This is a textbook on optimization and its applications in mathematical finance intended for use by students in MS programs in financial engineering. The book presents a wide variety of optimization methods in chapters that alternate with chapters on applications of the methods to problems in finance.
Optimization methods include linear programming (by both the simplex method and interior point methods), quadratic programming, nonlinear programming, integer programming, dynamic programming, stochastic programming, conic programming, and robust optimization. One notable omission with applications in portfolio optimization is mixed integer quadratic programming. Whole books have been written on each of these topics, so the presentation is necessarily concise. Students without previous exposure to optimization from an introductory operations research course are likely to be bewildered by the many methods presented here.
Applications of the optimization methods to finance include asset pricing and arbitrage, portfolio optimization, option pricing, asset-back securities, combinatorial auctions, and value at risk. Many of these applications, such as the use of quadratic programming in portfolio optimization, are well established. The authors have also presented some new and very interesting material on applications of conic optimization including the approximation of covariance matrices and robust optimization of financial problems with uncertain data.
The mathematical level of this book is appropriate for its intended audience with many examples and relatively few formal definitions and theorems. Most theorems are given without proof. Some of the exercises require the reader to use the mathematical theory while other exercises are computational. The authors provide information about software packages for solving the different types of optimization problems although none is provided with the book. Students are expected to find and use appropriate software in many of the exercises.
This book will be a useful textbook for students in financial engineering at the MS level. However, students will need either previous background in optimization or additional resources on optimization. There is also too much material in this book to reasonably cover in a one semester course. Instructors will have to carefully select the material to be presented. The book will also be of interest to researchers and graduate students in optimization who are interested in applications of optimization to financial problems.
Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems.
1. Introduction; 2. Linear programming: theory and algorithms; 3. LP models: asset/liability cash flow matching; 4. LP models: asset pricing and arbitrage; 5. Nonlinear programming: theory and algorithms; 6. NLP volatility estimation; 7. Quadratic programming: theory and algorithms; 8. QP models: portfolio optimization; 9. Conic optimization tools; 10. Conic optimization models in finance; 11. Integer programming: theory and algorithms; 12. IP models: constructing an index fund; 13. Dynamic programming methods; 14. DP models: option pricing; 15. DP models: structuring asset backed securities; 16. Stochastic programming: theory and algorithms; 17. SP models: value-at-risk; 18. SP models: asset/liability management; 19. Robust optimization: theory and tools; 20. Robust optimization models in finance; Appendix A. Convexity; Appendix B. Cones; Appendix C. A probability primer; Appendix D. The revised simplex method; Bibliography; Index.