Free ebooks ==> www.Ebook777.com www.Ebook777.com P1: TIX/XYZ JWBK527-fm P2: ABC JWBK527-Morini September 2, 2011 19:37 Printer: Yet to come Free ebooks ==> www.Ebook777.com www.Ebook777.com P1: TIX/XYZ JWBK527-fm P2: ABC JWBK527-Morini September 2, 2011 19:37 Printer: Yet to come Understanding and Managing Model Risk i P1: TIX/XYZ JWBK527-fm P2: ABC JWBK527-Morini September 2, 2011 19:37 Printer: Yet to come For other titles in the Wiley Finance series please see www.wiley.com/finance ii P1: TIX/XYZ JWBK527-fm P2: ABC JWBK527-Morini September 2, 2011 19:37 Printer: Yet to come Free ebooks ==> www.Ebook777.com Understanding and Managing Model Risk A Practical Guide for Quants, Traders and Validators Massimo Morini A John Wiley & Sons, Ltd., Publication www.Ebook777.com iii P1: TIX/XYZ JWBK527-fm P2: ABC JWBK527-Morini September 2, 2011 19:37 Printer: Yet to come This edition first published 2011 Copyright © 2011 John Wiley & Sons, Ltd Registered Office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com The rights of Massimo Morini to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved 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 or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher Wiley publishes in a variety of electronic formats and by print-on-demand Some material included with standard print versions of this book may not be included in e-books or in print-on-demand If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com For more information about Wiley products, visit www.wiley.com Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The publisher is not associated with any product or vendor mentioned in this book This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Library of Congress Cataloging-in-Publication Data: Understanding and managing model risk : a practical guide for quants, traders and validators / [edited by] Massimo Morini – 1st ed p cm – (Wiley finance series) Includes bibliographical references and index ISBN 978-0-470-97761-3 (hardback) Risk management Risk management–Mathematical models I Morini, Massimo HD61.U53 2011 332.64 5–dc23 2011031397 ISBN 978-0-470-97761-3 (hbk), ISBN 978-1-119-96085-0 (ebk), ISBN 978-0-470-97774-3 (ebk), ISBN 978-0-470-97775-0 (ebk) A catalogue record for this book is available from the British Library Set in 10/12pt Times by Aptara Inc., New Delhi, India Printed and bound by CPI Group (UK) Ltd, Croyden, CRO 4YY iv P1: TIX/XYZ JWBK527-fm P2: ABC JWBK527-Morini September 2, 2011 19:37 Printer: Yet to come Contents Preface xi Acknowledgements PART I xix THEORY AND PRACTICE OF MODEL RISK MANAGEMENT Understanding Model Risk 1.1 What Is Model Risk? 1.1.1 The Value Approach 1.1.2 The Price Approach 1.1.3 A Quant Story of the Crisis 1.1.4 A Synthetic View on Model Risk 1.2 Foundations of Modelling and the Reality of Markets 1.2.1 The Classic Framework 1.2.2 Uncertainty and Illiquidity 1.3 Accounting for Modellers 1.3.1 Fair Value 1.3.2 The Liquidity Bubble and the Accountancy Boards 1.3.3 Level 1, 2, go? 1.3.4 The Hidden Model Assumptions in ‘vanilla’ Derivatives 1.4 What Regulators Said After the Crisis 1.4.1 Basel New Principles: The Management Process 1.4.2 Basel New Principles: The Model, The Market and The Product 1.4.3 Basel New Principles: Operative Recommendations 1.5 Model Validation and Risk Management: Practical Steps 1.5.1 A Scheme for Model Validation 1.5.2 Special Points in Model Risk Management 1.5.3 The Importance of Understanding Models 3 17 22 22 30 38 38 40 41 42 48 49 51 52 53 54 59 60 Model Validation and Model Comparison: Case Studies 2.1 The Practical Steps of Model Comparison 2.2 First Example: The Models 2.2.1 