Alexander Carol Unified Therory SLV

8/9/2019 Alexander Carol Unified Therory SLV

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StochasticLocalVolatility

CAROLALEXANDER

ICMA Centre, University of Reading

LEONARDOM.NOGUEIRA

ICMA Centre, University of Reading and Banco Central do Brasil

First

version:September2004Thisversion:January2008

ABSTRACT

There

aretwouniquevolatilitysurfacesassociatedwithanyarbitrage-freesetofstandard

Europeanoptionprices,theimpliedvolatilitysurfaceandthelocalvolatilitysurface.Several

papershavediscussed


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thestochasticdifferentialequationsforimpliedvolatilities

thatareconsistentwiththeseoptionpricesbutthestaticanddynamic

no-arbitrageconditionsarecomplex,mainlyduetothelarge(oreveninfinite)dimensions

ofthestateprobabilityspace.Theseno-arbitrageconditionsarealsoinstrument-specificandhavebeenspecifiedforsomesimpleclassesofoptions.However,theproblemis

easiertoresolve


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whenwespecifystochasticdifferentialequationsfor

localvolatilitiesinstead.Andtheoptionpricesandhedgeratiosthatare

obtainedbymakinglocalvolatilitystochasticareidenticaltothoseobtainedbymaking

instantaneousvolatilityorimpliedvolatilitystochastic.Afterprovingthatthereisaone-to-onecorrespondencebetweenthestochasticimpliedvolatilityandstochasticlocalvolatilityapproaches,we

deriveasimple


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dynamicno-arbitrageconditionforthestochasticlocal

volatilitymodelthatismodel-specific.Theconditionisveryeasytocheck

inlocalvolatilitymodelshavingonlyafewstochasticparameters.

JEL

Classification:G13,C16Keywords:Localvolatility,stochasticvolatility,unifiedtheoryofvolatility,localvolatilitydynamics.

CorrespondingAddress:ICMACentre,UniversityofReading,PO

Box242,Reading,


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UnitedKingdom,RG66BATel:+44(0)1183

786675Fax:+44(0)1189314741

E-mails:[email protected] and [email protected]

Acknowledgements:We

aremostgratefulforveryusefulcommentsonanearlierdraftfromBruno

DupireofBloomberg,NewYork;HyungsokAhnofNomuraBank,London;EmanuelDermanofColumbiaUniversity,NewYork;JacquesPezieroftheICMACentre;and

NicoleBrangerof


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theUniversityofMunster.

Electronic copy available at: http://ssrn.com/abstract=1107685


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I.INTRODUCTIONOptionpricingmodelsseek

topriceandhedgeexoticandpath-dependentoptionsconsistentlywiththemarket

pricesofsimpleEuropeancallsandputs.Twomainstrandsofresearchhave

beendevelopedinaprolificliterature:stochasticvolatilityasinHullandWhite(1987),SteinandStein(1991),Heston(1993)andmanyothers,wherethe

varianceorvolatility


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ofthepriceprocessisstochastic;and

localvolatilityasinDupire(1994),DermanandKani(1994)andRubinstein

(1994)wherearbitrage-freeforwardvolatilitiesareelocked-infbytradingoptionstoday.Theterm

elocalvolatilityfhassubsequentlybeenextendedtocoveranydeterministicvolatilitymodelwhereforwardvolatilitiesareafunctionoftimeandassetprice.Fora

detailedreviewof


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thesemodels,seeJackwerth(1999),Skiadopoulos(2001),

Bates(2003),Psychoyiosetal.(2003)andNogueira(2006).

Stochastic

andlocalvolatilitymodelshavebeenregardedasalternativeandcompetingapproachesto

thesameunobservablequantity,theinstantaneousvolatilityoftheunderlyingasset.Butitisonlywhenonetakesarestrictedviewofvolatilitydynamicsthat

theyappearto


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bedifferent.Undermoregeneralassumptionsthe

twoapproachesyieldidenticalclaimpricesandhedgeratios.Thispaperunifies

thetwoapproachesintoasingletheorybyprovingthatalocalvolatility

modelwithstochasticparametersyieldsidenticalimpliedvolatilitydynamicstothoseofSchonbucherfs(1999)marketmodelofstochasticimpliedvolatilities.Wethusprovideamodel-based

proofofthe


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unifiedtheoryofvolatilityofDupire(1996).

Bymodelingparameterdynamicsunderno-arbitrageconditionswearealso

implicitlymodelingthedynamicsofthelocalvolatilitysurface,theimpliedvolatilitysurface

andtheimpliedprobabilitydistribution.Henceourapproachcanberegardedasanalternativetodirectlymodelingthedynamicsoflocalvolatilities(Dupire,1996,Derman

andKani,1998),


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impliedvolatilities(Schonbucher,1999,Braceetal.,

2001,Ledoitetal.,2002)andimpliedprobabilities(PanigirtzoglouandSkiadopoulos,2004).

Severalrecentpapersexamineno-arbitrageconditionsinthestochasticimpliedvolatility

framework.Bydefiningameasureunderwhichtheunderlyingpriceandtheimpliedvolatilitiesaremartingales,Zilber(2007)appliesthefundamentaltheoremofarbitrageto

provideaconstructive


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proofoftheexistenceofmarketmodels

ofimpliedvolatilitiesthatadmitneitherstaticnordynamicarbitrage.Theabsence

ofdynamicarbitrageplacesconstraintsonthestateprobabilityspace,andfurthercomplex

conditionsforabsenceofstaticarbitragearerequired,asspecifiedbyCarrandMadan(2005)orDavisandHobson(2007).However,SchweizerandWisseltransform

themarketmodel


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forimpliedvolatilitiesintoamarketmodel

forforwardimpliedvolatilities(Schweizerand

Electronic copy available at: http://ssrn.com/abstract=1107685


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Wissel,2008)orforlocalimplied

volatilities(SchweizerandWissel,2007)andprovidedthesearenon-negativetherecan

benostaticarbitrage.Thistransformationsimplifiestheproblem,andtheyshowhow

absenceofdynamicarbitrageyieldsrestrictionsonthedrifttermsintheforwardimpliedvolatilitymodel.Theseconditionsareanalogoustothedriftrestrictionsin

theHJMmarket


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modelforinterestrates(Heathetal,

1992).Foraparticularchoiceofvolatilitycoefficients,anarbitrage-freemodel

onlyexistsifthedriftcoefficients,whichdependonthevolatilitycoefficients,satisfy

thedriftconditions.Thedriftconditionisspecifiedforsomeparticularinstrumentssuchasstandardcalloptions,poweroptionsandthelogcontract.However,generalizing

theseresultsto


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derivedrift

restrictionswhenimplied

volatilitieswithdifferentmaturitiesand strikesaremodeledinthesameframeworkis

acomplexproblembecausetheadmissiblestatespacehasacomplicatedstructure.

