Hedging of the european option in discete time under proportional transaction costs

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1、MathMethOperRes(2004)59:315–328DOI10.1007/s001860300323HedgingoftheEuropeanoptionindiscretetimeunderproportionaltransactioncostsMarekKocin´skiKatedraEkonometriiiInformatyki,WydziałEkonomiczno-Rolniczy,SzkołaGło´wnaGospodarstwaWiejskiegoul.Nowoursynowska166,02-787Warszawa,Poland(

2、e-mail:kocinski@delta.sggw.waw.pl)Manuscriptreceived:April2003/Finalversionreceived:September2003Abstract.InthepaperhedgingoftheEuropeanoptioninadiscretetimefinancialmarketwithproportionaltransactioncostsisstudied.Itisshownthatforacertainclassofoptionsthesetofportfolioswhichallow

3、tohedgeanoptioninadiscretetimemodelwithaboundedsetofpossiblechangesinastockpriceisthesameasthesetofsuchportfolios,underassumptionthatthestockpriceevolutionisgivenbyasuitableCRRmodel.Keywords:Europeanoption,Self-financingstrategy,Hedging,Transactioncosts1991MathematicsSubjectClass

4、ification:Primary90A12;Secondary93E201IntroductionInthepaperweconsidertheproblemofhedgingoftheEuropeanoptioninadiscretetimemarketmodel.Althoughtheproblemofhedgingofcontingentclaimsindiscretetimeunderproportionaltransactioncostswasstudiedinmanypapers(see[1]–[12])itappearstobenontr

5、ivialtoapplytotherealmarkettheresultswhichwereobtainedforageneralmodel(see[3],[4],[11]).FromcalculationpointofviewasocalledCox-Ross-Rubinstein(CRR)modelisveryconvenientsinceinthisparticularmodelitiseasytogettheexactvalueofthepriceofanoptionaswellasthesetofportfolioswhichallowtoh

6、edgetheoption(seeforinstance[1],[2],[9],[10]).OntheotherhandtheCox-Ross-Rubinsteinmodelseemstobetoosimpletobeaproperdescriptionoftherealstockpriceevolution.InthispaperhoweveritisshownthatforaspecialclassofoptionswhichincludespopularcalloptionthesetofportfolioswhichallowtohedgeaR

7、esearchsupportedbygrantPBZKBNno.016/P03/99316M.Kocin´skicontingentclaiminaquitegeneralmodelofthestockpriceprocessisthesameasintheCox-Ross-Rubinsteinmodel.TheresultthereforeseemstobeinterestingforpractitionerssinceitjustifiestheuseoftheCRRmodelap-proachtopricederivativesforacertai

8、nclassofoptions.Thepossibilityofreducingofthemo