Analytically Tractable Stochastic Stock Price Models

370 Pages·2012·2.199 MB·English

by  Archil Gulisashvili

Springer Finance
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Archil Gulisashvili
Analytically
Tractable
Stochastic Stock
Price Models
ArchilGulisashvili
DepartmentofMathematics
OhioUniversity
Athens,OH,USA
ISBN978-3-642-31213-7 ISBN978-3-642-31214-4(eBook)
DOI10.1007/978-3-642-31214-4
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012945072
MathematicsSubjectClassification(2010): 91Gxx,91G80,91B25,91G20
JELClassification: GO2,G13
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To Olga,Alex,and Misha
Preface
Thisbookfocusesprimarilyonapplicationsofmathematicalanalysistostockprice
models with stochastic volatility and more general stochastic asset price models.
The central objective of the book is to characterize limiting behavior of several
important functions associated with such models, e.g., stock price densities, call
andputpricingfunctions,andimpliedvolatilities.
Stock price models with stochastic volatility have been developed in the last
decadestoimprovepricingandhedgingperformanceoftheclassicalBlack–Scholes
modelandtoaccountforcertainimperfectionsinit.Themainshortcomingofthe
Black–Scholes model is its constant volatility assumption. Statistical analysis of
stockmarketdatashowsthatthevolatilityofastockisatime-dependentquantity.
Moreover,itexhibitsvariousrandomfeatures.Stochasticvolatilitymodelsaddress
thisrandomnessbyassumingthatboththestockpriceandthevolatilityarestochas-
ticprocessesaffectedbydifferentsourcesofrisk.UnliketheBlack–Scholesmodel,
stock price models with stochastic volatility explain such stylized facts as the im-
pliedvolatilitysmileandskew.Theycanalsoincorporatetheleverageeffect,thatis,
thetendencyofthevolatilityofthestocktoincreasewhenthestockpricedecreases.
Stochastic volatility models reflect the leverage effect by imposing the restriction
thatthestockpriceandthevolatilityarenegativelycorrelated.
Animportantprobleminmathematicalfinanceistodescribetheasymptoticbe-
havior of the stock price density in a stochastic volatility model. Once we have a
goodunderstandingofhowthisdensitychanges,wecanestimatemanyotherchar-
acteristicsofthemodel,forexample,leftandrighttailsofstockreturndistributions,
optionpricingfunctions,andimpliedvolatilities.Asymptoticformulasfordistribu-
tiontailsofstockreturnsinastochasticvolatilitymodelcanbeusedtoanalyzehow
well the model addresses the tail risk. In financial practice, the tail risk is defined
astheprobabilitythatstockreturnswillmovemorethanthreestandarddeviations
beyondthemean.TheBlack–Scholesmodelunderestimatesthetailrisk,sincethe
probability of extreme variations of stock returns in this model is negligible. This
follows from the fact that distribution tails of stock returns in the Black–Scholes
modeldecaylikeGaussiandensityfunctions.Inthepresentbook,weobtainsharp
asymptotic formulas with relative error estimates for stock price densities in three
vii
viii Preface
popular stock price models with stochastic volatility: the Hull–White model, the
Stein–Steinmodel,andtheHestonmodel.Theseformulasshowthatfortheabove-
mentionedmodels,thestockpricedistributionshaveParetotypetails,thatistosay,
thetailsdecaylikeregularlyvaryingfunctions.Asaconsequence,theHull–White,
Stein–Stein,andHestonmodelsestimatetheprobabilityofabruptdownwardmove-
mentsofstockprices(disastrousscenarios)betterthantheBlack–Scholesmodel.
Theimpliedvolatilityassociatedwiththecallpricingfunctioninastochasticas-
setpricemodelmaybepoeticallydescribedasthereflectionofthisfunctioninthe
Black–Scholesmirror.OnecanobtaintheimpliedvolatilitybyinvertingtheBlack–
Scholescallpricingfunctionandcomposingtheinversefunctionwiththecallpric-
ingfunctionofourinterest.Asubstantialpartofthepresentbookdiscussesmodel
freeasymptoticformulasfortheimpliedvolatilityatextremestrikesingeneralas-
setpricemodels.Someofthereasonswhysuchasymptoticformulasareimportant
arethefollowing.Ontheonehand,theseformulashelptocheckwhetherthegiven
stochasticassetpricemodelproducesaskewedvolatilitypatternoftenobservedin
realmarkets.Ontheotherhand,sincetheimpliedvolatilityatextremestrikesisas-
sociatedwithout-of-the-moneyandin-the-moneyputandcalloptions,theanalysis
of the implied volatility for large and small strikes quantifies the expectations and
fearsofinvestorsofpossiblelargeupwardordownwardmovementsinassetprices.
Notethatbuyingout-of-themoneyputoptionshasbeenapopularhedgingstrategy
againstnegativetailrisk.
The text is organized as follows. The main emphasis in Chaps. 1–7 is on spe-
cialstochasticvolatilitymodels(theHull–White,Stein–Stein,andHestonmodels).
