Priorliterature - PDF document

Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels Outlineofthistalk Motivationforfractionalmodels Empiricalvolatilitystatistics FractionalBrownianmotion(fBm) Priorfractionalmodelsofvolatility Adata-drivenstochasticvolatilitymodel:fBergomi FractionalSteinSteinandfractionalSABRmodels Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels MotivationI:Betterttingstochasticvolatilitymodels Conventionalstochasticvolatilitymodelsgeneratevolatilitysurfacesthatareinconsistentwiththeobservedvolatilitysurface. Instochasticvolatilitymodels,theATMvolatilityskewisconstantforshortdatesandinverselyproportionaltoTforlongdates. Empirically,wendthatthetermstructureofATMskewisproportionalto1=Tforsome01=2overaverywiderangeofexpirations. Theconventionalsolutionistointroducemorevolatilityfactors,asforexampleintheDMRandBergomimodels. Onecouldimaginethepower-lawdecayofATMskewtobetheresultofadding(oraveraging)manysub-processes,eachofwhichischaracteristicofatradingstylewithaparticulartimehorizon. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels Power-lawsfromaveraging:Atoyexample Thefollowingexample,adaptedfrom[ComteandRenault],illustrateshowpower-lawbehaviorcanemergefromtheaveragingofshortmemoryprocesses. ConsiderthefollowingOUprocess(Xt=logtsay)indexedby:Xt()=Zt0e�(t�s)dWs:ThenXtN(0;()2)with()2=Rt0e�2(t�s)ds: Consideramultiplicityofsuchprocesseswithgamma-distributed.Explicitly,p�()=�1e�= �()forsome�0and�0. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels Then,theaverageXN(0;2)with2=Z10p�()Zt0e�2(t�s)dds=Zt01 [1+2(t�s)]dsandXt=Zt0dWs [1+(t�s)]=2: Thus,averagingshortmemoryvolatilityprocesses(withexponentialkernels)overdierenttimescalescangenerateavolatilityprocesswithapower-lawkernel Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels MotivationII:Power-lawscalingofthevolatilityprocess Aseparatebut(presumably)relatedreasonforconsideringfractionalvolatilitymodelsisthatthetimeseriesofrealizedvolatilityexhibitspower-lawscaling. TheOxford-ManInstituteofQuantitativeFinancemakeshistoricalrealizedvariance(RV)estimatesfreelyavailableathttp://realized.oxford-man.ox.ac.uk.Theseestimatesareupdateddaily. UsingdailyRVestimatesasproxiesforinstantaneousvariance,wemayinvestigatethetimeseriespropertiesofvtempirically. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels SPXrealizedvariancefrom2000to2014 Figure2:KRVestimatesofSPXrealizedvariancefrom2000to2014. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels ThevarianceofSPXRV LetvRtdenotetherealizedvarianceofSPXondayt. WeconsidertwomeasuresofvarianceofRVoverthetimeinterval
1: 1 V1(
):=h(vRt+
�vRt)2i 2 V2(
):=h(log(vRt+
)�log(vRt))2i WendthatV2(
)=A
2HwithH0:14andA0:38: 1h
idenotesasampleaverage. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels Stationarityoflog(vRt) Supposelog(vRt)ismean-squarestationaryforlarget(aswecertainlybelieve). Then,V2(
)=h(log(vRt+
)�log(vRt))2i4M2whereM2=var(logvR)1. Ifso,wemusthaveV2(
)const.as
!1.Power-lawscalingofV2(
)canholdonlyuptosomelongtimescale. Ahand-wavingestimateofthistimescaleusingA~
2H4M2gives~
24years. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels VarianceofRVdierencesvs
Figure3:Log-logplotsofV1(
)andV2(
)respectively.V2(
)wins! Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels FractionalBrownianmotion(fBm) FractionalBrownianmotion(fBm)fWHt;t2RgistheuniqueGaussianprocesswithmeanzeroandautocovariancefunctionEhWHtWHsi=1 2njtj2H+jsj2H�jt�sj2HowhereH2(0;1)iscalledtheHurstindexorparameter. Inparticular,whenH=1=2,fBmisjustBrownianmotion. IfH�1=2,incrementsarepositivelycorrelated. IfH1=2,incrementsarenegativelycorrelated. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels RepresentationsoffBm ThereareinnitelymanypossiblerepresentationsoffBmintermsofBrownianmotion.Forexample,with =1 2�H, Mandelbrot-VanNess WHt=CHZt�1dWs (t�s) �Z0�1dWs (�s) : wherethechoiceCH=s 2H�(3=2�H) �(H+1=2)�(2�2H)ensuresthatEhWHtWHsi=1 2nt2H+s2H�jt�sj2Ho: Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels AnotherrepresentationoffBm DeneKH(t;s)=CHF ;� ;1� ;1�t s1 (t�s) :whereF(
