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Albanese C., Campolieti G. Advanced Derivatives Pricing and Risk

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Albanese C., Campolieti G. Advanced Derivatives Pricing and Risk Advanced Derivatives Pricing and Risk Management ThisPageIntentionallyLeftBlank ADVANCED DERIVATIVES PRICING AND RISK MANAGEMENT Theory, Tools and Hands-On Programming Application Claudio Albanese and Giuseppe Campolieti AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Elsevier Academic Press 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper. Copyright © 2006, Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Application Submitted British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 13: 978-0-12-047682-4 ISBN 10: 0-12-047682-7 The content of this book is presented solely for educational purposes. Neither the authors nor Elsevier/Academic Press accept any responsibility or liability for loss or damage arising from any application of the material, methods or ideas, included in any part of the theory or software contained in this book. The authors and the Publisher expressly disclaim all implied warranties, including merchantability or fitness for any particular purpose. There will be no duty on the authors or Publisher to correct any errors or defects in the software. For all information on all Elsevier Academic Press publications visit our Web site at www.books.elsevier.com Printed in the United States of America 050607080910987654321 Working together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org Contents Preface xi PART I Pricing Theory and Risk Management 1 CHAPTER 1 Pricing Theory 3 1.1 Single-Period Finite Financial Models 6 1.2 Continuous State Spaces 12 1.3 Multivariate Continuous Distributions: Basic Tools 16 1.4 Brownian Motion, Martingales, and Stochastic Integrals 23 1.5 Stochastic Differential Equations and Ito’sˆ Formula 32 1.6 Geometric Brownian Motion 37 1.7 Forwards and European Calls and Puts 46 1.8 Static Hedging and Replication of Exotic Pay-Offs 52 1.9 Continuous-Time Financial Models 59 1.10 Dynamic Hedging and Derivative Asset Pricing in Continuous Time 65 1.11 Hedging with Forwards and Futures 71 1.12 Pricing Formulas of the Black–Scholes Type 77 1.13 Partial Differential Equations for Pricing Functions and Kernels 88 1.14 American Options 93 1.14.1 Arbitrage-Free Pricing and Optimal Stopping Time Formulation 93 1.14.2 Perpetual American Options 103 1.14.3 Properties of the Early-Exercise Boundary 105 1.14.4 The Partial Differential Equation and Integral Equation Formulation 106 CHAPTER 2 Fixed-Income Instruments 113 2.1 Bonds, Futures, Forwards, and Swaps 113 2.1.1 Bonds 113 2.1.2 Forward Rate Agreements 116 v vi Contents 2.1.3 Floating Rate Notes 116 2.1.4 Plain-Vanilla Swaps 117 2.1.5 Constructing the Discount Curve 118 2.2 Pricing Measures and Black–Scholes Formulas 120 2.2.1 Stock Options with Stochastic Interest Rates 121 2.2.2 Swaptions 122 2.2.3 Caplets 123 2.2.4 Options on Bonds 124 2.2.5 Futures–Forward Price Spread 124 2.2.6 Bond Futures Options 126 2.3 One-Factor Models for the Short Rate 127 2.3.1 Bond-Pricing Equation 127 2.3.2 Hull–White, Ho–Lee, and Vasicek Models 129 2.3.3 Cox–Ingersoll–Ross Model 134 2.3.4 Flesaker–Hughston Model 139 2.4 Multifactor Models 141 2.4.1 Heath–Jarrow–Morton with No-Arbitrage Constraints 142 2.4.2 Brace–Gatarek–Musiela–Jamshidian with No-Arbitrage Constraints 144 2.5 Real-World Interest Rate Models 146 CHAPTER 3 Advanced Topics in Pricing Theory: Exotic Options and State-Dependent Models 149 3.1 Introduction to Barrier Options 151 3.2 Single-Barrier Kernels for the Simplest Model: The Wiener Process 152 3.2.1 Driftless Case 152 3.2.2 Brownian Motion with Drift 158 3.3 Pricing Kernels and European Barrier Option Formulas for Geometric Brownian Motion 160 3.4 First-Passage Time 168 3.5 Pricing Kernels and Barrier Option Formulas for Linear and Quadratic Volatiltiy Models 172 3.5.1 Linear Volatility Models Revisited 172 3.5.2 Quadratic Volatility Models 178 3.6 Green’s Functions Method for Diffusion Kernels 189 3.6.1 Eigenfunction Expansions for the Green’s Function and the Transition Density 197 3.7 Kernels for the Bessel Process 199 3.7.1 The Barrier-Free Kernel: No Absorption 199 3.7.2 The Case of Two Finite Barriers with Absorption 202 3.7.3 The Case of a Single Upper Finite Barrier with Absorption 206 3.7.4 The Case of a Single Lower Finite Barrier with Absorption 