Table Of ContentListed Volatility
and Variance
Derivatives
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Listed Volatility
and Variance
Derivatives
A Python-based Guide
DR. YVES J. HILPISCH
Thiseditionfirstpublished2017
©2017YvesHilpisch
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Contents
Preface xi
PARTONE
IntroductiontoVolatilityandVariance
CHAPTER1
Derivatives,VolatilityandVariance 3
1.1 OptionPricingandHedging 3
1.2 NotionsofVolatilityandVariance 6
1.3 ListedVolatilityandVarianceDerivatives 7
1.3.1 TheUSHistory 7
1.3.2 TheEuropeanHistory 8
1.3.3 VolatilityofVolatilityIndexes 9
1.3.4 ProductsCoveredinthisBook 10
1.4 VolatilityandVarianceTrading 11
1.4.1 VolatilityTrading 11
1.4.2 VarianceTrading 13
1.5 PythonasOurToolofChoice 14
1.6 QuickGuideThroughtheRestoftheBook 14
CHAPTER2
IntroductiontoPython 17
2.1 PythonBasics 17
2.1.1 DataTypes 17
2.1.2 DataStructures 20
2.1.3 ControlStructures 22
2.1.4 SpecialPythonIdioms 23
2.2 NumPy 28
2.3 matplotlib 34
2.4 pandas 38
2.4.1 pandasDataFrameclass 39
2.4.2 Input-OutputOperations 45
2.4.3 FinancialAnalyticsExamples 47
2.5 Conclusions 53
v
vi CONTENTS
CHAPTER3
Model-FreeReplicationofVariance 55
3.1 Introduction 55
3.2 SpanningwithOptions 56
3.3 LogContracts 57
3.4 StaticReplicationofRealizedVarianceandVarianceSwaps 57
3.5 ConstantDollarGammaDerivativesandPortfolios 58
3.6 PracticalReplicationofRealizedVariance 59
3.7 VSTOXXasVolatilityIndex 65
3.8 Conclusions 67
PARTTWO
ListedVolatilityDerivatives
CHAPTER4
DataAnalysisandStrategies 71
4.1 Introduction 71
4.2 RetrievingBaseData 71
4.2.1 EUROSTOXX50Data 71
4.2.2 VSTOXXData 74
4.2.3 CombiningtheDataSets 76
4.2.4 SavingtheData 78
4.3 BasicDataAnalysis 78
4.4 CorrelationAnalysis 83
4.5 ConstantProportionInvestmentStrategies 87
4.6 Conclusions 93
CHAPTER5
VSTOXXIndex 95
5.1 Introduction 95
5.2 CollectingOptionData 95
5.3 CalculatingtheSub-Indexes 105
5.3.1 TheAlgorithm 106
5.4 CalculatingtheVSTOXXIndex 114
5.5 Conclusions 118
5.6 PythonScripts 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
CHAPTER6
ValuingVolatilityDerivatives 129
6.1 Introduction 129
6.2 TheValuationFramework 129
6.3 TheFuturesPricingFormula 130
Contents vii
6.4 TheOptionPricingFormula 132
6.5 MonteCarloSimulation 135
6.6 AutomatedMonteCarloTests 141
6.6.1 TheAutomatedTesting 141
6.6.2 TheStorageFunctions 145
6.6.3 TheResults 146
6.7 ModelCalibration 153
6.7.1 TheOptionQuotes 154
6.7.2 TheCalibrationProcedure 155
6.7.3 TheCalibrationResults 160
6.8 Conclusions 163
6.9 PythonScripts 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
CHAPTER7
AdvancedModelingoftheVSTOXXIndex 179
7.1 Introduction 179
7.2 MarketQuotesforCallOptions 179
7.3 TheSRJDModel 182
7.4 TermStructureCalibration 183
7.4.1 FuturesTermStructure 184
7.4.2 ShiftedVolatilityProcess 190
7.5 OptionValuationbyMonteCarloSimulation 191
7.5.1 MonteCarloValuation 191
7.5.2 TechnicalImplementation 192
7.6 ModelCalibration 195
7.6.1 ThePythonCode 196
7.6.2 ShortMaturity 199
7.6.3 TwoMaturities 201
7.6.4 FourMaturities 203
7.6.5 AllMaturities 205
7.7 Conclusions 209
7.8 PythonScripts 210
7.8.1 srjd fwd calibration.py 210
7.8.2 srjd_simulation.py 212
7.8.3 srjd_model_calibration.py 215
CHAPTER8
TermsoftheVSTOXXanditsDerivatives 221
8.1 TheEUROSTOXX50Index 221
8.2 TheVSTOXXIndex 221
8.3 VSTOXXFuturesContracts 223
8.4 VSTOXXOptionsContracts 224
8.5 Conclusions 225
viii CONTENTS
PARTTHREE
ListedVarianceDerivatives
CHAPTER9
RealizedVarianceandVarianceSwaps 229
9.1 Introduction 229
9.2 RealizedVariance 229
9.3 VarianceSwaps 235
9.3.1 DefinitionofaVarianceSwap 235
9.3.2 NumericalExample 235
9.3.3 Mark-to-Market 239
9.3.4 VegaSensitivity 241
9.3.5 VarianceSwapontheEUROSTOXX50 242
9.4 Variancevs.Volatility 247
9.4.1 SquaredVariations 247
9.4.2 AdditivityinTime 247
