Listed Volatility and Variance Derivatives - A Python-based Guide

Preface 1 I Introduction to Volatility and Variance 3 1 Derivatives, Volatility and Variance 5 1.1 Option Pricing and Hedging 5 1.2 Notions of Volatility and Variance 7 1.3 Listed Volatility and Variance Derivatives 8 1.3.1 The US History 8 1.3.2 The European History 10 1.3.3 Volatility of Volatility Indexes 11 1.3.4 Products Covered in this Book 12 1.4 Volatility and Variance Trading 12 1.4.1 Volatility Trading 13 1.4.2 Variance Trading 14 1.5 Python as Our Tool of Choice 15 1.6 Quick Guide Through Rest of the Book 15 2 Introduction to Python 19 2.1 Python Basics 19 2.1.1 Data Types 19 2.1.2 Data Structures 21 2.1.3 Control Structures 23 2.1.4 Special Python Idioms 24 2.2 NumPy 27 2.3 matplotlib 32 2.4 pandas 36 2.4.1 pandas Data Frame class 36 2.4.2 Input-Output Operations 40 2.4.3 Financial Analytics Examples 43 2.5 Conclusions 48 3 Model-Free Replication of Variance 49 3.1 Introduction 49 3.2 Spanning with Options 49 3.3 Log Contracts 50 3.4 Static Replication of Realized Variance and Variance Swaps 51 3.5 Constant Dollar Gamma Derivatives and Portfolios 51 3.6 Practical Replication of Realized Variance 52 3.7 VSTOXX as Volatility Index 57 3.8 Conclusions 59 II Listed Volatility Derivatives 61 4 Data Analysis and Strategies 63 4.1 Introduction 63 4.2 Retrieving Base Data 63 4.2.1 EURO STOXX 50 Data 63 4.2.2 VSTOXX Data 65 4.2.3 Combining the Data Sets 67 4.2.4 Saving the Data 68 4.3 Basic Data Analysis 69 4.4 Correlation Analysis 72 4.5 Constant Proportion Investment Strategies 77 4.6 Conclusions 82 5 VSTOXX Index 83 5.1 Introduction 83 5.2 Collecting Option Data 84 5.3 Calculating the Sub-Indexes 91 5.3.1 The Algorithm 91 5.4 Calculating the VSTOXX Index 98 5.5 Conclusions 101 5.6 Python Scripts 103 5.6.1 index-collect-option-data.py 103 5.6.2 index-subindex-calculation.py 107 5.6.3 index-vstoxx-calculation.py 110 6 Valuing Volatility Derivatives 113 6.1 Introduction 113 6.2 The Valuation Framework 113 6.3 The Futures Pricing Formula 114 6.4 The Option Pricing Formula 115 6.5 Monte Carlo Simulation 118 6.6 Automated Monte Carlo Tests 123 6.6.1 The Automated Testing 123 6.6.2 The Storage Functions 126 6.6.3 The Results 128 6.7 Model Calibration 133 6.7.1 The Option Quotes 133 6.7.2 The Calibration Procedure 134 6.7.3 The Calibration Results 138 6.8 Conclusions 141 6.9 Python Scripts 141 6.9.1 srd-functions.py 141 6.9.2 srd-simulation-analysis.py 145 6.9.3 srd-simulation-results.py 148 6.9.4 srd-model-calibration.py 151 7 Advanced Modeling of the VSTOXX Index 155 7.1 Introduction 155 7.2 Market Quotes for Call Options 155 7.3 The SRJD Model 158 7.4 Term Structure Calibration 159 7.4.1 Futures Term Structure 159 7.4.2 Shifted Volatility Process 163 7.5 Option Valuation by Monte Carlo Simulation 164 7.5.1 Monte Carlo Valuation 165 7.5.2 Technical Implementation 165 7.6 Model Calibration 169 7.6.1 The Python Code 169 7.6.2 Short Maturity 171 7.6.3 Two Maturities 173 7.6.4 Four Maturities 175 7.6.5 All Maturities 176 7.7 Conclusions 181 7.8 Python Scripts 181 7.8.1 srjd-fwd-calibration.py 181 7.8.2 srjd-simulation.py 183 7.8.3 srjd-model-calibration.py 185 8 Terms of the VSTOXX and its Derivatives 191 8.1 The EURO STOXX 50 Index 191 8.2 The VSTOXX Index 192 8.3 VSTOXX Futures Contracts 192 8.4 VSTOXX Options Contracts 193 8.5 Conclusions 195 III Listed Variance Derivatives 197 9 Realized Variance and Variance Swaps 199 9.1 Introdution 199 9.2 Realized Variance 199 9.3 Variance Swaps 204 9.3.1 Definition of a Variance Swap 204 9.3.2 Numerical Example 205 9.3.3 Mark-to-Market 208 9.3.4 Vega Sensitivity 209 9.3.5 Variance Swap on the EURO STOXX 50 211 9.4 Variance vs. Volatility 216 9.4.1 Squared Variations 216 9.4.2 Additivity in Time 216 9.4.3 Static Hedges 218 9.4.4 Broad Measure of Risk 218 9.5 Conclusions 218 10 Variance Futures at Eurex 219 10.1 Introduction 219 10.2 Variance Futures Concepts 220 10.2.1 Realized Variance 220 10.2.2 Net Present Value Concepts 220 10.2.3 Traded Variance Strike 224 10.2.4 Traded Futures Price 224 10.2.5 Number of Futures 225 10.2.6 Par Variance Strike 225 10.2.7 Futures Settlement Price 225 10.3 Example Calculation for a Variance Future 225 10.4 Comparison of Variance Swap and Future 230 10.5 Conclusions 233 11 Trading and Settlement 235 11.1 Introduction 235 11.2 Overview of Variance Futures Terms 235 11.3 Intraday Trading 236 11.4 Trade Matching 239 11.5 Different Traded Volatilities 239 11.6 After the Trade Matching 241 11.7 Further Details 243 11.7.1 Interest Rate Calculation 243 11.7.2 Market Disruption Events 243 11.8 Conclusions 244 IV DX Analytics 245 12 DX Analytics An Overview 247 12.1 Introduction 247 12.2 Modeling Risk Factors 248 12.3 Modeling Derivatives 250 12.4 Derivatives Portfolios 253 12.4.1 Modeling Portfolios 253 12.4.2 Simulation and Valuation 255 12.4.3 Risk Reports 256 12.5 Conclusions 257 13 DX Analytics Square-Root Diffusion 259 13.1 Introduction 259 13.2 Data Import and Selection 259 13.3 Modeling the VSTOXX Options 262 13.4 Calibration of the VSTOXX Model 264 13.5 Conclusions 269 13.6 Python Scripts 269 13.6.1 dx-srd-calibration.py 269 14 DX Analytics Square-Root Jump Diffusion 275 14.1 Introduction 275 14.2 Modeling the VSTOXX Options 275 14.3 Calibration of the VSTOXX Model 279 14.4 Calibration Results 283 14.4.1 Calibration to 1 Maturity 283 14.4.2 Calibration to 2 Maturities 283 14.4.3 Calibration to 5 Maturities 285 14.4.4 Calibration without Penalties 285 14.5 Conclusions 288 14.6 Python Scripts 288 14.6.1 dx-srjd-calibration.py 288 Bibliography 303 Index 305