Tài liệu Alternative Investments and Stratagies doc

7373tpPath.indd 1 5/19/10 3:29:55 PM

May 13, 2010 10:6 WSPC/SPI-B913 b913-fm FA

This page intentionally left blank

World Scientific

7373tpPath.indd 2 5/19/10 3:29:56 PM

Library of Congress Cataloging-in-Publication Data

Alternative investments and strategies / edited by Rüdiger Kiesel, Matthias Scherer & Rudi Zagst.

p. cm.

ISBN-13: 978-9814280105

ISBN-10: 9814280100

1. Investments--Moral and ethical aspects. 2. Portfolio management--Moral and ethical aspects.

I. Kiesel, Rüdiger, 1962– II. Scherer, Matthias. III. Zagst, Rudi, 1961–

HG4515.13.A498 2010

332.6--dc22

2010013167

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

For photocopying of material in this volume, please pay a copying fee through the Copyright

Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to

photocopy is not required from the publisher.

Typeset by Stallion Press

Email: [email protected]

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,

electronic or mechanical, including photocopying, recording or any information storage and retrieval

system now known or to be invented, without written permission from the Publisher.

Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd.

Published by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Printed in Singapore.

Juliet - Alternative Investments.pmd 1 8/2/2010, 6:18 PM

May 13, 2010 10:6 WSPC/SPI-B913 b913-fm FA

PREFACE

Asset allocation investigates the optimal division of a portfolio among different asset

classes. Standard theory involves the optimal mix of risky stocks, bonds, and cash

together with various subdivisions of these asset classes. Underlying this is the insight

that diversification allows for achieving a balance between risk and return: by using

different types of investment, losses may be limited and returns are made less volatile

without losing too much potential gain.

These insights are made precise using the benchmark theory of mathematical

finance, the Black-Scholes-Merton theory, based on Brownian motion as the driving

noise process for risky asset prices. Here, the distributions of financial returns of the

risky assets in a portfolio are multivariate normal, thus relating to the standard mean￾variance portfolio theory of Markowitz with its risk-return paradigm as above.

Recent years have seen many empirical studies shedding doubt on the Black￾Scholes-Merton model, and motivating various alternative modeling approaches,

which were able to reproduce the stylized facts of asset returns (such as heavy tails and

volatility clustering) much better. Also, various new asset classes and specific financial

tools for achieving better diversification have been created and entered the investment

universe.

This book combines academic research and practical expertise on these new (often

called alternative) assets and trading strategies in a unique way. We include the prac￾titioners’ viewpoint on new asset classes as well as academic research on modeling

approaches, for new asset classes. In particular, alternative asset classes such as power

forward contracts, forward freight agreements, and investment in photovoltaic facil￾ities are discussed in detail, both on a stand-alone basis and with a view to their

effects on diversification in combination with classical asset. We also analyse credit￾related portfolio instruments and their effect in achieving an optimal asset allocation.

In this context, we highlight aspects of financial structures which may sometimes be

neglected, such as default risk of issuer in case of certificates or the role that model

v

May 13, 2010 10:6 WSPC/SPI-B913 b913-fm FA

vi Preface

risk plays within asset allocation problems. This leads naturally to the use of robust

asset allocation strategies.

Extending the classical mean-variance portfolio setting, we include dynamic port￾folio strategies and illustrate different portfolio protection strategies. In particular, we

compare the benefits of such strategies and investigate conditions under which Con￾stant Proportion Portfolio Insurance (CPPI) may be prefered to Option-Based Portfolio

Insurance (OBPI) and vice versa. We also contribute to the understanding of gap risk

by analyzing this risk for CPPI and Constant Proportion Debt Obligations (CPDO) in

a sophisticated modeling framework. Such analyses are supplemented and extended

by an investigation of the optimality of hedging approaches such as variance-optimal

hedging and semistatic variants of classical hedging strategies.

Many of the articles can serve as guides for the implementation of various models.