9.4.3 StaticHedges 250
9.4.4 BroadMeasureofRisk 250
9.5 Conclusions 250
CHAPTER10
VarianceFuturesatEurex 251
10.1 Introduction 251
10.2 VarianceFuturesConcepts 252
10.2.1 RealizedVariance 252
10.2.2 NetPresentValueConcepts 252
10.2.3 TradedVarianceStrike 257
10.2.4 TradedFuturesPrice 257
10.2.5 NumberofFutures 258
10.2.6 ParVarianceStrike 258
10.2.7 FuturesSettlementPrice 258
10.3 ExampleCalculationforaVarianceFuture 258
10.4 ComparisonofVarianceSwapandFuture 265
10.5 Conclusions 268
CHAPTER11
TradingandSettlement 269
11.1 Introduction 269
11.2 OverviewofVarianceFuturesTerms 269
11.3 IntradayTrading 270
11.4 TradeMatching 274
11.5 DifferentTradedVolatilities 275
11.6 AftertheTradeMatching 277
11.7 FurtherDetails 279
11.7.1 InterestRateCalculation 279
11.7.2 MarketDisruptionEvents 280
11.8 Conclusions 280
Contents ix
PARTFOUR
DXAnalytics
CHAPTER12
DXAnalytics–AnOverview 283
12.1 Introduction 283
12.2 ModelingRiskFactors 284
12.3 ModelingDerivatives 287
12.4 DerivativesPortfolios 290
12.4.1 ModelingPortfolios 292
12.4.2 SimulationandValuation 293
12.4.3 RiskReports 294
12.5 Conclusions 296
CHAPTER13
DXAnalytics–Square-RootDiffusion 297
13.1 Introduction 297
13.2 DataImportandSelection 297
13.3 ModelingtheVSTOXXOptions 301
13.4 CalibrationoftheVSTOXXModel 303
13.5 Conclusions 308
13.6 PythonScripts 308
13.6.1 dx srd calibration.py 308
CHAPTER14
DXAnalytics–Square-RootJumpDiffusion 315
14.1 Introduction 315
14.2 ModelingtheVSTOXXOptions 315
14.3 CalibrationoftheVSTOXXModel 320
14.4 CalibrationResults 325
14.4.1 CalibrationtoOneMaturity 325
14.4.2 CalibrationtoTwoMaturities 325
14.4.3 CalibrationtoFiveMaturities 325
14.4.4 CalibrationwithoutPenalties 331
14.5 Conclusions 332
14.6 PythonScripts 334
14.6.1 dx srjd calibration.py 334
Bibliography 345
Index 347
Preface
Volatility and variance trading has evolved from something opaque to a standard tool in
today’sfinancialmarkets.Themotivesfortradingvolatilityandvarianceasanassetclass
ofitsownarenumerous.Amongothers,itallowsforeffectiveoptionandequityportfoliohedg-
ingandriskmanagementaswellasstraightoutspeculationonfuturevolatility(index)move-
ments.Thepotentialbenefitsofvolatility-andvariance-basedstrategiesarewidelyaccepted
byresearchersandpractitionersalike.
Withregard toproducts itmainlystartedout around 1993 withover-the-counter (OTC)
varianceswaps.Ataboutthesametime,theChicagoBoardOptionsExchangeintroducedthe
VIXvolatilityindex.Thisindexstillservestoday–afterasignificantchangeinitsmethod-
ology – as the underlying risk factor for some of the most liquidly traded listed derivatives
inthisarea.Thelistingofsuchderivativesallowsforamorestandardized,costefficientand
transparentapproachtovolatilityandvariancetrading.
This book covers some of the most important listed volatility and variance derivatives
with a focus on products provided by Eurex. Larger parts of the content are based on the
EurexAdvancedServicestutorialserieswhichusePythontoillustratethemainconceptsof
volatilityandvarianceproducts.IamgratefulthatEurexallowedmetousethecontentsofthe
tutorialseriesfreelyforthisbook.
Pythonhasbecomenotonlyoneofthemostwidelyusedprogramminglanguagesbutalso
oneofthemajortechnologyplatformsinthefinancialindustry.Itismorelikeaplatformsince
thePythonecosystemprovidesawealthofpowerfullibrariesandpackagesusefulforfinancial
analyticsandapplicationbuilding.Italsointegrateswellwithmanyothertechnologies,like
thestatisticalprogramming language R,usedinthefinancial industry. Youcanfind linksto
allPythonresourcesunderhttp://lvvd.tpq.io.
IthankMichaelSchwedforprovidingpartsofthePythoncode.Ialsothankmyfamily
foralltheirloveandsupportovertheyears,especiallymywifeSandraandourchildrenLilli
andHenry.IdedicatethisbooktomybeloveddogJil.Imissyou.
Yves
Voelklingen,Saarland,April2016
xi
