Bartholomew-Biggs Nonlinear Optimization with Financial Applications

1. Auflage 2006
ISBN: 978-0-387-24149-4
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 276 Seiten, eBook

This instructive book introduces the key ideas behind practical nonlinear optimization, accompanied by computational examples and supporting software. It combines computational finance with an important class of numerical techniques.

Zielgruppe

Research

Autoren/Hrsg.

Bartholomew-Biggs, Michael

Inhaltsverzeichnis

List of Figures

List of Tables

Preface 1: PORTFOLIO OPTIMIZATION

1. Nonlinear optimization

2. Portfolio return and risk

3. Optimizing two-asset portfolios

4. Minimimum risk for three-asset portfolios

5. Two- and three-asset minimum-risk solutions

6. A derivation of the minimum risk problem

7. Maximum return problems

2: ONE-VARIABLE OPTIMIZATION

1. Optimality conditions

2. The bisection method

3. The secant method

4. The Newton method

5. Methods using quadratic or cubic interpolation

6. Solving maximum-return problems

3: OPTIMAL PORTFOLIOS WITH N ASSETS

1. Introduction

2. The basic minimum-risk problem

3. Minimum risk for specified return

4. The maximum return problem

4: UNCONSTRAINED OPTIMIZATION IN N VARIABLES
1. Optimality conditions
2. Visualising problems in several variables
3. Direct search methods
4. Optimization software and examples

5: THE STEEPEST DESCENT METHOD
1. Introduction

2. Line searches
3. Convergence of the steepest descent method
4. Numerical results with steepest descent
5. Wolfe’s convergence theorem
6. Further results with steepest descent

6: THE NEWTON METHOD
1. Quadratic models and the Newton step

2. Positive definiteness and Cholesky factors
3. Advantages and drawbacks of Newton’s method
4. Search directions from indefinite Hessians
5. Numerical results with the Newton method

7: QUASINEWTON METHODS

1. Approximate second derivative information

2. Rauk-two updates for the inverse Hessian
3. Convergence of quasi-Newton methods
4. Numerical results with quasi-Newton methods
5. The rank-one update for the inverse Hessian
6. Updating estimates of the Hessian

8: CONJUGATE GRADIENT METHODS

1. Conjugate gradients and quadratic functions
2. Conjugate gradients and general functions
3. Convergence of conjugate gradient methods
4.Numerical results with conjugate gradients
5. The truncated Newton method

9: OPTIMAL PORTFOLIOS WITH RESTRICTIONS
1. Introduction

2. Transformations to exclude short-selling
3. Results from Minrisk2u and Maxret2u

4. Upper and lower limits on invested fractions

10: LARGER-SCALE PORTFOLIOS
1. Introduction

2. Portfolios with increasing numbers of assets
3. Time-variation of optimal portfolios
4. Performance of optimized portfolios

11: DATA-FITTING AND THE GAUSS-NEWTON METHOD
1. Data fitting problems
2. The Gauss-Newton method
3. Least-squares in time series analysis
4. Gauss-Newton applied to time series
5. Least-squares forms of minimum-risk problems
6. Gauss-Newton applied to Minrisk1 and Minrisk2

12: EQUALITY CONSTRAINED OPTIMIZATION
1. Portfolio problems with equality constraints
2. Optimality conditions
3. A worked example

4. Interpretation of Lagrange multipliers
5. Some example problems

13: LINEAR EQUALITY CONSTRAINTS
1. Equality constrained quadratic programming
2. Solving minimum-risk problems as EQPs
3. Reduced-gradient methods
4. Projected gradient methods
5. Results with methods for linear constraints

14: PENALTY FUNCTION METHODS
1. Introduction
2. Penalty functions
3. The Augmented Lagrangian
4. Results with P-SUMT and AL-SUMT
5. Exact penalty functions

15: SEQUENTIAL QUADRATIC PROGRAMMING
1. Introduction
2. Quadratic/linear models
3. SQP methods based on penalty functions
4. Results with AL-SQP

5. SQP line searches and the Maratos effect

16: FURTHER PORTFOLIO PROBLEMS
1. Including transaction costs

2. A re-balancing problem
3. A sensitivity problem

17: INEQUALITY CONSTRAINED OPTIMIZATION
1. Portfolio problems with inequality constraints
2. Optimality conditions
3. Transforming inequalities to equalities

4. Transforming inequalities to simple bounds
5. Example

Leseproben

Preface (p. XVI)

This book has grown out of undergraduate and postgraduate lecture courses given at the University of Hertfordshire and the University of Bergamo. Its pri- mary focus is on numerical methods for nonlinear optimization. Such methods can be applied to many practical problems in science, engineering and manage- ment: but, to provide a coherent secondary theme, the applications considered here are all drawn from financial mathematics. (This puts the book in good company since many classical texts in mathematics also dealt with commer- cial arithmetic.) In particular, the examples and case studies are concerned with portfolio selection and with time-series problems such as fitting trend- lines and trend-channels to market data.

The content is intended to be suitable for final-year undergraduate students in mathematics (or other subjects with a high mathematical or computational content) and exercises are provided at the end of most sections. However the book should also be useful for postgraduate students and for other researchers and practitioners who need a foundation for work involving development or application of optimization algorithms.

It is assumed that readers have an un- derstanding of the algebra of matrices and vectors and of the Taylor and Mean Value Theorems in several variables. Prior experience of using computational methods for problems such as solving systems of linear equations is also de- sirable, as is familiarity with iterative algorithms (e.g., Newton's method for nonlinear equations in one variable).

The approach adopted in this book is a blend of the practical and theoretical. A description and derivation is given for most of the currently popular methods for continuous nonlinear optimization. For each method, important conver- gence results are outlined (and we provide proofs when it seems instructive to do so). This theoretical material is complemented by numerical illustrations which give a flavour of how the methods perform in practice.

It is not always obvious where to draw the line between general descriptions of algorithms and the more subtle and detailed considerations relating to re- search issues. The particular themes and emphases in this book have grown out of the author's experience at the Numerical Optimization Centre (NOC). This was established in 1968 at the Hatfield College of Technology (predeces- sor of the University of Hertfordshire) as a centre for research in optimization techniques.

Since its foundation, the NOC has been engaged in algorithm de- velopment and consultancy work (mostly in the aerospace industry). The NOC staff has included, at various times, Laurence Dixon, Ed Hersom, Joanna Go- mulka, Sean McKeown and Zohair Maany who have all made contributions to the state-of-the-art in fields as diverse as quasi-Newton methods, sequential quadratic programming, nonlinear least-squares, global optimization, optimal control and automatic differentiation.

The computational results quoted in this book have been obtained using a Fortran90 module called SAMPO. This is based on the NOC's OPTIMA library - a suite of subroutines for different types of minimization problem. The name SAMPO serves as an acronym for Software And Methods for Portfolio Optimization. (However, it is also worth mentioning that The Sampo appears in Finnish mythology as a magical machine which grinds out endless supplies of corn, salt and gold.