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 μ 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 Δ 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 α 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