Equity hybrid derivatives

Equity Hybrid Derivatives MARCUS OVERHAUS ´ ANA BERMUDEZ HANS BUEHLER ANDREW FERRARIS CHRISTOPHER JORDINSON AZIZ LAMNOUAR John Wiley & Sons, Inc www.ebook3000.com ´ Copyright c 2007 by Marcus Overhaus, Aziz Lamnouar, Ana Bermudez, Hans Buehler, Andrew Ferraris, and Christopher Jordinson All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the Web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books For more information about Wiley products, visit our Web site at www.wiley.com Library of Congress Cataloging-in-Publication Data: [et al.] Equity hybrid derivatives / Marcus Overhaus p cm — (Wiley finance series) Includes bibliographical references and index ISBN-13: 978-0-471-77058-9 (cloth) ISBN-10: 0-471-77058-2 (cloth) Derivative securities Convertible securities I Overhaus, Marcus II Title III Series HG6024.A3E684 2006 332.64 57—dc22 2006005369 Printed in the United States of America 10 Contents Preface ix PART ONE Modeling Volatility CHAPTER Theory 1.1 Concepts of Equity Modeling 1.1.1 The Forward 1.1.2 The Shape of Dividends to Come 1.1.3 European Options on the Pure Stock Process 1.2 Implied Volatility 1.2.1 Sticky Volatilities 1.3 Fitting the Market 1.3.1 Arbitrage-Free Option Price Surfaces 1.3.2 Implied Local Volatility 1.3.3 European Payoffs 1.3.4 Fitting the Market with Discrete Martingales 1.4 Theory of Replication 1.4.1 Replication in Diffusion-Driven Markets CHAPTER Applications 10 11 13 16 16 17 21 23 27 30 35 2.1 Classic Equity Models 2.1.1 Heston 2.1.2 SABR 2.1.3 Scott’s Exponential Ornstein-Uhlenbeck Model 2.1.4 Other Stochastic Volatility Models 2.1.5 Extensions of Heston’s Model 2.1.6 Cliquets 2.1.7 Forward-Skew Propagation 2.2 Variance Swaps, Entropy Swaps, Gamma Swaps 2.2.1 Variance Swaps 2.2.2 Entropy Swaps 2.2.3 Gamma Swaps 2.3 Variance Swap Market Models 2.3.1 Finite Dimensional Parametrizations 2.3.2 Examples 2.3.3 Fitting to the Market 35 35 43 45 45 46 49 52 56 58 68 69 71 76 79 83 iii www.ebook3000.com iv CONTENTS PART TWO Equity Interest Rate Hybrids CHAPTER Short-Rate Models 3.1 Introduction 3.2 Ornstein-Uhlenbeck Models 3.3 Calibrating to the Yield Curve 3.3.1 Hull-White Model 3.3.2 Generic Ornstein-Uhlenbeck Models 3.4 Calibrating the Volatility 3.4.1 Hull-White/Vasicek 3.4.2 Generic Ornstein-Uhlenbeck Models 3.5 Pricing Hybrids 3.5.1 Finite Differences 3.5.2 Monte Carlo 3.6 Appendix: Least-Squares Minimization 3.6.1 Newton-Raphson Method 3.6.2 Broyden’s Method CHAPTER Hybrid Products 4.1 The Effects of Assuming Stochastic Rates 4.2 Conditional Trigger Swaps 4.3 Target Redemption Notes 4.3.1 Structure 4.3.2 Back-Testing 4.3.3 Valuation Approach 4.3.4 Hedging 4.4 Convertible Bonds 4.4.1 Introduction 4.4.2 The Governing Equation 4.4.3 Detailed Specification of the Model 4.4.4 Analytical Solutions for a Special CB 4.5 Exchangeable Bonds 4.5.1 The Valuation PDE 4.5.2 Coordinate Transformations for Numerical Solution CHAPTER Constant Proportion Portfolio Insurance 5.1 Introduction to Portfolio Insurance 5.2 Classical CPPI 5.3 Restricted CPPI 5.3.1 Constraints on the Investment Level 5.3.2 Constraints on the Floor 5.3.3 An Example Structure 91 91 94 95 95 98 100 101 104 105 106 107 109 110 110 112 112 115 118 118 120 123 127 128 128 131 134 137 138 138 140 145 145 146 149 149 149 151 v Contents 5.4 Options on CPPI 5.4.1 The Pricing 5.4.2 Delta, Gamma, and Vega Exposures 5.4.3 Hedging 5.5 Nonstandard CPPIs 5.5.1 Complex Fee Structures 5.5.2 Dynamic Gearing 5.5.3 Perpetual CPPI 5.5.4 Flexi-Portfolio CPPI 5.5.5 Off-Balance-Sheet CPPI 5.6 