)isGauss'shypergeometricfunction.Then,fBmcanalsoberepresentedas: Molchan-Golosov WHt=Zt0KH(t;s)dWs: TheMandelbrot-VanNessrepresentationusestheentirehistoryoftheBrownianmotionfWs;stg. TheMolchan-GolosovrepresentationusesonlythehistoryoftheBrownianmotionfromtime0. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels Why\fractional"? Denotethedierentiationoperatord dtbyD.ThenD�1f(t)=Zt0f(s)ds:TheCauchyformulaforrepeatedintegrationgivesforanyintegern�0,D�nf(t)=Zt01 n!(t�s)n�1f(s)ds:Thegeneralizationofthisformulatorealgivesthedenitionofthefractionalintegral:D�f(t)=1 �()Zt0(t�s)�1f(s)ds:NoteinparticularthatD0f(t)=f(t). Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels ComteandRenault:FSVmodel [ComteandRenault]wereperhapsthersttomodelvolatilityusingfractionalBrownianmotion. Intheirfractionalstochasticvolatility(FSV)model,dSt St=tdZtdlogt=�(logt�)dt+ d^WHt(4)with^WHt=Zt0(t�s)H�1=2 �(H+1=2)dWs;1=2H1andE[dWtdZt]=dt. TheFSVmodelisageneralizationoftheHull-Whitestochasticvolatilitymodel. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels Integralformulation Solving(4)formallygivest=exp+e�t(log0�)+ Zt0e�(t�s)d^WHs:(5) H�1=2toensurelong-memory. Stationarityisachievedwiththeexponentialkernele�(t�s)atthecostofintroducinganexplicittimescale�1. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels FSVcovariance Deneyt=logt.Wehavecov(yt;yt+
)/Z0�1esdsZ
�1e(s0�
)ds0js�s0j2H�2:ThenE(yt+
�yt)2=2var[yt]�2cov(yt;yt+
)wherecov(yt;yt+
)/e�k
2k2HZk
0eudu u2�2H+e�k
2k2H�(2H�1)+ek
2k2HZ+1k
e�udu u2�2H: Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels VilelaMendes Anempiricalstudyofthescalingofvolatilityestimatesby[VilelaMendesandOliveira]motivatestheirdata-reconstructedmodel:dSt St=tdZtlog(t)=+k nWHt�WHt�o(6)orequivalently,t=exp+k Zt01s�t�dWHs: WenotethatthismodellooksverysimilartotheComte-Renaultmodel(5). TheindicatorfunctionkernelactsliketheexponentialkernelinFSVtoforcestationarityforlongtimes. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels FeaturesofthefractionalBergomimodel Theforwardvariancecurvet(u)=E[vujFt]=vtexpZt;u+22ZutK(u;s)2ds:dependsonthehistoricalpathfWs;stgoftheBrownianmotionsinceinception(t=�1say). ThefractionalBergomimodelisnon-Markovian:E[vujFt]6=E[vujvt]: However,giventhe(innite)statevectort(u),whichcaninprinciplebecomputedfromoptionprices,thedynamicsofthemodelarewell-determined. Inpractice,thereisabid-oerspreadandwedon'thaveoptionpricesforallstrikesandexpirations. Thereisinherentmodelrisk! Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels Re-interpretationoftheconventionalBergomimodel Aconventionaln-factorBergomimodelisnotself-consistentforanarbitrarychoiceoftheinitialforwardvariancecurvet(u). t(u)=E[vujFt]shouldbeconsistentwiththeassumeddynamics. ViewedfromtheperspectiveofthefractionalBergomimodelhowever: Theinitialcurvet(u)re ectsthehistoryfWs;stgofthedrivingBrownianmotionuptotimet. Theexponentialkernelsintheexponentof(2)approximatemorerealisticpower-lawkernels. Theconventionaltwo-factorBergomimodelisthenjustiedinpracticeasatractableMarkovianengineeringapproximationtoamorerealisticfractionalBergomimodel. