Utility-based hedging and pricing with a nontraded

NonlinearAnalysis71(2009)e1952e1969ContentslistsavailableatScienceDirectNonlinearAnalysisjournalhomepage:,AnnaGerardibaDipartimentodiScienze,Facolta'diEconomia,Universita'diChieti-Pescara,I-65127-Pescara,ItalybDipartimentodiIngegneriaElettrica,Facolta'diIngegneria,Universita'dell'Aquila,I-67100-L'Aquila,ItalyarticleinfoMSC:91B7060J7591B2491B1693E20Keywords:UtilitymaximizationMinimalentropymeasureJumpdiffusionsabstractOptimalstrategiesforhedgingaclaimonanontradedassetXareanalyzed.TheclaimisvaluedandhedgedinanexponentialutilitymaximizationframeusingacorrelatedtradedassetS.Thetradedassetisdescribedasageometricjumpprocessandthenontradedassetasajump-diffusionprocesshavingcommonjumptimeswithS.TheclassicaldynamicprogrammingapproachleadstocharacterizingthevaluefunctionasasolutiontotheHamiltonJacobiBellmanequation.Closedformformulasforthevaluefunction,intermsofanewprobabilitymeasureQequivalenttotherealworldprobabilitymeasureP,andfortheoptimalinvestmentstrategyaregiven.Admissibilityfortheoptimalstrategyisdiscussedand,viaadualityresult,anexplicitexpressionforthedensityoftheminimalentropymartingalemeasure(MEMM)isprovided.Closedformformulasarealsogivenforthewriter'sindifferencepriceintermsofQandtheMEMM.'2009ElsevierLtd.Allrightsreserved.1.IntroductionAconsiderablepartofthevastdevelopmentinMathematicalFinanceoverthepasttwodecadeswasdeterminedbytheapplicationsofstochasticmodels(i.e.theoryofstochasticprocesses,optimalstochasticcontrol,stochasticdifferentialequa-tionsandconvexanalysis).Aclassicalproblemineconomictheoryistheproblemofaninvestorwhomaximizeshisexpectedutilityofterminalwealthinacontinuoustimestochasticsecuritymarket.Thefundamentalstochasticmodelofoptimalin-vestmentwasfirstintroducedbyMerton[1,2]andlaterstudiedbyseveralauthors(seeforexample[38]andthereferencestherein).Mostoftheliteratureonthissubjectisbasedontheassumptionthatunderlyingassetspricesfollowadiffusion-typeprocess,thecontributionofthispaperistoconsideramarketwhereassetspricesmayexhibitajumpingbehaviour.Theimportanceoftheutilitymaximizationproblemisalsorevealedinthetheoryofderivativespricinginincompletemarkets,wheretheagent'spreferenceshavetobegivenconsideration,sinceriskcannotbecompletelyhedged.Aderivativeorcontingentclaimisafinancialinstrumentwhosevaluedependsontheevolutionofanunderlyingprocess,suchasacommodity,astockindex,theinterestrateor,inthecaseofweatherclaims,temperatures,rainfall,snowfall.Thewell-knownBlackandScholesoptionpricingtheoryisheavilybasedonthecompletenessofthemarket,whereeverycontingentclaimcanbereplicatedbyaself-financingdynamicportfoliostrategy.Inthiscase,onecanreducetozerotheriskofthederivativebyasuitablestrategy.Inincompletemarkets,suchas,forexample,marketswithtransactioncosts,marketsmodeledbymarkedpointprocesseswithinfinitenumberofmarksorinthepresenceofanontradeableasset,thisisnotpossibleandonehastochoosesomeapproachtoevaluatethederivatives.Duringthepastyears,derivativeswithnontradedassetshavebeenattractinganeverincreasinginterestandnounifiedmethodhasbeenconsideredfortheirevaluation.Asinothercasesofderivativespricinginincompletemarketsarathersuccessfulmethodhasbeenproventobetheso-calledutilityapproach.Correspondingauthor.Tel.:[phone].E-mailaddress:[e-mail](C.Ceci).0362-546X/$seefrontmatter'2009ElsevierLtd.Allrightsreserved.doi:10.1016/j.na.2009.02.105C.Ceci,A.Gerardi/NonlinearAnalysis71(2009)e1952e1969e1953Inthispaper,wechoosethisapproachforevaluatingandhedgingclaimswrittenonnontradedassetsinamarketwhereassetspricesmayexhibitajumpingbehaviour.Anagentexpectstoreceiveorpayoutaclaimonanontradedassetandtradesonacorrelatedassettomanagehisrisk.Theseproblemsoccurofteninpractice.Examplesincludeoptionsonbasketsofstockswherethebasketisilliquid,derivativesonassetsonwhichtradingisnotpermitted(executivestockoptions)orweatherderivatives(see[9]andthereferencestherein).Amongtheminsurancederivativesdependingonaexternalphysicalriskprocess,suchasatemperatureinalowdimensionalclimatemodel[10].Withtheadventofintradayinformationonfinancialsecuritypricequotes,aresearchinfinancehasbeendevotedtomodelswithjumps[1115,3335].Onaverysmalltimescaleactualpricesdonotreallychangecontinuouslyovertime,butratheratdiscreterandompointsintimeinreactiontotradeortosignificantnewinformations.Inthispaperweconsiderapriceevolutionmodelthatpossesthesefeatures,namelyageometricmarkedpointprocessSandaEuropeanoptionwrittenonanontradedassetXcorrelatedwithS.Thehedgingandpricingproblemsareanalyzedfromtheviewpointoftheoptionseller.Afterreceivingthepremium,hehastohedgetheriskofthederivativeinvestinginabondandinatradedriskyassetScorrelatedwithX.Wesupposethatheisriskaversewithconstantabsoluteriskaversion(exponentialutility).In[9,7]thisproblemhasbeenstudiedwhentradedandnontradedassetarecorrelateddiffusionprocesses.Herein,asdescribedinSection2,thetradedriskyassetSismodeledasageometricmarkedpointprocessandthenontradedassetXasajumpdiffusion.TheprocessesSandXaresup