Bycontrast,thederivationofno-arbitrageconditionsinastochasticlocalvolatilityframeworkisrelativelystraightforward,becausethestaticno-arbitrageconditionissimplythat

localvolatilitiesare


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non-negativeandtheadmissiblestatespaceis

relativelysimple.1Inthispaperwederiveasingledriftconditionfor

thestochasticlocalvolatilitymodeltobearbitrage-free,andspecifytheconditionfor

somespecificexamplesofstochasticlocalvolatilitymodels.RecentlyasimilarapproachhasbeenconsideredbyWissel(2007),whodefineselocalimpliedvolatilitiesfasthe

discreteanalogueof


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thelocalvolatilityfunctionderivedfromarbitrage-free

pricesofstandardEuropeanoptionsgivenbyDupirefsequation(Dupire,1994).The

maindifferencebetweenthisandourapproachisthatwe

specify

stochasticdynamicsfortheparameters ofthelocalvolatilityfunction,whereasWisseluseslocalimpliedvolatilitiestoparameterizeEuropeanoptionpricesandthenspecifiesstochasticdynamics

forthesevolatilities


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directly.

Theremainderofthis

paperisasfollows:SectionIIreviewstheinter-relationshipbetweenlocalvolatility

andstochasticvolatility.SectionIIIintroducesthestochasticlocalvolatility(SLV)model,derives

theno-arbitragedriftconditionandexaminesthisconditionforsomespecificstochasticlocalvolatilitymodels;SectionIVderivesthelocalvariance,impliedprobabilityandimplied

volatilitydynamicsthat


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areconsistentwiththeSLVclaimprice

dynamics.InparticularitprovesthedualitybetweentheSLVmodeland

themarketmodelofimpliedvolatilitiesintroducedbySchonbucher(1999);SectionVsummarizes

andconcludes.

1Also,whereasimpliedvolatilitiesareinstrument-specific,beingdefinedastheinversepriceofastandardEuropeanoption,alocalvolatility

surfacecanbe


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definedwithoutreferencetotradableinstruments.


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II.UNIFIEDTHEORYOFVOLATILITYThis

sectionprovidesanon-technicalsummaryofthetheoryofunifiedvolatilitythat

wasfirstintroducedbyDupire(1996),andaformalintroductiontotheconcept

ofstochasticlocalvolatility.

II.1GyongyfstheoremThelocalvolatilitymodelcanbeunderstoodfromtheearlyworkofGyongy(1986).Supposethat

Xt isareal-valued


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one-dimensionalItoprocessstartingatX0=

0withdynamics:

dXt=(t,)dt+(t,)dBt(1)where(B)kx1isak-dimensionalWiener

processontheprobabilityspace(N,A,P),thepossibly

t

randomcoefficients(t,)and((t,))satisfythe

regularityconditionsof


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anItoprocessand

1xk

I.denotesdependenceonsomearbitraryvariables.Inparticular,

supposethatT(t,)ispositive.Gyongy(1986)provedthatthereexists

anotherstochasticprocessX.twhichisasolutionofthestochasticdifferentialequation:

.

.

...

=(,t)dt

+v(


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,)dBt,X=

0(2)

dXbtXtX

tt0

withnon-random coefficientsb andv definedby:

def

bt(

,x)=E.(t,)Xt=x.

P

.

1(3)

T

v(t,x)


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def=(E.(t,)Xt=

x.)2

P

.

andwhichadmitsthesame marginalprobabilitydistributionasthatofXt foreveryt >

0.Thatis,foreveryItoprocessofthetype(1)thereisadeterministicprocess(2)thatemimicsfthemarginaldistributionofXt forevery

t.


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NowweestablishthelinkbetweenGyongyfs

resultandlocalvolatility.Supposek =1anddefine

1

X=lnSS0),(t,)=.22(t,)

and(t,)=(t,).Then(1)becomes:

t(t

1

dlnS=(.2

(t,))dt


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+(t,)dB

t2t

andbyItofslemma:

dS

t

=dt+(t,)dBt(4)

St

whichisthestochasticdifferentialequationforafinancialassetSt withpossiblystochasticvolatility.DenotebyS anarbitraryrealizationofSt for

somet.


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0andputx=ln(SS

0).Using(3)wehave:


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v2(t,x)=v2

(t,ln(SS=E.t,StL2tS

0))2()=S.def=(,)P

.

bt(,x)=bt(,ln(SS0))=EP..122(t,

)St=


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S=.22L(tS

)

.1,

.

..

Replacingtheseinto(2)withXt=ln(S.tS0)

,S.0=S0andusingItofslemma,weobtain:

.

=dt+L(tS)t(5)

dSt


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,.tdB.

.

StwhichisthestochasticdifferentialequationofS.t

withthedeterministic localvolatilityL(,).

tS

ThusS.tinthelocalvolatilitymodel(5)hasthesamemarginaldistributionasSt in(4)foreveryt.

Besides,asthere

isaone-to-one


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relationshipbetweenrisk-neutralmarginalprobabilitiesandthe

pricesofstandardEuropeanoptions(BreedenandLitzenberger,1978),bothmodels(4)

and(5)producethesamepricesforsimplecallsandputsaftera

measurechangefromP totherisk-neutral

measure.

II.2DupirefsequationForeveryarbitrage-freemodeloftheform(4)thereisa

correspondinglocalvolatility


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model,suchas(5).Inparticular,Dupire

(1994)showedthatthelocalvolatilityin(5)isuniqueandis

alsogivenbythesquarerootof

2

.f0

f0.2f0

2L(tS,)t=T,S=2.+(r.qK+qfK(6)

=K)


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0.2

.TK

.Kwheref0=f0(,

KT)is

thepriceofastandardEuropeanoptionwithstrikeK andmaturityT at

timet =0whentheunderlyingassetpriceisS0andwhenthelocalvolatilityiscalibrated;r andqdenoterespectivelytherisk-freeinterest

rateandthe


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dividendyieldforequities(ortheircounterparts

forothermarkets)bothassumedconstantandcontinuously-compounded.

Localvolatility

modelsarewidelyusedinpracticebecausetheyenablefastandaccuratepricing

ofexoticclaimswhenonlymarginaldistributionsarerequired.However,itwouldbeamistaketointerpretlocalvolatilityasacompleterepresentationofthe

truestochasticprocess


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drivingtheunderlyingassetprice.Localvolatility

ismerelyasimplificationthatispracticallyusefulfordescribingaprice

processwithnon-constantvolatility.Moreprecisely,althoughthemarginaldistributionsarethesame

atthetimewhenthelocalvolatilityiscalibrated,clearlySt andS.tdonotfollowthesamedynamics(cf.(4)and(5))henceoptions

priceswillhave


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differentdynamicsundereachmodel,andhedge

ratioscandiffersubstantially.Therefore,forpricingandhedgingsome


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exoticoptions,theperfectfitto

thecurrentvanillaoptionspricesprovidedbythelocalvolatility

theorymaynotbeenoughandadeeperunderstandingofthedynamicsof

St isrequired.

