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

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

ISBN: 978-1-119-01163-7 April 2015 224 Pages

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## Description

**A unique, in-depth guide to options pricing and valuing their greeks, along with a four dimensional approach towards the impact of changing market circumstances on options**

How to Calculate Options Prices and Their Greeks is the only book of its kind, showing you how to value options and the greeks according to the Black Scholes model but also how to do this without consulting a model. You'll build a solid understanding of options and hedging strategies as you explore the concepts of probability, volatility, and put call parity, then move into more advanced topics in combination with a four-dimensional approach of the change of the P&L of an option portfolio in relation to strike, underlying, volatility, and time to maturity. This informative guide fully explains the distribution of first and second order Greeks along the whole range wherein an option has optionality, and delves into trading strategies, including spreads, straddles, strangles, butterflies, kurtosis, vega-convexity , and more. Charts and tables illustrate how specific positions in a Greek evolve in relation to its parameters, and digital ancillaries allow you to see 3D representations using your own parameters and volumes.

The Black and Scholes model is the most widely used option model, appreciated for its simplicity and ability to generate a fair value for options pricing in all kinds of markets. This book shows you the ins and outs of the model, giving you the practical understanding you need for setting up and managing an option strategy.

• Understand the Greeks, and how they make or break a strategy

• See how the Greeks change with time, volatility, and underlying

• Explore various trading strategies

• Implement options positions, and more

Representations of option payoffs are too often based on a simple two-dimensional approach consisting of P&L versus underlying at expiry. This is misleading, as the Greeks can make a world of difference over the lifetime of a strategy. How to Calculate Options Prices and Their Greeks is a comprehensive, in-depth guide to a thorough and more effective understanding of options, their Greeks, and (hedging) option strategies.

## Table of contents

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