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels FromfBergomitofSABR VarianceislognormalinthefBergomimodelandthusvolatilityisalsolognormal. Thenrewrite(10)asu=^t(u)EZutK(u;s)dWs(11)whereE(X)isthestochasticexponentialofXandtheforwardvolatilityisgivenby^t(u)=E[ujFt]=E[p vujFt]=p t(u)exp�2 2ZutK(u;s)2ds: Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels FromfBergomitofSABR TheformalsolutionofthefractionalBergomimodelmaythenbewrittenasST=S0exp(ZT0tdZt�1 2ZT02tdt)=S0E ZT0^0(t)EZt0K(t;s)dWsdZt!: ^0(t)istypicallyaslowly-varyingfunctionoftsowemaywriteSTS0E(T)ZT0EZt0K(t;s)dWsdZt(12)with(T)2=1 TRT0^0(t)2dt: Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels ThefractionalSABR(fSABR)model SettingK(t;s)=KH(t;s)(theMolchan-Golosovkernel),weidentify(12)asthesolutionofthefollowingfSABRmodel:dSt St=tdZtt=(T)EWHtwheredWtdZt=dt. fSABRisthenaturalfBmextensionoftheSABRmodelof[Haganetal.](with=1). IfH1=2,thevarianceofvolatilitygrowssublinearlygeneratinganaturaltermstructureofATMvolatilityskew. ThefSABRstochasticvolatilitymodelisnon-Markovian,justlikethefractionalBergomimodel. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels ThefractionalSteinandStein(fSS)model Asanevenmoretractablealternative,considerthefollowingmodel:dSt St=tdZtt=0+WHt:againwithdWtdZt=dt. InthefSSmodel,thevolatilitytisanormalrandomvariable. Again,ifH1=2,thevarianceofvolatilitygrowssublinearlythusvolatilityis\mean-reverting". ThefSSmodelcanbeconsideredatoyversionofthemorerealisticfSABRmodelwheremorequantitiesofinterestareexplicitlycomputable. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels SimulationoffSSandfSABRmodels First,foreachMonteCarlopath,generatethecorrelatedBrownianincrements
Wtand
Zt. Giventhe
Wt,theWHtareconstructed,forexampleusingtheCholeskydecompositionmethod. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels SPXsmilesinthefSABRmodel InFigure9,weshowhowthefSABRmodelgeneratesverygoodtstotheSPXoptionmarketasof04-Feb-2010,adaywhentheATMvolatilitytermstructurehappenedtobe at. fSABRparameterswere:=0:235,=0:2=0:235,H=1=4,=�0:7. Noteinparticularthatwehaveobtainedagoodttothewholevolatilitysurfaceusingamodelwithveryfewparameters! Moreover,HandcanbexedfromtheVIXmarket. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels Summary Varianceofthelogofrealizedvarianceexhibitsclearpower-lawscaling. LongmemoryofRVisrejectedbythedata. Theresultingdata-drivenmodel(fBergomi)isanon-MarkoviangeneralizationoftheBergomimodel. TheconventionalMarkovianBergomimarketmodelcanbeviewedasanaccurateapproximationtofBergomi. ThefSABRmodel,anaturalgeneralizationoftheSABRmodel,isatractableapproximationtothefBergomimodel. fSABRtsobservedsmilesandskewsremarkablywell. ThevalueoftheHurstexponentHisxedbythetermstructureofVIXat-the-moneyimpliedvolatility. Motivation fBm Priorliterature ThefBergomimodel fSSandfSABRmodels References TorbenGAndersen,TimBollerslev,FrancisXDiebold,andHeikoEbens,Thedistributionofrealizedstockreturnvolatility,JournalofFinancialEconomics61(1)43{76(2001). LorenzoBergomi,SmiledynamicsII,RiskMagazine67{73(October2005). LorenzoBergomiandJulienGuyon,Thesmileinstochasticvolatilitymodels,SSRN(2011). FabienneComte,LaureCoutinandEricRenault,Anefractionalstochasticvolatilitymodels,AnnalsofFinance8(2{3),337{378(2012). FabienneComteandEricRenault,Longmemorycontinuoustimemodels,JournalofEconometrics73(1)101{149(1996). FabienneComteandEricRenault,Longmemoryincontinuous-timestochasticvolatilitymodels,MathematicalFinance829{323(1998). LaurentDecreusefond,StochasticintegrationwithrespecttoVolterraprocesses,Annalesdel'InstitutHenriPoincare(B)ProbabilityandStatistics41(2)123{149(2005). JimGatheral,TheVolatilitySurface:APractitioner'sGuide,WileyFinance(2006). PHagan,DKumar,ALesniewski,andDWoodward,Managingsmilerisk,WilmottMagazine,84{108,(September2002). RuiVilelaMendesandMariaJo~aoOliveira,,Adata-reconstructedfractionalvolatilitymodel,arXivavailableathttp://arxiv.org/abs/math/0602013v2(2006).