In addition, we also present state-of-the-art models and explain modern tools from

financial mathematics, such as Markov-Switching models, time-changed Lévy models,

variants of lognormal approximations, and copula structures.

This books combines a unique mix of authors.Also many of our students improved

the outcome of the project with critical and insightful comments. Particular thanks goes

to Georg Grüll, Peter Hieber, Julia Kraus, Matthias Lutz, Jan-Frederik Mai, Kathrin

Maul, Kevin Metka, Daniela Neykova, Johannes Rauch,Andreas Rupp, Daniela Selch,

and Christofer Vogt.

R. Kiesel, M. Scherer, and R. Zagst

May 13, 2010 10:6 WSPC/SPI-B913 b913-fm FA

CONTENTS

Preface v

Part I. Alternative Investments

Chapter 1. Socially Responsible Investments 3

Sven Hroß, Christofer Vogt and Rudi Zagst

1.1 Introduction ................................. 4

1.2 Recent Research on SRI .......................... 5

1.3 How Sustainable is Sustainability? ..................... 6

1.3.1 Description of the Dataset ..................... 6

1.3.2 Introduction to Markov Transition Matrices ............ 6

1.3.3 Results of Markov Transition Matrices .............. 7

1.4 SRI in Portfolio Context .......................... 8

1.4.1 Description of the Dataset and Statistical Properties ....... 8

1.4.2 Markov-Switching Model . . . . . . . . . . . . . . . . . . . . . 11

1.4.3 Fitting the Model Parameters . . . . . . . . . . . . . . . . . . . 11

1.4.4 Simulation of Returns . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.5 Portfolio Optimization Models . . . . . . . . . . . . . . . . . . 13

1.4.6 Definition of Investor Types . . . . . . . . . . . . . . . . . . . . 15

1.4.7 Optimal Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 2. Listed Private Equity in a Portfolio Context 21

Philipp Aigner, Georg Beyschlag, Tim Friederich,

Markus Kalepky and Rudi Zagst

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Defining Private Equity Categories . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Financing Stages . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.2 Divestment Strategies . . . . . . . . . . . . . . . . . . . . . . . 24

vii

May 13, 2010 10:6 WSPC/SPI-B913 b913-fm FA

viii Contents

2.2.3 Type of Financing . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.4 Classification of Private Equity Fund Investments . . . . . . . . 26

2.2.4.1 Venture capital funds . . . . . . . . . . . . . . . . . . . 26

2.2.4.2 Buyout funds . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.4.3 Leveraged buyouts (LBO) . . . . . . . . . . . . . . . . 27

2.3 Investment Possibilities — One Asset, Many Classes . . . . . . . . . . . 28

2.3.1 Direct Investments . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.2 Private Equity Funds . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2.1 Key players . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.3 Cash Flow Structure of a Private Equity Fund . . . . . . . . . . . 31

2.3.4 Fund-of-Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.4.1 Structure of a private equity fund-of-funds . . . . . . . 32

2.3.4.2 Advantages . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.4.3 Disadvantages . . . . . . . . . . . . . . . . . . . . . . 33

2.3.5 Publicly Traded Private Equity . . . . . . . . . . . . . . . . . . 33

2.3.6 Secondary Transactions . . . . . . . . . . . . . . . . . . . . . . 34

2.3.6.1 Types of secondary transactions . . . . . . . . . . . . . 34

2.3.6.2 Buyer’s motivation . . . . . . . . . . . . . . . . . . . . 35

2.4 Private Equity as Alternative Asset Class

in an Investment Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.1 Characteristics of LPE Return Series . . . . . . . . . . . . . . . 36

2.4.2 Modeling Return Series with Markov-Switching Processes . . . . 37

2.4.2.1 Markov–Switching models . . . . . . . . . . . . . . . 37

2.4.2.2 Fitting the parameters . . . . . . . . . . . . . . . . . . 39

2.4.2.3 Simulation of return paths . . . . . . . . . . . . . . . . 40

2.4.3 Listed Private Equity in Asset Allocation . . . . . . . . . . . . . 40

2.4.3.1 Performance measurement . . . . . . . . . . . . . . . . 40

2.4.3.2 Portfolio optimization frameworks . . . . . . . . . . . 42

2.4.3.3 Definition of investor types . . . . . . . . . . . . . . . 43

2.4.3.4 Optimization of portfolios . . . . . . . . . . . . . . . . 44

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Chapter 3. Alternative Real Assets in a Portfolio Context 51