Forinstance,iftheetruefrisk-neutralpricedynamicsfollowedastochasticvolatilitymodelsuchas:

dSt

=r.


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qdt+YtdBSt

(7)

,

()()t

dY

=(tY,)dt+(tY,)dWdBWt

=dt

tttt

thesamelocalvolatilityfunctionwouldbeobtainedfrombothDupirefsequation(6)and:

2L


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(tS,)=E.2(Yt)

St=S.(8)

.

wheretheexpectationistakenundertherisk-neutralmeasure.SeeDupire(1996)and

DermanandKani(1998)fortheproofofthisresult.Butgiventhat(Yt)isstochastic,thelocalvolatilitywill

also

bestochastic.


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II.3Towardsatheoryofstochastic

localvolatilityStochasticvolatility,whichindeedappearedintheliteraturebeforelocal

volatility,isbasedonempiricalevidencethatvolatilitydisplayscharacteristicsofastochastic

processonitsown.2Thereisvastevidence,mostlyfromS&P500indexoptions,thatoptionpricesaredrivenbyatleastone

randomfactorother


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thantheassetprice.Forinstance,Bakshi

et al. (2000)findthatintradayoptionpricesdonotalwaysfollowsthemovements

oftheunderlyingprice,whichisinconsistentwiththeperfectcorrelationfromdeterministic

volatilitymodels.BuraschiandJackwerth(2001)findthatonecannotspanthepricingkernelusingonlytwoassets,andsorejectdeterministicvolatilitymodelsin

favourofmodels


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thatincorporateadditionalriskfactors.Covaland

Shumway(2001)findabnormalreturnsforcallsandputsandconcludethat

thisshouldbecausedbyatleastoneadditionalriskfactor.

Atypicalstochasticvolatilitymodeldefinesthepriceprocessasin(7),wheretheinstantaneousvolatilityisafunctionofthestochasticprocessYt

,possiblycorrelated


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withtheunderlyingprice.

For

instance,HullandWhite(1987)define(Yt)=YtwhereYt is

lognormallydistributedbut

theirmodeldoesnotallowformeanreversion,

sovolatilitycangrowindefinitely.InthemodelsbyScott(1987)andSteinandStein(1991)Yt ismeanrevertingbutitisalsonormaland

cangonegative


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ifmeanreversionisnotstrong.Hence

(Yt)isconvenientlychosentomaparealnumber

intoapositivenumber.InthestrongGARCHdiffusionofNelson(1990)

Yt followsageometric,

2Seee.g.Schwert(1989),Ghyselsetal.(1996),Fouqueetal.(2000),Psychoyiosetal.(2003)andGatheral(2005).


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mean-revertingprocessandintheHeston

(1993)andBallandRoma(1994)models,Yt hasanon-centralchi-square

distributionandispositiveandmeanreverting.Inallthemodelsabovewith

the

exceptionofHestonfs,thecorrelationbetweenpriceandvolatilityiszero,althoughextensionstocorrelatedvolatilityprocesseshavebeenderivedinmany

cases.Fora


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reviewoftypicalstochasticvolatilityprocessesand

theirdistributionalassumptions,seePsychoyioset al. (2003).

Stochasticvolatilitymodelshave

thedrawbackthatthevolatilityprocessitselfisnotobservable.Itcanonly

beestimatedfromhistoricaldataorcalibratedtocurrentoptionprices.But,eitherwaytheresultingprocessisstilldependentontheparticularstructureassumed

for(Y


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andYt .Thus,

t)

supportedbythewidespreaduseoftheBlack-Scholesmodel,severalauthorshave

insteadmodelledthebehaviourofthe(observable)Black-Scholesimpliedvolatilityovertimeand

verifiedempiricallytheexistenceofmultipleriskfactorsdrivingoptionprices.SeeSkiadopouloset al. (1999),Alexander(2001),ContanddaFonseca(2002),Contet al. (2002),Fengleret al. (2003),

Hafner(2004)and


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Fengler(2005).

Thesefindingsmotivated

theclassofestochasticimpliedvolatilityfmodels,inwhichastochasticprocess

isexplicitlyassumedfortheBlack-Scholesimpliedvolatilityofvanillaoptionsandused

topriceandhedgeexoticoptions,withthestochasticprocessfortheinstantaneousvolatilityfollowingfromno-arbitragearguments.Infact,sinceforanyvanillaoption

thereisone


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andonlyoneimpliedvolatilitythatis

consistentwiththeoptionprice,itfollowsthatimpliedvolatilitiesareas

observableasoptionprices,sothatstandardeconometrictechniquesmaybeusedto

searchforrepresentationsofvolatilitythatarebothtractableandrealistic.Well-knownmembersofthisclassincludethe

modelsofSchonbucher(1999),Ledoit

andSanta-Clara(1998)


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andBraceet al (2001).

Butthe

Black-Scholesimpliedvolatilityisnottheonlyformofvolatilitythatis

observable.Inparticular,foranycompletesetofarbitrage-freepricesofvanillacalls

andputs,Dupire(1994)hasshownthatthereisauniquelocalvolatilitysurfaceandthatthissurfacecanbecomputeddirectlyfromoptionprices

usingequation(6)


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above.Canthedynamicsoflocalvolatilities

alsobemodelleddirectlyandusedtoderivearbitrage-freepricesandhedge

ratiosforexoticoptionsthatareconsistentwiththeevolutionoflocalvolatility

overtime?DermanandKani(1998)haveaddressedthisquestionandproposedaestochasticlocalvolatilityfmodelbasedonstochasticimpliedtrees.Theyusedtrees

because,whenworking


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incontinuoustime,onehastodeal

withachallengingno-arbitrageconditionforthedriftofthelocal

volatilityprocess.


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Inthispaperwederivea

continuousstochasticlocalvolatilitymodelthatistractable,intuitive,arbitrage-freeandthat

helpsexplaintheallegedeinstabilityfoflocalvolatilitysurfacesovertime.Aswe

shallsee,thisisachievedbyassumingaparametricfunctionalformforlocalvolatilityandmodellingthetimeevolutionofthelocalvolatilityparametersunder

no-arbitrageconditions.Our


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approachisanalogoustothatofa

stochasticimpliedvolatility(SIV)model.InatypicalSIVmodel,oneuses

theBlack-Scholesmodeltocomputetheimpliedvolatilityofvanillaoptionsforeach

dateinthesampleandappliesstandardeconometricmethods,suchasprincipalcomponentanalysisandARIMAmodels,toproducearealisticdynamicsforimpliedvolatilities.