CPPI as an Underlying 5.7 Other Issues Related to the CPPI 5.7.1 Liquidity Issues (Hedge Funds) 5.7.2 Assets Suitable for CPPIs 5.8 Appendixes 5.8.1 Appendix A 5.8.2 Appendix B 5.8.3 Appendix C 152 152 152 152 153 153 154 154 155 156 158 158 158 158 159 159 160 161 PART THREE Equity Credit Hybrids CHAPTER Credit Modeling 167 6.1 Introduction 6.2 Background on Credit Modeling 6.2.1 Structural Approach 6.2.2 Reduced-Form Approach 6.3 Modeling Equity Credit Hybrids 6.3.1 Dynamics of the Hazard Rate 6.3.2 Model Choice 6.4 Pricing 6.4.1 Credit Default Swap 6.4.2 Credit Default Swaption 6.4.3 European Call 6.5 Calibration 6.5.1 Stripping of Hazard Rate 6.5.2 Calibration of the Hazard Rate Process 6.5.3 Calibration of the Equity Volatility 6.5.4 Discussion 6.6 Introduction of Discontinuities 6.6.1 The New Framework 6.6.2 Dynamics of the Survival Probability 6.6.3 Pricing of European Options 6.6.4 Fourier Pricing 6.7 Equity Default Swaps 6.7.1 Modeling Equity Default Swaps www.ebook3000.com 167 167 168 171 175 175 176 180 180 181 184 186 186 187 188 188 188 189 189 190 194 196 198 vi CONTENTS 6.7.2 Single-Name EDSs in a Deterministic Hazard Rate Model 6.8 Conclusion 198 203 PART FOUR Advanced Pricing Techniques CHAPTER Copulas Applied to Derivatives Pricing 7.1 Introduction 7.2 Theoretical Background of Copulas 7.2.1 Definitions 7.2.2 Measures of Dependence 7.2.3 Copulas and Stochastic Processes 7.2.4 Some Popular Copulas 7.3 Factor Copula Framework 7.4 Applications to Derivatives Pricing 7.4.1 Equity Derivatives: The Altiplano 7.4.2 Credit Derivatives: Basket and Tranche Pricing 7.5 Conclusion CHAPTER Forward PDEs and Local Volatility Calibration 8.1 Introduction 8.1.1 Local and Implied Volatilities 8.1.2 Dupire’s Formula and Its Problems 8.1.3 Dupire-like Formula in Multifactor Models 8.2 Forward PDEs 8.3 Pure Equity Case 8.4 Local Volatility with Stochastic Interest Rates 8.5 Calibrating the Local Volatility 8.6 Special Case: Vasicek Plus a Term Structure of Equity Volatilities CHAPTER Numerical Solution of Multifactor Pricing Problems Using Lagrange-Galerkin with Duality Methods 9.1 Introduction 9.2 The Modeling Framework: A General D-factor Model 9.2.1 Strong Formulation of the Linear Problem: Partial Differential Equations 9.2.2 Truncation of the Domain and Boundary Conditions 9.2.3 Strong Formulation of the Nonlinear Problem: Partial Differential Inequalities 9.2.4 Weak Formulation of the Nonlinear Problem: Variational Inequalities 207 207 207 207 209 211 213 217 218 218 223 228 229 229 229 231 232 233 235 238 242 244 248 248 250 251 253 254 256 vii Contents 9.3 Numerical Solution of Partial Differential Inequalities (Variational Inequalities) 9.3.1 A Duality (or Lagrange Multiplier) Method 9.4 Numerical Solution of Partial Differential Equations (Variational Equalities): Classical Lagrange-Galerkin Method 9.4.1 Semi-Lagrangian Time Discretization: Method of Characteristics 9.4.2 Space Discretization: Galerkin Finite Element Method 9.4.3 Order of Classical Lagrange-Galerkin Method 9.5 Higher-Order Lagrange-Galerkin Methods 9.5.1 Crank-Nicolson Characteristics/Finite Elements 9.6 Application to Pricing of Convertible Bonds 9.6.1 Numerical Solution 9.6.2 Numerical Results 9.7 Appendix: Lagrange Triangular Finite Elements 9.7.1 Lagrange Triangular Finite Elements 9.7.2 Coefficients Matrix and Independent Term in Two Dimensions CHAPTER 10 American Monte Carlo 259 260 262 262 265 270 271 272 279 280 280 285 285 287 297 10.1 10.2 10.3 10.4 Introduction Broadie and Glasserman Regularly Spaced Restarts