Contents
0.1 Why a Second Edition . . . . . . . . . . . . . . . . . . . . . . . . xvii
0.2 What This Book Is Not About . . . . . . . . . . . . . . . . . . . xix
0.3 Structure of the Book . . . . . . . . . . . . . . . . . . . . . . . . xx
0.4 The New Sub-Title . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
I Foundations 1
1 Theory and Practice of Option Modelling 3
1.1 Introduction and Plan of the Chapter . . . . . . . . . . . . . . . 3
1.2 The Role of Models in Derivatives Pricing . . . . . . . . . . . . . 3
1.2.1 What Are Models For? . . . . . . . . . . . . . . . . . . . . 3
1.2.2 The Fundamental Approach . . . . . . . . . . . . . . . . . 5
1.2.3 The Instrumental Approach . . . . . . . . . . . . . . . . . 7
1.2.4 A Conundrum (or, ”What is Vega Hedging For?”) . . . . 9
1.3 The E¢cient Market Hypothesis and Why It Matters for Option
Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 The Three Forms of the EMH . . . . . . . . . . . . . . . . 10
1.3.2 Pseudo-Arbitrageurs in Crisis . . . . . . . . . . . . . . . . 11
1.3.3 Model Risk for Traders and Risk Managers . . . . . . . . 12
1.3.4 The Parable of the Two Volatility Traders . . . . . . . . . 13
1.4 Market Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 Di¤erent Users of Derivatives Models . . . . . . . . . . . . 16
1.4.2 In-Model and Out-of-Model Hedging . . . . . . . . . . . . 16
1.5 The Calibration Debate . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.1 Historical versus Implied Calibration . . . . . . . . . . . . 20
1.5.2 The Logical Underpinning of the Implied Approach . . . . 21
1.5.3 Are Derivatives Markets Informationally E¢cient? . . . . 23
1.5.4 Back to Calibration . . . . . . . . . . . . . . . . . . . . . 29
1.5.5 A Practical Recommendation . . . . . . . . . . . . . . . . 29
1.6 Across-Markets Comparison of Pricing and Modelling Practices . 30
1.7 Using Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Option Replication 35
2.1 The Bedrock of Option Pricing . . . . . . . . . . . . . . . . . . . 35
2.2 The Analytic (PDE) Approach . . . . . . . . . . . . . . . . . . . 36
2.2.1 The Assumptions . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.2 The Portfolio-Replication Argument (Deterministic Volatility)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.3 The Market Price of Risk with Deterministic Volatility . . 39
2.2.4 Link with Expectations - the Feynman-Kac Theorem . . . 41
2.3 Binomial replication . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.1 First approach - Replication Strategy . . . . . . . . . . . 43
2.3.2 Second approach - ‘Naive Expectation?. . . . . . . . . . . 45
2.3.3 Third Approach - ‘Market Price of Risk?. . . . . . . . . . 47
2.3.4 A Worked-Out Example . . . . . . . . . . . . . . . . . . . 49
2.3.5 Fourth Approach - Risk-Neutral Valuation . . . . . . . . . 51
2.3.6 Pseudo-Probabilities . . . . . . . . . . . . . . . . . . . . . 52
2.3.7 Are the Quantities ? and ? Really Probabilities? . . . . 53
2.3.8 Introducing Relative Prices . . . . . . . . . . . . . . . . . 55
2.3.9 Moving to a Multi-Period Setting . . . . . . . . . . . . . . 57
2.3.10 Fair Prices as Expectations . . . . . . . . . . . . . . . . . 60
2.3.11 Switching Numeraires and Relating Expectations Under
Di¤erent Measures . . . . . . . . . . . . . . . . . . . . . . 62
2.3.12 Another Worked-Out Example . . . . . . . . . . . . . . . 65
2.3.13 Relevance of the Results . . . . . . . . . . . . . . . . . . . 66
2.4 Justifying the Two-State Branching Procedure . . . . . . . . . . 67
2.4.1 How To Tell a Jump When You Seen One . . . . . . . . . 67
2.5 The Nature of the Transformation between Measures: Girsanov’s
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.5.1 An Intuitive Argument . . . . . . . . . . . . . . . . . . . . 71
2.5.2 A Worked-Out Example . . . . . . . . . . . . . . . . . . . 73
2.6 Switching Between the PDE, the Expectation and the Binomial-