Likewise,inthe


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stochasticlocalvolatility(SLV)modeldescribedin

thenextsectionweassumethataparametriclocalvolatilityiscalibrated

totheobservedpricesofvanillaoptionsateachdateinthesample

and,again,mayapplysimpleeconometricmethodstoproducerealisticdynamicsforthelocalvolatilityparameters.

Infact,thesemodelsaremorethan

analogous:weshow


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thattheyareequivalent.Thatis,we

provethat,undertheassumptionthattheSLVmodelholds,onecan

derivethesameimpliedvolatilitydynamicsasobservedforSIVmodels.Thisis

animportantresultbutitisnotreallyasurprise.Sinceimpliedvolatilitiesandlocalvolatilitiesarederivedfromthesameoptionprices,theirdynamics

shouldbeconsistent


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ifthemarketisarbitrage-free.Nevertheless,it

isstillworthwhilecheckingthatourtheoryisindeedself-consistent.

TheSLVmodelisnotamerespecialcaseoftheclassof

SIVmodels.WeproveatheoremwhichshowsthatverifyingasingledriftconditionisenoughtoguaranteeabsenceofarbitrageintheSLVmodel.

Bycontrast,the


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SIVmodelimposesadriftconditionwhose

coefficientsdependontheparticularoptionstrikeandmaturity,andverifyingthat

thesecoefficientsareconsistentacrossdifferentstrikesandmaturitiesisnotstraightforward,as

SchweizerandWissel(2008)havedemonstrated.

II.4StochasticlocalvolatilityThissubsectionformalizestheconceptofstochasticlocalvolatilityandprovidesfurthermotivation

forthemodel


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derivedinSectionIII.FollowingGyongy(1986),

Dupire(1994)andourdiscussionabove,itisalwayspossibletodefine

localvolatilitydynamicsthatmimicthemarginalprobabilitydistributionoftheunderlyingasset

price.Thesedynamics,undertherisk-neutralmeasure,couldbewrittenas:

.

.

dSt=(r.qdt

)+,)


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dB(9)

(tS.

.Ltt

St


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withr andq asaboveand,in

theabsenceofarbitrage,thelocalvolatilityfunctionL(tS

,)uniquely

determinedaccordingto(6).

However,since

findingL(tSrequiresacontinuumoftradedoptionprices,directcomputation

,)ofthelocalvolatilityfunctionisproblematic.3Thelocal

volatilitysurfacecan


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beveryirregularandsensitivetothe

interpolationmethodsusedbetweenquotedoptionpricesandtheirextrapolationto

boundaryvalues(seee.g.BouchouevandIsakov(1997,1999)andAvellanedaet

al. (1997)).Consequentlymostrecentworkonlocalvolatilityhasintroducedavarietyofparametricformsforlocalvolatilityfunctionsinwhichtheparameters

12


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t=(,,...,n)

t

arecalibratedtothemarketpricesofvanillaoptionsat

everyt.If0denotesthecalibratedvaluesof

theparameters

attime0theunderlyingassetpriceprocessassumedatthistimeis:

.

..

dSt(r.

qdt)+


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L(,t,0)dB

t(10)

=tSS.t

Then,when

themodelisre-calibratedattimet1>t0,wehave:

.

dS.t=(r.)+(tS,.t,1)dB.(11)

qdtLt

St


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andthiswilldifferfrom(10)unless

1=0.Whilstlocalvolatilitymodelsassumeparametersare

constant,inpracticeparametersarenotconstantandchangeeachtimethe

modelisre-calibrated.Thisisbecause(10)isnotnecessarilythetruemodeldespitetheperfectfittooptionspricesitcouldachieveattime

0.


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Tosummarize,localvolatilitymodelsassumea

deterministicinstantaneousvolatilityforthepriceprocessandthisimpliesastatic

localvolatility.4Thatis,theforwardvolatilitiesareobtainedbycalibrationtocurrent

marketpricesandinthelocalvolatilityframeworktheyshouldberealizedwithcertainty.Yetthisassumptionisnotnecessary.BothDupire(1996)andDerman

andKani(1998)


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recognizethisandthusdefinethelocal

varianceastheconditionalexpectationofastochasticvariance,suchas:

2L(tS,)=E.2(tS,t,)

St=S,A0..attime0(12)

Q

3ThereisaversionofDupirefsequationusingimpliedvolatilities

thatismore


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stable.Seee.g.Fengler(2005,ch.3).

4Weassumethatthelocalvolatilityisnotafunctionof

thecurrentassetpriceS0otherwiseitcannotbestatic.


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whereEQ denotestheexpectationunderthe

impliedrisk-neutralprobabilityQ,i.e.thedistributionthatisconsistentwithcurrent

pricesofEuropeanputsandcalls(seeJackwerth,1999).ThefiltrationA0includes

allinformationuptotime0anddenotesallsourcesofuncertaintythatmayinfluence

theinstantaneousvolatilityprocessotherthanthe

assetprice.The


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instantaneousvariancein(12)isverygeneral

becausecanbeanyarbitrage-freesetofcontinuousstochasticprocesses.In

particular,thisdefinitionoflocalvarianceisconsistentwithanyunivariatediffusionstochastic

volatilitymodel.Forthisreason,Dupire(1996)named(12)theeunifiedtheoryofvolatilityf.

Notethattakingtheexpectationin(12)ignoresthe

residualuncertaintyfrom


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anditsinfluenceontheinstantaneous

volatility.Thisuncertaintyistransferredtothelocalvolatilitysurfaceitself.That

is,althoughlocally(i.e.ateachcalibration)thesurfaceisindeed

adeterministicfunctionoft andS,thatsurfacehasstochasticdynamics.Thisexplainswhythelocalvolatilitysurfacecanbeveryunstableonre-calibration:the

uncertaintyfrom


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doesnotjustdisappearfromthemodel.

