How to Calculate Options Prices and Their Greeks: Exploring the Black Scholes Model from Delta to Vega

Praise for How to Calculate Options Prices and Their Greeks

"Clearly written and easy to read. The book covers all the essential topics and has many very interesting approaches. In my current job, liaising with derivatives trading houses and as a former options trader, I can conclude that the book demonstrates that the Netherlands still functions as an expert area for the options business. How to Calculate Options Prices and Their Greeks has everything in it to become the standard reference book on options. Highly recommended!"
Rogier Elsenburg, Head of Benelux & France Huxley Banking & Financial Services

"Pierino Ursone′s How to Calculate Options Prices and their Greeks provides a long overdue, comprehensive, and at the same time digestible update to the literature exploring options pricing and strategy. His writing is clear, concise, and straight–forward; unlocking the Holy Grail for the inexperienced and even the most experienced options trader. Ursone liberally illustrates each chapter with graphics and acknowledges the Black Scholes Model′s mathematical and theoretical nderpinnings in plain, straight–forward language that makes new understanding of the complexity of options trading possible for both the novice and the expert. Ursone′s thorough explanation of the practical application of first and second order greeks goes beyond the simplistic, two–dimensional strategies most traders will find when trying to educate themselves about options. He uses his vast experience and knowledge to act as expert, trader, tactician, and teacher, guiding readers through the markets and allowing them to look inside as they could never before."
Matt Daen, Options Specialist

"Finally a holistic guide to options! This practical guide is essential for those seeking an understanding of option derivatives. Beyond merely informative, this handbook for the practitioner details the finer points of the use of options as tools for trading and price–risk management. No professional should be without it."
Dr Tom James, Commodity Market Expert & Author

"An easy to read handbook for investors who wish to thoroughly understand, trade and manage an options portfolio. Pierino explains the characteristics and behaviour of options in a pragmatic way, coming from a trading perspective. Very helpful for private and professional investors with a goal of running a complex derivatives portfolio in a controlled manner."
Jelle Elzinga, Former Global Management Committee Optiver Group

Preface ix

Chapter 1 Introduction 1

Chapter 2 The Normal Probability Distribution 7

Standard deviation in a financial market 8

The impact of volatility and time on the standard deviation 8

Chapter 3 Volatility 11

The probability distribution of the value of a Future after one year of trading 11

Normal distribution versus log–normal distribution 11

Calculating the annualised volatility traditionally 15

Calculating the annualised volatility without 17

Calculating the annualised volatility applying the 16% rule 19

Variation in trading days 20

Approach towards intraday volatility 20

Historical versus implied volatility 23

Chapter 4 Put Call Parity 25

Synthetically creating a Future long position, the reversal 29

Synthetically creating a Future short position, the conversion 30

Synthetic options 31

Covered call writing 34

Short note on interest rates 35

Chapter 5 Delta 37

Change of option value through the delta 38

Dynamic delta 40

Delta at different maturities 41

Delta at different volatilities 44

20 80 Delta region 46

Delta per strike 46

Dynamic delta hedging 47

The at the money delta 50

Delta changes in time 53

Chapter 6 Pricing 55

Calculating the at the money straddle using

Black and Scholes formula 57

Determining the value of an at the money straddle 59

Chapter 7 Delta II 61

Determining the boundaries of the delta 61

Valuation of the at the money delta 64

Delta distribution in relation to the at the money straddle 65

Application of the delta approach, determining the delta of a call spread 68

Chapter 8 Gamma 71

The aggregate gamma for a portfolio of options 73

The delta change of an option 75

The gamma is not a constant 76

Long term gamma example 77

Short term gamma example 77

Very short term gamma example 78

Determining the boundaries of gamma 79

Determining the gamma value of an at the money straddle 80

Gamma in relation to time to maturity,

volatility and the underlying level 82

Practical example 85

Hedging the gamma 87

Determining the gamma of out of the money options 89

Derivatives of the gamma 91

Chapter 9 Vega 93

Different maturities will display different volatility regime changes 95

Determining the vega value of at the money options 96

Vega of at the money options compared to volatility 97

Vega of at the money options compared to time to maturity 99

Vega of at the money options compared to the underlying level 99

Vega on a 3–dimensional scale, vega vs maturity and vega vs volatility 101

Determining the boundaries of vega 102

Comparing the boundaries of vega with the boundaries of gamma 104

Determining vega values of out of the money options 105

Derivatives of the vega 108

Vomma 108

Chapter 10 Theta 111

A practical example 112

Theta in relation to volatility 114

Theta in relation to time to maturity 115

Theta of at the money options in relation to the underlying level 117

Determining the boundaries of theta 118

The gamma theta relationship 120

Theta on a 3–dimensional scale, theta vs maturity and theta vs volatility 125

Determining the theta value of an at the money straddle 126

Determining theta values of out of the money options 127

Chapter 11 Skew 129

Volatility smiles with different times to maturity 131

Sticky at the money volatility 133

Chapter 12 Spreads 135

Call spread (horizontal) 135

Put spread (horizontal) 137

Boxes 138

Applying boxes in the real market 139

The Greeks for horizontal spreads 140

Time spread 146

Approximation of the value of at the money spreads 148

Ratio spread 149

Chapter 13 Butterfly 155

Put call parity 158

Distribution of the butterfly 159

Boundaries of the butterfly 161

Method for estimating at the money butterfly values 163

Estimating out of the money butterfly values 164

Butterfly in relation to volatility 165

Butterfly in relation to time to maturity 166

Butterfly as a strategic play 166

The Greeks of a butterfly 167

Straddle strangle or the Iron fly 171

Chapter 14 Strategies 173

Call 173

Put 174

Call spread 175

Ratio spread 176

Straddle 177

Strangle 178

Collar (risk reversal, fence) 178

Gamma portfolio 179

Gamma hedging strategies based on Monte Carlo scenarios 180

Setting up a gamma position on the back of prevailing kurtosis in the market 190

Excess kurtosis 191

Benefitting from a platykurtic environment 192

The mesokurtic market 193

The leptokurtic market 193

Transition from a platykurtic environment towards a leptokurtic environment 194

Wrong hedging strategy: Killergamma 195

Vega convexity/Vomma 196

Vega convexity in relation to time/Veta 202

Index 205

PIERINO URSONE has extensive option trading experience. He began his career as a Market Maker with Optiver, an international market maker that trades on all of the world′s major financial markets. Afterwards, Ursone ran his own option trading company on the Dutch options exchange in Amsterdam, and after nine years in equity options, he entered the Energy commodity market, trading options on a proprietary basis.