Contents
Preface xxi
0.1 Why a Second Edition? xxi
0.2 What This Book Is Not About xxiii
0.3 Structure of the Book xxiv
0.4 The New Subtitle xxiv
Acknowledgements xxvii
I Foundations 1
1 Theory and Practice of Option Modelling 3
1.1 The Role of Models in Derivatives Pricing 3
1.1.1 What Are Models For? 3
1.1.2 The Fundamental Approach 5
1.1.3 The Instrumental Approach 7
1.1.4 A Conundrum (or, ‘What is Vega Hedging For?’) 8
1.2 The Efficient Market Hypothesis and Why It Matters for Option Pricing 9
1.2.1 The Three Forms of the EMH 9
1.2.2 Pseudo-Arbitrageurs in Crisis 10
1.2.3 Model Risk for Traders and Risk Managers 11
1.2.4 The Parable of the Two Volatility Traders 12
1.3 Market Practice 14
1.3.1 Different Users of Derivatives Models 14
1.3.2 In-Model and Out-of-Model Hedging 15
1.4 The Calibration Debate 17
1.4.1 Historical vs Implied Calibration 18
1.4.2 The Logical Underpinning of the Implied Approach 19
1.4.3 Are Derivatives Markets Informationally Efficient? 21
1.4.4 Back to Calibration 26
1.4.5 A Practical Recommendation 27
1.5 Across-Markets Comparison of Pricing and Modelling Practices 27
1.6 Using Models 30
2 Option Replication 31
2.1 The Bedrock of Option Pricing 31
2.2 The Analytic (PDE) Approach 32
2.2.1 The Assumptions 32
2.2.2 The Portfolio-Replication Argument (Deterministic Volatility) 32
2.2.3 The Market Price of Risk with Deterministic Volatility 34
2.2.4 Link with Expectations – the Feynman–Kac Theorem 36
2.3 Binomial Replication 38
2.3.1 First Approach – Replication Strategy 39
2.3.2 Second Approach – ‘Na¨ıve Expectation’ 41
2.3.3 Third Approach – ‘Market Price of Risk’ 42
2.3.4 A Worked-Out Example 45
2.3.5 Fourth Approach – Risk-Neutral Valuation 46
2.3.6 Pseudo-Probabilities 48
2.3.7 Are the Quantities π1 and π2 Really Probabilities? 49
2.3.8 Introducing Relative Prices 51
2.3.9 Moving to a Multi-Period Setting 53
2.3.10 Fair Prices as Expectations 56
2.3.11 Switching Numeraires and Relating Expectations Under
Different Measures 58
2.3.12 Another Worked-Out Example 61
2.3.13 Relevance of the Results 64
2.4 Justifying the Two-State Branching Procedure 65
2.4.1 How To Recognize a Jump When You See One 65
2.5 The Nature of the Transformation between Measures: Girsanov’s Theorem 69
2.5.1 An Intuitive Argument 69
2.5.2 A Worked-Out Example 70
2.6 Switching Between the PDE, the Expectation and the Binomial
Replication Approaches 73
3 The Building Blocks 75
3.1 Introduction and Plan of the Chapter 75
3.2 Definition of Market Terms 75
3.3 Hedging Forward Contracts Using Spot Quantities 77
3.3.1 Hedging Equity Forward Contracts 78
3.3.2 Hedging Interest-Rate Forward Contracts 79
3.4 Hedging Options: Volatility of Spot and Forward Processes 80
3.5 The Link Between Root-Mean-Squared Volatilities and the
Time-Dependence of Volatility 84
3.6 Admissibility of a Series of Root-Mean-Squared Volatilities 85
3.6.1 The Equity/FX Case 85
3.6.2 The Interest-Rate Case 86
3.7 Summary of the Definitions So Far 87
3.8 Hedging an Option with a Forward-Setting Strike 89
3.8.1 Why Is This Option Important? (And Why Is it Difficult
to Hedge?) 90
3.8.2 Valuing a Forward-Setting Option 91
3.9 Quadratic Variation: First Approach 95
3.9.1 Definition 95
3.9.2 Properties of Variations 96
3.9.3 First and Second Variation of a Brownian Process 97
3.9.4 Links between Quadratic Variation and
T
t σ(u)2 du 97
3.9.5 Why Quadratic Variation Is So Important (Take 1) 98