Volatility and Correlation2ndEditionThe Perfect Hedger and the FoxRiccardo RebonatoVolatility and Correlation2ndEditionVolatility and Correlation2ndEditionThe Perfect Hedger and the FoxRiccardo RebonatoPublished 2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,West Sussex PO19 8SQ, EnglandTelephone (+44) 1243 779777Copyright 2004 Riccardo RebonatoEmail (for orders and customer service enquiries): cs-books@wiley.co.ukVisit our Home Page on www.wileyeurope.comor www.wiley.comAll Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted inany form or by any means, electronic, mechanical, photocopying,recording, scanning or otherwise, except underthe terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by theCopyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission inwriting of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, JohnWiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed topermreq@wiley.co.uk,or faxed to (+44) 1243 770620.This publication is designed to provide accurate and authoritative information in regard to the subject mattercovered. It is sold on the understandingthat the Publisher is not engaged in rendering professional services. Ifprofessional advice or other expert assistance is required, the services of a competent professional should besought.Other Wiley Editorial OfcesJohn Wiley & Sons Inc., 111 River Street, Hoboken,NJ 07030, USAJossey-Bass, 989 Market Street, San Francisco, CA 94103-1741,USAWiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, GermanyJohn Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, AustraliaJohn Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1Wiley also publishesits booksin a variety of electronic formats. Some content that appearsin print may not be available in electronic books.Library of Congress Cataloging-in-Publication DataRebonato, Riccardo.Volatility and correlation: the perfect hedger and the fox/RiccardoRebonato 2nd ed.p. cm.Rev. ed. of: Volatility and correlation in the pricing of equity. 1999.Includes bibliographicalreferences and index.ISBN 0-470-09139-8(cloth: alk. paper)1. Options (Finance) Mathematical models. 2. Interest ratefutures Mathematical models. 3. Securities Prices Mathematical models.I. Rebonato, Riccardo. Volatility and correlation in the pricing of equity.II. Title.HG6024.A3R432004332.64

53 dc222004004223British Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryISBN 0-470-09139-8Typeset in 10/12 Times by Laserwords Private Limited, Chennai, IndiaPrinted and bound in Great Britain by TJ International,Padstow, CornwallThis book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production.To my parentsTo RosamundContentsPreface xxi0.1 Why a Second Edition? xxi0.2 What This Book Is Not About xxiii0.3 Structure of the Book xxiv0.4 The New Subtitle xxivAcknowledgements xxviiI Foundations 11 Theory and Practice of Option Modelling 31.1 The Role of Models in Derivatives Pricing 31.1.1 What Are Models For? 31.1.2 The Fundamental Approach 51.1.3 The Instrumental Approach 71.1.4 A Conundrum (or, What is Vega Hedging For?) 81.2 The Efcient Market Hypothesis and Why It Matters for Option Pricing 91.2.1 The Three Forms of the EMH 91.2.2 Pseudo-Arbitrageurs in Crisis 101.2.3 Model Risk for Traders and Risk Managers 111.2.4 The Parable of the Two Volatility Traders 121.3 Market Practice 141.3.1 Different Users of Derivatives Models 141.3.2 In-Model and Out-of-Model Hedging 151.4 The Calibration Debate 171.4.1 Historical vs Implied Calibration 181.4.2 The Logical Underpinning of the Implied Approach 191.4.3 Are Derivatives Markets Informationally Efcient? 211.4.4 Back to Calibration 261.4.5 A Practical Recommendation 27viiviii CONTENTS1.5 Across-Markets Comparison of Pricing and Modelling Practices 271.6 Using Models 302 Option Replication 312.1 The Bedrock of Option Pricing 312.2 The Analytic (PDE) Approach 322.2.1 The Assumptions 322.2.2 The Portfolio-Replication Argument (Deterministic Volatility) 322.2.3 The Market Price of Risk with Deterministic Volatility 342.2.4 Link with Expectations the FeynmanKac Theorem 362.3 Binomial Replication 382.3.1 First Approach Replication Strategy 392.3.2 Second Approach Nave Expectation 412.3.3 Third Approach Market Price of Risk 422.3.4 A Worked-Out Example 452.3.5 Fourth Approach Risk-Neutral Valuation 462.3.6 Pseudo-Probabilities 482.3.7 Are the Quantities1and2Really Probabilities? 492.3.8 Introducing Relative Prices 512.3.9 Moving to a Multi-Period Setting 532.3.10 Fair Prices as Expectations 562.3.11 Switching Numeraires and Relating Expectations UnderDifferent Measures 582.3.12 Another Worked-Out Example 612.3.13 Relevance of the Results 642.4 Justifying the Two-State Branching Procedure 652.4.1 How To Recognize a Jump When You See One 652.5 The Nature of the Transformation between Measures: Girsanovs Theorem692.5.1 An Intuitive Argument 692.5.2 A Worked-Out Example 702.6 Switching Between the PDE, the Expectation and the BinomialReplication Approaches 733 The Building Blocks 753.1 Introduction and Plan of the Chapter 753.2 Denition of Market Terms 753.3 Hedging Forward Contracts Using Spot Quantities 773.3.1 Hedging Equity Forward Contracts 783.3.2 Hedging Interest-Rate Forward Contracts 793.4 Hedging Options: Volatility of Spot and Forward Processes 80CONTENTS ix3.5 The Link Between Root-Mean-Squared Volatilities and theTime-Dependence of Volatility 843.6 Admissibilityof a Series of Root-Mean-Squared Volatilities 853.6.1 The Equity/FX Case 853.6.2 The Interest-Rate Case 863.7 Summary of the Denitions So Far 873.8 Hedging an Option with a Forward-Setting Strike 893.8.1 Why Is This Option Important? (And Why Is it Difcultto Hedge?) 903.8.2 Valuing a Forward-Setting Option 913.9 Quadratic Variation: First Approach 953.9.1 Denition 953.9.2 Properties of Variations 963.9.3 First and Second Variation of a Brownian Process 973.9.4 Links between Quadratic Variation and Tt(u)2du 973.9.5 Why Quadratic Variation Is So Important (Take 1) 984 Variance and Mean Reversion in the Real and the Risk-Adjusted Worlds 1014.1 Introduction and Plan of the Chapter 1014.2 Hedging a Plain-Vanilla Option: General Framework 1024.2.1 Trading Restrictions and Model Uncertainty:Theoretical Results 1034.2.2 The Setting 1044.2.3 The Methodology 1044.2.4 Criterion for Success 1064.3 Hedging Plain-Vanilla Options: Constant Volatility 1064.3.1 Trading the Gamma: One Step and Constant Volatility 1084.3.2 Trading the Gamma: Several Steps and Constant Volatility 1144.4 Hedging Plain-Vanilla Options: Time-Dependent Volatility 1164.4.1 Views on Gamma Trading When the Volatility is TimeDependent 1164.4.2 Which View Is the Correct One? (and the FeynmanKacTheorem Again) 1194.5 Hedging Behaviour In Practice 1214.5.1 Analysing the Replicating Portfolio 1214.5.2 Hedging Results: the Time-Dependent Volatility Case 1224.5.3 Hedging with the Wrong Volatility 1254.6 Robustness of the Black-and-Scholes Model 1274.7 Is the Total Variance All That Matters? 1304.8 Hedging Plain-Vanilla Options: Mean-Reverting Real-World Drift 131x CONTENTS4.9 Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again 1354.9.1 The CrouhyGalai Set-Up 1355 Instantaneous and Terminal Correlation 1415.1 Correlation, Co-Integration and Multi-Factor Models 1415.1.1 The Multi-Factor Debate 1445.2 The Stochastic Evolution of Imperfectly Correlated Variables 1465.3 TheRoleof Terminal CorrelationintheJoint Evolutionof StochasticVariables 1515.3.1 Dening Stochastic Integrals 1515.3.2 Case 1: European Option, One Underlying Asset 1535.3.3 Case 2: Path-Dependent Option, One Asset 1555.3.4 Case 3: Path-Dependent Option, Two Assets 1565.4 Generalizing the Results 1625.5 Moving Ahead 164II Smiles Equity and FX 1656 Pricing Options in the Presence of Smiles 1676.1 Plan of the Chapter 1676.2 Background and Denition of the Smile 1686.3 Hedging with a Compensated Process: Plain-Vanilla and Binary Options 1696.3.1 Delta- and Vega-Hedging a Plain-Vanilla Option 1696.3.2 Pricing a European Digital Option 1726.4 Hedge Ratios for Plain-Vanilla Options in the Presence of Smiles 1736.4.1 The Relationship Between the True Call Price Functionaland the Black Formula 1746.4.2 Calculating the Delta Using the Black Formula and theImplied Volatility 1756.4.3 Dependence of Implied Volatilities on the Strike and theUnderlying 1766.4.4 Floating and Sticky Smiles and What They Imply about Changesin Option Prices 1786.5 Smile Tale 1: Sticky Smiles 1806.6 Smile Tale 2: Floating Smiles 1826.6.1 Relevance of the Smile Story for Floating Smiles 1836.7 When Does Risk Aversion Make a Difference? 1846.7.1 Motivation 1846.7.2 The Importance of an Assessment of Risk Aversionfor Model Building 1856.7.3 The Principle of Absolute Continuity 186CONTENTS xi6.7.4 The Effect of Supply and Demand 1876.7.5 A Stylized Example: First Version 1876.7.6 A Stylized Example: Second Version 1946.7.7 A Stylized Example: Third Version 1966.7.8 Overall Conclusions 1966.7.9 The EMH Again 1997 Empirical Facts About Smiles 2017.1 What is this Chapter About? 2017.1.1 Fundamental and Derived Analyses 2017.1.2 A Methodological Caveat 2027.2 Market Information About Smiles 2037.2.1 Direct Static Information 2037.2.2 Semi-Static Information 2047.2.3 Direct Dynamic Information 2047.2.4 Indirect Information 2057.3 Equities 2067.3.1 Basic Facts 2067.3.2 Subtler Effects 2067.4 Interest Rates 2227.4.1 Basic Facts 2227.4.2 Subtler Effects 2247.5 FX Rates 2277.5.1 Basic Facts 2277.5.2 Subtler Effects 2277.6 Conclusions 2358 General Features of Smile-Modelling Approaches 2378.1 Fully-Stochastic-Volatility Models 2378.2 Local-Volatility (Restricted-Stochastic-Volatility) Models 2398.3 JumpDiffusion Models 2418.3.1 Discrete Amplitude 2418.3.2 Continuum of Jump Amplitudes 2428.4 VarianceGamma Models 2438.5 Mixing Processes 2438.5.1 A Pragmatic Approach to Mixing Models 2448.6 Other Approaches 2458.6.1 Tight Bounds with Known Quadratic Variation 2458.6.2 Assigning Directly the Evolution of the Smile Surface 2468.7 The Importance of the Quadratic Variation (Take 2) 246xii CONTENTS9 The Input Data: Fitting an Exogenous Smile Surface 2499.1 What is This Chapter About? 2499.2 Analytic Expressions for Calls vs Process Specication 2499.3 Direct Use of Market Prices: Pros and Cons 2509.4 Statement of the Problem 2519.5 Fitting Prices 2529.6 Fitting Transformed Prices 2549.7 Fitting the Implied Volatilities 2559.7.1 The Problem with Fitting the Implied Volatilities 2559.8 Fitting the Risk-Neutral Density Function General 2569.8.1 Does It Matter if the Price Density Is Not Smooth? 2579.8.2 Using Prior Information (Minimum Entropy) 2589.9 Fitting the Risk-Neutral Density Function: Mixture of Normals 2599.9.1 Ensuring the Normalization and Forward Constraints 2619.9.2 The Fitting Procedure 2649.10 Numerical Results 2659.10.1 Description of the Numerical Tests 2659.10.2 Fitting to Theoretical Prices: Stochastic-Volatility Density 2659.10.3 Fitting to Theoretical Prices: VarianceGamma Density 2689.10.4 Fitting to Theoretical Prices: JumpDiffusion Density 2709.10.5 Fitting to Market Prices 2729.11 Is the TermCSReally a Delta? 