# Volatility

## Practical Options Theory

Wiley Finance Editions

1. Edition November 2018

208 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-119-50161-9**

Gain a deep, intuitive and technical understanding of practical options theory

The main challenges in successful options trading are conceptual, not mathematical. Volatility: Practical Options Theory provides financial professionals, academics, students and others with an intuitive as well as technical understanding of both the basic and advanced ideas in options theory to a level that facilitates practical options trading. The approach taken in this book will prove particularly valuable to options traders and other practitioners tasked with making pricing and risk management decisions in an environment where time constraints mean that simplicity and intuition are of greater value than mathematical formalism.

The most important areas of options theory, namely implied volatility, delta hedging, time value and the so-called options greeks are explored based on intuitive economic arguments alone before turning to formal models such as the seminal Black-Scholes-Merton model. The reader will understand how the model free approach and mathematical models are related to each other, their underlying theoretical assumptions and their implications to level that facilitates practical implementation.

There are several excellent mathematical descriptions of options theory, but few focus on a translational approach to convert the theory into practice. This book emphasizes the translational aspect, while first building an intuitive, technical understanding that allows market makers, portfolio managers, investment managers, risk managers, and other traders to work more effectively within--and beyond--the bounds of everyday practice.

* Gain a deeper understanding of the assumptions underlying options theory

* Translate theoretical ideas into practice

* Develop a more accurate intuition for better time-constrained decision making

This book allows its readers to gain more than a superficial understanding of the mechanisms at work in options markets. Volatility gives its readers the edge by providing a true bedrock foundation upon which practical knowledge becomes stronger.

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, sigmaimplied 11

1.5.2 Probabilities and Breakevens 15

1.5.3 Implied Volatility and Realized Volatility 15

1.5.4 Realized Volatility, ¯sigmarealized 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, sigmai) >= 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, Theta 50

3.1.1 Overnight Theta for an ATM Option 51

3.1.2 Dependence of Theta(St, t, sigmai) on S¯t 52

3.1.3 Dependence of Theta(St, t, sigmai) on t 60

3.2 Trader's Summary 65

CHAPTER 4 The Basic Greeks: Gamma 67

4.1 Gamma, Gamma 68

4.2 Gamma and Time Decay 70

4.3 Traders' Gamma, Gammatrader 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, ¯sigmaimplied 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, sigmaimplied