In Chap. 1, we consider stochastic processes, which play the role of volatility in
these models, i.e., geometric Brownian motion, Ornstein–Uhlenbeck process, and
Cox–Ingersoll–Ross process (Feller process). Chapter 2 introduces general corre-
latedstockpricemodelswithstochasticvolatility.Italsodiscussesrisk-neutralmea-
suresinsuchmodels.Chapter3isconcernedwithrealizedvolatilityandmixingdis-
tributions.Foranuncorrelatedstochasticvolatilitymodel,themixingdistributionis
thelawof therealizedvolatility,whileforcorrelatedmodels,mixingdistributions
aredefinedasjointdistributionsofvariouscombinationsofthevarianceofthestock
price,theintegratedvolatility,andtheintegratedvariance.Chapter4considersin-
tegral transforms of mixing distribution densities, and provides explicit formulas
for the stock price density in terms of mixing distributions. In Chap. 5 we prove
a Tauberian theorem for the two-sided Laplace transform, and also Abelian theo-
remsforfractionalintegralsandforintegraloperatorswithlog-normalkernels.The
TauberiantheoremisusedinChap.5tocharacterizetheasymptoticsofmixingdis-
tributions by inverting their Laplace transforms approximately, while the Abelian
theoremforfractionalintegralsisahelpfultoolinthestudyofmixingdistributions
intheHull–Whitemodel.InChap.6weprovideasymptoticformulaswitherrores-
timatesforthestockpricedistributiondensitiesintheHull–White,Stein–Stein,and
Heston models. For the correlated Heston model the proof of the asymptotic for-
mulaisbasedonaffineprinciples,whileintheabsenceofcorrelationanalternative
proof of the asymptotic formula is given. In the latter proof the Abelian theorem
for integral operators with log-normal kernels plays an important role. Finally, in
Preface ix
Chap.7weincludeashortexpositionofthetheoryofregularlyvaryingfunctions.
ThischapteralsoconsidersParetotypedistributionsandtheirapplications.
The second part of the book (Chaps. 8–11) is devoted to general call and put
pricing functions in no-arbitrage setting and to the Black–Scholes implied volatil-
ity.InthebeginningofChap.8weproveacharacterizationtheoremforcallpricing
functions,andattheendofthischapterweestablishsharpasymptoticformulaswith
error estimates for the call pricing functions in the Hull–White, Stein–Stein, and
Heston models. Chapter 8 also presents an analytical proof of the Black–Scholes
calloptionpricingformula,whichisarguablythemostfamousformulaofmathe-
matical finance. Chapter 9 introduces the notion of implied volatility (or “smile”)
and provides model free asymptotic formulas for the implied volatility at extreme
strikes. One more topic discussed in Chap. 9 concerns certain symmetries hidden
in option pricing models. The contents of Chap. 10 can be guessed from its title
“More Formulas for Implied Volatility”. It is shown in this chapter that R. Lee’s
momentformulasfortheimpliedvolatilityandthetail-wingformulasduetoS.Be-
naim and P. Friz can be derived from more general results established in Chap. 9.
Chapter10alsopresentsanimportantresultobtainedbyE.RenaultandN.Touzi,
whichcanbeshortlypresentedasfollows:“Theabsenceofcorrelationbetweenthe
stockpriceandthevolatilityimpliessmile”.ThelastsectionofChap.10dealswith
J. Gatheral’sSVIparameterizationof impliedvariance.SVI parameterizationpro-
vides a good approximation to implied variance observed in the markets and also
to implied variance used in stochastic volatility models. Finally, in Chap. 11 we
studyimpliedvolatilityinmodelswithoutmomentexplosions.Hereweshowthat
V.V. Piterbarg’s conjecture concerning the limiting behavior of implied volatility
in models without moment explosions is true in a modified form. Chapter 11 also
studies smile asymptotics in various special models, e.g., the displaced diffusion
model,theconstantelasticityofvariancemodel,SV1andSV2modelsintroduced
byL.C.G.RogersandL.A.M.Veraart,andthefinitemomentlog-stablemodelsde-
velopedbyP.CarrandL.Wu.
Wewillnextmakeabriefcomparisonbetweenthepresentbookandthefollow-
ing related books: [Lew00, Gat06, H-L09], and [FPSS11]. It is easy to check that
although the books on the previous list and the present book have the same main
heroes(stochasticvolatilitymodels,stockpricedensities,optionpricingfunctions,
and implied volatilities), they differ substantially from each other with respect to
thechoiceofspecialtopics.Forexample,thebookbyA.Lewis[Lew00]dealswith
variousmethodsofoptionpricingunderstochasticvolatility,andthetopicscovered
in [Lew00] include the volatility of volatility series expansions, volatility explo-
sions, and related corrections in option pricing formulas. The book by J. Gatheral
[Gat06]isarichsourceofinformationonimpliedandlocalvolatilitiesinstochastic
stock price models and on the asymptotic and dynamic behavior of volatility sur-
faces. In particular, the book [Gat06] discusses early results on smile asymptotics
forlargeandsmallstrikes.ThebookbyP.Henry-Labordère[H-L09]usespowerful
methodsofdifferentialgeometryandmathematicalphysicstostudytheasymptotic
behaviorofimpliedvolatilityinlocalandstochasticvolatilitymodels.Forinstance,
heatkernelexpansionsinRiemannianmanifoldsandSchrödingersemigroupswith
x Preface
Katoclasspotentialsplayanimportantrolein[H-L09].ThebookbyJ.-P.Fouque,
G.Papanicolaou,R.Sircar,andK.Sølna[FPSS11]isdevotedtopricingandhedg-
ingoffinancialderivativesinstochasticvolatilitymodels.In[FPSS11],regularand
singular perturbation techniques are used to study small parameter asymptotics of
option pricing functions and implied volatilities. The authors of [FPSS11] obtain
firstandsecondorderapproximationstoimpliedvolatilityinsingle-factorandmul-
tifactorstochasticvolatilitymodels,andexplainhowtousetheseapproximationsto
calibratestochastic volatilitymodels and price more complexderivative contracts.