The Credit Default Swap 2.2.2 Structural First-Passage Models 63 63 65 66 67 v P1: TIX/XYZ JWBK527-fm P2: ABC JWBK527-Morini vi September 2, 2011 19:37 Printer: Yet to come Contents 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.2.3 Reduced-Form Intensity Models 2.2.4 Structural vs Intensity: Information First Example: The Payoff Gap Risk in a Leveraged Note The Initial Assessment 2.4.1 First Test: Calibration to Liquid Relevant Products 2.4.2 Second Test: a Minimum Level of Realism The Core Risk in the Product 2.5.1 Structural Models: Negligible Gap Risk 2.5.2 Reduced-Form Models: Maximum Gap Risk A Deeper Analysis: Market Consensus and Historical Evidence 2.6.1 What to Add to the Calibration Set 2.6.2 Performing Market Intelligence 2.6.3 The Lion and the Turtle Incompleteness in Practice 2.6.4 Reality Check: Historical Evidence and Lack of it Building a Parametric Family of Models 2.7.1 Understanding Model Implications Managing Model Uncertainty: Reserves, Limits, Revisions Model Comparison: Examples from Equity and Rates 2.9.1 Comparing Local and Stochastic Volatility Models in Pricing Equity Compound and Barrier Options 2.9.2 Comparing Short Rate and Market Models in Pricing Interest Rate Bermudan Options Stress Testing and the Mistakes of the Crisis 3.1 Learning Stress Test from the Crisis 3.1.1 The Meaning of Stress Testing 3.1.2 Portfolio Stress Testing 3.1.3 Model Stress Testing 3.2 The Credit Market and the ‘Formula that Killed Wall Street’ 3.2.1 The CDO Payoff 3.2.2 The Copula 3.2.3 Applying the Copula to CDOs 3.2.4 The Market Quotation Standard 3.3 Portfolio Stress Testing and the Correlation Mistake 3.3.1 From Flat Correlation Towards a Realistic Approach 3.3.2 A Correlation Parameterization to Stress the Market Skew 3.4 Payoff Stress and the Liquidity Mistake 3.4.1 Detecting the Problem: Losses Concentrated in Time 3.4.2 The Problem in Practice 3.4.3 A Solution From Copulas to Real Models 3.4.4 Conclusions 3.5 Testing with Historical Scenarios and the Concentration Mistake 3.5.1 The Mapping Methods for Bespoke Portfolios 3.5.2 The Lehman Test 3.5.3 Historical Scenarios to Test Mapping Methods 3.5.4 The Limits of Mapping and the Management of Model Risk 3.5.5 Conclusions 69 72 74 77 77 78 81 82 82 85 85 86 86 87 88 93 95 99 99 105 111 111 112 113 116 118 118 119 122 124 125 126 131 136 137 139 145 150 151 152 156 157 164 168 P1: TIX/XYZ JWBK527-fm P2: ABC JWBK527-Morini September 2, 2011 19:37 Printer: Yet to come Contents Preparing for Model Change Rates and Funding in the New Era 4.1 Explaining the Puzzle in the Interest Rates Market and Models 4.1.1 The Death of a Market Model: August 2007 4.1.2 Finding the New Market Model 4.1.3 The Classic Risk-free Market Model 4.1.4 A Market Model with Stable Default Risk 4.1.5 A Market with Volatile Credit Risk 4.1.6 Conclusions 4.2 Rethinking the Value of Money: The Effect of Liquidity in Pricing 4.2.1 The Setting 4.2.2 Standard DVA: Is Something Missing? 4.2.3 Standard DVA plus Liquidity: Is Something Duplicated? 4.2.4 Solving the Puzzle 4.2.5 Risky Funding for the Borrower 4.2.6 Risky Funding for the Lender and the Conditions for Market Agreement 4.2.7 Positive Recovery Extension 4.2.8 Two Ways of Looking at the Problem: Default Risk or Funding Benefit? The Accountant vs the Salesman 4.2.9 Which Direction for Future Pricing? vii 171 171 173 174 178 182 192 200 201 204 206 207 207 208 209 210 211 214 PART II SNAKES IN THE GRASS: WHERE MODEL RISK HIDES Hedging 5.1 Model Risk and Hedging 5.2 Hedging and Model Validation: What is Explained by P&L Explain? 