III.STOCHASTICLOCALVOLATILITY(SLV)MODELIII.1ModelspecificationSuppose

thetrueassetpriceprocessfollowsageometricBrownianmotionwithsomearbitrary

stochasticvolatilityundertherisk-neutralmeasure:

dSt

=(r.)+tS,t,)t

qdt

(dB(13)


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St

Therisk-freerater and

dividendyieldq areassumedconstantandthevolatilityprocessis,forthe

moment,onlyassumedtobeboundedandcontinuoussothatSt isavalid

Itoprocess.Nowassumethatalocalvolatilitymodeliscalibratedateachtimeu .0andproducestherisk-neutraldynamicsforallt .u given

by:


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dS.t

qdt

(t)t

.=(r.)+LtS

,.,udB.(14)St

Hence,allthe

uncertaintyinisassumedtobecapturedbythedynamicsoftheparametersofthelocalvolatilitymodel.5Thisway,thelocalvolatilityis

regardedasa


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functionoft,St andthestochasticprocess

uofparametersthatarenotfixeduntiltheyarecalibrated

tothemarketat

somefuturetimeu .0.Withthis

definitionthelocalvolatilityfunctionhasanimplicitdependence

5Thisisnotastrongassumption.Itisanalogoustoassumingthatimplied

volatilitydynamicscapture


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allimperfectionsoftheBlack-Scholesmodel.Bates

(2003)arguesthatdailyre-calibrationisawayofhidingtheimperfections

ofamodelsothatitalwaysfitsthedata.Theseimperfectionswould

thenexpressthemselvesinthedynamicsofparameters.


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onthefuturevaluesofthe

parameterssothatthelocalvolatilitybecomesstochastic.Thestandardlocalvolatility

modelcanbeviewedasarestrictedformofthismodelinthe

specialcasethattheparametersareconstant.

Next,assumethattherisk-neutraldynamicsforeachparameterin(14)areasfollows:

iiii


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dt=i(t,t)dt

+i(t,t)dWt(15)

ii2

iidW=iS,(,t,t)dB

t+1.,(,,)dZ(16)

tStS

tiSttt

where

ijij

dW,W


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=t,,dt

dZi,B=0(17)

ij,(

tt)

t

t

ij

for

i, j I{1,2cn}.Hereisthecorrelationbetweenvariationsinand,and,isthe

ij,ttiS

i

correlationbetweenvariations


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intandSt respectively.Thedynamics(15)

arealsoassumedtosatisfytheregularityconditionsforanItoprocess.6

Wecallthemodeldefinedby(13).(17)thestochasticlocalvolatility

(SLV)model.

Bytakingthelimitof(12)whent 0inthecontextofdynamics(13)and(14),itiseasyto

showthatL


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(0,S0,0)=(0,S0,)

,sothattheinstantaneousvolatilityattime0isuniquelygiven

bythe

localvolatilitycalibratedattime0,whentheasset

priceisS0andtheparametersare0.Thisresultcanbegeneralizedtoanycalibrationtimeu .0as(,,)=(uS

,).Thus,


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sincethelocal

uS,

Luuu

volatilitymodel(14)iscalibratedforall

u,theinstantaneousvolatilityin(13)atanytimet iseasilyobtainedfrom

the(possiblyanalytical)localvolatilityfunctionbysetting:

(,,)=L(tS,t)

tSt,t

.(18)


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Sincethevalueoft is

discoveredonlywhenthelocalvolatilitymodeliscalibratedattimet,

itfollowsthat(tS,isstochastic,asitshouldbeby

definition.

,t)

III.2ClaimpricedynamicsandhedgeratiosWenowderivethepricedynamicsforageneralcontingentclaim

onS underthe


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SLVmodel.Foranytimet .0

denotebyft=f(,t,t)themodel

priceoftheclaimthatiscalibratedattimet.

tS

6Jumpsintheinstantaneousvolatility,ifany,canbemodelledbyaddingPoissonjumpstothedynamicsoftheparametersofthelocal

volatilitymodel.In


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thiscase,ourtheoreticalfindingsneedto

beextendedaccordinglyforjumps.


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Sincet containsthefutureparametersof

alocalvolatilitymodel,theclaimpricemustsatisfythefollowingpartial

differentialequationateachtimet >0:7

tfft2f

22t

+r.qS+1S=rf(19)()t2t2t

tStSt

where


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=(tStt

,

,).Notethat(19)onlyholdslocally,assumingthemodel

isre-calibratedateachtimet.

Theorem1

In

themodel(13).(17),therisk-neutraldynamicsofthecontingentclaimpricearegivenby:

.ff.

f

tt2


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ti

df=rfdt+

.S+i.dB+1.idZ

(20)

tiS

tti,tiiS,

t

Sii

.tt.t

andfortheabsenceofarbitragethefollowingdriftcondition

mustbesatisfied


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foreveryt >0:

.

ft2ft2ft.

+S+

12=0(21)

iti

,

.iiSi

ij,ijij.iSj

.ttttt.

We


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nowconsidertheclaimpricedynamicsfor

somespecialcasesofthemodel.Case1:If(t,i)=

0"iwehavethestandardrisk-neutraldynamicsofadeterministicvolatility

it

model.Inotherwords,whenthelocalvolatilityparametersarenon-stochastic:

ft

df=rfdt+

(,,


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)SdB

tS

tttttt

St

Case2:Ift,i

)0and

,,,i)=1thennonewsource

ofuncertaintyisintroduced,

((tSi

itiStt

so(20)becomes:

.ftft.

df=rf


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dt+S+dB

tt.tii.t

Si

.

tt.

wherethediffusioncoefficientismodifiedbuttheclaim

isredundant(i.e.itcanbereplicated).

Case3:If(t,i0and,tS,i)=0"ithen

theclaimprice


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dynamicswillbedrivenbya

it)iS(,ttmulti-factormodel:

fftti

df=rfdt+SdB+

idZ

ttttit

Stit

7Withinalocalvolatilitymodeltheparametersareassumedconstant,hence

theclaimprice


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canbeexpressedasafunctionof

t andSt only.Then(19)followsfromanapplicationofItofslemmaand

therisk-neutralityargument.


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Theorem2

Thefirst

andsecondordersensitivitiesofacontingentclaimpricef(,

,withrespecttoS,and

tS)

thefirstordersensitivitytotimet,aregivenby:

i,f

fniS

=+

i


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Si=1Sn2n

2

fiS.f1fjjS

,f.

=+i,2.+

(22)

2.iiij.

Si=1SSSj=1S

..f.

.


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nfiiS,

1

=+i.i.(r.q)+

2

i,.

iS.

ti=1

.

Thesecondtermontherighthandsideof(22)isanadjustmentfactorthatdependsonthe

correlationbetween

movementsofeach


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parameterandmovementsoftheassetprice

S.Ineffect,wespliteachhedgeratiointotwoparts:a

sensitivityderivedfromthestandardview(i.e.calibratedtothesmileata

fixedpointintime)andanadjustmentfactorthatdependsonthedynamicsofthestochasticparameters.

