目次

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     5 1.3.1 Understanding Option Premiums   6 1.3.2 Relation between Premium and Breakeven  7 1.4 Strike Conventions      8 1.5 What is Volatility? 9 1.5.1 Implied Volatility, simplied     9 1.5.2 Probabilities and Breakevens     13 1.5.3 Implied Volatility and Realised Volatility   13 1.5.4 Realised Volatility, srealised    14 1.6 Trader’s Summary 17 2 Understanding OptionsWithout a Model    19 2.1 Vanilla Options  19 2.1.1 Option Payoffs      20 2.2 Making Assumptions     21 2.3 Understanding Vt with Economic Assumptions    21 2.4 Delta and Delta Hedging     23 2.5 The Value Function      24 2.6 Defining Delta 25 2.7 Understanding Delta      26 2.8 Delta as the Probability of an In–The–Money Expiry   29 2.9 Applying Delta as the Probability of an ITM Expiry in Practical Trading  33 2.10 Constructing Vt  34 2.10.1 Jensen’s Inequality 35 2.10.2 Trading Intuition Behind Jensen’s Inequality   36 2.10.3 American Options     37 2.10.4 Gradient of Vt     37 2.10.5 Drawing Vt      37 2.11 Option Deltas 39 2.12 A Note on Forwards      39 2.13 Put–Call Parity 41 2.14 Trader’s Summary 43 3 The Basic Greeks: Theta 45 3.1 Theta, q  46 3.1.1 Overnight Theta for an ATM option    47 3.1.2 Dependence of q(St ; t;si) on St    48 3.1.3 Dependence of q(St ; t;si) on t    56 3.2 Trader’s Summary 60 4 The Basic Greeks: Gamma      61 4.1 Gamma, G  62 4.2 Gamma and Time Decay     63 4.3 Traders’ Gamma, Gtrader      64 4.4 Gamma–Time Decay Trade–offs In More Detail   64 4.5 PnL Explain  66 4.5.1 Example: Gamma, Time Decay and PnL Explain for a 1 week Option      66 4.6 Delta Hedging and PnL Variance     69 4.7 Transaction Costs 71 4.8 Daily PnL Explain 71 4.9 The Gamma Profile      73 4.9.1 Gamma and Spot     73 4.9.2 Gamma and Implied Volatility    74 4.9.3 Gamma and Time     75 4.9.4 Total Gamma.     76 4.10 Trader’s Summary 76 5 The Basic Greeks: Vega      79 5.1 Vega  80 5.2 Understanding Vega via the PDF     81 5.3 Understanding Vega via Gamma Trading   81 5.4 Vega of an ATMS Option across Tenors    82 5.5 Vega and Spot 82 5.6 Dependence of Vega on Implied Volatility    85 5.7 Vega Profiles Applied in Practical Options Trading   85 5.8 Vega and PnL Explain     87 5.9 Trader’s Summary 87 6 Implied Volatility and Term Structure    89 6.1 Implied Volatility, simplied     90 6.2 Term Structure 94 6.3 Flat Vega and Weighted Vega Greeks     94 6.3.1 Flat Vega 94 6.3.2 Weighted Vega      95 6.3.3 Beta Weighted Vega      97 6.4 Forward Volatility, Forward Variance and Term Volatility  97 6.4.1 Calculating Implied Forward Volatility    99 6.5 Building a Term Structure Model using Daily Forward Volatility 100 6.6 Setting Base Volatility Using a 3 Parameter GARCH Model 102 6.6.1 Applying the 3 Parameter Model   104 6.6.2 Limitations of GARCH    105 6.6.3 Risk Management Using the 3 Parameter Model   106 6.6.4 Empirical GARCH estimation    106 6.7 Volatility Carry and Forward Volatility Agreements   107 6.7.1 Volatility Carry in the GARCH model   108 6.7.2 Common Pitfalls in Volatility Carry Trading   108 6.8 Trader’s Summary 109 7 Vanna, Risk Reversal and Skewness     111 7.1 Risk Reversal 112 7.2 Skewness 114 7.3 Delta Space  116 7.4 Smile in Delta Space      117 7.5 Smile Vega  119 7.5.1 Smile Vega Notionals     121 7.6 Smile Delta  122 7.6.1 Considerations Relating to Smile Delta    123 7.7 Trader’s Summary 124 8 Volgamma, Butterfly and Kurtosis    125 8.1 The Butterfly Strategy     126 8.2 Volgamma and Butterfly      127 8.3 Kurtosis  128 8.4 Smile  129 8.5 Butterflies and Smile Vega     130 8.6 Trader’s Summary 131 9 Black–Scholes–Merton Model     133 9.1 The Log–normal Diffusion Model     133 9.2 The BSM Partial Differential Equation (PDE)   134 9.3 Feynman–Kac. 137 9.4 Risk Neutral Probabilities     138 9.5 Probability of Exceeding the Breakeven in the BSM model  139 9.6 Trader’s Summary 139 10 The Black–Scholes Greeks     141 10.1 Spot Delta, Dual Delta and Forward Delta    141 10.1.1 Spot Delta 141 10.1.2 The ATM Strike and the Delta Neutral Straddle  143 10.1.3 Dual Delta 144 10.1.4 Forward Delta     144 10.2 Theta  145 10.3 Gamma  147 10.4 Vega  147 10.5 Vanna   148 10.6 Volgamma  148 10.7 Trader’s Summary 148 11 Predictability and Mean Reversion    149 11.1 The Past and the Future      149 11.2 Empirical Analysis      150 A Probability   155 A.1 Probability Density Functions (PDFs)    155 A.1.1 Discrete Random Variables and PMFs   155 A.1.2 Continuous Random Variables and PDFs   156 A.1.3 Normal and Lognormal Distributions   157 B Calculus  161 Glossary   163 References  167