Replication Approaches . . . . . . . . . . . . . . . . . . . . . . . 75
3 The Building Blocks 77
3.1 Introduction and Plan of the Chapter . . . . . . . . . . . . . . . 77
3.2 De…nition of Market Terms . . . . . . . . . . . . . . . . . . . . . 77
3.3 Hedging Forward Contracts Using Spot Quantities . . . . . . . . 80
3.3.1 Hedging Equity Forward Contracts . . . . . . . . . . . . . 80
3.3.2 Hedging Interest-Rate Forward Contracts . . . . . . . . . 81
3.4 Hedging Options: Volatility of Spot and Forward Processes . . . 82
3.5 The Link Between Root-Mean-Squared Volatilities and the Time-
Dependence of Volatility . . . . . . . . . . . . . . . . . . . . . . . 87
3.6 Admissibility of a Series of Root-Mean-Squared Volatilities . . . 88
3.6.1 The Equity/FX Case . . . . . . . . . . . . . . . . . . . . . 88
3.6.2 The Ineterest-Rate Case . . . . . . . . . . . . . . . . . . . 89
3.7 Summary of the De…nitions So Far . . . . . . . . . . . . . . . . . 90
3.8 Hedging an Option with a Forward-Setting Strike . . . . . . . . 92

3.8.1 Why Is This Option Important? (And Why Is it Di¢cult
to Hedge?) . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.8.2 Valuing a Forward-Setting Option . . . . . . . . . . . . . 94
3.9 Quadratic Variation: First Approach . . . . . . . . . . . . . . . . 97
3.9.1 De…nition . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.9.2 Properties of Variations . . . . . . . . . . . . . . . . . . . 99
3.9.3 First and Second Variation of a Brownian Process . . . . 99
3.9.4 Links between Quadratic Variation and RT
t ¾(u)2du . . . 100
3.9.5 Why Quadratic Variation Is So Important (Take 1) . . . . 100
4 Variance and Mean Reversion in the Real and the Risk-Adjusted
Worlds 103
4.1 Introduction and Plan of the Chapter . . . . . . . . . . . . . . . 103
4.2 Hedging a Plain-Vanilla Option: General Framework . . . . . . . 104
4.2.1 Trading Restrictions and Model Uncertainty: Theoretical
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.2 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2.3 The Methodology . . . . . . . . . . . . . . . . . . . . . . . 107
4.2.4 Criterion for success . . . . . . . . . . . . . . . . . . . . . 108
4.3 Hedging Plain-Vanilla Options: Constant-Volatility . . . . . . . . 109
4.3.1 Trading the Gamma: One Step and Constant Volatility . 110
4.3.2 Trading the Gamma: Several Steps and Constant Volatility115
4.4 Hedging Plain-Vanilla Options: Time-Dependent Volatility . . . 116
4.4.1 Views on Gamma Trading When the Volatility is Time-
Dependent . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4.2 Which View Is the Correct One? (and the Feynman-Kac
Theorem Again) . . . . . . . . . . . . . . . . . . . . . . . 120
4.5 Hedging Behaviour In Practice . . . . . . . . . . . . . . . . . . . 122
4.5.1 Analyzing the Replicating Portfolio . . . . . . . . . . . . . 122
4.5.2 Hedging Results: the Time-Dependent Volatility Case . . 123
4.5.3 Hedging with the Wrong Volatility . . . . . . . . . . . . . 125
4.6 Robustness of the Black-and-Scholes Model . . . . . . . . . . . . 126
4.7 Is the Total Variance All That Matters? . . . . . . . . . . . . . . 128
4.8 Hedging Plain-Vanilla Options: Mean-Reverting Real-World Drift 129
4.9 Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again131
4.9.1 The Crouhy-Galai Set-Up . . . . . . . . . . . . . . . . . . 132
5 Instantaneous and Terminal Correlation 137
5.1 Correlation, Co-Integration and Multi-Factor Models . . . . . . . 137
5.1.1 The Multi-Factor Debate . . . . . . . . . . . . . . . . . . 140
5.2 The Stochastic Evolution of Imperfectly Correlated Variables . . 142
5.3 The Role of Terminal Correlation in the Joint Evolution of Stochastic
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.3.1 De…ning Stochastic Integrals . . . . . . . . . . . . . . . . 146
5.3.2 First case: European Option, One Underlying Asset . . . 148
5.3.3 Case 2: Path-Dependent Option, One Asset . . . . . . . . 150