Wolfgang Mader, Sven Treu and SebastianWillutzky

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Overview on Alternative Real Assets . . . . . . . . . . . . . . . . . . . . 52

3.3 Modeling Photovoltaic Investments . . . . . . . . . . . . . . . . . . . . 53

3.3.1 General Approach . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.2 Definition of the Investment Project . . . . . . . . . . . . . . . . 54

May 13, 2010 10:6 WSPC/SPI-B913 b913-fm FA

Contents ix

3.3.3 Modeling of Risk Factors . . . . . . . . . . . . . . . . . . . . . 56

3.3.3.1 Economic factors . . . . . . . . . . . . . . . . . . . . 56

3.3.3.2 Non-economic factors . . . . . . . . . . . . . . . . . . 57

3.3.3.3 Historical analysis of monthly global irradiance . . . . 58

3.3.3.4 Monte Carlo analysis of yearly global irradiance . . . . 61

3.4 Photovoltaic Investments in a Portfolio Context . . . . . . . . . . . . . . 63

3.4.1 Setting the Portfolio Context . . . . . . . . . . . . . . . . . . . . 63

3.4.2 Including Photovoltaic Investments in a Portfolio . . . . . . . . . 64

3.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Chapter 4. The Freight Market and Its Derivatives 71

Rüdiger Kiesel and Patrick Scherer

4.1 Introduction: the Freight Market . . . . . . . . . . . . . . . . . . . . . . 72

4.1.1 Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.2 Cargo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.3 Routes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Freight Rates: What Drives the Market? . . . . . . . . . . . . . . . . . . 74

4.2.1 Demand for Shipping Capacity . . . . . . . . . . . . . . . . . . 75

4.2.2 Supply of Shipping Capacity . . . . . . . . . . . . . . . . . . . 76

4.2.3 Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Freight Derivatives: Hedging or Speculating? . . . . . . . . . . . . . . . 77

4.3.1 Forward Freight Agreement . . . . . . . . . . . . . . . . . . . . 77

4.3.2 Freight Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4 Explanatory Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4.1 Explanatory Power . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.2 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4.3 Selection Algorithm “Top Five” . . . . . . . . . . . . . . . . . . 83

4.4.4 Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5 Predicting Freight Spot and Futures Rates . . . . . . . . . . . . . . . . . 86

4.6 The Backtesting Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Chapter 5. On Forward Price Modeling in Power Markets 93

Fred Espen Benth

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2 HJM Approach to Power Forward Pricing . . . . . . . . . . . . . . . . . 95

5.3 Power Forwards and Approximation by Geometric

Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

May 13, 2010 10:6 WSPC/SPI-B913 b913-fm FA

x Contents

5.3.1 A Geometric Brownian Motion Dynamics

by Volatility Averaging . . . . . . . . . . . . . . . . . . . . . . . 101

5.3.2 A Geometric Brownian Motion Dynamics

by Moment Matching . . . . . . . . . . . . . . . . . . . . . . . 103

5.3.3 The Covariance Structure Between Power Forwards . . . . . . . 106

5.3.4 The Distribution of a Power Forward . . . . . . . . . . . . . . . 108

5.3.5 Numerical Analysis of the Power Forward Distribution . . . . . . 110

5.4 Pricing of Options on Power Forwards . . . . . . . . . . . . . . . . . . . 114

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 6. Pricing Certificates Under Issuer Risk 123

Barbara Götz, Rudi Zagst and Marcos Escobar

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.3 Pricing of Certificates Under Issuer Risk . . . . . . . . . . . . . . . . . . 126

6.3.1 Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.3.2 Index Certificates . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.3.3 Participation Guarantee Certificates . . . . . . . . . . . . . . . . 132

