition, general theory (no sophisticated mathematical exposition of theory and proofs), and application of the methods presented. This is exactly what the author is aiming for, and will suit undergraduates, postgraduates and practitioners who need a basic knowledge of nonlinear optimization. As such the book is very useful. No specific background in numerical analysis or optimization is needed.

Chapters 1 and 2 are quite introductory: the author tries to build simple finance intuition on portfolio theory with two or three assets. Focusing on the asset portfolio example, simple concepts of optimization are covered. Chapter 3 replicates calculations from previous chapters but here the author considers portfolios with N assets. Chapter 4 gives the appropriate background of unconstrained optimization problems with nvariables. The algorithms for solving such problems are given in the following four chapters, on the steepest descent method, the Newton method, Quasi-Newton methods, and conjugate gradient methods respectively. For each of the methods the theory, numerical algorithm, and numerical results are presented along with several worked examples.

Chapter 9 considers optimal portfolios with restrictions such as short-selling. Chapter 10 considers problems of optimization in optimal portfolios with an increasing number of variables (assets) and time variation. Solutions for such problems are explained further in chapter 11 where data-fitting and the Gauss-Newton method are considered. Time series analysis is also considered as a least squares solutions and as a Gauss-Newton solution. Chapter 12 re-expresses the unconstrained problems presented so far in the book, as constrained optimization problems. A general discussion of constrained problems is given along with several examples for the purposes of intuition. Additional methods for problem solving are presented in chapters 13-15 which cover reduced gradient methods, project gradient methods, penalty function methods, and sequential quadratic programming.

Chapter 16 considers additional problems in portfolio theory such as transaction costs, re-balancing, and sensitivity. Chapters 17-21 provide the background and algorithms for inequality constrained optimization. General theoretical aspects with some worked examples are provided in chapter 17, while the other chapters cover numerical computation and examples. There are quite a number of worked examples which really enable easier understanding of the concepts.

Chapter 22 covers re-balancing in a more detailed manner than what was considered in prior parts of the book. The book ends with a chapter on global unconstrained optimization and how it can be used in portfolio theory.

The book is written in a very easy-to-read style which is extremely suitable for an undergraduate student. For postgraduate students or experienced practitioners this will be light reading. Exercises are provided at the end of each section, which makes it easy to follow the topics covered. Some exercises ask for proofs, while some are applications-based. The book contains numerous computational examples demonstrating the methods covered, and the author provides software for the computational examples. The software is useful if one wants to examine the numerical calculations in detail but it is not required for sound understanding of the text.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She holds a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance, more precisely in statistical mehods for credit and market risk. Apart from the academic work, she works in consulting for financial institutions.

Date Received: 
Saturday, January 1, 2005
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Michael Bartholomew-Biggs
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Ita Cirovic Donev

Portfolio Optimization - One-variable optimization - Optimal portfolios with n assets - Unconstrained optimization in n variables - The steepest descent method - The Newton method - Quasi-Newton methods - The conjugate gradient method - Optimal portfolios with restrictions - Larger-scale portfolio problems - Data-fitting and the Gauss-Newton method - Equality constrained optimization - Methods for linear equality constraints - Penalty function methods - Sequential quadratic programming - Further portfolio problems - Inequality constrained optimization - Extending equality-constraint methods to inequalities - Barrier function methods - Interior point methods - Data-fitting using inequality constraints - Portfolio re-balancing - Global unconstrained optimization

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Monday, August 29, 2005