Inside Volatility Arbitrage : The Secrets of Skewness

272 Pages·2005·4.96 MB·English

by  Alireza Javaheri

Inside Volatility
Arbitrage
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Inside Volatility
Arbitrage
The Secrets of Skewness
ALIREZA JAVAHERI
John Wiley & Sons, Inc.
Copyright©2005byAlirezaJavaheri.Allrightsreserved
PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey
PublishedsimultaneouslyinCanada
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LibraryofCongressCataloging-in-PublicationData
Javaheri, Alireza.
Insidevolatilityarbitrage : thesecretsofskewness / AlirezaJavaheri.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN0-471-73387-3(cloth)
1. Stocks–Proces–Mathematicalmodels. 2. Stochasticprocesses. I.
Title.
HG4636.J382005
332.63’222’0151922–dc22 2005004696
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
Contents
Illustrations ix
Acknowledgments xv
Introduction xvii
Summary xvii
ContributionsandFurtherResearch xxiii
DataandPrograms xxiv
CHAPTER1
TheVolatilityProblem 1
Introduction 1
TheStockMarket 2
TheStockPriceProcess 2
HistoricVolatility 3
TheDerivativesMarket 4
TheBlack-ScholesApproach 5
TheCox-Ross-RubinsteinApproach 6
JumpDiffusionandLevel-DependentVolatility 7
JumpDiffusion 8
Level-DependentVolatility 10
LocalVolatility 14
TheDupireApproach 14
TheDerman-KaniApproach 17
StabilityIssues 18
CalibrationFrequency 19
StochasticVolatility 20
StochasticVolatilityProcesses 20
GARCHandDiffusionLimits 21
ThePricingPDEUnderStochasticVolatility 24
TheMarketPriceofVolatilityRisk 25
TheTwo-FactorPDE 26
TheGeneralizedFourierTransform 27
TheTransformTechnique 27
SpecialCases 28
TheMixingSolution 30
TheRomano-TouziApproach 30
v
vi CONTENTS
AOne-FactorMonteCarloTechnique 32
TheLong-TermAsymptoticCase 34
TheDeterministicCase 34
TheStochasticCase 35
ASeriesExpansiononVolatility-of-Volatility 37
Pure-JumpModels 40
VarianceGamma 40
VarianceGammawithStochasticArrival 43
VarianceGammawithGammaArrivalRate 45
CHAPTER2
TheInferenceProblem 46
Introduction 46
UsingOptionPrices 49
DirectionSet(Powell)Method 49
NumericTests 50
TheDistributionoftheErrors 50
UsingStockPrices 54
TheLikelihoodFunction 54
Filtering 57
TheSimpleandExtendedKalmanFilters 59
TheUnscentedKalmanFilter 62
Kushner’sNonlinearFilter 65
ParameterLearning 67
ParameterEstimationviaMLE 81
Diagnostics 95
ParticleFiltering 98
ComparingHestonwithOtherModels 120
ThePerformanceoftheInferenceTools 127
TheBayesianApproach 144
UsingtheCharacteristicFunction 157
IntroducingJumps 158
PureJumpModels 168
Recapitulation 184
ModelIdentification 185
ConvergenceIssuesandSolutions 185
CHAPTER3
TheConsistencyProblem 187
Introduction 187
TheConsistencyTest 189
TheSetting 190
Contents vii
TheCross-SectionalResults 190
RobustnessIssuesfortheCross-SectionalMethod 190
Time-SeriesResults 193
FinancialInterpretation 194
ThePesoTheory 197
Background 197
NumericResults 199
TradingStrategies 199
SkewnessTrades 200
KurtosisTrades 200
DirectionalRisks 200
AnExactReplication 202
TheMirrorTrades 203
AnExampleoftheSkewnessTrade 203
MultipleTrades 208
HighVolatility-of-VolatilityandHighCorrelation 209
Non-GaussianCase 213
VGSA 215
AWordofCaution 218
ForeignExchange,FixedIncome,andOtherMarkets 219
ForeignExchange 219
FixedIncome 220
References 224
Index 236

Illustrations
Figures
1.1 The SPX Historic Rolling Volatility from 2000/01/03 to
2001/12/31. 4
1.2 The SPX Volatility Smile on February 12, 2002 with
Index=$1107.50,1Monthand7MonthstoMaturity. 8
1.3 The CEV Model for SPX on February 12, 2002 with
Index=$1107.50,1MonthtoMaturity. 11
1.4 The BCG Model for SPX on February 12, 2002 with
Index=$1107.50,1MonthtoMaturity. 12
1.5 The GARCH Monte Carlo Simulation with the Square-
Root Model for SPX on February 12, 2002 with
Index=$1107.50,1MonthtoMaturity. 24
1.6 TheSPXimpliedsurfaceasof03/09/2004. 31
1.7 Mixing Monte Carlo Simulation with the Square-Root
Model for SPX on February 12, 2002 with Index =
$1107.50,1Monthand7MonthstoMaturity. 33
1.8 Comparing the Volatility-of-Volatility Series Expansion
withtheMonteCarloMixingModel. 38
1.9 Comparing the Volatility-of-Volatility Series Expansion
withtheMonteCarloMixingModel. 39
1.10 Comparing the Volatility-of-Volatility Series Expansion
withtheMonteCarloMixingModel. 39
1.11 TheGammaCumulativeDistributionFunctionP(a(cid:1)x)for
VariousValuesoftheParametera. 42
1.12 TheModifiedBesselFunctionofSecondKindforaGiven
Parameter. 42
1.13 TheModifiedBesselFunctionofSecondKindasaFunction
oftheParameter. 43
2.1 The S&P500 Volatility Surface as of 05/21/2002 with
Index=1079.88. 51
2.2 Mixing Monte Carlo Simulation with the Square-Root
Model for SPX on 05/21/2002 with Index=$1079.88,
Maturity 08/17/2002 Powell (direction set) optimization
methodwasusedforleast-squarecalibration. 51
ix