Preface xi
Part One Introduction to Volatility and Variance
Chapter 1 Derivatives, Volatility and Variance 3
1.1 Option Pricing and Hedging 3
1.2 Notions of Volatility and Variance 6
1.3 Listed Volatility and Variance Derivatives 7
1.3.1 The US History 7
1.3.2 The European History 8
1.3.3 Volatility of Volatility Indexes 9
1.3.4 Products Covered in this Book 10
1.4 Volatility and Variance Trading 11
1.4.1 Volatility Trading 11
1.4.2 Variance Trading 13
1.5 Python as Our Tool of Choice 14
1.6 Quick Guide Through the Rest of the Book 14
Chapter 2 Introduction to Python 17
2.1 Python Basics 17
2.1.1 Data Types 17
2.1.2 Data Structures 20
2.1.3 Control Structures 22
2.1.4 Special Python Idioms 23
2.2 NumPy 28
2.3 matplotlib 34
2.4 pandas 38
2.4.1 pandas DataFrame class 39
2.4.2 Input-Output Operations 45
2.4.3 Financial Analytics Examples 47
2.5 Conclusions 53
Chapter 3 Model-Free Replication of Variance 55
3.1 Introduction 55
3.2 Spanning with Options 56
3.3 Log Contracts 57
3.4 Static Replication of Realized Variance and Variance Swaps 57
3.5 Constant Dollar Gamma Derivatives and Portfolios 58
3.6 Practical Replication of Realized Variance 59
3.7 VSTOXX as Volatility Index 65
3.8 Conclusions 67
Part Two Listed Volatility Derivatives
Chapter 4 Data Analysis and Strategies 71
4.1 Introduction 71
4.2 Retrieving Base Data 71
4.2.1 EURO STOXX 50 Data 71
4.2.2 VSTOXX Data 74
4.2.3 Combining the Data Sets 76
4.2.4 Saving the Data 78
4.3 Basic Data Analysis 78
4.4 Correlation Analysis 83
4.5 Constant Proportion Investment Strategies 87
4.6 Conclusions 93
Chapter 5 VSTOXX Index 95
5.1 Introduction 95
5.2 Collecting Option Data 95
5.3 Calculating the Sub-Indexes 105
5.3.1 The Algorithm 106
5.4 Calculating the VSTOXX Index 114
5.5 Conclusions 118
5.6 Python Scripts 118
5.6.1 index collect option_data.py 118
5.6.2 index_subindex_calculation.py 123
5.6.3 index_vstoxx_calculation.py 127
Chapter 6 Valuing Volatility Derivatives 129
6.1 Introduction 129
6.2 The Valuation Framework 129
6.3 The Futures Pricing Formula 130
6.4 The Option Pricing Formula 132
6.5 Monte Carlo Simulation 135
6.6 Automated Monte Carlo Tests 141
6.6.1 The Automated Testing 141
6.6.2 The Storage Functions 145
6.6.3 The Results 146
6.7 Model Calibration 153
6.7.1 The Option Quotes 154
6.7.2 The Calibration Procedure 155
6.7.3 The Calibration Results 160
6.8 Conclusions 163
6.9 Python Scripts 163
6.9.1 srd_functions.py 163
6.9.2 srd simulation analysis.py 167
6.9.3 srd simulation results.py 171
6.9.4 srd model calibration.py 174
Chapter 7 Advanced Modeling of the VSTOXX Index 179
7.1 Introduction 179
7.2 Market Quotes for Call Options 179
7.3 The SRJD Model 182
7.4 Term Structure Calibration 183
7.4.1 Futures Term Structure 184
7.4.2 Shifted Volatility Process 190
7.5 Option Valuation by Monte Carlo Simulation 191
7.5.1 Monte Carlo Valuation 191
7.5.2 Technical Implementation 192
7.6 Model Calibration 195
7.6.1 The Python Code 196
7.6.2 Short Maturity 199
7.6.3 Two Maturities 201
7.6.4 Four Maturities 203
7.6.5 All Maturities 205
7.7 Conclusions 209
7.8 Python Scripts 210
7.8.1 srjd fwd calibration.py 210
7.8.2 srjd_simulation.py 212
7.8.3 srjd_model_calibration.py 215
Chapter 8 Terms of the VSTOXX and its Derivatives 221
8.1 The EURO STOXX 50 Index 221
8.2 The VSTOXX Index 221
8.3 VSTOXX Futures Contracts 223
8.4 VSTOXX Options Contracts 224
8.5 Conclusions 225
Part Three Listed Variance Derivatives
Chapter 9 Realized Variance and Variance Swaps 229
9.1 Introduction 229
9.2 Realized Variance 229
9.3 Variance Swaps 235
9.3.1 Definition of a Variance Swap 235
9.3.2 Numerical Example 235
9.3.3 Mark-to-Market 239
9.3.4 Vega Sensitivity 241
9.3.5 Variance Swap on the EURO STOXX 50 242
9.4 Variance vs. Volatility 247
9.4.1 Squared Variations 247
9.4.2 Additivity in Time 247
9.4.3 Static Hedges 250
9.4.4 Broad Measure of Risk 250
9.5 Conclusions 250
Chapter 10 Variance Futures at Eurex 251
10.1 Introduction 251
10.2 Variance Futures Concepts 252
10.2.1 Realized Variance 252
10.2.2 Net Present Value Concepts 252
10.2.3 Traded Variance Strike 257
10.2.4 Traded Futures Price 257
10.2.5 Number of Futures 258
10.2.6 Par Variance Strike 258
10.2.7 Futures Settlement Price 258
10.3 Example Calculation for a Variance Future 258
10.4 Comparison of Variance Swap and Future 265
10.5 Conclusions 268
Chapter 11 Trading and Settlement 269
11.1 Introduction 269
11.2 Overview of Variance Futures Terms 269
11.3 Intraday Trading 270
11.4 Trade Matching 274
11.5 Different Traded Volatilities 275
11.6 After the Trade Matching 277
11.7 Further Details 279
11.7.1 Interest Rate Calculation 279
11.7.2 Market Disruption Events 280
11.8 Conclusions 280
Part Four DX Analytics
Chapter 12 DX Analytics - An Overview 283
12.1 Introduction 283
12.2 Modeling Risk Factors 284
12.3 Modeling Derivatives 287
12.4 Derivatives Portfolios 290
12.4.1 Modeling Portfolios 292
12.4.2 Simulation and Valuation 293
12.4.3 Risk Reports 294
12.5 Conclusions 296
Chapter 13 DX Analytics - Square-Root Diffusion 297
13.1 Introduction 297
13.2 Data Import and Selection 297
13.3 Modeling the VSTOXX Options 301
13.4 Calibration of the VSTOXX Model 303
13.5 Conclusions 308
13.6 Python Scripts 308
13.6.1 dx srd calibration.py 308
Chapter 14 DX Analytics - Square-Root Jump Diffusion 315
14.1 Introduction 315
14.2 Modeling the VSTOXX Options 315
14.3 Calibration of the VSTOXX Model 320
14.4 Calibration Results 325
14.4.1 Calibration to One Maturity 325
14.4.2 Calibration to Two Maturities 325
14.4.3 Calibration to Five Maturities 325
14.4.4 Calibration without Penalties 331
14.5 Conclusions 332
14.6 Python Scripts 334
14.6.1 dx srjd calibration.py 334
Bibliography 345
Index 347