2759.12 Fitting the Risk-Neutral Density Function:The Generalized-Beta Approach 2779.12.1 Derivation of Analytic Formulae 2809.12.2 Results and Applications 2879.12.3 What Does This Approach Offer? 29110Quadratic Variation and Smiles 29310.1 Why This Approach Is Interesting 29310.2 The BJN Framework for Bounding Option Prices 29310.3 The BJN Approach Theoretical Development 29410.3.1 Assumptions and Denitions 29410.3.2 Establishing Bounds 29710.3.3 Recasting the Problem 29810.3.4 Finding the Optimal Hedge 29910.4 The BJN Approach: Numerical Implementation 30010.4.1 Building a Traditional Tree 30110.4.2 Building a BJN Tree for a Deterministic Diffusion 30110.4.3 Building a BJN Tree for a General Process 30410.4.4 Computational Results 307CONTENTS xiii10.4.5 Creating Asymmetric Smiles 30910.4.6 Summary of the Results 31110.5 Discussion of the Results 31210.5.1 Resolution of the CrouhyGalai Paradox 31210.5.2 The Difference Between Diffusions and JumpDiffusionProcesses: the Sample Quadratic Variation 31210.5.3 How Can One Make the Approach More Realistic? 31410.5.4 The Link with Stochastic-Volatility Models 31410.5.5 The Link with Local-Volatility Models 31510.5.6 The Link with JumpDiffusion Models 31510.6 Conclusions (or, Limitations of Quadratic Variation) 31611Local-Volatility Models: the Derman-and-Kani Approach 31911.1 General Considerations on Stochastic-Volatility Models 31911.2 Special Cases of Restricted-Stochastic-Volatility Models 32111.3 The Dupire, Rubinstein and Derman-and-Kani Approaches 32111.4 Greens Functions (ArrowDebreu Prices) in the DK Construction 32211.4.1 Denition and Main Properties of ArrowDebreu Prices 32211.4.2 Efcient Computation of ArrowDebreu Prices 32411.5 The Derman-and-Kani Tree Construction 32611.5.1 Building the First Step 32711.5.2 Adding Further Steps 32911.6 Numerical Aspects of the Implementation of the DK Construction 33111.6.1 Problem 1: Forward Price Greater ThanS(up) or SmallerThanS(down) 33111.6.2 Problem 2: Local Volatility Greater Than12|S(up) S(down)| 33211.6.3 Problem 3: Arbitrariness of the Choice of the Strike 33211.7 Implementation Results 33411.7.1 Benchmarking 1: The No-Smile Case 33411.7.2 Benchmarking 2: The Time-Dependent-Volatility Case 33511.7.3 Benchmarking 3: Purely Strike-Dependent Implied Volatility 33611.7.4 Benchmarking 4: Strike-and-Maturity-Dependent ImpliedVolatility 33711.7.5 Conclusions 33811.8 Estimating Instantaneous Volatilities from Prices as an Inverse Problem 34312Extracting the Local Volatility from Option Prices 34512.1 Introduction 34512.1.1 A Possible Regularization Strategy 34612.1.2 Shortcomings 34612.2 The Modelling Framework 34712.3 A Computational Method 349xiv CONTENTS12.3.1 Backward Induction 34912.3.2 Forward Equations 35012.3.3 Why Are We Doing Things This Way? 35212.3.4 Related Approaches 35412.4 Computational Results 35512.4.1 Are We Looking at the Same Problem? 35612.5 The Link Between Implied and Local-Volatility Surfaces 35712.5.1 Symmetric (FX) Smiles 35812.5.2 Asymmetric (Equity) Smiles 36112.5.3 Monotonic (Interest-Rate) Smile Surface 36812.6 Gaining an Intuitive Understanding 36812.6.1 Symmetric Smiles 36912.6.2 Asymmetric Smiles: One-Sided Parabola 37012.6.3 Asymmetric Smiles: Monotonically Decaying 37212.7 What Local-Volatility Models Imply about Sticky and Floating Smiles 37312.8 No-Arbitrage Conditions on the Current Implied Volatility Smile Surface 37512.8.1 Constraints on the Implied Volatility Surface 37512.8.2 Consequences for Local Volatilities 38112.9 Empirical Performance 38512.10 Appendix I: Proof that2Call(St,K,T,t )K2= (ST)|K38613Stochastic-Volatility Processes 38913.1 Plan of the Chapter 38913.2 Portfolio Replication in the Presence of Stochastic Volatility 38913.2.1 Attempting to Extend the Portfolio Replication Argument 38913.2.2 The Market Price of Volatility Risk 39613.2.3 Assessing the Financial Plausibility of39813.3 Mean-Reverting Stochastic Volatility 40113.3.1 The OrnsteinUhlenbeck Process 40213.3.2 The Functional Form Chosen in This Chapter 40313.3.3 The High-Reversion-Speed, High-Volatility Regime 40413.4 Qualitative Features of Stochastic-Volatility Smiles 40513.4.1 The Smile as a Function of the Risk-Neutral Parameters 40613.5 The Relation Between Future Smiles and Future Stock Price Levels 41613.5.1 An Intuitive Explanation 41713.6 Portfolio Replication in Practice: The Stochastic-Volatility Case 41813.6.1 The Hedging Methodology 41813.6.2 A Numerical Example 42013.7 Actual Fitting to Market Data 42713.8 Conclusions 436CONTENTS xv14JumpDiffusion Processes 43914.1 Introduction 43914.2 The Financial Model: Smile Tale 2 Revisited 44114.3 Hedging and Replicability in the Presence of Jumps: FirstConsiderations 44414.3.1 What Is Really Required To Complete the Market? 44514.4 Analytic Description of JumpDiffusions 44914.4.1 The Stock Price Dynamics 44914.5 Hedging with JumpDiffusion Processes 45514.5.1 Hedging with a Bond and the Underlying Only 45514.5.2 Hedging with a Bond, a Second Option and the Underlying 45714.5.3 The Case of a Single Possible Jump Amplitude 46014.5.4 Moving to a Continuum of Jump Amplitudes 46514.5.5 Determining the FunctiongUsing the Implied Approach 46514.5.6 Comparison with the Stochastic-Volatility Case (Again) 47014.6 The Pricing Formula for Log-Normal Amplitude Ratios 47014.7 The Pricing Formula in the Finite-Amplitude-Ratio Case 47214.7.1 The Structure of the Pricing Formula for Discrete JumpAmplitude Ratios 47414.7.2 Matching the Moments 47514.7.3 Numerical Results 47614.8 The Link Between the Price Density and the Smile Shape 48514.8.1 A Qualitative Explanation 49114.9 Qualitative Features of JumpDiffusion Smiles 49414.9.1 The Smile as a Function of the Risk-Neutral Parameters 49414.9.2 Comparison with Stochastic-Volatility Smiles 49914.10 JumpDiffusion Processes and Market Completeness Revisited 50014.11 Portfolio Replication in Practice: The JumpDiffusion Case 50214.11.1A Numerical Example 50314.11.2Results 50414.11.3Conclusions 50915VarianceGamma 51115.1 Who Can Make Best Use of the VarianceGamma Approach? 51115.2 The VarianceGamma Process 51315.2.1 Denition 51315.2.2 Properties of the Gamma Process 51415.2.3 Properties of the VarianceGamma Process 51415.2.4 Motivation for VarianceGamma Modelling 51715.2.5 Properties of the Stock Process 51815.2.6 Option Pricing 519xvi CONTENTS15.3 Statistical Properties of the Price Distribution 52215.3.1 The Real-World (Statistical) Distribution 52215.3.2 The Risk-Neutral Distribution 52215.4 Features of the Smile 52315.5 Conclusions 52716Displaced Diffusions and Generalizations 52916.1 Introduction 52916.2 Gaining Intuition 53016.2.1 First Formulation 53016.2.2 Second Formulation 53116.3 Evolving the Underlying with Displaced Diffusions 53116.4 Option Prices with Displaced Diffusions 53216.5 Matching At-The-Money Prices with Displaced Diffusions 53316.5.1 A First Approximation 53316.5.2 Numerical Results with the Simple Approximation 53416.5.3 Rening the Approximation 53416.5.4 Numerical Results with the Rened Approximation 54416.6 The Smile Produced by Displaced Diffusions 55316.6.1 How Quickly is the Normal-Diffusion Limit Approached? 55316.7 Extension to Other Processes 56017No-Arbitrage Restrictions on the Dynamics of Smile Surfaces 56317.1 A Worked-Out Example: Pricing Continuous Double Barriers 56417.1.1 Money For Nothing: A Degenerate Hedging Strategyfor a Call Option 56417.1.2 Static Replication of a Continuous Double Barrier 56617.2 Analysis of the Cost of Unwinding 57117.3 The Traders Dream 57517.4 Plan of the Remainder of the Chapter 58117.5 Conditions of No-Arbitrage for the Stochastic Evolution of Future SmileSurfaces 58217.5.1 Description of the Market 58217.5.2 The Building Blocks 58417.6 Deterministic Smile Surfaces 58517.6.1 Equivalent Descriptions of a State of the World 58517.6.2 Consequences of Deterministic Smile Surfaces 58717.6.3 Kolmogorov-Compatible Deterministic Smile Surfaces 58817.6.4 Conditions for the Uniqueness of Kolmogorov-CompatibleDensities 589CONTENTS xvii17.6.5 Floating Smiles 59117.7 Stochastic Smiles 59317.7.1 Stochastic Floating Smiles 59417.7.2 Introducing Equivalent Deterministic Smile Surfaces 59517.7.3 Implications of the Existence of an EquivalentDeterministic Smile Surface 59617.7.4 Extension to Displaced Diffusions 59717.8 The Strength of the Assumptions 59717.9 Limitations and Conclusions 598III Interest Rates Deterministic Volatilities 60118Mean Reversion in Interest-Rate Models 60318.1 Introduction and Plan of the Chapter 60318.2 Why Mean Reversion Matters in the Case of Interest-Rate Models 60418.2.1 What Does This Mean for Forward-Rate Volatilities? 60618.3 A Common Fallacy Regarding Mean Reversion 60818.3.1 The Grain of Truth in the Fallacy 60918.4 The BDT Mean-Reversion Paradox 61018.5 The Unconditional Variance of the Short Rate in BDT theDiscrete Case 61218.6 The Unconditional Variance of the Short Rate in BDTtheContinuous-Time Equivalent 61618.7 Mean Reversion in Short-Rate Lattices: Recombining vs Bushy Trees 61718.8 Extension to More General Interest-Rate Models 62018.9 Appendix I: Evaluation of the Variance of the Logarithm of theInstantaneous Short Rate 62219Volatility and Correlation in the LIBOR Market Model 62519.1 Introduction 62519.2 Specifying the Forward-Rate Dynamics in the LIBOR Market Model 62619.2.1 First Formulation: Each Forward Rate in Isolation 62619.2.2 Second Formulation: The Covariance Matrix 62819.2.3 Third Formulation: Separating the Correlation from theVolatility Term 63019.3 Link with the Principal Component Analysis 63119.4 Worked-Out Example 1: Caplets and a Two-Period Swaption 63219.5 Worked-Out Example 2: Serial Options 63519.6 Plan of the Work Ahead 636xviii CONTENTS20Calibration Strategies for the LIBOR Market Model 63920.1 Plan of the Chapter 63920.2 The Setting 63920.2.1 A Geometric Construction: The Two-Factor Case 64020.2.2 Generalization to Many Factors 64220.2.3 Re-Introducing the Covariance Matrix 64220.3 Fitting an Exogenous Correlation Function 64320.4 Numerical Results 64620.4.1 Fitting the Correlation Surface with a Three-Factor Model 64620.4.2 Fitting the Correlation Surface with a Four-Factor Model 65020.4.3 Fitting Portions of the Target Correlation Matrix 65420.5 Analytic Expressions to Link Swaption and Caplet Volatilities 65920.5.1 What Are We Trying to Achieve? 65920.5.2 The Set-Up 65920.6 Optimal Calibration to Co-Terminal Swaptions 66220.6.1 The Strategy 66221Specifying the Instantaneous Volatility of Forward Rates 66721.1 Introduction and Motivation 66721.2 The Link between Instantaneous Volatilitiesand the Future Term Structure of Volatilities 66821.3 A Functional Form for the Instantaneous Volatility Function 67121.3.1 Financial Justication for a Humped Volatility 67221.4 Ensuring Correct Caplet Pricing 67321.5 Fitting the Instantaneous Volatility Function: Imposing TimeHomogeneity of the Term Structure of Volatilities 67721.6 Is a Time-Homogeneous Solution Always Possible? 67921.7 FittingtheInstantaneousVolatilityFunction: TheInformationfromtheSwaption Market 68021.8 Conclusions 68622Specifying the Instantaneous Correlation Among Forward Rates 68722.1 Why Is Estimating Correlation So Difcult? 68722.2 What Shape Should We Expect for the Correlation Surface? 