IMT Institute for Advanced Studies, Lucca Lucca, Italy Deterministic Shift Extension of Affine Models

158 Pages·2016·2.49 MB·English

IMT Institute for Advanced Studies, Lucca
Lucca, Italy
Deterministic Shift Extension of Affine Models
for Variance Derivatives
PhDPrograminComputerDecisionandSystemScience,
curriculum: ManagementScience
XXVIIICycle
By
Gabriele Pompa
2015

Program Coordinator: Prof. Rocco De Nicola, IMT Institute for Advanced
StudiesLucca
Supervisor: Prof. FabioPammolli,IMTInstituteforAdvancedStudiesLucca
Supervisor: Prof. RobertoReno`,UniversityofVerona
ThedissertationofGabrielePompaiscurrentlyunderreview.
IMT Institute for Advanced Studies, Lucca
2015

Contents
Abstract ix
1 AffineModels: preliminaries 3
1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 VIXandVIXderivatives 17
2.1 Markets: definitionsandempiricalfacts . . . . . . . . . . . . . . . . 18
2.1.1 VIXIndex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.2 VIXFutures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.3 VIXOptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Models: standaloneandconsistentapproach . . . . . . . . . . . . . . 27
2.2.1 StandalonemodelsofVIX . . . . . . . . . . . . . . . . . . . . 27
2.2.2 ConsistentmodelsofS&P500andVIX . . . . . . . . . . . . . 33
3 TheHeston++model 54
3.1 PricingVIXderivativeswiththeHeston++model . . . . . . . . . . 56
3.1.1 Modelspecification . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1.2 Nestedmodels . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.3 SPXandVIXderivativespricing . . . . . . . . . . . . . . . . 60
3.2 Ageneraldisplacedaffineframeworkforvolatility . . . . . . . . . 65
3.2.1 AffinemodelingofVIXindex . . . . . . . . . . . . . . . . . . 72
3.2.2 AffinemodelingofVIXderivatives . . . . . . . . . . . . . . 76
4 TheHeston++model: empiricalanalysis 79
4.1 Empiricalanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Calibrationresults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
vii
4.2.1 Impactoftheshort-term . . . . . . . . . . . . . . . . . . . . . 100
4.2.2 AnalysiswithFellerconditionimposed . . . . . . . . . . . . 111
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A Mathematicalproofsandaddenda 120
A.1 ConditionalcharacteristicfunctionsofHmodels . . . . . . . . . . . 120
A.2 ProofofProposition4: CH++(K,t,T) . . . . . . . . . . . . . . . . . 123
SPX
A.3 ProofofProposition5: VIXH++ . . . . . . . . . . . . . . . . . . . . 123
t
A.4 ProofofProposition6: FH++(t,T)andCH++(K,t,T) . . . . . . . . 123
VI(cid:104)X (cid:12) (cid:105)VIX
A.5 Proofofproposition9: EQ (cid:82)T X ds(cid:12)F . . . . . . . . . . . . . . . 125
t s (cid:12) t
A.6 Proof of proposition 11: F (t,T) and C (K,t,T) under the
VIX VIX
displacedaffineframework . . . . . . . . . . . . . . . . . . . . . . . 128
A.7 Affinityconservationunderdisplacementtransformationofinstan-
taneousvolatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
References 139
viii
Abstract
The growing demand for volatility trading and hedging has lead to-
day to a liquid market for derivative securities written on it, which
made these instruments a widely accepted asset class for trading, di-
versifyingandhedging. Thisgrowingmarkethasconsistentlydriven
the interest of both practitioner and academic researchers, which can
find in VIX and derivatives written on it a valuable source of infor-
mations on S&P500 dynamics, over and above vanilla options. Their
popularitystemsfromthenegativecorrelationbetweenVIXandSPX
index,whichmaketheseinstrumentsidealtotakeapurepositionon
thevolatilityoftheS&P500withoutnecessarilytakingapositiononits
direction. InthisrespectfuturesonVIXenablethetradertoexpressa
visionofthemarketsfuturevolatilityandcalloptionsonVIXofferpro-
tectionfrommarketdownturnsinaclear-cutway. Fromthetheoreti-
calpointofview,thishasleadtotheneedofaframeworkforconsis-
tentlypricingvolatilityderivativesandderivativesontheunderlying,
thatistheneedtodesignmodelsabletofittheobservedcross-section
ofoptionpricesofbothmarketsandproperlypriceandhedgeexotic
products. TheconsistentpricingofvanillaoptionsonS&P500andfu-
turesandoptionsonVIXisarequirementofprimaryimportancefor
amodeltoprovideanaccuratedescriptionofthevolatilitydynamics.
Sinceequityandvolatilitymarketsaredeeplyrelated,butatthesame
time show striking differences, the academic debate around the rele-
vantfeaturesshouldamodelincorporateinordertobecoherentwith
bothmarketsisstillongoing.Inthisthesisweleverageonthegrowing
literatureconcerningthedevelopingofmodelsforconsistentlypricing
volatility derivatives and derivatives on the underlying and propose
the Heston++ model, which is an affine model belonging to the class
of models analyzed by Duffie et al. (2000) with a multi-factor volatil-
ity dynamics and a rich jumps structure both for price and volatility.
The multi-factor Heston (1993) structure enables the model to better
ix
capture VIX futures term structures along with maturity-dependent
smilesofoptions. Moreover, bothcorrelatedandidiosyncraticjumps
inpriceandvolatilityfactorshelpinreproducingthepositivesloping
skewofoptionsonVIX,thankstoanincreasedleveloftheskewness
ofVIXdistributionsubsumedbythemodel.Thekeyfeatureofourap-
proachistoimposeanadditivedisplacement,inthespiritofBrigoand
Mercurio(2001),ontheinstantaneousvolatilitydynamicswhich,act-
ingaslowerboundforitsdynamics,noticeablyhelpsincapturingthe
termstructureofvolatility. Bothincreasingthefittotheat-the-money
termstructureofvanillaoptions,asalreadypointedoutinPacatietal.
(2014),andremarkablycapturingthedifferentshapesexperiencedby
thetermstructureoffuturesonVIX.Moreover,weproposeageneral
affine framework which extends the affine volatility frameworks of
Leippold et al. (2007), Egloff et al. (2010) and Branger et al. (2014) in
whichtherisk-neutraldynamicsoftheS&P500indexfeaturesseveral
diffusive and jump risk sources and two general forms of displace-
mentcharacterizethedynamicsoftheinstantaneousvarianceprocess,
which is affine in the state vector of volatility factors. The instanta-
neous volatility is modified according to a general affine transforma-
tion in which both an additive and a multiplicative displacement are
imposed,thefirstsupportingitsdynamics,thesecondmodulatingits
amplitude. WecalibratetheHeston++modeljointlyandconsistently
on the three markets over a sample period of two years, with over-
all absolute (relative) estimation error below 2.2% (4%). We analyze
thedifferentcontributionsofjumpsinvolatility. Weaddtwosources
ofexponentialupwardjumpsinoneofthetwovolatilityfactors. We
first add them separately as an idiosyncratic source of discontinuity
(as in the SVVJ model of Sepp (2008b)) and then correlated and syn-
chronized with jumps in price (as in the SVCJ model of Duffie et al.
(2000)).Finally,weletthetwodiscontinuitysourcesacttogetherinthe
full-specified model. For any model considered, we analyze the im-
pactofactingadisplacementtransformationonthevolatilitydynam-
ics. Inaddition,weperformtheanalysisrestrictingfactorparameters
freedom to satisfy the Feller condition. Our empirical results show a
decisiveimprovementinthepricingperformanceovernon-displaced
x