On the Hedging of American Options in Discrete Tim

arXiv:math/0502189v1[math.PR]9Feb2005OntheHedgingofAmericanOptionsinDiscreteTimeMarketswithProportionalTransactionCostsBrunoBOUCHARD∗LaboratoiredeProbabilit´esetMod`elesAl´eatoiresCNRS,UMR7599Universit´eParis6andCRESTe-mail:[e-mail]redeProbabilit´esetMod`elesAl´eatoiresCNRS,UMR7599Universit´eParis7e-mail:[e-mail]:January2005AbstractInthisnote,weconsiderageneraldiscretetimefinancialmarketwithpro-portionaltransactioncostsasinKabanovandStricker[4],Kabanovetal.[5],Kabanovetal.[6]andSchachermayer[10].Weprovideadualformulationforthesetofinitialendowmentswhichallowtosuper-hedgesomeAmericanclaim.WeshowthatthisextendstheresultofChalasaniandJha[1]whichwasob-tainedinamodelwithconstanttransactioncostsandriskyassetswhichevolveonafinitedimensionaltree.Wealsoprovidefairlygeneralconditionsunderwhichtheexpectedformulationintermsofstoppingtimesdoesnotwork.Keywords:Sumofrandomconvexcones,Transactioncosts,Americanoption.MSCClassification(2000):91B28,60G40.1IntroductionWeconsideradiscretetimefinancialmarketwithproportionaltransactioncosts.Thesemarketshavealreadybeenwidelystudied.Inparticular,aproofofthefundamentaltheoremofassetpricingwasgiveninKabanovandStricker[4]inthecaseoffiniteΩandfurtherdevelopedinKabanovetal.[5],[6],R´asonyi[8],Schachermayer[10]amongothers.Inthesepapers,asuper-replicationtheoremisalsoprovidedforEuropeancontingentclaims.TheaimofthispaperistoextendthistheoremtoAmericanoptions.Itiswellknownthat,forfrictionlessmarkets,thesuper-replicationpriceofanAmericanclaimadmitsadualrepresentationintermsofstoppingtimes.However,itisprovedinChalasaniandJha[1]that,inmarketswithfixedproportionaltransactioncostsandwithassetsevolvingona∗TheauthorswouldliketothankY.Kabanovforfruitfuldiscussionsonthesubject.1finitedimensionaltree,thisformulationdoesnotholdanymore.Intheirsetting,theyshowthatacorrectdualformulationcanbeobtainedifwereplacestoppingtimesbyrandomizedstoppingtimes,whichamountstoworkwithwhattheycall“approximatemartingalenode-measures”.Inthispaper,weprovideanewdualformulationforthepriceofAmericanoptionwhichworksinthegeneralframeworkofC-valuedprocessesasintroducedinKabanovetal.[5],andextendsthedualformulationofChalasaniandJha[1].Therestofthepaperisorganizedasfollows.InSection2,wedescribethemodelandgivethedualformulation.ThelinkbetweenourformulationandtheoneobtainedbyChalasaniandJha[1]isexplainedinSection3.Section4isdevotedtocounter-examples.TheproofofthedualformulationisprovidedinSection5.2Modelandmainresult2.1ProblemformulationSetT={0,...,T}forsomeT∈N\{0}andlet(Ω,F,P)beacompleteprobabilityspaceendowedwithafiltrationF=(Ft)t∈T.Inallthispaper,inequalitiesinvolvingrandomvariableshavetobeunderstoodintheP−a.s.sens.WeassumethatFT=FandthatF0istrivial.Givenanintegerd≥1,wedenotebyKthesetofC-valuedprocessesKsuchthatRd+\{0}⊂int(Kt)forallt∈T.1FollowingthemodelizationofKabanovetal.[6],foragivenK∈Kandx∈Rd,wedefinetheprocessVx,ξbyVx,ξt:=x+tXs=0ξs,t∈T,whereξbelongstoA(K)={ξ∈L0(Rd;F)s.t.ξt∈−Ktforallt∈T},and,forarandomsetE⊂RdP−a.s.andG⊂F,L0(E;F)(resp.L0(E;G))isthecollectionofF-adaptedprocesses(resp.G-measurablevariables)withvaluesinE.Thefinancialinterpretationisthefollowing:xistheinitialendowmentinnumberofphysicalunitsofsomefinancialassets,ξtistheamountofphysicalunitsofassetswhichisexchangedattand−Ktisthesetofaffordableexchangesgiventherelativepricesoftheassetsandtheleveloftransactioncosts.Beforetogoon,weillustratethismodelizationthroughanexample(seealsoSection3,KabanovandStricker[4]andKabanovetal.[6]).1Here,wefollowKabanovetal.[6]andsaythatasequenceofset-valuedmappings(Kt)t∈TisaC-valuedprocessifthereisacountablesequenceofRd-valuedF-adaptedprocessesXn=(Xnt)t∈Tsuchthat,foreveryt∈T,P−a.s.onlyafinitebutnon-zeronumberofXntisdifferentfromzeroandKt=cone{Xnt,n∈N}.ThismeansthatKtisthepolyhedralconegeneratedbytheP−a.s.finiteset{Xnt,n∈NandXnt6=0}.2Example2.1Letusaconsideracurrencymarketwithdassetswhosepriceprocessismodelledbythe(0,∞)d-valuedF-adaptedprocessS.LetMd+denotethesetofd-dimensionalsquarematriceswithnon-negativeentries.LetλbeaMd+-valuedF-adaptedprocessandconsidertheC-valuedprocess(Kt)t∈TdefinedbyKt(ω)=x∈Rd:∃a∈Md+s.t.xi+Xj6=i≤daji−aijπijt(ω)≥0∀i≤d,whereπijt:=(Sjt/Sit)(1+λijt)foralli,j≤dandt∈T.Intheaboveformulation,aijstandsforthenumberofunitsofassetjobtainedbysellingaijπijtunitsofassetsi.λijtisthecoefficientofproportionaltransactioncostspaidinunitsofassetiforatransferfromassetitoassetj.Ifξt∈−Kt,thenwecanfindsomefinancialtransfersηt=(ηijt)i,j≤d∈L0(Md+;Ft)suchthatξit≤Xj6=i≤dηjit−ηijtSjtSit(1+λijt),i≤d,i.e.theglobalchangeintheportfoliopositionistheresultofsingleexchanges,ηijt,betweenthedifferentfinancialassets,afterpossiblythrowingawaysomeunitsoftheseassets.TherandomsetKtdenotestheso-calledsolvencyregion,i.e.Vt∈Ktmeansthat,uptoanimmediatetransferξt,Vtcanbetransformedintoaportfoliowithnoshort-position˜Vt=Vt+ξt∈Rd+.Observethatwecanassume,withoutlossofgenerality,that(1+λikt)(1+λkjt)≥(1+λijt)i,j,k≤d,t∈T.Indeed,ifthisconditionisnotsatisfiedthenany“optimal”strategywouldinduceaneffectivetransactioncostequalto˜