68822.3 Features of the Simple Exponential Correlation Function 68922.4 Features of the Modied Exponential Correlation Function 69122.5 Features of the Square-Root Exponential Correlation Function 69422.6 Further Comparisons of Correlation Models 69722.7 Features of the SchonmakersCoffey Approach 69722.8 Does It Make a Difference (and When)? 698CONTENTS xixIV Interest Rates Smiles 70123How to Model Interest-Rate Smiles 70323.1 What Do We Want to Capture? A Hierarchy of Smile-ProducingMechanisms 70323.2 Are Log-Normal Co-Ordinates the Most Appropriate? 70423.2.1 Dening Appropriate Co-ordinates 70523.3 Description of the Market Data 70623.4 Empirical Study I: Transforming the Log-Normal Co-ordinates 71523.5 The Computational Experiments 71823.6 The Computational Results 71923.7 Empirical Study II: The Log-Linear Exponent 72123.8 Combining the Theoretical and Experimental Results 72523.9 Where Do We Go From Here? 72524(CEV) Processes in the Context of the LMM 72924.1 Introduction and Financial Motivation 72924.2 Analytical Characterization of CEV Processes 73024.3 Financial Desirability of CEV Processes 73224.4 Numerical Problems with CEV Processes 73424.5 Approximate Numerical Solutions 73524.5.1 Approximate Solutions: Mapping to Displaced Diffusions 73524.5.2 Approximate Solutions: Transformation of Variables 73524.5.3 Approximate Solutions: the PredictorCorrector Method 73624.6 Problems with the PredictorCorrector Approximation for the LMM 74725Stochastic-Volatility Extensions of the LMM 75125.1 Plan of the Chapter 75125.2 What is the Dog and What is the Tail? 75325.3 Displaced Diffusion vs CEV 75425.4 The Approach 75425.5 Implementing and Calibrating the Stochastic-Volatility LMM 75625.5.1 Evolving the Forward Rates 75925.5.2 Calibrating to Caplet Prices 75925.6 Suggestions and Plan of the Work Ahead 76426The Dynamics of the Swaption Matrix 76526.1 Plan of the Chapter 76526.2 Assessing the Quality of a Model 76626.3 The Empirical Analysis 76726.3.1 Description of the Data 76726.3.2 Results 768xx CONTENTS26.4 Extracting the Model-Implied Principal Components 77626.4.1 Results 77826.5 Discussion, Conclusions and Suggestions for Future Work 78127Stochastic-Volatility Extensionof the LMM: Two-Regime Instantaneous Volatility 78327.1 The Relevance of the Proposed Approach 78327.2 The Proposed Extension 78327.3 An Aside: Some Simple Properties of Markov Chains 78527.3.1 The Case of Two-State Markov Chains 78727.4 Empirical Tests 78827.4.1 Description of the Test Methodology 78827.4.2 Results 79027.5 How Important Is the Two-Regime Feature? 79827.6 Conclusions 801Bibliography 805Index 813Preface0.1 Why a Second Edition?This second edition is, in reality, virtually a whole new book. Approximately 80% of thematerial has been added, fully reworked or changed. Let me explain why I have felt thatundertaking such a task was needed.Someof themessages of therst editionhave, to alarge extent, becomeaccepted inthetradingcommunity(andperhapsthersteditionofthisbookplayedasmallroleinthis process). Let me mention a few. It is now more widely understood, for instance, thatjust recoveringtodaysmarket pricesofplain-vanillaoptionsisanecessarybut bynomeanssufcientcriterionforchoosingagoodmodel. Asaconsequence,themodellingemphasis has gradually shifted away from the ability of a model to take an accurate snap-shotoftodaysplain-vanillaoptionmarket, towardspredictinginareasonablyaccurateway the future smile.To give another example, it is now generally recognized that what matters for pricingis the terminal andnot just the instantaneous correlationamongthe state variables.Therefore traders now readily acknowledge that time-dependent instantaneous volatilitiescanbeveryeffectiveincreatingde-correlationamonginterestrates.Asacorollary,theonce commonly held view that one needs very-high-dimensional models to price complexinterest-rate instruments has been challenged and proven to be, if not wrong, certainly anoverstatement of the truth.Moving to more general pricing considerations, it is now acknowledged that the market-completeness assumption should be invoked to obtain the powerful results it allows (e.g.theuniquenessofthefairpriceandofthehedgingstrategyorthereplicabilityofanarbitraryterminal payoff) onlyif nanciallyjustiable, not just becauseit makes themodellingeasy. So, most tradersnowrecognizethat claimingthat, say, local-volatilitymodelsaredesirablebecause theyallowacomplete-market frameworktoberetainedsquarely puts thecart before thehorse. The relevant questionis whether a given marketis truly complete (or completable), not whether a given model assumes it to be so.As these ideas have become part of the received wisdom as to how models should beused, I have felt that other issues, perhaps not so relevant when the rst edition appeared,now need to be looked at more carefully. For instance, I think that the distinction betweenwhat I call in my book the fundamental and the instrumental approaches to option pricinghas not received the attention it deserves. Different types of traders use models in differentways, and for different purposes. The question should therefore at least be asked whetherthesameclass of models canreallysimultaneouslyservetheneeds of bothtypes ofxxixxii PREFACEtrader. Is there any such thing as the best model for the plain-vanilla trader and for theexotictrader? Arethosefeatures thatmakeamodel desirablefor theformer necessarilyappealing to the latter?Linked to this is the practical and theoretical importance for option pricing of the jointpracticesofvegahedginganddailymodelre-calibration. Ibelievethatthesetwonear-universal practices have not been analysed as carefully as they should be. Yet I think thatthey lie at the heart of option pricing, and that they should inuence at a very deep levelthe choice of a pricing model.Another reasonfor updatingthe original workis that interest-ratesmiles wereaninteresting second-order effect when I was writing the rst edition. They have now becomeanessential ingredient of termstructuremodelling, andthe consensus of the tradingcommunity is beginning to crystallize around a sufciently well-established methodology,that it makes sense to present a coherent picture of the eld.More generally, outside the interest-rate arena traders have encountered great difcul-tiesinttingmarket smilesinanancially-convincingandnumerically-robust mannerstartingfromaspecicationof theprocessfor theunderlying. Asaconsequencetheyhave become increasingly interested in trying to model directly the evolution of the smilesurface. Isthisasoundpractice?Canit betheoreticallyjustied, orisit just apracti-tionerslegerdemain?Ihadonlyhintedat theseissuesintherst edition, but theyaregiven a much fuller treatment in the present work.Apart fromtheimmediateapplications, thesedevelopmentshavegivenrisetosomeimportant questions, suchas: What is moreimportant tomodel, thedynamics of theunderlying, or theevolutionof thesmilesurface(i.e. of theassociatedoptions)?; Canwe(shouldwe)alwaysassumethatthechangesinoptionpricescanbederivedfromastochastic process that can simultaneously account for the evolution of the underlying?; Ifthis were not the case, is it really possible (and practicable) to set up arbitrage strategiesto exploit this lack of coherence?The last question brings me naturally to another aspect of option pricing that I questionmore explicitly in this second edition, namely the reliance on the informational efciencyofmarketsimplicit inthecommonlyusedcalibrationandhedgingpractices. Thistopicislinkedtothepopularand,thesedays,trendytopicofbehaviouralnance.Idiscussat several pointsinthissecondeditionwhyIthinkthatoneshouldatleastquestiontheclassical rational-investor, efcient-market paradigm when it comes to option pricing.AnothertopicthatIemphasizemorestronglyinthissecondeditionisthefollowing.In the post Black-and-Scholes era the perfect-replication idea has become the bedrock ofoption pricing. In a nutshell: If we can replicate perfectly, we dont have to worry aboutaversion to risk. All models,of course, are wrong, and thereal question isnot whethertheyaretrue insomemetaphysical sensebut whether theyareuseful. Lookedat inthis light, the perfect-replication model has been immensely useful for the rst-generationof optionproducts. I feel, however, that, whenit comestosomeof theproductsthataretradedtoday, thedichotomousdistinctionbetweencompletemarkets, wherepayoffreplicationshouldalwaysbepossible, andincompletemarkets, wherenoself-nancinghedging strategy can recover with certainty a derivatives payoff, might be fast approachingits best-before date. I make an argument as to why this is the case throughout this newedition, but especially in Part II.The more one looks into a certain subject, the simpler the overarching structure beginsto appear. I think that, by working with pricing models for close to 15 years, I have come0.2 WHAT THIS BOOK IS NOT ABOUT xxiiito see certain underlying principles and regularities that can greatly help in understandingwhat different modellingapproachesoffer. So, simplifyinggreatly, muchofmyunder-standingoftheimpactofcorrelationandvolatilityonoptionpricingcanbecondensedasfollows. Oneinsightisthatdifferentcomplex(i.e.smile-producing)modelscanbelooked at as machines that produce different types of stochasticity for the quadratic vari-ationofthe(logarithmic)pricechanges, andthat thenatureofthestochasticityofthisquantity is a crucial quantity in understanding the pricing and the degree of replicabilityof complexoptions. Thesecondinsight is that, for all their apparent complexityanddiversity of forms, what stochastic-parameter models really provide is just a mechanismfor correlating the future stochastic quadratic variations with the future realizations of theunderlying(s). If we focus on this aspect we can quickly recover sight of the pricing woodwithout our view being obscured by the modelling trees (or bushes). The third insight isthat, much as traders, for the reasons hinted at above, would like to prescribe directly theevolution of smile surfaces, process-based approaches give the safest guarantee that theywill be able do so avoiding arbitrage. In a way, there is not much more to my book thanthis. I do not expect these rather Delphic statements to be clear at this stage, but they willhopefully become so as the book unfolds.0.2 What This Book Is Not AboutOnereviewbyareaderofanearlierbookofminecomplainedthat hefoundthebooktoo narrowly focused on the modelling of interest-rate derivatives. Given that the title ofthebookinquestionwasInterest-Rate Models,myrstreactionhadbeentothinkthat,atleast, Ishouldhavebeenabsolvedfromthesinsofdeceptionandambiguity. Ihaverealized, however, that the writer should not assume that a title is always self-explanatory.So, toavoidfuturecomplaints,here are afew indicationsof what thissecond editionisnot about.First ofall, Idonot deal withthestatistical estimationofvolatilityandcorrelationusingmarketdata.Often,butnotalways,Iwillassumethattheimpliedroutehasbeenfollowed, wherebyavolatilityor correlationinput isderivedfromthepricesofplain-vanilladerivatives. However, I donot accept thispracticeuncritically, anddiscussatlength