6.3.4 Bonus Guarantee Certificates . . . . . . . . . . . . . . . . . . . 134

6.3.5 Discount Certificates . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3.6 Bonus Certificates . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Chapter 7. Asset Allocation with Credit Instruments 147

Barbara Menzinger, Anna Schlösser and Rudi Zagst

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.2 Simulation Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.3 Framework for Total Return Calculation . . . . . . . . . . . . . . . . . . 153

7.4 Optimization Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.4.1 Mean-Variance Optimization . . . . . . . . . . . . . . . . . . . 156

7.4.2 CVaR Optimization . . . . . . . . . . . . . . . . . . . . . . . . 157

7.5 Model Calibration and Simulation Results . . . . . . . . . . . . . . . . . 157

7.5.1 Mean-Variance Approach . . . . . . . . . . . . . . . . . . . . . 162

7.5.2 Conditional Value at Risk . . . . . . . . . . . . . . . . . . . . . 164

7.5.3 Comparison of Selected Optimal Portfolios . . . . . . . . . . . . 167

7.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 170

Chapter 8. Cross Asset Portfolio Derivatives 175

Stephan Höcht, Matthias Scherer and Philip Seegerer

May 13, 2010 10:6 WSPC/SPI-B913 b913-fm FA

Contents xi

8.1 Introduction to Cross Asset Portfolio Derivatives . . . . . . . . . . . . . 175

8.1.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . 176

8.2 Collateralized Obligations . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.3 A Comparison of CFO with CTSO . . . . . . . . . . . . . . . . . . . . . 179

8.3.1 Structural Features of CFO . . . . . . . . . . . . . . . . . . . . 179

8.3.2 Structural Features of CTSO . . . . . . . . . . . . . . . . . . . . 181

8.3.3 The Different Risks . . . . . . . . . . . . . . . . . . . . . . . . 181

8.3.4 Correlation of Tail Events in CTSO . . . . . . . . . . . . . . . . 181

8.4 Pricing Cross Asset Portfolio Derivatives . . . . . . . . . . . . . . . . . 182

8.4.1 Pricing Trigger Swaps . . . . . . . . . . . . . . . . . . . . . . . 182

8.4.2 Pricing nth-to-Trigger Baskets . . . . . . . . . . . . . . . . . . . 183

8.4.3 Pricing CTSO . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8.4.4 Modeling Approaches . . . . . . . . . . . . . . . . . . . . . . . 185

8.4.4.1 The structural approach . . . . . . . . . . . . . . . . . 185

8.4.4.2 The copula approach . . . . . . . . . . . . . . . . . . . 186

8.4.5 An Example for an nth-to Trigger Basket . . . . . . . . . . . . . 188

8.4.5.1 A pricing exercise of Example 3

(structural approach) . . . . . . . . . . . . . . . . . . 188

8.4.5.2 A pricing exercise of Example 3

(copula approach) . . . . . . . . . . . . . . . . . . . . 189

8.4.5.3 Resulting model spreads . . . . . . . . . . . . . . . . . 190

8.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Part II. Alternative Strategies

Chapter 9. Dynamic Portfolio Insurance Without Options 201

Dominik Dersch

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

9.2 Simple Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.2.1 Buy-and-Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.2.2 Stop-Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

9.2.3 The Bond Floor Strategy . . . . . . . . . . . . . . . . . . . . . . 204

9.2.4 Plain Vanilla CPPI . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.3 Historical Simulation I . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