III.3AbsenceofArbitrageTheabsence

ofstaticarbitrage


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isguaranteedprovidedthelocalvolatilitiesare

nevernegative,asinSchweizerandWissel(2008).Sowenowconsider

theconditionsforabsenceofdynamicarbitrageintheSLVmodel.Alocal

volatilitymodelisarbitrage-freeundertheassumptionofastaticlocalvolatilitysurface,i.e.whenthevolatilityateachnodeofatreeforthe

assetpriceis


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knownwithcertainty.Butsincethissurface

isnotfoundtobestaticinpractice,theabsenceofdynamic

arbitragerequirementimposesrestrictionsonlocalvolatilitydynamics.Butweshouldviewthis

inthecontextofconstantparameterstochasticvolatilitymodels.Itisusuallythatcasethatsuchamodelisearbitrage-freefintheory.Howeverifthe

calibratedparameterschange


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overtime,whichis

usually

thecase,thereisnothing toensurethatthemodelisarbitrage-freein

practice.

[Insertsectiononexistenceofarbitrage-freeprices].

In

particular,thedriftcondition(21)mustholdforany claim,henceitplacesastrongconstrainton

thedynamicsofthemodelparameters.For

instance,ifi


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=0i thenalsoi=

0i.Thusifthe

volatilitysurfacemovesat

allitdoessostochastically.


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[DiscussotherconditionsforspecificSLV

models]

III.4SimilarApproachesDupire(1996)andDermanandKani

(1998)proposemodelingthedynamicsoflocalvarianceforeachfuturepriceK and

timeT directly.Forinstance:

dS=(r.)+t

qdt()dB(23)

SdKT()

,(,


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)KTtS)dWi

(24)

2,tS,=aKTtSdt+

bi,(,

In(23)(t)representsthestochastic

instantaneousvolatilityconsistentwiththelocalvariancedynamics(24)anditcanbederivedfromtheintegralformof(24):

22i

t=


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t,S+a,

udu+buSudWu

( )()St

,(00)tSt,(uS())

tSt,(,())i()

tt

00

where(t)=,).Therefore,if

a,andbi


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,areknownforeveryK andT,

thenthe

(tS

St,KTKT

stochastic

volatility(t)isfullyspecifiedintermsofthelocalvolatilitysurface

todaySt,(t0,S0)anditsdynamicsovertime.Notethat(24)holdsforeachpair(,)

KTinthelocal

variancesurfaceand


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thatthesamevectorofWienerprocesses

Wdrivesthedynamicsofthewholesurface,

iii

accordingtothecoefficientsa,andb,.Forinstance,if

b,=bforevery(,thenany

KTKT

KTKT)

shockcausesaparallelshiftinthelocal

variancesurface.Since


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thereareaninfinitenumberof

i

equations(24)afunctionalformforaandeach

bKTmayberequiredinpractice.

KT,,

Thisapproach

isattractivebecauseitmodelslocalvolatilitydirectlyanditisgeneralenoughtocaptureanycontinuousdynamicsforvolatility.However,westillneedto

imposearestriction


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onthedrifttermsaKTtoavoid

arbitrageopportunitiesand,althoughthedependenceofthedriftson

,K andT allowsenoughfreedomtodefinearisk-neutralmeasure,theresulting

expressionisrathercomplex.Solvingtheno-arbitrageconditionisnottrivialincontinuoustime,soDermanandKani

(1998)employstochasticimpliedtrees,

anextensionto


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theiroriginalimpliedtrinomialtrees.Bycontrast,

modelingthedynamicsoftheparametersasintheSLVmodelis

tractableandcanbeeasilyputinpracticebecausethevectort ofparameters

isobservable.

[InsertreviewofnewpaperbyWissel,2007]


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IV.EQUIVALENTAPPROACHESInthissection

wefirstderivetheimpliedprobabilityandlocalvarianceSDEsthatare

equivalenttothestochasticlocalvolatilitymodel(13).(17).Thenweprove

thatourmodelisequivalenttothemarketmodelofimpliedvolatilities.

IV.1ImpliedprobabilityandlocalvariancedynamicsNowconsiderthemodelfs

implieddistributionat


115/196

timeT.Thisisdefinedasthe

risk-neutralmarginalprobabilitydistributionoftheassetpriceST)thatis

consistentwiththemarketpricesofliquid

(optionsexpiringat

timeT.BreedenandLitzenberger(1978)showthatthiscanbeobtainedfromasimpledifferentiationofthevanillaoptionpricewithrespecttoK:

(


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.)2fKT,)

rT

t(

T()=e2(25)

tS

S=K

K

T

wheret(S)

istheimpliedrisk-neutraldensityoftheassetpriceS attimeT >t.Dupire(1996)

showsthat(25)istheundiscountedpriceof

aninfinitesimalbutterfly


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spreadcenteredatK,andhenceTt

(K)isamartingalewhoserisk-neutraldynamicsundertheSLVmodelare

givenby:

..2

d

=.S+iiS,i.dB+i1.iS,dZi(26).Si.i

iwhere


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wehaveassumedanarbitrage-freemarketand

usedtheresult(20).8

Likewise,applyingItofslemmatothe

localvariance(6)undertheassumption(21)weobtain:

V..

GG..V..GG..

dV=

SdB.S+dt+

dW.S


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+dt(27)

i

,i..i.i,jijj..

..

iSi.iS,

S..Si..i

..Sj..2f(,)

KT2

whereG(KT,)=ln2andV(KT,)=

(tS,)


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t=T,S=Kisamartingaleunderthe

K-strike

KLT-maturityeforwardrisk-adjustedfmeasure,i.e.themeasuredefined

bythepriceofaninfinitesimalbutterflyspreadcentredatK asnumeraire.See

DermanandKani(1998)orFengler(2005,section3.8).

8ItisinterestingtocomparethiswiththedynamicsofPanigirtzoglouandSkiadopoulos

(2004),whopropose


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asimplemean-revertingmodelfortheentire

impliedrisk-neutraldistributionbasedprincipalcomponentanalysis.Butthecorrespondenceisnot

one-to-onesincetheSLVmodelhasmoreparameters.


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IV.2StochasticimpliedvolatilitydynamicsDenote

theBlack-Scholes(B-S)priceattimet ofastandardEuropeanoptionwith

strikeK and

maturityT whentheassetpriceisS andtheimplied

volatilityis,by:

KT

BSBS

fKT=f,(tS,,KT)

,

KT,


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ThemarketimpliedvolatilityM,

(tSisthatsuchthattheB-Smodelpriceequalsthe

observed

KT,)marketpriceoftheoption.Likewise,when

thelocalvolatilitymodeliscalibratedtoamarket

impliedvolatilitysurfaceateachtimet themodelimpliedvolatilityL,(tS,

KT,


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)isdefinedby

equating

thelocalvolatilitypricetotheB-Sprice:

LBS

Lf(tS,)=f(tS,,,(tS,))

,,(28)

KT,KT,KT

NotethatM,tSandL,(tS,will

bethesame


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forallK andT ifthelocalvolatility

modelis

,,

KT()KT)

abletofitmarketpricesexactly.Yet,unfortunatelythisishardlythecase

foraparametricmodel.