its limitations. Also, much as this book is not about statistical techniques to estimatevolatilityandcorrelations,Idogivealotofimportancetothecongruencebetweentheoutputs of a model and the available empirical evidence. I therefore present in some detailmarket information about the behaviour of smile surfaces.I do notcover in thissecond editiontheissues of jumps,volatilityand correlation inthe context of credit derivatives. The topic is important, the interest in the eld is growingrapidlyandthereisalotofexcitingresearchactivity.Ithink, however,thattheareaisstilltoouidtoallowforasynthesiswithreasonablechancesofstillbeingusefulbythe timethissecond editionreaches theshelves. Also,in thecase of credit derivatives Iam beginning to doubt more and more the applicability of the possibly-imperfect-but-still-very-goodpayoff-replicationparadigmthatisatthebasisofcurrentderivativespricing.Perhaps when replication is so imperfect we should be embracing different ways of pricingoptions, possiblyalongthelines of theno-too-good-deal approach(see, for example,Cochrane, 2001), which weakens the requirement of no arbitrage.xxiv PREFACEHaving chosen not to deal with credit derivatives, I have decided not to treat measuresof dependence more sophisticated than pairwise correlation. Copula techniques are there-forenot covered. Thisisapity, but, withthecurrent draft runningat over800pages,I hadtodraw thelinesomewhere. For similarreasons,co-integration techniques,whichcan be particularly relevant in the case of interest-rate derivatives, are treated in this bookonly in passing.0.3 Structure of the BookDespite theextensivereworkings and extensions,thestructure of thesecond editionhasnot changed radically. There are now four parts. The rst deals with a Black world withoutsmiles.(Thingsdoget morecheerful later on.) Therst chapter isimportant,because ithighlights my philosophical approach to option pricing, and justies why I give a lot ofimportance to certain aspects (such as the future dynamics of smile surfaces) and relativelylittle to others (such as what the true process for the underlying is). With the foundationsrmlyestablished,Iplacemyselfinadiffusivesettingwithdeterministicvolatility,andexplore in detail the role of volatility in arriving at perfect payoff replication. I stress thedifferencebetweenvolatilityandvariance, anddiscusstheroleofmeanreversion. Theinterplay between time-dependent volatility and terminal de-correlation is the topic of thelast chapter in Part I.Part II deals with smiles in the equity and FX worlds. First I review relevant empiricalinformation about smiles, and I show how one can go from noisy market quotes of plain-vanilla option prices to a nice and smooth smile volatility surface. I then revisit the conceptofquadraticvariation, whichservesasal rougetolinkthefollowingchapters.Theseare devoted to local stochastic-volatility, general stochastic-volatility, jumpdiffusion andvariance-gamma processes. Part II is concluded by an important chapter that discusses ifandtowhatextentonecandispensewithanexplicit specicationofamodel, andcandirectly prescribe the dynamics of the smile surface.InPart IIIIreverttoaworldwithoutsmiles, butIfocusoninterestrates. SincetheLIBORmarket model issimplyaset of no-arbitrageconditionsgivenavolatilityandcorrelation structure, it has pride of place in this part of the book.Part IV extendsthesettingusedfor thedeterministic-volatilityLIBOR marketmodelinordertoaccount forsmilesinananciallymotivatedandcomputationallytractablemanner. Inorder of increasingcomplexityI deal withCEVprocesses, withdiffusivestochasticvolatilityandwithMarkov-chainprocesses. Thethreeapproachesarenestedwithin each other and afford an increasingly convincing picture of the empirically observ-able interest-rate dynamics.0.4 The New SubtitleFinally, abrief comment about thenewsubtitle. It is clearlyapunonthetitleof abookbyIsaiahBerlin(TheHedgehogandtheFox), who, inturn, borrowedit fromaRussianproverb: Thefoxknowsmanytricks, but thehedgehogknowsonebigtrick.As I understand the proverb, it means that, much as the fox might be cunning and have arich bag of tricks, knowing one simple but powerful trick can be just as effective. In thecase of the hedgehog the trick is rolling itself up in a ball of quills. For the perfect hedger,0.4 THE NEW SUBTITLE xxvthe big trick is perfect payoff replication, and, simple as the trick might be, it has provenextremely powerful and versatile. As mentioned above, one of the points I will try to makein my book is that we might be getting close to the point where we have squeezed all thepossiblemileageout oftheperfect-payoff-replicationtrick, andwemight soonhavetobegin to behave more like foxes (and introduce new, non-perfect-replication-based tricksinto the game).AcknowledgementsItisapleasuretothankLorenzoLieschforcarefulreadingofanearlierversionofthemanuscript,and for useful comments.I havegreatly beneted from discussionswithDrMark Joshi, Dr Dherminder Kainth, Dr Sukhdeep Mahal and Dr Jan Kwiatkowski. I wouldliketothankthemallfortheirsuggestions, andforpointingoutseveralwaysinwhichthissecondeditioncouldbeimproved. I remainindebtedtoDr EmanueleAmeriofordiscussions relating to the rst edition of this book. All remaining errors are mine.xxviiPart IFoundationsChapter 1Theory and Practice of OptionModelling1.1 The Role of Models in Derivatives Pricing1.1.1 What Are Models For?Theideathat thepriceof anancial instrument might bearrivedat usingacomplexmathematical formula is relatively new, and can be traced back to the Black-and-Scholes(1973) formula.1Of course, formulaewereusedbeforethenfor pricingpurposes, forinstance in order to convert the price of a bond into its gross redemption yield. However,theseearly(preBlack-and-Scholes)formulaebyandlargeprovidedaverytransparenttransformationfromoneset of variablestoanother, anddidnot carryalongaheavybaggageof model assumptions. TheBlack-and-Scholesformulachangedall that, andwenowliveinaworldwhereitisacceptedthatthevalueofcertainilliquidderivativesecurities can be arrived at on the basis of a model (the acceptance of this is the basis ofthe practice of marking-to-model).Themodels that developedfromthefamilytreethat has Black-and-Scholes at itsroots shared the common assumptions that the estimation of the drift (growth rate, trend)component of thedynamicsof therelevant nancial driver wasnot relevant toarriveatthepriceofthederivativeproduct. Thisinsight directlyfollowsfromtheconceptofpayoff replication, and is discussed in detail in this book in Chapter 2.In order to implement these models practitioners paid more and more attention to, andbegantocollect, directempiricalmarketdataataveryatomistic(oftentransactional)level. Thiswasdoneforseveral reasons: forinstance, forassessingthereasonablenessofamodelsassumptions, orforseekingguidanceinthedevelopment ofnewmodels,orforestimatingtheinputsofexistingmodels. Theveryavailabilityofthiswealthofinformation, however, suggestednewopportunities. Perhaps, embeddedinthesedata,there could be information about the market microstructure that could provide informationnot only about the volatility of a price series, but also about its short-term direction.1Parts of this chapter have been adapted from Rebonato (2004) and from Rebonato (2003b)34 CHAPTER 1 THEORY AND PRACTICE OF OPTION MODELLINGAgain, the practicewas not strictlynew, since the idea of predictingfuturepricemovements from their past history (chartism in a generalized sense) pre-dated Black-and-Scholes probably by decades. Yet these earlier approaches (which, incidentally, never wonacademicrespectability), weretypicallybasedon, at most, dailyobservations, andpur-ported to make predictions over time-scales of weeks and months. The new, transactional-level data, onthe other hand, weremade upof millions of observations, sometimescollected (as in the case of FX trades) minutes or seconds apart. The availability of thesedata made possible the calibration of predictive models, which try to anticipate stock pricemovements over time-scales sometimes as short as a few minutes.Thiswasjust thetypeof dataandmodelsthat manyof thenew, and, intheearly2000s, immensely popular, hedge funds required in order to try to get an edge over anever-growing competition (in 2001 one new hedge fund was being launched every weekincontinental Europealone). Thesehedgefunds andtheproprietarytradingdesks ofinternationally active banks therefore become the users and developers of a second breedofmodels, whichdifferedfromthemembersoftheBlack-and-Scholesfamilybecausethey were explicitly trying to have a predictive directional power. Unlike the early rathercrudechartist approaches, thesenewmodelsemployedverycomplexandsophisticatedmathematical techniques, and, if theywerenot beingroutinelypublishedinacademicjournals, it had more to do with the secretive nature of the associated research than withany lack of intellectual rigour.Twotypesofmodelhadthereforedevelopedandcoexistedbytheendofthe1990s:models as predictors (the hedge-fund models) and models as payoff-replication engines(the derivatives models). With some caveats, the distinction was clear, valid and unam-biguous. It is the derivatives models that are the subject of this book.Havingsaidthat, recent developmentsinthederivativesindustryhaveincreasinglyblurredthisonce-clear-cut distinction.Productshaveappeared whosepayoff dependstorstorderon,say,thecorrelationbetweendifferentequityindices(e.g.basketoptions),or onthecorrelationbetweenFXratesand/or betweendifferent-currencyyieldcurves(e.g. power-reverse-dual swaps), or onpossiblydiscontinuousmovesincredit spreadsand default correlations (e.g. tranched credit derivatives). The Black-and-Scholes-inspiredreplication paradigm remains the prevalent approach when trying to price these new-breedmodels. Yet their value depends to rst order on quantities poorly hedgeable and no easiertopredictthandirectionalmarkettrends.Theunderlyingmodelmightwellassumethattheseinput quantities aredeterministic (as is normallythecasefor correlations), butthis does not take away the fact that they are difcult to estimate, that they render payoffreplication very complex, if not impossible, and that their real-world realizations inuenceto a very large extent the variability of the option-plus-hedge portfolio. The result of thisstate of affairs is that, once the best hedging portfolio has been put in place, there remainsan unavoidable variance of return at expiry from the complex option and its hedges.Sureenough, perfect replicationcannot beexpectedevenfor thesimplest options:marketsarenot frictionless, tradingcannot becontinuous, bidoffer spreadsdoexist,etc.YettherobustnessoftheBlack-and-Scholes formula(discussedatseveralpointsinthisbook)ensuresthattheterminalvariabilityoftheoveralltotalportfolioisrelativelylimited. The difference, however, between the early, relatively simple option payoffs andthe new, more complex products, while in theory only a matter of degrees, is in practicelargeenoughtoquestionthevalidityofpricesobtainedonthebasisofthereplicationapproach (i.e. assuming that one can effectively hedge all the sources of uncertainty).1.1 THE ROLE OF MODELS IN DERIVATIVES