asset allocation under threat of a crash

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1、CHAPTER67assetallocationunderthreatofacrashInthisChapter...•howtoallocateyourmoneytoaninvestmentwhenyouareworriedaboutamarketcrash67.1INTRODUCTIONThefollowingbyRalfKorn1andmyselfwasoriginallypublishedintheInternationalJournalofTheoreticalandAppliedFinance5171–188(2002).Weconsiderthedeterminationo

2、foptimalportfoliosunderthethreatofacrash.Ourmainassumptionisthatupperboundsforboththecrashsizeandthenumberofcrashesoccurringbeforethetimehorizonaregiven.Wemakenoprobabilisticassumptiononthecrashsizeorthecrashtimedistribution.Theoptimalstrategiesinthepresenceofacrashpossibilityarecharacterizedbyab

3、alanceproblembetweeninsuranceagainstthecrashandgoodperformanceinthecrash-freesituation.Explicitsolutionsforthelog-utilitycasearegiven.Ourmainfindingisthatconstantportfoliosarenolongeroptimalones.Itiswellknownthattheclassicallognormalstock/Black–Scholesmodelisnotabletoexplainlargejumpsinstockprices

4、appearinginreal-worldsecuritymarkets.Inparticular,itdoesnotcontainthepossibilityofacrashofthestockprices.Althoughthesecrashesarerareevents,theydooccurinreallife;theOctober1987crashbeingthelargestinrecentmemory.Therefore,thereisalongtraditionofmodelingjumpsinstockprices.Aseeminglyobviouscandidateo

5、faclassofsuitablestochasticprocessesisthatofjump-diffusionprocesses(seeMerton(1976)foranearlyreference).However,jump-diffusionapproachesonlyleadtostrategiesthathedgeacrashsituationinthemeanwhichisnorealprotectionagainsttheconsequencesofajumpatall.Inparticular,aninvestorfollowingsuchastrategywills

6、ufferlargelossesduringacrash.Asacontrast,byimplementingthestrategythatwewillproposeaninvestorneednotbeinfearofacrashasourstrategygiveshimfullprotectioninasensethatwillbemademorepreciseinthefollowingsections.AnalternativeformodelingcrashesisgiveninHuaandWilmott(1997).There,theunderlyingassumptions

7、arethatboththemaximalnumberofcrashesinagiventimeintervalandthebiggest1RalfKorn,FachbereichMathematik,UniversitatKaiserslautern,Germany¨korn@mathematik.uni-kl.de1062PartFiveadvancedtopicspossiblesizeofthecrashesareknown