Preface xiii Acknowledgments xv About the Author xvii CHAPTER 1 Volatility and Options 1 1.1 What Is an Option? 1 1.2 Options Are Bets on Volatility 3 1.3 Option Premiums and Breakevens 6 1.3.1 Understanding Option Premiums 6 1.3.2 Relation Between Premium and Breakeven 7 1.4 Strike Conventions 9 1.5 What Is Volatility? 10 1.5.1 Implied Volatility, implied 11 1.5.2 Probabilities and Breakevens 15 1.5.3 Implied Volatility and Realized Volatility 15 1.5.4 Realized Volatility, realized 16 1.6 Trader's Summary 19 CHAPTER 2 Understanding Options Without a Model 21 2.1 Vanilla Options 21 2.1.1 Option Payoffs 22 2.2 Making Assumptions 23 2.3 Understanding Vt with Economic Assumptions 24 2.4 Delta and Delta Hedging 25 2.5 The Value Function 26 2.6 Defining Delta 27 2.7 Understanding Delta 30 2.8 Delta as the Probability of an In-the-Money Expiry 32 2.9 Applying Delta as the Probability of an ITM Expiry in Practical Trading 37 2.10 Constructing Vt 38 2.10.1 Jensen's Inequality: Vt = V(St, t, i) max(St K, 0) 40 2.10.2 Trading Intuition Behind Jensen's Inequality 40 2.10.3 American Options 41 2.10.4 Gradient of Vt 42 2.10.5 Drawing Vt 42 2.11 Option Deltas 44 2.12 A Note on Forwards 45 2.13 Put-Call Parity 46 2.14 Trader's Summary 48 CHAPTER 3 The Basic Greeks: Theta 49 3.1 Theta, ? 50 3.1.1 Overnight Theta for an ATM Option 51 3.1.2 Dependence of ?(St, t, i) on St 52 3.1.3 Dependence of ?(St, t, i) on t 60 3.2 Trader's Summary 65 CHAPTER 4 The Basic Greeks: Gamma 67 4.1 Gamma, ? 68 4.2 Gamma and Time Decay 70 4.3 Traders' Gamma, ?trader 70 4.4 Gamma-Time Decay Trade-offs in More Detail 71 4.5 PnL Explain 73 4.5.1 Example: Gamma, Time Decay, and PnL Explain for a 1-Week Option 73 4.6 Delta Hedging and PnL Variance 76 4.7 Transaction Costs 78 4.8 Daily PnL Explain 79 4.9 The Gamma Profile 81 4.9.1 Gamma and Spot 81 4.9.2 Gamma and Implied Volatility 82 4.9.3 Gamma and Time 83 4.9.4 Total Gamma 84 4.10 Trader's Summary 84 CHAPTER 5 The Basic Greeks: Vega 87 5.1 Vega 88 5.2 Understanding Vega via the PDF 89 5.3 Understanding Vega via Gamma Trading 89 5.4 Vega of an ATMS Option Across Tenors 90 5.5 Vega and Spot 91 5.6 Dependence of Vega on Implied Volatility 94 5.7 Vega Profiles Applied in Practical Options Trading 95 5.8 Vega and PnL Explain 96 5.9 Trader's Summary 97 CHAPTER 6 Implied Volatility and Term Structure 99 6.1 Implied Volatility, implied 100 6.2 Term Structure 104 6.3 Flat Vega and Weighted Vega Greeks 104 6.3.1 Flat Vega 105 6.3.2 Weighted Vega 106 6.3.3 Beta-Weighted Vega 108 6.4 Forward Volatility, Forward Variance, and Term Volatility 108 6.4.1 Calculating Implied Forward Volatility 110 6.5 Building a Term Structure Model Using Daily Forward Volatility 111 6.6 Setting Base Volatility Using a Three-Parameter GARCH Model 114 6.6.1 Applying the Three-Parameter Model 116 6.6.2 Limitations of GARCH 117 6.6.3 Risk Management Using the Three-Parameter Model 118 6.6.4 Empirical GARCH Estimation 118 6.7 Volatility Carry and Forward Volatility Agreements 119 6.7.1 Volatility Carry in the GARCH Model 120 6.7.2 Common Pitfalls in Volatility Carry Trading 121 6.8 Trader's Summary 121 CHAPTER 7 Vanna, Risk Reversal, and Skewness 123 7.1 Risk Reversal 125 7.2 Skewness 127 7.3 Delta Space 129 7.4 Smile in Delta Space 130 7.5 Smile Vega 132 7.5.1 Smile Vega Notionals 134 7.6 Smile Delta 135 7.6.1 Considerations Relating to Smile Delta 136 7.7 Trader's Summary 137 CHAPTER 8 Volgamma, Butterfly, and Kurtosis 139 8.1 The Butterfly Strategy 140 8.2 Volgamma and Butterfly 141 8.3 Kurtosis 142 8.4 Smile 143 8.5 Butterflies and Smile Vega 144 8.6 Trader's Summary 145 CHAPTER 9 Black-Scholes-Merton Model 147 9.1 The Log-normal Diffusion Model 148 9.2 The BSM Partial Differential Equation (PDE) 148 9.3 Feynman-Kac 152 9.4 Risk-Neutral Probabilities 153 9.5 Probability of Exceeding the Breakeven in the BSM Model 154 9.6 Trader's Summary 155 CHAPTER 10 The Black-Scholes Greeks 157 10.1 Spot Delta, Dual Delta, and Forward Delta 157 10.1.1 Spot Delta 157 10.1.2 The ATM Strike and the Delta-Neutral Straddle 159 10.1.3 Dual Delta 160 10.1.4 Forward Delta 161 10.2 Theta 161 10.3 Gamma 163 10.4 Vega 164 10.5 Vanna 164 10.6 Volgamma 165 10.7 Trader's Summary 165 CHAPTER 11 Predictability and Mean Reversion 167 11.1 The Past and the Future 167 11.2 Empirical Analysis 168 APPENDIX A Probability 173 A.1 Probability Density Functions (PDFs) 173 A.1.1 Discrete Random Variables and PMFs 173 A.1.2 Continuous Random Variables and PDFs 174 A.1.3 Normal and Log-normal Distributions 176 APPENDIX B Calculus 179 Glossary 181 References 183 Index 185 Implied Volatility, implied