Amoredetailedcomparisonshowsthatalargepartofthematerialappearinginthe
presentbookisnotcoveredbythebooksonthelist.Moreover,tothebestoftheau-
thor’sknowledge,manyoftheresultsdiscussedinthepresentbookhaveneverbeen
published before in book form. These results include sharp asymptotic formulas
witherrorestimatesforstockpricedensities,optionpricingfunctions,andimplied
volatilitiesinspecialstochasticvolatilitymodels,andsharpmodelfreeasymptotic
formulasforimpliedvolatilities.
This book is aimed at a variety of people: researchers in the field of financial
mathematics,professionalmathematiciansinterestedinapplicationsofmathemati-
cal analysis to finance, and advanced graduate students thinkingof a career in ap-
pliedanalysisorfinancialmathematics.Itisassumedthatthereaderisfamiliarwith
basicdefinitionsandfactsfromprobabilitytheory,stochasticdifferentialequations,
asymptoticanalysis,andcomplexanalysis.Thebookdoesnotaspiretocomplete-
ness,severalimportanttopicsrelatedtoitscontentshavebeenomitted.Forexample,
smallandlargematurityasymptoticsofimpliedvolatility,affinemodels,localmar-
tingaleoptionpricingmodels,orapplicationsofgeometricmethodstothestudyof
impliedvolatilityarenotdiscussedinthebook.Thereadercanfindselectedrefer-
encestopublicationsonthemissingsubjectsinthesections“NotesandReferences”
thatconcludeeachchapter,orsearchthebibliographyattheendofthebookforad-
ditionalreading.
Athens,Ohio ArchilGulisashvili
December2011
Acknowledgements
Thisbookwasfirstthoughtofasalecturenotesbook.Itsearlyversionswerebased
on the contents of several summer courses on selected topics in financial mathe-
maticsthatIgaveatBielefeldUniversity.However,thebookunderwentsignificant
transformationsduringmyworkonit.Ithasgrowninsize,anditsfinalversionlooks
morelikearesearchmonographthanalecturenotesbook.Iexpressmygratitudeto
MichaelRöcknerforextendingtometheinvitationstovisittheInternationalGrad-
uate College “Stochastics and Real World Models” at Bielefeld University in the
summersof2007–2011andforgivingmetheopportunitytospendseveralmonths
inaverystimulatingmathematicalenvironment.Ialsothankallthemembersofthe
graduatecollegeandthestochasticanalysisgroupintheDepartmentofMathemat-
icsatBielefeldUniversityfortheirhospitality,andallthestudentswhoattendedmy
coursesatBielefeldUniversity.
ThepresentbookwouldnotexistwithoutEliasM.Stein.Ithasbeenaprivilege
andapleasureworkingwithhimonaprojectconcerningclassicalstochasticvolatil-
itymodels.Asubstantialpartofthisbookhasstronglybenefitedfromhisimpact,
andmanyresultsobtainedinourjointpapersarecoveredinthebook.Iexpressmy
deepgratitudetoE.M.Steinforhisfriendship,collaboration,andencouragement.
Iamindebtedtomanyofmyfriendsandcolleagueswhowereofgreathelptome
inmyworkonthisbook.IwouldliketosingleoutPeterLaurenceforhisfriendship,
advice,valuablemathematicalandbibliographicalinformation,andstimulatingdis-
cussions on some of the topics covered in the book. Peter Friz and Sean Violante
forimportantremarksconcerningthecontentsofthebook.PeterFriz,StefanGer-
hold,andStephanSturmfortheirmuchenjoyableandfruitfulcollaborationonthe
joint project concerning stock price density asymptotics in the correlated Heston
model.Theresultsofthisworkareincludedinthebook.RenéCarmonaandRon-
nieSircarforinterestingconversationsandimportantadviceduringmyearlywork
onstochasticvolatilitymodelsandtheimpliedvolatility.RogerLeeforbeingvery
generousinsharingwithmehisprofoundknowledgeofstochasticvolatilitymod-
els and smile asymptotics. Josep Vives for his friendship and collaboration on the
jointpaperdevotedtostochasticvolatilitymodelswithjumps.Jean-PierreFouque,
Josef Teichmann, and Martin Keller-Ressel for always interesting and informative
conversations.
xi