A Time Series Approach to Option Pricing: Models, Methods and Empirical Performances

202 Pages·2015·3.676 MB·English

by  Christophe Chorro, Dominique Guégan, Florian Ielpo (auth.)

Christophe Chorro
Dominique Guégan
Florian Ielpo
A Time Series
Approach to
Option Pricing
Models, Methods and Empirical
Performances
A Time Series Approach to Option Pricing

Christophe Chorro • Dominique Guégan (cid:129)
Florian Ielpo
A Time Series Approach
to Option Pricing
Models, Methods and Empirical
Performances
123
ChristopheChorro FlorianIelpo
DominiqueGuégan LombardOdierDarierHentsch&Cie
MaisondesSciencesEconomiques GeneJve
UniversitéParis1PanthéonSorbonne Switzerland
Centred’EconomiedelaSorbonne
Paris
France
ISBN978-3-662-45036-9 ISBN978-3-662-45037-6(eBook)
DOI10.1007/978-3-662-45037-6
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014957876
©Springer-VerlagBerlinHeidelberg2015
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Notation
SetsandSpaces
; Emptyset
N Positiveintegers
Z Integers
Q Rationalnumbers
R Realnumbers
R Positiverealnumbers
C
R(cid:2) Strictlypositiverealnumbers
C
Œa;b(cid:2) Closedinterval
(cid:2)a;bΠOpeninterval
fa;:::;bg Integersbetweenaandb
.(cid:3);A;P/ Probabilityspace
Lp.(cid:3);A;P/ Realrandomvariableswithfinite
pordermoment
Functions
Argmax Argumentofthemaximum
Id Identityfunction
ex Exponentialofx
log.x/ Logarithmofx
.x/ Max.x;0/
C
1 .x/ 1ifx 2A,0otherwise
A
Re.x/ Realpartofx
K Besselfunctionofthethirdkind
(cid:4)
N DistributionfunctionofaN.0;1/
J.ˆ/ Jacobiandeterminantofˆ
Convergence
! Pointwiseconvergence
! Convergencealmostsure
a:s
! ConvergenceinLp.(cid:3);A;P/
Lp
v
vi Notation
! Convergenceindistribution
D
limsup,liminf limitsuperior,limitinferior
ProbabilityandFinance
EŒX(cid:2) Expectation
EŒXjG(cid:2) Conditionalexpectation
VarŒX(cid:2) Variance
CovŒX;Y(cid:2) Covariance
skŒX(cid:2) Skewness
kŒX(cid:2) Kurtosis
(cid:5) Autocovariance
X
(cid:6) Autocorrelation
X
dQ Radon–NikodymderivativeofQw.r.t.P
dP
(cid:7).X Iu(cid:2)t/ (cid:7)-Algebrageneratedby.X /
u u u(cid:3)t
XQ DiscountedvalueofX
t t
G Momentgeneratingfunction
Qem Elliott–Madanequivalentmartingalemeasure
Qess Esscherequivalentmartingalemeasure
Qqess QuadraticEsscherequivalentmartingalemeasure
ProbabilityDistributions
D.0;1/ Arbitrarydistributionwithmean0andvariance1
G .a;b/ Gammadistribution
a
GH.(cid:4);˛;ˇ;ı;(cid:8)/ Generalizedhyperbolicdistribution
GIG.(cid:4);(cid:9); / GeneralizedinverseGaussiandistribution
MN.(cid:10);(cid:8) ;(cid:8) ;(cid:7) ;(cid:7) / MixtureoftwoGaussiandistributions
1 2 1 2
N.m;(cid:7)2/ Gaussiandistributionwithmeanmandvariance(cid:7)2
P.(cid:4)/ Poissondistribution
GARCHModels
APARCH AsymmetricpowerARCH
ARCH Autoregressiveconditionalheteroscedasticity
EGARCH ExponentialGARCH
EWMA Exponentiallyweightedmovingaverage
GJR Glosten,JagannathanandRunkleGARCH
HN HestonandNandimodel
IGARCH IntegratedGARCH
Abbreviations
AD Andersen–Darling
AO Arbitrageopportunity
AAO Absenceofarbitrageopportunity
AARPE Averageabsoluterelativepricingerrors
ACF Autocorrelationfunction
Notation vii
ACVF Autocovariancefunction
ARMA Autoregressivemovingaverage
BFGS Broyden–Fletcher–Goldfarb–Shannoalgorithm
cadlag Rightcontinuouswithleftlimits
CCAPM Consumptioncapitalassetpricingmodel
EMS Empiricalmartingalesimulation
FFT FastFouriertransform
i.i.d Independentandidenticallydistributed
KS Kolmogorov–Smirnov
LRNVR Locallyriskneutralvaluationrelationship
MA Movingaverage
ML Maximumlikelihood
NIC Newsimpactcurve
PCA Principalcomponentanalysis
QML Quasimaximumlikelihood
REC Recursiveestimationmethod
RNVR Riskneutralvaluationrelationship
RMSE Rootmeansquarederror
SDF Stochasticdiscountfactor
S&P500 StandardandPoor’s500equityindex
VIX Chicago Board Options Exchange market volatility
index

Contents
1 Introduction .................................................................. 1
References..................................................................... 8
2 TheTimeSeriesToolboxforFinancialReturns.......................... 11
2.1 StylizedFacts............................................................ 11
2.1.1 Stationarity,ErgodicityandAutocorrelation.................. 12
2.1.2 Time-VaryingVolatilityandLeverageEffects ................ 15
2.1.3 SemiFat-TailedDistributions.................................. 16
2.2 SymmetricGARCHModels............................................ 18
2.2.1 FromLineartoNon-linearModels ............................ 18
2.2.2 Definitions ...................................................... 21
2.2.3 StationarityProperties.......................................... 23
2.2.4 CovarianceStructureoftheSquares........................... 27
2.2.5 WhyWeNeedMore:KurtosisandAsymmetry
inaGARCH(1,1)Model....................................... 31
2.3 AsymmetricExtensions................................................. 33
2.3.1 GJRModel...................................................... 35
2.3.2 EGARCHModel................................................ 37
2.3.3 APARCHModel................................................ 39
2.3.4 ConcludingRemarks ........................................... 41
2.4 ConditionalDistributionofReturns.................................... 42
2.4.1 GeneralizedHyperbolicDistributions......................... 42
2.4.2 MixtureofTwoGaussianDistributions ....................... 45
2.4.3 SomePracticalRemarks........................................ 47
2.5 GARCHinMean........................................................ 48
2.6 DealingwiththeEstimationChallenge................................ 49
2.6.1 MaximumLikelihood .......................................... 50
2.6.2 QuasiMaximumLikelihood ................................... 51
2.6.3 RecursiveEstimation........................................... 52
2.6.4 EmpiricalFiniteSamplePropertiesoftheThree
EstimationMethodologies ..................................... 53
ix