The SABR/LIBOR Market Model Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives

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1 The SABR/LIBOR Market Model Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives Riccardo Rebonato Kenneth McKay and Richard White A John Wiley and Sons, Ltd., Publication

3 The SABR/LIBOR Market Model

5 The SABR/LIBOR Market Model Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives Riccardo Rebonato Kenneth McKay and Richard White A John Wiley and Sons, Ltd., Publication

6 This edition first published , 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 The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 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 also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. 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. ISBN A catalogue record for this book is available from the British Library. Set in 10/12pt Times New Roman by Laserwords Private Ltd, Chennai, India. Printed in England by Antony Rowe Ltd, Chippenham, Wiltshire

7 Contents Acknowledgements xi 1 Introduction 1 I The Theoretical Set-Up 7 2 The LIBOR Market Model Definitions The Volatility Functions Separating the Correlation from the Volatility Term The Caplet-Pricing Condition Again The Forward-Rate/Forward-Rate Correlation The Simple Exponential Correlation The Multiplicative Correlation Possible Shapes of the Doust Correlation Function The Covariance Integral Again 21 3 The SABR Model The SABR Model (and Why it is a Good Model) Description of the Model The Option Prices Given by the SABR Model Special Cases ATM Options The Normal Case (β = 0) The Log-Normal Case (β = 1) Qualitative Behaviour of the SABR Model Dependence on σ0 T Dependence on β Dependence on ρ Dependence on ν The Link Between the Exponent, β, and the Volatility of Volatility, ν 35 v

8 vi CONTENTS 3.7 Volatility Clustering in the (LMM)-SABR Model The Market Analysis of σ0 T (β = 0.5) Analysis of ν T (β = 0.5) Analysis of ρ T (β = 0.5) How Do We Know that the Market has Chosen β = 0.5? The Problems with the SABR Modely Log-Normality of the Volatility Process Problems with the (Stochastic) CEV Process 47 4 The LMM-SABR Model The Equations of Motion The Nature of the Stochasticity Introduced by Our Model A Simple Correlation Structure A More General Correlation Structure Observations on the Correlation Structure The Volatility Structure What We Mean by Time Homogeneity The Volatility Structure in Periods of Market Stress A More General Stochastic Volatility Dynamics Calculating the No-Arbitrage Drifts Preliminaries Standard LIBOR and LIBOR in Arrearsy LIBOR in Arrears: The Volatility Drift The Drifts in the General Case of Several Forward Ratesy Volatility Drifts in the Swap Measure 75 II Implementation and Calibration 79 5 Calibrating the LMM-SABR Model to Market Caplet Prices The Caplet-Calibration Problem Choosing the Parameters of the Function, g( ), and the Initial Values, k0 T Choosing the Parameters of the Function h( ) Choosing the Exponent, β, and the Correlation, φ SABR Results Calibration in Practice: Implications for the SABR Model Looking at Caplets in Isolation Looking at Caplets and Swaptions Together Implications for Model Choice 99 6 Calibrating the LMM-SABR Model to Market Swaption Prices The Swaption Calibration Problem 101

9 CONTENTS vii 6.2 Swap Rate and Forward Rate Dynamics Approximating the Instantaneous Swap Rate Volatility, S t Approximating the Initial Value of the Swap Rate Volatility, 0 (First Route) Approximating 0 (Second Route) and the Volatility of Volatility of the Swap Rate, V Approximating the Swap-Rate/Swap-Rate-Volatility Correlation, R SABR Approximating the Swap Rate Exponent, B Results Comparison between Approximated and Simulation Prices Comparison between Parameters from the Approximations and the Simulations Conclusions and Suggestions for Future Work Appendix: Derivation of Approximate Swap Rate Volatility Appendix: Derivation of Swap-Rate/Swap-Rate-Volatility Correlation, R SABR Appendix: Approximation of ds t /S t Calibrating the Correlation Structure Statement of the Problem Creating a Valid Model Matrix First Strategy, Stage 1: Diagonalize P First Strategy, Stage 2: Analytic Optimization of c i Second Strategy: Optimizing over Angles A Case Study: Calibration Using the Hypersphere Method Which Method Should One Choose? Appendix 138 III Empirical Evidence The Empirical Problem Statement of the Empirical Problem What Do We Know from the Literature? Data Description Distributional Analysis and Its Limitations What is the True Exponent β? Appendix: Some Analytic Results Estimating the Volatility of the Forward Rates Expiry Dependence of Volatility of Forward Rates Direct Estimation Looking at the Normality of the Residuals Maximum-Likelihood and Variations on the Theme 171