The Longstaff and Schwartz Algorithm 10.4.1 The Algorithm 10.4.2 Example: A Call Option with Monthly Bermudan Exercise 10.5 Accuracy and Bias 10.5.1 Extension: Regressing on In-the-Money Paths 10.5.2 Linear Regression 10.5.3 Other Regression Schemes 10.5.4 Upper Bounds 10.6 Parameterizing the Exercise Boundary 297 299 299 301 301 303 305 306 308 310 310 311 Bibliography 313 Index 323 www.ebook3000.com Preface Equity hybrid derivatives are a very young class of structures which have drawn a lot of attention over the past two years for many different reasons Equity hybrid derivatives combine all existing, and therefore established, asset classes like equity, credit, interest rate, foreign exchange, and commodity derivatives Hence, they present a very interesting challenge to combining different modeling techniques and thereby forming a solid hybrid model framework This is why we have again decided to publish a book entirely concerned with this very interesting topic Hybrid derivatives are a strategic and profitable business that every serious top-tier investment bank needs to offer to its client base and are therefore an integral part of its derivatives business In this volume, we have not tried to write an introductory text: we have assumed some prior familiarity with mathematics and finance Part One of this book gives insight into different volatility models (Heston, SABR etc) and their applications to equity markets It also contains some very recent developments such as variance swap market models Part Two gives a brief review of short rate models and their incorporation into equity-interest rate hybrid structures Important examples are discussed, such as the conditional trigger swap (CTS), convertible bonds, and the very popular CPPI structures Part Three contains a thorough introduction to credit modeling and its importance to equity-credit hybrid derivative structures Pricing and calibration techniques are also discussed in detail, and important examples like the EDS (equity default swap) are given Part Four is dedicated to advanced pricing techniques applied to various hybrid and callable structures We start with copulas applied to equity and credit derivatives (Altiplanos and default baskets), then discuss forward PDEs and local volatility calibration techniques and their application to equity-rate hybrids This is followed by a thorough presentation of numerical solutions for multi-factor pricing problems, including an important example, the convertible bond Finally, we conclude with an exposition of American Monte Carlo techniques for derivative pricing We would like to offer our special thanks to Professor Alexander Schied for careful reading of the manuscript and valuable comments We would also like to express our gratitude to Kenji Felgenhauer, Eric Bensoussan, Peter Carr, and Maria Noguieras The Authors London February 2006 ix PART One Modeling Volatility www.ebook3000.com ... www.ebook3000.com Preface Equity hybrid derivatives are a very young class of structures which have drawn a lot of attention over the past two years for many different reasons Equity hybrid derivatives combine... Popular Copulas 7.3 Factor Copula Framework 7.4 Applications to Derivatives Pricing 7.4.1 Equity Derivatives: The Altiplano 7.4.2 Credit Derivatives: Basket and Tranche Pricing 7.5 Conclusion CHAPTER... THREE Equity Credit Hybrids CHAPTER Credit Modeling 167 6.1 Introduction 6.2 Background on Credit Modeling 6.2.1 Structural Approach 6.2.2 Reduced-Form Approach 6.3 Modeling Equity Credit Hybrids