Wenowderiveanexplicitrelationshipbetweenthestochasticlocalvolatilitypricedynamicsandtheevolutionofthemodel

impliedvolatility.This


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provesthattheSLVmodelhasimplied

volatilitydynamicsthatareidenticaltothosespecifiedbySchonbucher(1999)for

amarketmodelofimpliedvolatilities.Forthisweshallneedthefollowing

notationfortheB-Spriceandvolatilitysensitivities:

BSBS2BS

fff

BS,KT,

BS


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KTBSKT,

tS

,(tS,(2

(,)=;,

)=;tS,,)=

KT,,KT,KT

,

KT,KT,KT

tSS

(29)BS2BS2BSKTfKT,

BSf


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,BSKT,BS

f

KT(tS,,,)=;KT(,,,

)=;KTtS,,)

,KT,tSKT2,(

,KT=

S

andthefollowingnotationforthelocalvolatilitypricesensitivities:

LL

2L


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fff

L,

KT,L

KTLKT,

(tS,

)=;,tS,)=;,,)

KT,

KT(,,(tS=(30)

,KT

tSS2

Lemma1

Themodelimpliedvolatility

KTL


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,(tS,hasthefollowing

sensitivitiestot,S and:

,)


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LLBSLLBSLL

..1f

KT,,KT,

KT,KT,

KT,KT,KT,KT

=;=;=

BSBSiBSi

tS

KT,KT,KT

,


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2LLBSLBS

1.......

...

,BSKT,

KT,LBSKT

,KTBS,KT

=......2+....

KT,,KT,

KT,


133/196

KT

2BS.BS

..

.BS..

S..

KT,..KT,...KT...

,

2L2LLLBS

..

..

ff...

K,TKT

1,KT,


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1,KTBSBS,

KT

=..KT,+,....

iBSiBSi.KTBS

.S.....

SKT,.KT,..KT..

,.2L.2

LLL.


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f..2f

f

,,BS1KT,,

KT1KT

KT

=.j.KT.BS..

ijBSi,..ij

..

KT,.,

KT.

..

Thenext


136/196

lemmadescribesthepartialdifferentialequationthat

impliedvolatilitymustsatisfytobeconsistentwithanylocalvolatilitymodel.

Inthefollowingweusetheshorthandnotation

KT,

tStS,

=L(,,)and=(,),anddefined1andd2asinthe

B-Sformula:


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Lemma2

Themodel

impliedvolatilitymustsatisfythefollowingpartialdifferentialequation:

2

22.2d.

22.

dd..2.

.

1

+r.q.2S+2S

.2+


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12...+12=0

..

.

.

t.T

.t

.SS.S.(T.t)

..

NotethatthedifferentialequationinLemma2hasnopartialderivativeontheelementsof,eventhoughtheimplied

volatilityisnot


139/196

independentof.Henceevenwhenthe

localvolatilitysurfaceisstatictheimpliedvolatilitysurfacewillmoveover

time.However,problemsmayarisebecausethepermissiblemovementsaretoorestricted.

Nowweshowhowthedynamicsoftheentireimpliedvolatilitysurfacewillbegovernedbythesamestochasticfactorsasthosedrivingthe

localvolatilityand


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theoptionprice.Themainresultof

thissection,aTheorem,provestheequivalenceofthegeneralSLVmodel

withtheemarketmodelfofstochasticimpliedvolatilitiesspecifiedbySchonbucher(1999).

Lemma3

ThedynamicsofthemodelimpliedvolatilityintheSLVmodelaregivenby

n

d=


141/196

dt+SdB+i

dW(31)Si=1ii

where the drift term mustsatisfytT

(u)du


142/196

222

.

d12dd

212

()t=

12+.(32)

(T.t)T.t

Hereisrelatedtothecovariancebetweenimpliedvolatilityandassetpricemovements:

n

=S

+


143/196

iiS,iSi=1

and2isthevarianceoftheimpliedvolatilityprocess:

22nn

=+

.

ij(ij,jS)ij

,iS,

i=1j=1

andallpartial

derivativesof


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areasinLemma1.

Schonbucher(1999)expressedhisversionofthemarketmodelofimpliedvolatilities

intermsofuncorrelatedBrownianmotions.Thus,toproveequivalence,were-write(31)

usingonlyuncorrelatedBrownianmotionsasfollows:

Theorem3

ThedynamicsoftheSLVmodelimpliedvolatilitymaybewritten:

d=


145/196

dt+dB+njdZ

*j(33)

j=1

***

withanddefinedasinLemma3,andwheredBdZ=

dZdZ=0forijalmostsurely,and:

jij

2

=n

1.C


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jiiS,ij,i

i=j

whereCaretheelements

oftheCholeskydecompositionCofthecorrelationmatrixwith:

ij,

.n

Tij,,jS222

iS,

CC=,=

and=


147/196

+j.

ij

,1.21.2,)j=1

(iS,)(

jS

Apartfromminordifferencesinnotation,equation(33)isprecisely

thesameasequation(2.7)ofSchonbucher(1999)forthedynamicsofastochasticimpliedvolatilitywiththedrifttermgivenbyequation(3.7)of

thatpaper.Since


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Schonbucherfsisastochasticvolatilitymodel,this

corollaryprovesthatlocalandstochasticvolatilitymodelscanindeedbeunified

underasingleframework.

Thisresultalsoenablesonetospecify

theinstantaneouscorrelationbetweentheimpliedvolatilityandtheassetpricechangesas:

Cov(d,dS)Sdt

(34)

=


149/196

=

=

=

222n

,SVar(d)Var

(dS)dtSdt22

+j

j=1


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Hencetheimpliedvolatilityandasset

pricemovementswillhaveperfectcorrelation,of}1dependingonthesign

ofthecovariance,ifandonlyifj =0forall

j,i.e.whenthelocalvolatilitysurfaceisfixed.Inotherwords,theinstantaneousvolatilityisdeterministicifandonlyifvariations

in

impliedvolatilityand


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theassetpriceareperfectlycorrelated.

NotethatSchonbuchermodelstheimpliedvolatilitydynamicsforeachstrike

K andmaturityTseparately,whilstweprovidethedynamicsforallstrikesand

maturitiessimultaneously.Ifthereareoptionsfork strikesandm maturitiesinthemarkettheemarketmodelfspecifiesatleastmkdiffusions,oneforeachtraded

optionbecausethe


152/196

drifttermisoption-dependent,andassuringthat

thesediffusionsareconsistentandarbitrage-freeamongthemselvesisan

issuestillunderresearch.Ontheotherhand,theSLVapproachparameterizesthe

smilesurfacewithn


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assetpriceS.