PRICING 5So, in making the price for a complex derivative product, the trader will often have totake a directional market view on the realization of quantities such as correlations betweenFXratesandyieldcurves, default frequenciesor correlationsamongforwardratesindifferentcurrenciesand/orequityindices, orsub-sectorsthereof.Asaconsequence,thedistinction between predictive models (that explicitly require the ability to predict futuremarket quantities), and models as payoff replication machines (that are supposed to workwhatever thefuturerealizationsof themarket quantitieswill be) hasrecentlybecomeprogressivelyblurred.Thistopicisrevisitedinthenalsectionofthischapter,whereIarguethatonecouldmakeacaseforre-thinkingcurrent derivativespricingphilosophy,which still implicitly heavily relies on the existence of a replication strategy.1.1.2 The Fundamental ApproachIn option pricing there are at least two prevalent approaches (which I call in what followsthefundamental andinstrumental) todealingwithmodels. Thegeneral philosophythat underlies the rst can be described as follows. We begin by observing certain marketprices for plain-vanilla options. We assume that these prices are correct, in the sense thattheyembodyinthebest andmost completepossiblewayall therelevant informationavailableabout the stochastic process that drives the underlying(andpossibly, othervariables, suchas thestochasticvolatility;for simplicityI willconnethediscussiontotheunderlying, whichI will alsocall thestock). Webeginbypositingthat thistrueprocessis of aparticular form(say, ajumpdiffusion). Wecalculatewhat thepricesshould be if indeed our guess was correct. If the call prices derived using the model arenot correct, we conclude that we have not discovered the true process for the underlying.Iftheyarebetterthanthepricesproducedbyanothermodel(say, apurediffusion)wesaythatwehavereasontobelievethatthenewprocess(thejumpdiffusion)couldbeamoreaccuratedescriptionoftherealprocessfortheunderlyingthantheoldone(thepure diffusion).Alternatively, if the model has a large number of free parameters and we believe thatthe underlying process is correctly specied, we use all of the parameters describing thedynamicsoftheunderlyingtorecoverthemarketpricesoftheoptions. Thisiswhatisimplicitlydone, forinstance, withsomeimplementationsofthelocal volatilitymodels(see Chapters 11 and 12).Oragain,iftwomodelsgiveatofsimilarqualitytomarketpricesofplain-vanillaoptions, theconclusionisoftendrawnthat themodel that impliestheprocessfor theunderlyingmoresimilar towhat isstatisticallyobservedinrealityisthebetter one.The relevance of this distinctionis that, despite the similarityof the twomodels inreproducingthe plain-vanilla prices, better prices for complex options (i.e. prices inbetter agreement withthemarket practice) wouldbeobtainedif thesuperior processwere used.Thefundamentalapproachsoundsverysensible. Itis, however, underpinnedbyoneverystrongassumption: thetraderwhochoosesandcalibratesmodelsthiswayissub-scribing to the view that the market-created option prices must be fully consistent with thetrue, but a priori unknown, process for the underlying. The market, in other words, mustbeaperfect information-processingmachine, whichabsorbs alltherelevant informationabout the unknown process followed by the stock, and produces prices consistent witheach other (no arbitrage) and with this information set (informational efciency).6 CHAPTER 1 THEORY AND PRACTICE OF OPTION MODELLINGThis implicit assumption is very widespread: take, for instance, the practice of recover-ing all the observable option prices using a local-volatility model, discussed in Chapter 11.Even if weknew thetrueprocessof the underlyingtobeexactly adiffusionwithstate-dependent (local) volatility, it would only make sense to determine the shape of the localvolatilityfrom thetraded optionprices if we alsobelieved thatthese had been correctlycreated in the rst place on the basis of this model. We will see, however (see Chapter 11),thatthelocal-volatilitymodellingapproachwillrecoverbyconstructionanyexogenoussetof marketprices.Therefore, incarrying outthecalibrationweareimplicitlymakingtwo assumptions:1. that weknow, fromourknowledgeofnancialmarkets(asopposedtojust fromthemarket pricesofoptions)that thetrueprocessfortheunderlyingisindeedalocal-volatility diffusion; and2. that the market has fully incorporated this information in the price-making of plain-vanilla options.Inotherwords, byfollowingthisprocedurewedonot allowthepossibilitythat thetrue process was a local-volatility diffusion, but that the market failed to incorporate thisinformation in the prices of plain-vanilla options.Inrealityoptionprices are not exogenous natural phenomena, nor are theymadebyomniscient demi-gods withsupernatural knowledgeof thetrue processesfor thestochasticstatevariables. Optionpricesaremadebytraderswho, individually, mighthave littleor no idea about the true stochastic process for the underlying; who mightbeusingthepopularmodel ofthemonth; who, foravarietyofinstitutional constraints,might be afraid to use a model at odds with the current market practice (see Section 1.2.2);or who might be prevented from doing so by the limit structures in place at their tradinghouses.A strongbeliever inmarket efciency would counter thattheerrors of theindividualuninformed traders do not matter, in that they will either cancel each other out (if uncor-related), or will be eliminated (arbitraged away) by a superior and more unfettered traderwith the best knowledge about the true process. I discuss the implications for option pric-ingofthisstrongformofmarketefciencyinthenextsections, butthispositionmustbesquaredwithseveralempiricalobservationssuchas, forinstance, thefactthatsteepequitysmilessuddenlyappeared aftertheequitymarketcrashof1987 didtradersnotknowbeforetheeventthatthetrueprocesshadajumpcomponent?Anotherpuzzlingfact for the believer of the fundamental view is that a close-to-perfect t to the S&P500smilein2001canbeobtainedwithacubicpolynomial(seeAit-Sahalia(2002)):dowendit easier tobelievethat themarket knows about thetrueprocessfor theindex,prices options accordingly and when the prices obtained by using this procedure are con-verted into implied volatilities they magically lie on a cubic line? Or is it not simpler tospeculatethattradersquotepricesofplain-vanillaoptionswithamixtureofmodeluse,trading views about the future volatility and cubic interpolation across strikes?The matter cannot be settled with a couple of examples. For the moment I simply stressthat thecommonand, primafacie, verysensible(fundamental)approachtochoosingandcalibratingamodel that Ihavejust describedcanonlybejustiedifoneassumesthat theobserved prices of optionsreect in an informationallyefcient way everything1.1 THE ROLE OF MODELS IN DERIVATIVES PRICING 7that can be known about the true process. I will return to this topic in Section 1.4, whichdeals with the topic of calibration.1.1.3 The Instrumental ApproachAn alternative (instrumental) way to look at the choice of process for the underlying isto regard a given process specication as a tool not so much for driving the underlying,but for creatingpresent andfuture prices of options. Inthis approachit is thereforenatural tocomparehowthepricesof options (not of theunderlying) moveinrealityand inthemodel.Typicallythiscomparisonwillnotbemadeindollar terms,butusingtheimpliedvolatilitylanguage. Since, however, thereisaone-to-onecorrespondencebetweenimpliedvolatilitiesandoptionprices(foragivenvalueoftheunderlying)wecan indifferently use either language.So,traders,whomighthavenoideaaboutthetrueprocess for theunderlying,mightnone the less form ideas about how option prices (or, more likely, the associated impliedvolatilities) behaveover timeandindifferent market conditions: dotermstructureofvolatilities remain roughly similar, or do they change shape in totally unpredictable ways?Dosmiles for same-expiryoptions suddenlyappear andthenpermanentlyattenoutascalendartimegoesby, ordotheyapproximatelyretaintheirrelativesteepness?Doswaptionmatricesalwaysretainthesameshape, ordotheydisplayafewfundamentalmodes of deformation, among which they oscillate? How will the smile surface migrateas the underlying moves? Is it sticky or oating?These traders, who observe regularities in the implied volatilities (i.e. in the prices ofoptions) rather than in the underlying, tend to have relatively little interest in determiningthe true process for the stock price, and will instead prefer a process capable of producingthe desired features in the implied volatilities. The process for the underlying now becomesmore instrumental than fundamental: it is simply seen as a tool to obtain something else(i.e. thecorrectdynamicsfortheimpliedvolatilities). Indeed, someofthemost recentpricingapproacheshavetriedtodispensewiththespecicationof theprocessfor theunderlying altogether, and have directly prescribed the dynamics of the implied volatilitysurface. (See, for example, Schoenbucher (2000) and Samuel (2002).)2Could the process for the underlying be chosen with total disregard of the true processfor the underlying, as long as it reproduces the correct behaviour for the implied volatil-ities? This is unfortunately not the case, because, ultimately, the option trader will want,at theveryleast, todeltahedgeher positions, andthesuccessof theassociatedtrad-ing strategy will depend on the correct specication for thedynamics of the underlying.However, there is a considerable degree of robustness in the hedging process, at least aslong as certain important but rather broad features of the stock price process are capturedcorrectly. See the discussions in Sections 4.6, 13.6 and 14.11. Furthermore, if the optionprices and the process for the underlying were seriously incompatible, there would lie anobviousarbitragesomewhere, andthetraderwhototallydisregardedtheplausibilityofthestockdynamicswouldtheoreticallyexposeherselftotheriskofbecomingamoneymachinefor her fellowtraders. Inreality, however, real arbitragesarerather complextoput inplaceinpractice, andthethreat ofbeingat thereceivingendofanarbitragestrategy is often more theoretical than real. I discuss this topic further in Chapter 17 (see,in particular, Section 17.3, The Traders Dream).2The difculties and dangers in doing so are discussed in Chapter 17.8 CHAPTER 1 THEORY AND PRACTICE OF OPTION MODELLINGTherefore the second (instrumental) approach is more popular, and, explicitly or implic-itly, more widely followed in the market. It is also the conceptual framework that I prefer,and that I willpredominantlyfollow inthisbook.I willtryto justifythischoicein thischapter, and, asconcretesituationsandexamplesarise, inthecourseofthebook. It isimportant tokeepinmind, however, that neitherapproachiswithout conceptualblem-ishes: the rst requires an extreme faith in market efciency that I consider unwarranted;for the second to work one has to rely on a rather difcult-to-quantify robustness of thehedging process vis-` a-vistrading restrictions and mis-specications of the process for theunderlying.ThediscussionofwhatImeanbyrobustnessconstitutesoneoftherecurrent themesof this book. At this