Growth-optimal portfolios under transaction costs

arXiv:0707.3198v1[math.PR]21Jul2007Growth-optimalportfoliosundertransactioncostsJanPalczewski∗ŁukaszStettner†February1,2008AbstractThispaperstudiesaportfoliooptimizationprobleminadiscrete-timeMarkovianmodelofafinancialmarket,inwhichassetpricedynamicsde-pendonanexternalprocessofeconomicfactors.Therearetransactioncostswithastructurethatcovers,inparticular,thecaseoffixedplusproportionalcosts.Weprovethatthereexistsaself-financingtradingstrategymaximizingtheaveragegrowthrateoftheportfoliowealth.WeshowthatthisstrategyhasaMarkovianform.Ourresultisobtainedbylargedeviationsestimatesonempiricalmeasuresofthepriceprocessandbyageneralizationofthevanishingdiscountmethodtodiscontinuoustransitionoperators.Keywords:portfoliooptimization,transactioncosts,growthrate,logarithmicutility,Markovprocess,impulsivestrategy,vanishingdiscount1.IntroductionResearchersandpractitionershavelongbeenawarethatMarkovianmodelsofas-setpricedynamics,suchastheCox-Ross-RubinsteinmodelortheBlack-Scholesmodel,havesignificantdeficienciesrelatedtonon-stationarityofthefinancialmar-ket.Theyobservedthatthevolatilityandtheexpectedrateofreturnofassetpricesarenotconstantbutdependonaneconomicsituation,whichmaychangeoverlongertimespans.Asaremedy,theyintroducedadditionalprocessesmodelingvitalmarketvariables,suchasmarkettrendorpricevolatility.However,auni-fiedframeworkhasonlyrecenlybeenintroducedandhasattractedalotofinterest(seeeg.[5],[6],[12],[26],[25],[31]).Existingliteratureconcentratesmainlyoncontinuous-timediffusionmodels.Bieleckietal.[6]solveanassetmanagementproblemwhereeconomicfactors,asthoseadditionalmarketvariablesarecalled,formadiffusionthatisindependentoftheBrownianmotiongoverningtheprice∗SchoolofMathematics,UniversityofLeeds,LeedsLS29JT,UKandFacultyofMathematics,UniversityofWarsaw,Banacha2,02-097Warszawa,Poland(e-mail:[e-mail])†InstituteofMathematics,PolishAcademyofSciences,Sniadeckich8,00-950Warszawa,Poland,(e-mail:[e-mail]).1processandtheyaffectonlythedriftofthepriceprocess.FlemingandSheu[12]allowbothprocessestohavedependentBrownianmotionsbuttheirdiffusionsareofaspecialform.PalczewskiandStettner[26],though,assumeonlythatassetpricesandeconomicfactorsfollowonegeneralcontinuous-timeMarkovprocessandproveresultsconcerningoptimalportfolioselectionforinfinitetimedisountedperformancefunctionalundertransactioncosts.Inthepresentpaperwestudyaportfoliomanagementprobleminwhichper-formanceismeasuredbyanaveragegrowthrateoftheportfoliowealth.Weworkwithinadiscretetimeframeworkwhichallowsustoovercomelimitationsandtech-nicalitiesoftheexistingtheoryofcontinoustimeMarkovprocessesandimpulsivecontrol.Themarketconsistsofdassets,whosepricesare,ingeneral,interdepen-dent.Theirdynamicsareaffectedbyaprocessofeconomicfactors,whichisaMarkovprocessonaPolishspace(fordetailsseeSection2).Weassumethatas-setscannotgobankrupt(theirpricesarepositive).Weimposecostsofperformingtransactions.Thesecosts,inthesimplest,consistofafixedpart,independentofthetransaction,andaproportionalpart,dependingonthevolumeandthetypeofassetssoldorpurchased(see(4),(5)andthefollowingdiscussion).Thistypeoftransactioncostspreventscontinuoustradingincontinuous-timemodels(seee.g.[26])andemulatesexistingmarketmechanisms.Theframeworkofthispapercov-ersmoregeneraltransactioncostsstructuresaswell(seeSection6).PerformanceofaportfolioΠismeasuredbythefuntionalJ(Π)=liminfT→∞1TElnXΠ(T),(1)whereXΠ(T)isthewealthoftheportfolioΠattimeT.Thisfunctionalcom-putesanaveragegrowthrateoftheportfolioΠascanbeseenfromthefollowingreformulationoftheaboveformula:J(Π)=liminfT→∞1TET−1Xk=0lnXΠ(k+1)XΠ(k).(2)Theaimofthispaperistofindaportfoliothatmaximizesthevalueof(1).Thisisaninfinite-timecounterpartofthelogarithmicutilitymaximization,whichiswidelyusedintheeconomicandfinancialcommunity,whereoptimalportfoliosareref-eredtoaslog-optimalorgrowth-optimal.Forabroadertreatmentseetextbooks[10],[24].InmathematicalcontexttheresearchgoesbacktoKelly(see[21],[32])andhascontinuedindiscretetime([3])andcontinuoustime([1],[2])uptotoday([13],[17],[27]).Functional(1)canalsobeseenasarisksensitivefunctionalandtheliteratureisherebroadaswell([6],[22],[31]).Itshouldbestressedthatthemajorityofpapersconsiderscontinuoustimediffusionmodels,whereanoptimalstrategyisobtainedasasolutiontoanappropriateHJBequation,usuallyreformu-latedinavariationalform.Consequently,theresultsarebasedonasophisticatedtheoryofPDE’sandsolutionsusuallydonotusedirectlyprobabilisticpropertiesofthephenomenaunderstudy.Moreover,duetocomplexityofthestudiedPDEstheresultsareoftenofexistentialform.2Inthispaper,weapproachtheoptimizationproblem(1)fromaprobabilisticpointofview.Weprovethatthereexistsaself-financingportfoliostrategymax-imizingthegrowth-rate(1).WeshowthatthistradingstrategyhasaMarkovianform,i.e.aninvestmentdecisionattimetisbasedonlyonthestateofassetpricesandeconomicfactorsatt.Mainadditionstotheexistingtheoryaretransactioncostswithafixedtermandageneralformofdependenceofassetpricesoneco-nomicfactors.Asfarasweknowthereisnopaperthattreatsthistypeofproblemsinsuchgenerality.Ourstudydepends