V.CONCLUSIONS

Potentiallinksbetweenstochasticvolatilityandlocalvolatilitymodelswereidentifiedmany

yearsago,yetthesemodelshavebeendevelopedintwoseparatestrandsof

literature.Mostresearchonstochasticvolatilityhasspecifiedasinglefactordiffusionfortheinstantaneousvarianceorvolatilityoftheunderlyingasset;butresearchon

localvolatilitymodels


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hasassumedadeterministicinstantaneousvolatilityfunction

fortheunderlyingassetpricediffusion,withnoreferencetothedynamic

evolutionofvolatility.Bothapproacheswereincomplete,theformercapturingthedynamicproperties

ofvolatilitybutonlyinaone-dimensionalspace,thelatterfocusingonthemulti-dimensionalaspectsofvolatilitybutignoringitstime-evolution.However,recentdevelopmentson

diffusionsforimplied


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volatilityhaveextendedthestochasticvolatilityapproach

tobeconsistentwiththecross-sectionofimpliedvolatilitiesaswellas

theirdynamics.Toconcordwiththisview,thedeterministiclocalvolatilitymodel,which

impliesonlyadeterministicevolutionforimpliedvolatility,requiresgeneralization.

Wehaveshownthatthestochasticvolatilityandlocalvolatilityapproachescanbe

unifiedwithina


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generalframeworkanditisonlywhen

theseapproachestakearestrictedviewofvolatilitydynamicsthattheyappear

tobedifferent.FollowingDupire(1996)andDermanandKani(1998)weregard

thedeterministiclocalvolatilitymodelasmerelyaspecialcaseofamoregeneralstochasticlocalvolatilitymodel.Thatis,wedefinelocalvolatilityas

thesquareroot


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oftheconditionalexpectationofafuture

instantaneousvariancethatdependsonstochasticparametersofthelocalvolatility


158/196

functionaswellastheunderlying

price.Thuswehavemodeledthestochasticevolutionofalocallydeterministic

volatilitysurfaceovertimeandwehaveprovedanimportantgeneralresult:that

astochasticparametriclocalvolatilitymodelinducesimpliedvolatilitydynamicsthatareequivalenttothoseofamarketmodelforstochasticimpliedvolatilities.Hencethe

twomodelshave


159/196

identicalclaimprices.Explicitexpressionsforthe

delta,gammaandthetahedgeratiosareeasytoderiveinthe

SLVframework,andtheseareidenticaltothoseinthemarketmodelfor

impliedvolatilities.


160/196

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Journal of Theoretical and Applied Finance,4:3,403-438.Stein,E.

M.andJ.C.Stein(1991)StockPriceDistributionswithStochasticVolatility:anAnalyticApproach.Review of Financial Studies 4,727-752.

Wissel,J.(2007)Arbitrage-freeMarketModels

forOptionPricesh,


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Preprint, ETH Zurich.Availablefromhttp://www.nccr-finrisk.unizh.ch/wps.

Zilber,

A.(2007)AMarketModelforStochasticSmiles.Working paper.Availablefromhttp://ssrn.com/abstract=963572.


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APPENDIX

ProofofTheorem

1:ApplyingItofslemmatoaclaimpricef=f(

,t)

ttS,tgivesdynamics:

tfft

122ftf2f

2ft

ij

tit

df=dt+dS+dS+

d+


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d,iS+12

ijd,

tii

t

t

ttStS2

iti

S

ij

tttttt

Butusing(13)-(17):

.ff.

f

tt2


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ti

df=dt+

S+dB+1.dZ

tti

,.ti,

t.iSiSt

Siii

i

.tt.t

with

222

tff

f.f

f


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f.

22tt

t

t=+(r.qS)tt+12

S++SiS+12

t

t2.iiti,iij,ijij.tSSiSj

tt

.ttt


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tt.

Thedriftcondition

(21)followsfrom(19)andonnotingthatundertherisk-neutralprobability

the

driftoftheclaimpricemustbetherisk-freerate.

.

ProofofTheorem2:WhenmovementsinS andarecorrelated,wecanexpresseachi asa

functionoft,

S andZi sothat


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fromItofsformula:

ii2

i2iii

i.

2212.

d=+(

r.qS)+12S+2.dt+SdB+dZi

.

tS

S2ZS


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Z

.

i.

i

Equatingcoefficientsgives:

i2iiS

,i,

i

iS

==.

SSS2S2

i22i

=i1.,=0

ZiiSZi


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2

i

i,1

iS

=.

(r.q)+iS

i2i

,

tNowthechainrulegivesthefirstorderpricesensitivityas:

dffiff

i,


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=((tS,))=+

=+iS

f,

dSSiiSSiSiSimilarlythegammaandthetafollowusing:

dd

=((tS,,))=(f(tS,,)).

dSdt

ProofofLemma

1.Differentiate(28)


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withrespecttot,S andeachparameter

andapplythechainruleintheright-handsidewhenevernecessary.For

instance:


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LBSBSLLLBS

fff.

,

KTKT,

KT,KT,KT,KT,,KT

=+=

BS

SSSSKT

,

andsoforth..

Proof


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ofLemma2.SubtracttheBlack-ScholesPDE

from(19),apply(28)andLemma1,andusetherelationshipbetween

theBlack-Scholessensitivities:

1ddd

BSBSBS12

BSBS2BS

=2;=;=.KT,.

,KT,

KT,

KT,KT


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,KT

(T.t)

SST.t

ProofofLemma3.FromItofs

lemmaandusing(13)-(17),thedynamicsofthemodelimpliedvolatilityaregiven

by:

d=()tdt+SdB+iidWi

Si

222


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22

1

1

()t=+(r.qS)+

S++SiS+

22iiii,2ijijij,

tS


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SiiSij

UsingLemma1,thedriftexpands

to:

2L2L2L

221.ff

f.

+(r.qS)+1S+

+


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,+1

.S

22.iiiiSi2ij

ijij,.SSi

t

Sj

.1....

BS.

1.

.

.


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+

.

1

.i.S

iS

.BSBS.i,2.BSij

.ijij,

i.S.ij.

BS.BS

Next,usingthe


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driftcondition(21)andlemmas1and

2,thisre-arrangesto(32),withT >t,>0,

andand2asabove.Finally,ifisavalid

Itofsprocess,thentT(u)du


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andthefactthat:

22

dWdW=(+1.1.)dt=

dt.

jiS,,iS,ij,,

ijS

,jSij

as..

withdZdZdtandtheCholeskydecomposition(seee.g.Hafner,2004,section6.1.1).

ij


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ij,

9Notethat

theoptionpricesthatareconsistentwiththeimpliedvolatilitydynamics(31)

mustsatisfythesameno-arbitrageconditionsofLemma1.Besidesthis,there

isaninterestingsingularityonthedriftast T.However,thisisnotaproblemaslongasT(u)du t.


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t


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