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