stage I can give a brief account as follows. I will show (Sections 4.5and4.6)that, aslongasacertainquantity(thequadraticvariation)isdeterministicandknown, it makes relatively little difference for the success of the hedging programme howthe exact partitioning of this quantity actually occurs during the life of the option. I willthengoontoargue(Sections 13.6and14.11)that, ifthequadraticvariationisinsteadeither unknown or stochastic, the success of the hedging strategy will rely to a large extentonndingaportfoliowhosedependenceonthe(imperfectlyknown)realizationofthequadraticvariation issimilartothatof thecomplexproductthathas tobehedged.Thisobservation brings us naturally to the topic of vega hedging.1.1.4 A Conundrum (or, What is Vega Hedging For?)Suppose that a trader has a pricing model that describes the empirical statistical propertiesoftheprocessoftheunderlyingextremelywell, but that recoversthepricesoftradedoptions poorly. (By the way, according to the Efcient Market Hypothesis this could nothappen, but I postpone the discussion of this point until later sections.) The trader is alsoaware ofanothermodel,whichimpliesamuchlessrealisticprocessfortheunderlying,but whichreproducesthepresent andexpectedfutureimpliedvolatilitysurfacesverywell. What should the trader do? Which model should she use?The answer, I believe, is: It depends.If thetrader is aplain-vanilla optiontrader, sheshouldmakeuseof her superiorknowledge, and trade and set up dynamic delta-hedging strategies based on this knowledgeoftheprocessfortheunderlying. Shemight ndit difcult todoso, becausehertruemodel will prescribe different amounts of stock to be delta-neutral than the model adoptedbythemarketconsensus. Thereforeherpositionswillprobablynotappeardelta-neutralto therisk management function of her institution(unless she can exercise an unhealthyinuence on her middle ofce), and she might run against VaR or other limits. Also, sinceherbackofcewill presumablyuse, forsomethingasrelativelyliquidasplain-vanillaoptions,mark-to-market (rather thanmark-to-model),shewillbeabletorecognize littleornoprotassoonassheputsontheadvantageoustrade,andwillhavetorelyonthedifference between the true and wrong option values to trickle in during the life of theoption, as her trading strategy unfolds. (See the discussion in Section 1.2.4.) Despite theseconstraints, however,aplain-vanillatraderwillforfeithercompetitiveadvantageifsheslavishly follows the market in every price.If, ontheotherhand, sheisacomplex-derivativestrader, vegahedgingwill beforher atleastasimportantasdeltahedging(seethediscussioninSection 1.3.2astowhydelta hedgingbecomes relativelyless important for complex-derivativetraders ina1.2 THE EFFICIENT MARKET HYPOTHESIS 9nutshell, thisisbecauseof thehighcorrelationbetweentheerrorsinthedeltaof thecomplex product, and the errors in the delta of a well-chosen vega-hedging portfolio). Itisthereforecrucialforherthatthefuturevegare-hedgingcostspredictedbythemodelshouldbeassimilar aspossibletotheactual costsencounteredduringthelifeof theoption.These costs, in turn, willbe linked to the future prices of plain-vanillacalls (thefuture smile surface) that the wrong model recovers well by denition.Why doesnt the complex-derivatives trader, who knows the true process for the stockprice, dispense with vega hedging altogether, and simply engage in a correct delta-hedgingstrategy? She could only do this if the true process were such as to allow for market com-pleteness by trading just in the underlying, i.e. if any payoff could be exactly reproducedbytradingdynamicallyintheunderlying. Butthisisaveryspecial, andmostunlikely,case. In general, market incompleteness is the rule, not the exception, and a trader, evenarmedwiththeknowledgeofthetrueprocess, cannot hopethat thefuturevegaofthecomplex product will only be a function of the future realization of the stock price.Anal observation: successful optionproducts, whether plain-vanilla or complex,thrive if there is a strong customer demand for them (customer, in this sense, means coun-terpartiesfromoutsidethecommunityofprofessionaltraders).Thereforeoptiontradersdo not routinely make money by pitting their intellects against each other in a (zero-sum)war-gameof pricingmodels. It isfor themmuchmorereliableandprotabletodealwiththenon-tradingcommunity, byprovidingtheenduserswiththenancial payoffthey want (e.g. interest-rate protection, principal-protected products, yield enhancement,cheaper funding costs), and by exacting a compensation for the technological, intellectualand risk-management costs involved in providing this service. Given this trading reality,there are greater benets in being on the market smile, even when it is felt that a morerealistic model would not recover these market prices, than in standing alone at odds withthe market.1.2 The Efcient Market Hypothesis and Why It Mattersfor Option PricingImentionedintheprevioussectionthat theEfcient Market Hypothesis(EMH)hasadirect bearing on option pricing. In this section I discuss why this is the case. In order todosoitisimportanttoclarifywhatismeantbymarketefciency, andwhatconditionsmust bemet forit toprevail. Inparticular, Iwill stressthat rationalityofeachmarketplayerisnot requiredfortheEMHtoholdtrue, andthereforecriticismsofitsvaliditymust take a different, and subtler, route.1.2.1 The Three Forms of the EMHThe EMH can be formulated in forms of wider and wider applicability (see, for example,thetreatment inShleifer (2000), onwhichthis sectiondraws extensively). Themostradical form requires that all economic agents are fully informed and perfectly rational. Ifthey can all observe the same history (of prices, economic variables, political events, etc.),then they will all arrive at the same statistical conclusions about the real world, and willformpricesbydiscountingtheexpectedfuturecashowsfromasecurityatadiscountrate dependent on the undiversiable uncertainty of the security and on their risk aversion.10 CHAPTER 1 THEORY AND PRACTICE OF OPTION MODELLINGInthissensethevalueofthesecurityissaidtoembedall theinformationavailableinthemarket, itsvaluetobelinkedtofundamentals, andmarketstobeinformationallyefcient. All securities are fairly priced, excess returns over the riskless rate simply reectfaircompensationforexcessrisk, andve-dollarbanknotescannotbefoundlyingonthe pavement.A weaker form of market efciency (but one that arrives at the same conclusions) doesnot require all economic agents to be rational, but allows for a set of investors who pricesecuritiesonsentimentorwithimperfectinformation.Willthisaffect themarketprice?Actually, one can show that as long as the actions of the uninformed, irrational investorsare random (uncorrelated), their actions will cancel out and the market will clear at thesame prices that would obtain if all the agents were perfectly rational.Surely, however, the zero-correlation assumption is far too strong to be swallowed byanybody who has witnessed the recent dotcom mania. The very essence of bubbles, afterall, is that the actions of uninformed, or sentiment-driven, investors are just the oppositeofuncorrelated. Ifthisisthecase, thensupplyanddemand, ratherthanfundamentals,will determine theprice of a security. Is this theend of theefcient market hypothesis?Notquite.Lettherebeirrationalandco-ordinatedinvestors.Aslongastherealsoexistrational, well-informedagents whocanvaluesecurities onthebasis of fundamentalsand freely trade accordingly, priceanomalies willnotpersist.Thesepseudo-arbitrageurswillinfactbuytheirrationallycheapsecuritiesandsellthesentimentallyexpensiveones, andbysodoingwill drivethepricebacktothefundamentals. Whetherit isduetoirrationalityandsentiment or toanyother cause, inthisframeworkexcessdemandautomaticallycreatesextrasupply, andviceversa. Therefore, aslongasthesepseudo-arbitrageurs can freely take their positions, supply and demand will not affect equilibriumprices, thesewill againbebasedonthesuitablydiscountedexpectationoftheirfuturecash ows (the fundamentals) and the EMH still rules.It isimportanttostressthattheEMHisnotonlyintellectuallypleasing,buthas alsobeenextensivelytestedandhasemerged, byandlarge, vindicated. Theby-and-largequalier, however, is crucial for myargument. Inthemulti-trillionmarket of all thetraded securities, a theory that accounts for theprices of 99.9% of observed instrumentscan at the same time be splendidly successful, and yet leave up for grabs on the pavementenough ve-dollar notes to make a meaningful difference to the year-end accounts and theshare prices of many a nancial institution. This is more likely to be so if the instrumentsin question are particularly opaque. The possibility that the pseudo-arbitrageurs might notalways be able to bring prices in line with fundamentals should therefore be given seriousconsideration.1.2.2 Pseudo-Arbitrageurs in CrisisWhatcanpreventpseudo-arbitrageursfromcarryingouttheirtaskofbringingpricesinline with fundamentals?Tobeginwith, these pseudo-arbitrageurs (hedge funds, relative-valuetraders, etc.)often take positions not with their own money, but as agents of investors or shareholders(Shleifer (2000)). If the product is complex, and thus so is the model necessary to arriveat its price, the ultimate owners of the funds at risk might lack the knowledge, expertiseorinclinationtoassessthefairvalue,andwillhavetorelyontheiragentsjudgement.This trust, however, will not be extended for too long a period of time, and certainly not1.2 THE EFFICIENT MARKET HYPOTHESIS 11for many years. Therefore, the time-span over which securities are to revert to their fun-damental value must be relatively short (and almost certainly will not extend beyond thenext bonus date). If the supply-and-demand dynamics were such that the mis-priced instru-ment might move even more violently out of line with fundamentals, the position of thepseudo-arbitrageurwillswingintothered,andthetrust-me-I-am-a-pseudo-arbitrageurline might rapidly lose its appeal with the investors and shareholders.Another source of danger for relative-value traders is the existence of institutional andregulatoryconstraintsthat might forcetheliquidationof positions beforetheycanbeshown to be right: the EMH does not know about the existence of stop-loss limits, VaRlimits, concentration limits, etc.Poor liquidity,often compoundedwiththeabilityofthemarket toguessthepositionof a large relative-value player, also contributes to the difculties of pseudo-arbitrageurs.Considerforinstancethecaseofapseudo-arbitrageurwho, onthebasisofaperfectlysoundmodel,concluded thattraded equityimpliedvolatilitiesareimplausiblyhigh,andentered large short-volatility trades to exploit this anomaly. If the market became aware ofthese positions,and if, perhaps because of the institutionalconstraints mentioned above,thepseudo-arbitrageurhadtotrytounwindtheseshortpositionsbeforetheyhadcomein the money, the latter could experience a very painful short squeeze.Finally, very high information costs might act as a barrier to entry, or limit the numberofpseudo-arbitrageurs. Reliablemodelsrequireteamsofquantstodevisethem, scoresof programmers to implement them, powerful computers to run them and expensive datasources