A Call-Put Duality for Perpetual American Options

arXiv:math/0612648v1[math.PR]21Dec2006ACall-PutDualityforPerpetualAmericanOptionsAur´elienAlfonsiandBenjaminJourdainCERMICS,projetMATHFI,EcoleNationaledesPontsetChauss´ees,6-8avenueBlaisePascal,Cit´eDescartes,ChampssurMarne,77455Marne-la-vall´ee,France.e-mail:{alfonsi,jourdain}@cermics.enpc.frFebruary2,2008AbstractItiswellknown[5],[1]thatinmodelswithtime-homogeneouslocalvolatilityfunctionsandconstantinterestanddividendrates,theEuropeanPutpricesaretransformedintoEuropeanCallpricesbythesimultaneousexchangesoftheinterestanddividendratesandofthestrikeandspotpriceoftheunderlying.ThispaperinvestigatessuchaCallPutdualityforperpetualAmericanoptions.ItturnsoutthattheperpetualAmericanPutpriceisequaltotheperpetualAmericanCallpriceinamodelwhere,inadditiontothepreviousexchangesbetweenthespotpriceandthestrikeandbetweentheinterestanddividendrates,thelocalvolatilityfunctionismodified.WeprovethatequalityofthedualvolatilityfunctionsonlyholdsinthestandardBlack-Scholesmodelwithconstantvolatility.Thankstothesedualityresults,wedesignatheoreticalcalibrationprocedureofthelocalvolatilityfunctionfromtheperpetualCallandPutpricesforafixedspotpricex0.TheknowledgeofthePut(resp.Call)pricesforallstrikesenablestorecoverthelocalvolatilityfunctionontheinterval(0,x0)(resp.(x0,+∞)).Keywords:PerpetualAmericanoptions,Dupire’sformula,Call-PutDuality,Calibra-tionofvolatility,Optimalstopping.IntroductionInamodelwithlocalvolatilityfunctionς(t,x),interestrateranddividendrateδ(dSxt=ς(t,Sxt)SxtdWt+(r−δ)Sxtdt,t≥0Sx0=x(1)theinitialpriceh(T,y)=Ee−rT(y−SxT)+Call-putdualityforPerpetualAmericanOptions2oftheEuropeanPutoptionconsideredasafunctionofthematurityT0andtheStrikey0solvesDupire’spartialdifferentialequation[5]:(∂Th(T,y)=ς2(T,y)y22∂2yyh(T,y)+(δ−r)y∂yh(T,y)−δh(T,y),T,y0h(0,y)=(y−x)+,y0Oneeasilydeducesthatthefunctionh(T−t,y)for(t,y)∈[0,T]×R∗+satisfiesthepricingpartialdifferentialequationfortheCalloptionwithstrikexandmaturityTinthemodel(d¯Sy,Tt=ς(T−t,¯Sy,Tt)¯Sy,TtdWt+(δ−r)¯Sy,Ttdt,t∈[0,T]¯Sy,T0=y(2)withlocalvolatilityfunctionς(T−t,y),interestrateδanddividendrater.Thereforeh(T,y)=Ehe−δT(¯Sy,TT−x)+iandonededucesthefollowingCall-Putdualityrelationwhichisalsoaconsequenceof[1]∀T≥0,∀x,y0,Ee−rT(y−SxT)+=Ehe−δT(¯Sy,TT−x)+i.SinceitderivesfromDupire’sformula,thisCall-Putdualityequalityiscloselyrelatedtocalibrationissues.Oneremarksthatintheparticularcaseofatime-homogeneousvolatilityfunction(ς(t,x)=σ(x)),then¯Sy,Ttalsoevolvesaccordingtothesametime-homogeneousvolatilityfunction.Inthiswork,weareinterestedinderivingsuchaCall-PutdualityrelationinthecaseofAmericanoptionsandininvestigatingconsequencesintermsofcalibration.IntheBlack-Scholesmodelwithconstantvolatilityς(t,x)=σ,whenτdenotesaboundedstopping-timeofthenaturalfiltrationoftheBrownianmotion(Wt)t≥0,onehasEe−rτy−xeσWτ+(r−δ−σ22)τ+=Ee−δτeσWτ−σ22τye−σWτ+(δ−r+σ22)τ−x+=Ee−δτye−σWτ+(δ−r−σ22)τ−x+wherethesecondequalityfollowsfromGirsanovtheorem.Takingthesupremumoverallstopping-timesτsmallerthanTonededucesthatthepriceoftheAmericanPutoptionwithmaturityTisequaltothepriceoftheAmericanCalloptionwiththesamematurityuptothesimultaneousexchangebetweentheunderlyingspotpriceandthestrikeandbetweentheinterestanddividendrates.ExtensionsofthisresultwhentheunderlyingevolvesaccordingtotheexponentialofaL´evyprocesshavebeenobtainedin[7].Letusalsomentionthatanotherkindofdualityhasbeeninvestigatedin[13].But,toourknowledge,nostudyhasbeendevotedtothecaseofmodelswithlocalvolatilityfunctionslike(1).Inthepresentpaper,weconsiderthecaseofperpetual(T=+∞)Americanoptionsinmodelswithtime-homogeneouslocalvolatilityfunctionsς(t,x)=σ(x).Inthefirstpart,Call-putdualityforPerpetualAmericanOptions3werecoverwell-knownpropertiesoftheperpetualAmericancallandputpricingfunctionsbyextendinganapproachrecentlydevelopedbyBeibelandLerche[2]intheBlack-Scholescase.Thismakesthepaperself-contained.Inthesecondpart,weintroducetheframeworkusedintheremainingofthepaper.Inthethirdpartofthepaper,weconsidertheexerciseboundariesasfunctionsofthestrikevariableandcharacterizethemastheuniquesolutionsofsomenon-autonomousordinarydifferentialequations.Thefourthpartisdedicatedtoourmainresult.WeprovethattheperpetualAmericanPutpricesareequaltotheperpetualAmericanCallpricesinamodelwhere,inadditiontotheexchangesbetweenthespotpriceoftheunderlyingandthestrikeandbetweentheinterestanddividendrates,thevolatilityfunctionismodified.Wealsoderiveanexpressionofthismodifiedvolatilityfunction.NoticethatintheEuropeancasepresentedabove,time-homogeneousvolatilityfunctionsarenotmodified.Thefifthpartaddressescalibrationissues.Itturnsoutthatforagiveninitialvaluex00oftheunderlyingonerecoverstherestrictionofthetime-homogeneousvolatilityfunctionσ(x)to(0,x0](resp.[x0,+∞))fromtheperpetualPut(resp.Call)pricesforallstrikes.Inthelastpart,weshowthatatleastwhenδr,intheclassofvolatilityfunctionsanalyticinaneighbourhoodoftheorigin,theonlyonesinvariantbyourdualityresultaretheconstants.ThismeansthatthecaseofthestandardBlack-Scholesmodelpresentedaboveisveryspecific.Acknowledgements.WethankDamienLamberton(Univ.Marne-la-vall