Optimization Methods in Finance

The second edition of Optimization Methods in Finance comes 11 years after the successful first edition with a relevant extension to include material on mean-variance portfolio selection and multi-period models. Intended for master students in finance courses this textbook by Cornuéjols, Peña, and Tütüncü addresses a wide range of financial valuation and optimization problems with an effective and well-structured approach.

The book is characterized by an even treatment of relevant areas of financial theory and associated optimization and computational techniques. The authors employ an approach based first on the introduction of theory and algorithms for specific classes of optimization models and then with a focus on the associated financial models and applications.

All four parts into which the volume is structured have a similar framework. Part I includes introductory material with an overview of optimization models and linear programming (LP) techniques and related applications to asset-liability management (ALM) and asset pricing. Part II is devoted to single-period models and includes several chapters on optimization problem formulations with associated financial applications: from quadratic programming (QP) theory, with its related mean-variance (MV) portfolio model, to mixed integer programs (MIP) and stochastic programs (SP), together with related theory and applications to portfolio selection with cardinality constraints and to the formulation of risk-reward optimization problems based on different risk measures. A specific section is devoted to mean-Conditional Value-at-Risk (CVaR) portfolio optimization. Part III presents multi-period models and evolves from an introductory chapter (no 12) to the analysis of two relevant approaches for the formulation and solution of dynamic optimization problems: dynamic programming (DP, ch 13), multistage stochastic programming (MSP, ch 16) with associated applications to multi-period portfolio optimization (ch 14), binomial pricing (ch 15) and ALM (ch 16). Part IV, from ch 18–20, is devoted to other types of optimization problems, such as conic programming problems (ch 18), an introduction to the formulation and solution of robust optimization models (ch 19), and nonlinear programming problems instrumental to advanced valuation problems, for instance involving volatility surfaces for option pricing (ch 20).

Overall, the book provides a relevant and interesting overview of optimization techniques applied to quantitative finance and succeeds in clarifying the appropriate numerical and computational methods in a financial context motivated by practitioners’ views. This is an original and useful volume combining optimization models and finance for students with an operations research, finance or mathematics background.

The material devoted to linear and nonlinear programming concerning linear, quadratic, mixed-integer, conic, and general nonlinear programs is well conceived, with basic results and very valuable numerical examples in every chapter computed with commonly used programs (e.g. Microsoft Excel solvers and Matlab). The authors present throughout case studies and examples of modern software codes which can be adopted for a wide range of applications in finance. I found such an approach based on very accessible codes and easily interpretable algorithms a distinctive and positive feature of the volume. The authors clarify primal-dual relationships where appropriate and effectively link the analysis to specific financial problems.

With few refinements relative to the previous edition, the authors cover stochastic programming, dynamic programming and robust optimization approaches in chapters 10, 11, 13, 17 and 19, with applications to single and multi-period financial problems, including portfolio selection, asset-liability management, and asset pricing.

The material devoted to dynamic models and related financial applications is also a not-easy to find content in textbooks and the adopted approach, based on introductory examples, related problem formulations and exercises. This covers most of classical finance and is effective in conveying the richness of modern finance and the value and necessity of adopting a rigorous mathematical approach based on optimization techniques. Throughout the textbook, relevant theoretical results in finance are presented in an accessible way. Worth mentioning in this respect are the sections on the fundamental theorem of asset pricing, the capital asset pricing model, the one and two-fund separation theorems, market completeness, risk immunization, and hedging.

Eventually, upon reaching the end of this text, graduate students with different backgrounds will surely have obtained a clear and thorough understanding of optimization theory applied to financial problems. They will have gained in terms of problem formulation and solution initially from straightforward Excel-based examples and codes and subsequently from more complex and structured stochastic programming and second-order conic problems. The associated insights from optimization theory and finance this provides may be the most valuable feature of this textbook. The book can also be regarded as a good introduction to more advanced doctoral studies in operations research and financial optimization.

I complete this review by summarizing the main distinctive positive features of the volume and some areas in which improvements are possible.

Among the former:

• An effective and well-motivated (with examples and practical case-studies) introduction to optimization techniques adopted in key finance areas.

• A consistent and joint treatment of financial concepts and models with associated optimization frameworks and related, well-described algorithmic approaches, avoiding unnecessary complexity.

• An extended set of exercises, and when appropriate case studies, at the end of every chapter.

• A clear and methodologically well-conceived distinction between one-period static models and dynamic, multi-period models whose relevance is growing in financial practice. This comes with an accessible and compelling overview of current operational standards in the finance sector.

• The algorithmic view of the subject with several examples of pseudo-codes and software details for the optimization approach to given financial problems.

Areas of possible improvements can be related to:

• An overall not sufficiently accurate set of references at the end of the volume; surely those related to stochastic programming models lack some key and valuable contributions. Some very few I immediately noticed are indicated below (Constantinides and Malliaris 1995, Consigli and Dempster 1998, King 2002, Zenios 2008, Birge and Louveaux 2011, Mansini et al. 2014).

• The exercises at the end of each chapter are relevant, but they are included in the volume without a solution appendix or solution manual. This is at least my current assessment. I don't know about possible ongoing projects in this direction.

• The volume covers at an introductory level, yet pretty effectively, a wide range of financial problems providing, as previously pointed out, detailed and well-structured mathematical support. However, a pair of emerging topics, relevant at an MSc level, would be welcome in subsequent editions. Namely, material related to pricing and hedging in incomplete markets and liability-driven investments. The material devoted to stochastic programming applications could also be enriched, and a more extended section on risk-parity portfolios would be appropriate (Dupacova et al. 2000, Dempster et al. 2003).

Additional information

Notes on contributors

Giorgio Consigli

Giorgio Consigli is currently Professor of Applied Mathematics in Economics and Finance at the University of Bergamo (Italy). He holds a PhD in Mathematics and an MSc in Banking and Finance. He is currently Fellow of the UK Institute of Mathematics and its Applications (IMA) and Board member of the EURO WGs on Commodity and Financial modeling and of Stochastic Optimization.