to validate them.3The perceived market inefciency must therefore be sufcientlylarge not only to allow risk-adjusted exceptional prots after bidoffer spreads, but alsoto justify the initial investment.Inshort, becauseofalloftheabovethelifeofthepseudo-arbitrageurcanbe, ifnotnasty, brutish and short, at least unpleasant, difcult and fraught with danger. As a result,eveninthepresenceofasevereimbalanceofsupplyordemand, relative-valuetradersmight bemorereluctant tostepinandbringpricesinlinewithfundamentalsthantheEMH assumes.1.2.3 Model Risk for Traders and Risk ManagersI have mentioned the impact of risk management on trading practice and on price forma-tion. It is useful to explore this angle further.4WithintheEMHframeworkthegoals of tradersandriskmanagersarealigned: asuperiormodelwillbringthetraderacompetitiveadvantage,andwillberecognizedassuch by the market with very littletimelag. From this point of view, an accurate mark-to-market is purely a reection of the best information available, and true (fundamental)value, market price and model price all coincide. It therefore makes perfect sense to have asingle research centre, devoted to the study and implementation of the best model, whichwill servetheneedsof thefront-ofcetrader, of theriskmanager andof theproductcontroller justaswell.Lookedatfrom thisangle,modelriskissimplytheriskthatourcurrent model might not begoodenough, andcanbereducedbycreatingbetter and3When I was heading an interest-rate-derivatives trading desk I was half puzzled and half embarrassed whenI discovered that the harnessed power of the farm of parallel super-mini-computers in my small trading groupranked immediately after Los Alamos National Laboratory in computing power.4The following two sections have been adapted from Rebonato (2003b).12 CHAPTER 1 THEORY AND PRACTICE OF OPTION MODELLINGbettermodelsthat will trackthemonotonic, ifnot linear, improvement ofthemarketsinformational efciency.If we believe, however, that pseudo-arbitrageurs might in practice be seriously hinderedin bringing all prices in linewith fundamentals the interplay between true value, marketvalue and model value can be very different. Across both sides of the EMH divide there islittle doubt that, when it comes to trading-book instruments, what should be recorded forbooks-and-recordspurposesshouldbethebest guessforthepricethat agivenproductwouldfetchinthemarket. For theEMHsceptic, however, thereisnoguaranteethatthebestavailablemodel(i.e.themodelthatmostcloselypricestheinstrumentinlinewith fundamentals) should produce this price. Furthermore, there is no guarantee that themarket, insteadofswiftlyrecognizingtheerrorofitsways, might notstrayevenmoreseriously away from fundamentals.Ultimately, wheneveratraderentersaposition, hemustbelievethat, insomesense,the market is wrong. For the risk manager concerned about marking an option positionto model appropriately, on the other hand, the market must be right by denition. For theEMH believer the market can only be wrong for a short period of time (if at all), so thereis no real disconnect between the risk managers price, and the traders best price. For theEMHsceptic, ontheotherhand, thereisanirreconcilabletensionbetweenfront-ofcepricing and the risk-management model price.1.2.4 The Parable of the Two Volatility TradersToillustratefurther theoriginof thistension, let usanalyseastylizedbut instructiveexample.Two plain-vanillaoptiontraders (oneworkingfor Efcient Bank,andthesec-ond for Sceptical Bank) have carefully analysed the volatility behaviour of a certain stock,and both concluded that its level should be centred around 20%. The stock is not partic-ularlyliquid, andthebrokersquotes, obtainedwithirregularfrequency,donotdeviatesufcientlyfromthisestimatetowarrant enteringatrade. Oneday, however, withoutanythingnoticeablehavinghappenedinthemarket, animpliedvolatilityquoteof10%appears. Twohugeve-dollar notesarenowlyingontheoor,and bothtraders swiftlypickthemup. Bothtradersintendtocrystallizethevalueof themis-pricedoptionbyengaging in gamma trading, i.e. by buying the cheap option they will both end up longgamma and will dynamically hold a delta-neutralizing amount of stock, as dictated by theirmodel calibrated to 20% volatility. (See Section 4.3 for a discussion of gamma trading.)LifeiseasyfortheEfcientBanktrader.Shewillgotoherriskmanager,convincehim that the model used to estimate volatility is correct and argue that the informationallyefcient market will soonadjust the impliedvolatilityquote backto20%. This hasimportant implications. First of all, the prot from the trade can be booked immediately:thefront ofceandriskmanagement sharethesamestate-of-the-art model, andbothconcur thatthe price for theoption inlinewith fundamentals shouldbe obtained witha20%volatility. Furthermore,shouldanotherve-dollarbillappear onthepavement,thetrader at Efcient Bank will have every reason and incentive to pick it up again.Thecoincidenceofthefront-ofceandmiddle-ofcemodelshasyet anotherconse-quence. Thetraderworksonavolatility-arbitragedesk, andhermanagersarehappyforhertotakeaviewonvolatility, but not totakeasubstantial positionintheunder-lying.Theyhavetherefore granted hervery tightdeltalimits.This,however,creates noproblem, becauseherstrategyistobedelta-neutralateverypointintimeandtoenjoy1.2 THE EFFICIENT MARKET HYPOTHESIS 13thefact that shehasbought (at 10%) cheapconvexity (seeChapter 4). Crucially, inordertocrystallizethemodelprotsfromtrade,shewillengageinadynamichedgingstrategybasedonthesuperiormodel(calibratedwitha20%volatility),notonthetem-porarilyerroneousmarketmodel. Sincemiddleofceagainsharesthesamemodel, theriskmanagercalculatesthedeltaofthepositionexactlyinthesamewayasthetrader,andthereforeseesthewholeportfolioperfectlywithinthedesksdeltalimits(actually,fully delta neutral).LifeismuchharderforthetraderatScepticalBank.Shealsoworksonavolatility-arbitrage desk with tight delta limits,and her middle-ofce function also recognizes thatthemodel sheusesissoundandplausibleandconcursthat themarket must begoingthroughaphaseofsummermadness. Thesimilarities, however,virtuallyendhere.Herrisk-management functiondoesnot believethat asuperiormodel must beendorsedbythe market with effectively no delay, and therefore is not prepared to recognize the modelvalueimpliedbythe10%tradeasanimmediateprot. Amodel provisionwill besetaside. Sincethetraderwill not beabletobook(all)themodel prot upfront, shewillhave to rely on the prot dripping into the position over the life of the option as a resultoftradingthegamma. Thisprocesswill berelativelyslow, themoresothelongerthematurityoftheoption. Duringthisperiodthetraderisexposedtotheriskthatanotherrogue volatilityquote, say at 5%, mighteven create a negativemark-to-market for herposition. Her reaction to a second ve-dollar bill will therefore be considerably differentfrom that of her colleague at Efcient Bank. Furthermore, in order to carry out her gamma-tradingprogrammeshewouldliketobuyandselldeltaamountsofstockbasedonherbestestimateofthetruevolatility(20%).Middleofce,however,whohaveobservedthe10%trade, usesthemodelcalibratedwiththelowervolatilitytocalculatethedeltaexposureofthetrade, andthereforedoesnotregardherpositionasdelta-neutralatall.SheutilizesmoreVaRthan her colleague,mightsoonhitagainsther deltalimit,and,ifher trading performance is measured on the basis of VaR utilization, she will be deemedto be doing, on a risk-adjusted basis, more poorly than her colleague.This parablecouldbeexpandedfurther, but thecentral messageis clear: differentviewsaboutmarketefciencycangenerateverydifferentbehavioursandincentivesforotherwise identical traders. In the real world, however, nancial institutions are organizedmuchmorealongthelinesofScepticalBankthanofEfcientBank,andthiscreatesastrong disincentive for a trader to stray too much from the path of the commonly acceptedpricing model.The strength of this disincentive should not be underestimated. The story about EfcientBankandSceptical Bankmight havebeencontrivedandover-stylized, but areal-lifeexample can bring home the same point with greater force and clarity. As a trader entersa complex derivative transaction for which no transparent market prices are available, theproduct control function of her institution faces the problem of how to ascribe a value tothetrade. Commercialdataprovidersexist tofull thisneed. OnesuchmajorprovideractiveintheUnitedKingdomandintheUnitedStatesactsasfollows: thepricesofnon-visible trades are collected from the product control functionsof several participatinginstitutions; the outliers are excluded and an average price is created from the remainingquotes; information is then fed back to the contributing banks about the average price andabouthowmanystandarddeviationsawayfromtheconsensustheiroriginalquotewas.Ifthequotessubmittedbyaninstitutionareconsistentlyawayfromtheconsensus, theinstitutionisexpelled fromthe contributinggroup, andwill nolongersee the consensus14 CHAPTER 1 THEORY AND PRACTICE OF OPTION MODELLINGprices. Whenthishappens, theproduct control functionofthat bankwill nolongerbeabletodiscover thepriceof theopaquederivative, but simplyknowsthat thebankspricingmodel (evenif it might be, perhaps, abetterone) isawayfromtheindustryconsensus. Amodel reservewill havetobeappliedthat will typicallynegatetheextravalueascribedbythetradersmodel totheexoticproduct. Furthermore, sincethereisprimafacieevidencethatwherethetraderwouldliketomarkhertradesisawayfromthe market consensus, a limit on the maximum notional size of the offending positionsis likely to be imposed.Therefore, whiletodays models areindubitablymoreeffective(at least incertainrespects) than the early ones, I do not believe that the linear evolution paradigm, possiblyapplicabletosomeareasofphysics,5andaccordingtowhichlatermodelsarebetterinthesenseof beingcloser tothephenomenon,isnecessarily suitedtodescribingtheevolution of derivatives models. This real-life example shows that the disincentives againststraying away from market consensus (ultimately, the withdrawal of market information)can be even more powerful in practice than in the parable of the two traders. Models can,and do, evolve, but in a less unfettered manner than traditional linear accounts (and theEMH) assume.Model inertia is therefore certainly a very signicant feature to take into account whenanalysing models. There are however other aspects of market practice that have a profoundinuenceonhowmodelsaredeveloped,testedandused.Giventheirimportance,someof these are discussed below.1.3 Market Practice1.3.1 Different Users of Derivatives ModelsTo understand derivatives models it is essential to grasp how they are put to use. In general,a pricing model can be of interest to plain-vanilla-option traders, to relative-value tradersand to complex-derivatives traders. Relative-value and plain-vanilla traders are interestedinmodelsbecauseoftheirabilitytopredicthowoptionpricesshouldmoverelativetotheunderlying, andrelativetoeachother, givenacertainmoveintheunderlying. Forboth these classes of user, models should therefore have not just a descriptive, but also aprescriptivedimension.Thesituationisdifferent forcomplex-derivativetraders, whodonot