5.2.1 The Sceptical View 5.2.2 The Fundamentalist View and Black and Scholes 5.2.3 Back to Reality 5.2.4 Remarks: Recalibration, Hedges and Model Instability 5.2.5 Conclusions: from Black and Scholes to Real Hedging 5.3 From Theory to Practice: Real Hedging 5.3.1 Stochastic Volatility Models: SABR 5.3.2 Test Hedging Behaviour Leaving Nothing Out 5.3.3 Real Hedging for Local Volatility Models 5.3.4 Conclusions: the Reality of Hedging Strategies 219 219 221 222 222 224 226 228 229 231 232 238 241 Approximations 6.1 Validate and Monitor the Risk of Approximations 6.2 The Swaption Approximation in the Libor Market Model 6.2.1 The Three Technical Problems in Interest Rate Modelling 6.2.2 The Libor Market Model and the Swaption Market 6.2.3 Pricing Swaptions 6.2.4 Understanding and Deriving the Approximation 6.2.5 Testing the Approximation 6.3 Approximations for CMS and the Shape of the Term Structure 6.3.1 The CMS Payoff 6.3.2 Understanding Convexity Adjustments 243 243 245 245 247 250 253 257 264 265 266 P1: TIX/XYZ JWBK527-fm P2: ABC JWBK527-Morini September 2, 2011 19:37 Printer: Yet to come Free ebooks ==> www.Ebook777.com viii Contents 6.3.3 The Market Approximation for Convexity Adjustments 6.3.4 A General LMM Approximation 6.3.5 Comparing and Testing the Approximations 6.4 Testing Approximations Against Exact Dupire’s Idea 6.4.1 Perfect Positive Correlation 6.4.2 Perfect Negative Correlation 6.5 Exercises on Risk in Computational Methods 6.5.1 Approximation 6.5.2 Integration 6.5.3 Monte Carlo 267 269 271 276 278 280 283 283 285 285 Extrapolations 7.1 Using the Market to Complete Information: Asymptotic Smile 7.1.1 The Indetermination in the Asymptotic Smile 7.1.2 Pricing CMS with a Smile: Extrapolating to Infinity 7.1.3 Using CMS Information to Transform Extrapolation into Interpolation and Fix the Indetermination 7.2 Using Mathematics to Complete Information: Correlation Skew 7.2.1 The Expected Tranched Loss 7.2.2 Properties for Interpolation 7.2.3 Properties for Turning Extrapolation into Interpolation 293 295 295 298 298 Correlations 8.1 The Technical Difficulties in Computing Correlations 8.1.1 Correlations in Interest Rate Modelling 8.1.2 Cross-currency Correlations 8.1.3 Stochastic Volatility Correlations 8.2 Fundamental Errors in Modelling Correlations 8.2.1 The Zero-correlation Error 8.2.2 The 1-Correlation Error 303 303 305 307 312 315 316 319 Calibration 9.1 Calibrating to Caps/Swaptions and Pricing Bermudans 9.1.1 Calibrating Caplets 9.1.2 Understanding the Term Structure of Volatility 9.1.3 Different Parameterizations 9.1.4 The Evolution of the Term Structure of Volatility 9.1.5 The Effect on Early-Exercise Derivatives 9.1.6 Reducing Our Indetermination in Pricing Bermudans: Liquid European Swaptions 9.2 The Evolution of the Forward Smiles 323 324 325 326 329 332 334 10 When the Payoff is Wrong 10.1 The Link Between Model Errors and Payoff Errors 10.2 The Right Payoff at Default: The Impact of the Closeout Convention 10.2.1 How Much Will be Paid at Closeout, Really? 10.2.2 What the Market Says and What the ISDA Says 347 347 348 350 352 www.Ebook777.com 287 288 288 292 335 340 ... should be sought Library of Congress Cataloging-in-Publication Data: Understanding and managing model risk : a practical guide for quants, traders and validators / [edited by] Massimo Morini – 1st... Terraneo, Massimo Baldi, Francesco Lago, Stefano Martina, Alessandro Ravogli, Cristiano Maffi, Valeria Anzoino, Emiliano Carchen, Raffaele Lovero Marco Bianchetti and Andrea Prampolini are thanked for. .. regulations and human behaviour, and control for this in our management of model risk The fact that thousands of technical papers speak of very advanced models, and just a handful focus on model risk and