208 3.8 New Families of Analytical Pricing Formulas: “From x-Space to F-Space” 210 3.8.1 Transformation Reduction Methodology 210 3.8.2 Bessel Families of State-Dependent Volatility Models 215 3.8.3 The Four-Parameter Subfamily of Bessel Models 218 3.8.3.1 Recovering the Constant-Elasticity-of-Variance Model 222 3.8.3.2 Recovering Quadratic Models 224 Contents vii 3.8.4 Conditions for Absorption, or Probability Conservation 226 3.8.5 Barrier Pricing Formulas for Multiparameter Volatility Models 229 3.9 Appendix A: Proof of Lemma 3.1 232 3.10 Appendix B: Alternative “Proof” of Theorem 3.1 233 3.11 Appendix C: Some Properties of Bessel Functions 235 CHAPTER 4 Numerical Methods for Value-at-Risk 239 4.1 Risk-Factor Models 243 4.1.1 The Lognormal Model 243 4.1.2 The Asymmetric Student’s t Model 245 4.1.3 The Parzen Model 247 4.1.4 Multivariate Models 249 4.2 Portfolio Models 251 4.2.1 -Approximation 252 4.2.2 -Approximation 253 4.3 Statistical Estimations for -Portfolios 255 4.3.1 Portfolio Decomposition and Portfolio-Dependent Estimation 256 4.3.2 Testing Independence 257 4.3.3 A Few Implementation Issues 260 4.4 Numerical Methods for -Portfolios 261 4.4.1 Monte Carlo Methods and Variance Reduction 261 4.4.2 Moment Methods 264 4.4.3 Fourier Transform of the Moment-Generating Function 267 4.5 The Fast Convolution Method 268 4.5.1 The Probability Density Function of a Quadratic Random Variable 270 4.5.2 Discretization 270 4.5.3 Accuracy and Convergence 271 4.5.4 The Computational Details 272 4.5.5 Convolution with the Fast Fourier Transform 272 4.5.6 Computing Value-at-Risk 278 4.5.7 Richardson’s Extrapolation Improves Accuracy 278 4.5.8 Computational Complexity 280 4.6 Examples 281 4.6.1 Fat Tails and Value-at-Risk 281 4.6.2 So Which Result Can We Trust? 284 4.6.3 Computing the Gradient of Value-at-Risk 285 4.6.4 The Value-at-Risk Gradient and Portfolio Composition 286 4.6.5 Computing the Gradient 287 4.6.6 Sensitivity Analysis and the Linear Approximation 289 4.6.7 Hedging with Value-at-Risk 291 4.6.8 Adding Stochastic Volatility 292 4.7 Risk-Factor Aggregation and Dimension Reduction 294 4.7.1 Method 1: Reduction with Small Mean Square Error 295 4.7.2 Method 2: Reduction by Low-Rank Approximation 298 4.7.3 Absolute versus Relative Value-at-Risk 300 4.7.4 Example: A Comparative Experiment 301 4.7.5 Example: Dimension Reduction and Optimization 303 viii Contents 4.8 Perturbation Theory 306 4.8.1 When Is Value-at-Risk Well Posed? 306 4.8.2 Perturbations of the Return Model 308 4.8.2.1 Proof of a First-Order Perturbation Property 308 4.8.2.2 Error Bounds and the Condition Number 309 4.8.2.3 Example: Mixture Model 311 PART II Numerical Projects in Pricing and Risk Management 313 CHAPTER 5 Project: Arbitrage Theory 315 5.1 Basic Terminology and Concepts: Asset Prices, States, Returns, and Pay-Offs 315 5.2 Arbitrage Portfolios and the Arbitrage Theorem 317 5.3 An Example of Single-Period Asset Pricing: Risk-Neutral Probabilities and Arbitrage 318 5.4 Arbitrage Detection and the Formation of Arbitrage Portfolios in the N-Dimensional Case 319 CHAPTER 6 Project: The Black–Scholes (Lognormal) Model 321 6.1 Black–Scholes Pricing Formula 321 6.2 Black–Scholes Sensitivity Analysis 325 CHAPTER 7 Project: Quantile-Quantile Plots 327 7.1 Log-Returns and Standardization 327 7.2 Quantile-Quantile Plots 328 CHAPTER 8 Project: Monte Carlo Pricer 331 8.1 Scenario Generation 331 8.2 Calibration 332 8.3 Pricing Equity Basket Options 333 CHAPTER 9 Project: The Binomial Lattice Model 337 9.1 Building the Lattice 337 9.2 Lattice Calibration and Pricing 339 CHAPTER 10 Project: The Trinomial Lattice Model 341 10.1 Building the Lattice 341 10.1.1 Case 1 ( = 0) 342 10.1.2 Case 2 (Another Geometry with = 0) 343 10.1.3 Case 3 (Geometry with p+ = p−: Drifted Lattice) 343 10.2 Pricing Procedure 344 10.3 Calibration 346 10.4 Pricing Barrier Options 346 10.5 Put-Call Parity in Trinomial Lattices 347 10.6 Computing the Sensitivities 348 Contents ix CHAPTER 11 Project: Crank–Nicolson Option Pricer 349 11.1 The Lattice for the Crank–Nicolson Pricer 349 11.2 Pricing with Crank–Nicolson 350 11.3 Calibration 351 11.4 Pricing Barrier Options 352 CHAPTER 12 Project: Static Hedging of Barrier Options 355 12.1 Analytical Pricing Formulas for Barrier Options 355 12.1.1 Exact Formulas for Barrier Calls for the Case H ≤ K 355 12.1.2 Exact Formulas for Barrier Calls for the Case H ≥ K 356
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