9.4 Advanced Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

9.4.1 Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . 210

9.4.2 Transaction Filter . . . . . . . . . . . . . . . . . . . . . . . . . 210

9.4.3 Lock-in Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

May 13, 2010 10:6 WSPC/SPI-B913 b913-fm FA

xii Contents

9.4.4 Leverage and Constrain of Exposure . . . . . . . . . . . . . . . 212

9.4.5 Rebalancing Strategies for the Risky Portfolio . . . . . . . . . . 213

9.4.6 CPPI and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . 213

9.5 Historical Simulation II . . . . . . . . . . . . . . . . . . . . . . . . . . 214

9.5.1 Transaction Costs and Transaction Filter . . . . . . . . . . . . . 214

9.5.2 Lock-in Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

9.5.3 The Use of Leverage . . . . . . . . . . . . . . . . . . . . . . . . 220

9.5.4 CPPI on a Multi-Asset Risky Portfolio . . . . . . . . . . . . . . 222

9.6 Implement a Dynamic Protection Strategy with ETF . . . . . . . . . . . 223

9.7 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Chapter 10. How Good are Portfolio Insurance Strategies? 227

Sven Balder and Antje Mahayni

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

10.2 Optimal Portfolio Selection with Finite Horizons . . . . . . . . . . . . . 230

10.2.1 Problem (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

10.2.2 Problem (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

10.2.3 Problem (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

10.2.4 Comparison of Optimal Solutions . . . . . . . . . . . . . . . . . 238

10.3 Utility Loss Caused by Guarantees . . . . . . . . . . . . . . . . . . . . . 242

10.3.1 Justification of Guarantees and Empirical Observations . . . . . 242

10.3.2 Utility Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

10.4 Utility Loss Caused by Trading Restrictions

and Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

10.4.1 Discrete-Time CPPI . . . . . . . . . . . . . . . . . . . . . . . . 246

10.4.2 Discrete-Time Option-Based Strategy . . . . . . . . . . . . . . . 249

10.4.3 Comments on Utility Loss and Shortfall Probability . . . . . . . 250

10.5 Utility Loss Caused by Guarantees

and Borrowing Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 252

10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

Chapter 11. Portfolio Insurances, CPPI and CPDO, Truth or Illusion? 259

Elisabeth Joossens and Wim Schoutens

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

11.2 Credit Risk and Credit Default Swaps . . . . . . . . . . . . . . . . . . . 261

11.2.1 Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

11.2.2 Credit Default Swaps (CDS) . . . . . . . . . . . . . . . . . . . . 265

11.3 Portfolio Insurances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

May 13, 2010 10:6 WSPC/SPI-B913 b913-fm FA

Contents xiii

11.4 Modeling of CPPI Dynamics Using Multivariate

Jump-Driven Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

11.4.1 Multivariate Variance Gamma Modeling . . . . . . . . . . . . . 270

11.4.2 Swaptions on Credit Indices . . . . . . . . . . . . . . . . . . . . 273

11.4.2.1 Black’s model . . . . . . . . . . . . . . . . . . . . . . 273

11.4.2.2 The variance gamma model . . . . . . . . . . . . . . . 274

11.4.3 Spread Modeling by Correlated VG Processes . . . . . . . . . . 275

11.4.3.1 The pricing of CPPIs . . . . . . . . . . . . . . . . . . . 275

11.4.3.2 Gap risk . . . . . . . . . . . . . . . . . . . . . . . . . 279

11.5 Recent Developments for CPPI . . . . . . . . . . . . . . . . . . . . . . 281

11.5.1 Portfolio Insurance: The Extreme Value Approach

to the CPPI Method . . . . . . . . . . . . . . . . . . . . . . . . 282

11.5.2 VaR Approach for Credit CPPI . . . . . . . . . . . . . . . . . . 283

11.5.3 CPPI with Cushion Insurance . . . . . . . . . . . . . . . . . . . 284

11.6 A New Financial Instrument: Constant Proportion Debt Obligations . . . 285

11.6.1 The Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

11.6.2 CPDOs in the Spotlight . . . . . . . . . . . . . . . . . . . . . . 289