10 viii CONTENTS 9.5 Information About the Volatility from the Options Market Overall Conclusions Estimating the Correlation Structure What We are Trying to Do Some Results from Random Matrix Theory Empirical Estimation Descriptive Statistics The Forward-Rate/Forward-Rate Correlation Matrix The Forward-Rate/Volatility Correlation Block The Volatility/Volatility Correlation Matrix Signal and Noise in the Empirical Correlation Blocks The Forward-Rate/Forward-Rate Correlation Matrix The Volatility/Volatility Correlation Matrix The Forward-Rate/Volatility Correlation Block What Does Random Matrix Theory Really Tell Us? Calibrating the Correlation Matrices The Fitting Procedure Results How Much Information Do the Proposed Models Retain? Eigenvalues of the Correlation Blocks Eigenvalues of Differences in the Correlation Blocks Entropy Measures The Forward-Rate/Volatility Correlation Block 202 IV Hedging Various Types of Hedging Statement of the Problem Three Types of Hedging In- and Out-of-Model Hedging Functional-Dependence Hedging Definitions First-Order Derivatives with Respect to the Underlyings Delta Hedging Vega Hedging Second-Order Derivatives with Respect to the Underlyings Vanna and Volga Generalizing Functional-Dependence Hedging How Does the Model Know about Vanna and Volga? Choice of Hedging Instrument 220

11 CONTENTS ix 12 Hedging against Moves in the Forward Rate and in the Volatility Delta Hedging in the SABR-(LMM) Model Vega Hedging in the SABR-(LMM) Model (LMM)-SABR Hedging in Practice: Evidence from Market Data Purpose of this Chapter Notation Estimation of the Unobservable Volatility Tests of the Hedging Performance of the SABR Model Tests of the Hedging Performance of the LMM-SABR Model Hedging Results for the SABR Model Hedging Results for the LMM-SABR Model Conclusions Hedging the Correlation Structure The Intuition Behind the Problem Hedging the Forward-Rate Block Hedging the Volatility-Rate Block Hedging the Forward-Rate/Volatility Block Final Considerations Hedging in Conditions of Market Stress Statement of the Problem The Volatility Function The Case Study Hedging The Normal-to-Normal State Transition The Normal-to-Excited Transition Normal-to-Unknown Transition Starting from the Excited State Results Hedging Results for the Normal-to-Normal Transition Hedging Results for the Normal-to-Excited Transition Are We Getting Something for Nothing? 270 References 271 Index 275

13 Acknowledgements It is a pleasure to acknowledge the help provided by many colleagues and friends. In particular, the advice and suggestions of Paul Doust, Andrei Pogudin, Jian Chen, Raphael Albrecht, Dhermider Kainth and Michael Dogwood have been of great help. This book is much the better thanks to them. We are grateful to John Wiley for agreeing to publish this book, and for the enthusiasm they have shown for the project. Caitlin Cornish has been a most efficient and supportive commissioning editor. Finally, two of us (RR and KM) cannot help feeling some pangs of envy towards our third co-author, Richard. Unfortunately for us, but probably wisely for him, a few months into the project Richard decided to take a year off to tour the world with his girlfriend. We suspect that the pleasures of proofreading and reference checking may have played a part in making trekking through Siberia appear more attractive than it is normally cracked up to be. Be that as it may, his contribution to this book has been so important that, proofreading or no proofreading, he has earned full authorship, and we feel proud to have him as third co-author. (Just don t do this again, Richard.) xi

15 Chapter 1 Introduction All models are wrong, but some models are useful We present in this book a financially motivated extension of the LIBOR market model that reproduces for all strikes and maturities the prices of the plain-vanilla hedging instruments (swaptions and caplets) produced by the SABR model. In other words, our extension of the LIBOR market model accurately recovers in a financially motivated manner the whole of the SABR smile surface. As the SABR model has become the market standard for European options, just the recovery of the smile surface by a dynamic model could be regarded as a useful achievement in itself. However, we have tried to do more. As we have stressed in the opening sentences, we have tried to accomplish this task in a way that we consider financially justifiable. Our reason for insisting on financial reasonableness is not (just) an aesthetic one. We believe that the quality of a derivatives model should be judged not just on the basis of its ability to price today s hedging instruments, but also on the basis of the quality of the hedges it suggests. We believe that these hedges can be good only if the model is rooted in empirical financial reality. The empirical financial reality of relevance for the pricing and hedging of complex derivatives is the dynamics of the smile surface. We explain below why we believe that this is the case. We are therefore not just offering yet another model. We present a philosophy of option pricing that takes into account the realities of the industry needs (e.g., the need to calibrate as accurately as possible to the plain-vanilla reference hedging instruments, the need to obtain prices and hedges in reasonable time) while reproducing a realistic future evolution of the smile surface (our financial reality ). Until recently choosing between fitting today s prices very accurately and being respectful of financial reality (given our meaning of the term) entailed making hard choices. For instance, some approaches, such as local-volatility modelling (see, e.g., Dupire (1994), Derman and Kani (1994)), fulfilled (by construction) very well the first set of requirements (perfect fitting of today s smile). This made local volatility models very popular with some traders. Yet, the dynamics of the smile these models implied were completely wrong. Indeed, the SABR model, which constitutes the starting point for our extension, was introduced to remedy the wrong dynamics imposed by the local-volatility framework. 1

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