haveaccesstoreadily visible market prices for the structured products they trade in, and therefore requirethemodels tocreate theseprice, giventheobservablemarket inputs for theunder-lyingandtheplain-vanillaimpliedvolatilities. Sincecomplextraderswill, ingeneral,vega hedge their positions, exact recovery of the plain-vanilla hedging instruments thedescriptiveaspect of amodel becomes paramount.Therecovery of present and futureoptionpricesislinkedtothecurrentandfuturevegahedgingandtomodelcalibration.These very important practices are discussed in the next section.5Irealizethat, withthisstatement, Iamwalkingintoaphilosophical thicket, andthatsomephilosophersofsciencewoulddenyevenmodelsinfundamentalphysicsanyclaimofbeingtrueinanabsolutesense. Istay clear of this controversy, and the more mundane points I make about the evolution of derivatives modelsremain valid irrespective of whether an absolute or a social view of scientic progress is more valid.1.3 MARKET PRACTICE 151.3.2 In-Model and Out-of-Model HedgingPossibly no aspect of derivatives trading has a deeper-reaching impact on pricing than thejoint practices of out-of-model hedging and model recalibration. In-model hedging referstothepracticeof hedgingacomplexoptionbytakingpositionsindelta amountsoftraded instruments to neutralize the uncertainty from the stochasticdrivers of the processfor the underlying. In a Black-and-Scholes world, neutralizing the movements in an optionprice by buying a delta amount of stock is a classical example of in-model hedging.Out-of-model hedging is the taking of positions to neutralize the sensitivity of a com-plexproduct tovariationsininput quantitiesthatthe model assumesdeterministic(e.g.volatility). InaBlack-and-Scholesworld, vegahedgingisaprimeexampleof out-of-model hedging.Needlesstosay,out-of-modelhedgingisonconceptuallyrather shakyground:ifthevolatilityis deterministicandperfectlyknown, as manymodels usedtoarriveat thepriceassumeittobe,therewouldbenoneedtoundertakevegahedging. Furthermore,calculating the vega statistics means estimating the dependence on changes in volatility ofa price that has been arrived at assuming the self-same volatility to be both deterministicand perfectly known. Despite these logical problems, the adoption of out-of-model hedgingingeneral, andof vegahedginginparticular, is universal inthecomplex-derivativestradingcommunity. Thetrader whoengagesinthislogicallydubiousvegahedgingatinception of a trade can at least console herself as follows. If her model has been correctlycalibrated to the current market prices of the vega-hedging instruments, she will be addingto the original delta-neutral portfolio another self-nancing, delta-neutral and fairly-valuedportfolio of options. By so doing she will simply have exchanged part of her wealth fromcashintostockandfairly-pricedoptions, andthiscanhavenoimpact onthevalueofthecomplextrade(becauseher model was correctlycalibratedtothecurrent marketpricesofthehedginginstruments).Butwhataboutfuturevegare-hedgingtransactions?If, conditional on a future level for the underlying, these vega trades will be carried out inthe real world at the same future prices that the model ascribes to them today, once againthetrader has notionallyjustadded lotsof self-nancing, delta-neutral and fairly-valuedportfolios of forward-starting options. The economic effect of this is zero. This is no longertrue, however, if the model predicts today future re-hedging costs different from what willbeencounteredinreality. Ifsystematic, thisdifferencebetweentherealandtheoreticallevel offuturere-hedgingtransactionscanmakethewholestrategynon-self-nancing,and cause money to bleed in or (most likely) out of the traders account.Similarlyimportant, universal anddifcult tojustifytheoreticallyisthepracticeofre-calibrating a model to the current market plain-vanilla prices throughout the life of thecomplextrade. Letuslookatthispracticeinsomedetail. Becauseoftheneedtovegahedgeatthestartofthelifeofacomplextransaction,atrader willbeginbycalibratingher model in such a way as to recover the current (day-0) prices of all the options neededfor hedging.Oncethemodelhasbeen calibrated inthismanner,thepriceof acomplexderivative will be calculated, and the trader will begin the dynamic hedging strategy to becarried out until the option expiry. Let us now move a few days into the trade. On day 2,thesamemodel calibrationusedonday0will not ingeneralproducespot (i.e. day-2)plain-vanillaoptionpricesinlinewiththemarket. Thereforeif thefuturere-hedgingtransactions were carried out with the models parameter as per day-0s calibration, theirmodel priceswouldnot coincidewiththemarket prices. Toavoidthis, thetraderwillre-calibrate the model on the basis of these new benchmark option prices, and re-calculate16 CHAPTER 1 THEORY AND PRACTICE OF OPTION MODELLINGthepriceofthecomplexinstrumentonthebasisofthenewcalibration,assumed againto be valid until the option expiry. As an unrepentant sinner, therefore, every morning thetrader who re-calibrates a model admits that yesterdays calibration (and price) had beenwrong,yetmakesapricetoday(withthenewparameters)thatrestsontheassumptionthat the new calibration will be valid until the products expiry.Thesetwopracticesarecloselylinked. Inafriction-lessmarket, ifamodel didnothave to be re-calibrated during its life, future vega transactions would have no economicimpact. If a modelonly needed to be calibrated once and for all, itwould always implythe same future prices for plain-vanilla options in the same future attainable states of theworld. Therefore,contingent onaparticularrealizationofthestockpriceorrate, thesetradeswouldbetransactedatthefutureconditionalpricesfortheplain-vanillahedginginstruments implicit inthe day-0calibration. Exchanginginthe futureanamount ofmoney equal to the fair value (according to the model) of the future plain-vanilla optionsrequired for re-hedging has no economic effect today, and, as a consequence, would notaffect todays model price of the complex derivative. This is no longer true, however, if,in order torecover the future spotplain-vanillaprices, themodel has to be re-calibratedday after day.Clearly, no model will be able to predict exactly what the future re-hedging costs willbe(eveniftheBlack-and-Scholesapproachassumesthistobepossible). Itishoweverimportant touseamodel that, asmuchaspossible, knows about possiblefuturere-hedging costs and assigns to them the correct (risk-adjusted) probabilities. Since the truecost of an option is linked (in a completemarket, is equal) to thecost of the replicatingportfolio, the trader will therefore have to keep two possible sources of cost in mind: thehedgingcostsincurred atinception,and thoseencountered duringthelifeoftheoption.Asfor theinitial costs, thesecomefromin-model hedging(e.g. thecost of thedeltaamountofstock),andfromout-of-modelhedging.Ihavearguedthatthelatterhave,intheory and in the absence of bidoffer spreads, no economic effect today, since the traderwhohascorrectlycalibratedhermodel issimplybuyingat fairvalueaseriesoffairlypricedoptions. Indeed, neglectingagainbidoffer spreads, afterbookingtheseout-of-modelhedginginitialtrades,theP&Laccountofthetraderwilldisplaynochange.Forasmall number of products(suchas, for instance, Europeandigital options) aninitialportfolioisall thatisneeded tohedge exactly thecomplex tradeuntilexpiry.See, forinstance,thediscussioninChapter 17. Forthistypeoftraderecoveryoftodaysplain-vanilla prices is all that matters, and the trader does not have to worry whether the futureconditional option prices predicted by the model will be in line with reality or not. Thesepseudo-complex trades,however, are few, and,byand large, uninterestingvariationsontheplain-vanillatheme. For all bonadecomplextradessomedegreeof in- andout-of-modelvegare-hedging willalwayshavetobeundertaken.Themorethefuturevegatrades will be important, the more the trader will be sensitive to the correct prediction bythe model of the future conditional plain-vanilla option prices, and the less exact recoveryof todays prices becomes the only relevant criterion in assessing the quality of a model.Choosing good inputs to a model therefore means recovering todays prices in such away that tomorrows volatilities and correlations, as predicted by the model, will producefuture plain-vanilla option prices as similar as possible to what will be encountered in themarket. Given the joint practices of vega re-hedging and model re-calibration, the bestcalibration methodology is therefore the one that will require as little future re-estimationof the model parameters as possible.1.4 THE CALIBRATION DEBATE 17Looking at the problem in this light, one of the most important questions is: How shouldthe model inputs that will give rise to the more stable calibration and to the smallest re-hedging surprises be estimated?6Answering this fundamental question requires choosingthe source of information (statistical analysis or market implication) that can best servethe trading practice. This is therefore the topic of the next section.1.4 The Calibration DebateIn principle, to calibrate a model in order to price complex derivatives, one could followtwodistinct routes: onecouldprescribethewholereal-worlddynamicsforthedrivingfactor(s) (e.g. the stock price or the short rate) and for the associated risk premia. Giventhese inputs, the equilibrium prices for all assets (the underlying andthe derivatives) canbe obtained. This approach is called absolute pricing. Alternatively, one could assign thevolatilityandcorrelationfunctions(thecovariancestructure,forshort)ofthestochasticstate variables. On the basis of this much more limited information, the prices of optionsgiven the price of the underlying can be obtained. This approach is called relative pricing.Readers familiar with the Black-and-Scholes approach might nd that the rst (absolute)approach goes against the grain, since it fails to take advantage of the greatest strengthof relativepricing, i.e. theirrelevanceof thedifcult-to-estimatereal-worlddrift. Yet,intheinterest-ratearena, estimationof thereal-worlddynamics of thedrivingfactor(typically, the short rate) and a separate specication of the associated risk premium wasthe approach of choice for the rst term-structure models. I discuss the evolution of thesemodelsindetailinRebonato(2004), whereIexplainwhythispracticewasabandonedin favour of working directly in the risk-neutral measure.Whilebothroutesareinprinciplepossible, thesedaysforpracticalpricingpurposestherelative-pricingrouteis almost universallyadoptedfor derivatives. Therefore, thespecication of a relative-pricing, arbitrage-free model in a complete-market setting has incurrent tradingpractice becometantamounttoassigningthecovariance structureamongthestatevariables (or just thevolatility, if onlyonestochasticvariabledescribesthenancial universe).When this is combined with the market practices of out-of-model hedging and modelre-calibration discussed in the previous section, it produces some important consequences,which have a direct bearing on calibration. This is because any choice of volatilities andcorrelations will determine the model-implied future