11.6.3 Rating CPDOs Under VG Dynamics . . . . . . . . . . . . . . . 289

11.7 Comparison Between CPPI and CPDO . . . . . . . . . . . . . . . . . . 291

11.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

Chapter 12. On the Benefits of Robust Asset Allocation for CPPI Strategies 295

Katrin Schöttle and Ralf Werner

12.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

12.2 The Financial Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

12.2.1 The Basic Financial Market . . . . . . . . . . . . . . . . . . . . 297

12.2.2 The Riskless Asset . . . . . . . . . . . . . . . . . . . . . . . . . 298

12.2.3 The Risky Asset . . . . . . . . . . . . . . . . . . . . . . . . . . 298

12.2.4 Classical Mean–Variance Analysis . . . . . . . . . . . . . . . . 300

12.2.5 The Trading Strategy . . . . . . . . . . . . . . . . . . . . . . . . 302

12.3 The Standard CPPI Strategy . . . . . . . . . . . . . . . . . . . . . . . . 302

12.3.1 The Simple Case . . . . . . . . . . . . . . . . . . . . . . . . . . 303

12.3.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . 305

12.3.3 Shortfall Probability of CPPI Strategies . . . . . . . . . . . . . . 308

12.3.4 Improving CPPI Strategies . . . . . . . . . . . . . . . . . . . . . 310

12.3.5 CPPI Strategies Under Estimation Risk . . . . . . . . . . . . . . 313

12.4 Robust Mean–Variance Optimization and Improved CPPI Strategies . . . 316

12.4.1 Robust Mean–Variance Analysis . . . . . . . . . . . . . . . . . . 317

12.4.2 Uncertainty Sets Via Expert Opinions or Related Estimators . . . 317

May 13, 2010 10:6 WSPC/SPI-B913 b913-fm FA

xiv Contents

12.4.3 Uncertainty Sets Via Confidence Sets . . . . . . . . . . . . . . . 319

12.4.4 Usage and Implications for CPPI Strategies . . . . . . . . . . . . 321

12.4.5 CPPIs with Robust Asset Allocations . . . . . . . . . . . . . . . 323

12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

Chapter 13. Robust Asset Allocation Under Model Risk 327

Pauline Barrieu and Sandrine Tobelem

13.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

13.2 A Robust Approach to Model Risk . . . . . . . . . . . . . . . . . . . . . 329

13.2.1 The Absolute Ambiguity Robust Adjustment . . . . . . . . . . . 330

13.2.2 Relative Ambiguity Robust Adjustment . . . . . . . . . . . . . . 333

13.2.3 ARA Parametrization . . . . . . . . . . . . . . . . . . . . . . . 334

13.3 Some Definitions Relative to the Ambiguity-Adjusted

Asset Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

13.4 Empirical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

13.4.1 Portfolios Tested . . . . . . . . . . . . . . . . . . . . . . . . . . 337

13.4.2 Performance Measures . . . . . . . . . . . . . . . . . . . . . . . 339

13.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

13.4.3.1 Performances of the different models . . . . . . . . . . 341

13.4.3.2 SEU portfolio . . . . . . . . . . . . . . . . . . . . . . 342

13.4.3.3 Ambiguity robust portfolios . . . . . . . . . . . . . . . 342

13.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Chapter 14. Semi-Static Hedging Strategies for Exotic Options 345

Hansjörg Albrecher and Philipp Mayer

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

14.2 Hedging Path-Independent Options . . . . . . . . . . . . . . . . . . . . 347

14.2.1 Plain Vanilla Options with Arbitrary Strikes are Liquid . . . . . . 348

14.2.2 Finitely Many Liquid Strikes . . . . . . . . . . . . . . . . . . . 349

14.3 Hedging Barrier and Other Weakly Path Dependent Options . . . . . . . 350

14.3.1 Model-Dependent Strategies: Perfect Replication . . . . . . . . . 351

14.3.2 Model-Dependent Strategies: Approximations . . . . . . . . . . 357

14.3.3 Model-Independent Strategies: Robust Strategies . . . . . . . . . 359

14.4 Hedging Strongly Path-Dependent Options . . . . . . . . . . . . . . . . 361

14.4.1 Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . 362

14.4.2 Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

14.5 Case Study: Model-Dependent Hedging of Discretely

Sampled Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

14.6 Conclusion and Future Research . . . . . . . . . . . . . . . . . . . . . . 370