Table Of ContentIMT 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
Description:Their popularity stems from the negative correlation between VIX and SPX index, which make factors help in repro- ducing the positive sloping skew of options on VIX, thanks to an increased level. 1 capturing the term structure of volatility expressed both through the ATM term structure of vanilla
