Optimal Portfolios: Stochastic Models for Optimal Investment and Risk Management in Continuous Time (Hardcover)~ Ralf Korn





Review"recommended to those who want to obtain a quick overview about methods of portfolio theory." -- Metrika, 1999

Product DescriptionFocuses on the construction of optimal investment strategies in a security market model where the prices follow diffusion processes. Beginning with presenting the complete Black-Scholes type model, the book moves on to incomplete models and models including constraints and transaction costs. The methods and models presented include the stochastic control method of Merton, the martingale method of Cox-Huang and Karatzas et al, the log optimal method of Cover and Jamshidian, the value-preserving model of Hellwig, and so forth. Stress is laid on rigorous mathematical presentation and clear economics interpretation while technicalities are kept to a minimum.

Product Details Hardcover: 338 pages Publisher: World Scientific Publishing Company (September 1997) Language: English ISBN-10: 9810232152 ISBN-13: 978-9810232153

CONTENTS
Preface ..................................................................................................................... v
Some Guidelines and General Notations ............................................................ ix
Chapter 1. Introduction and Discrete-Time Models ........................................... 1
1.1. General and Historic Remarks: A Short Survey,
1.2. Mean-Variance Analysis in a One-Period Model:
The Markowitz-Approach,
1.3. More on One-Period and Discrete-Time Approaches to
Portfolio Selection
1.4. Criticisms and Limitations of Discrete-Time Models

Chapter 2. The Continuous-Time Market Model .............................................. 15
2.1. The Security Price Processes 15
2.2. The Wealth Process and the Actions of a Small Investor 21
2.3. Completeness of the Market Model and the Growth-Optimum Portfolio 25
2.4. Option Pricing: A Short Introduction 29
2.5. Convergence of Discrete Markets to Continuous Markets 34
Chapter 3. The Continuous-Time Portfolio Problem ........................................ 37
3.1. Introduction and Formulation of the Problem 37
3.2. Comparison between Stochastic Control and Martingale
Method - A Preview via a Simple Discrete Example 41
3.3. The Stochastic Control Method to Solve the Portfolio Problem 48

3.5. The Martingale Method Revisited a: Pliska’s Version 82
3.6. An Application: “Minimising the Difference to the Terminal Wealth
of a Richer Investor?” 92
3.4. The Martingale Approach to the Continuous-Time Portfolio Problem
Chapter 4. Constrained Continuous-Time Problems ...................................... 101
4.1. Portfolio Problems with Constraints 101
4.2. A Dual Method to Solve Portfolio Problems with Constraints on
the Terminal Wealth 101

4.3. A Continuous-Time Mean-Variance Problem
4.4. Portfolio Problems with Constrained Strategies
4.5. Some Examples of Constrained Problems

Chapter 5. Portfolio Optimisation in the Presence of Transaction Costs ..... 151
5.1. Optimal Life-Time Consumption with Proportional Transaction Costs 15 1
5.2. Impulse Control Methods and Portfolio Optimisation with Strictly
Positive Transaction Costs 171
5.3. Maximising the Growth Rate under Fixed Transaction Costs 206
Chapter 6. Non-Utility Based Portfolio Selection Models ............................... 223
6.1. Universal Portfolios: The Discrete-Time Model 223
6.2. Asymptotically Optimal Portfolios and Universal Portfolios in
Continuous Time 240
6.3. Optimal Cash Management in Equity Index Tracking with
Transaction Costs 259
6.4. Value Preserving Portfolio Strategies 279
Appendix .............................................................................................................. 295
A. Normal Distribution, Conditional Expectation, Stochastic Processes 295
B. Introduction to Stochastic Integrals and the Ito-Calculus 303
C. Controlled Stochastic Differential Equations 320
D. Laplace’s Method of Integration 328
E. Optimisation of Convex Functionals 328
References ...................................................................................................... : ..... 331
Index .................................................................................................................... 337