Analysis, Geometry, and Modeling in Finance 13

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1、Chapter12PortfolioOptimizationandBellman-Hamilton-JacobiEquationAbstractPricingandHedgingderivativesproductsisessentiallyaproblemofportfoliooptimization.Onceameasureofriskhasbeenchosen,thepricecanbedenedasthemeanvalueoftheprotandloss(P&L)andthebesthedgingstrategyistheoptimal

2、controlwhichminimizestherisk.IntheBlack-Scholesmodel,theonlysourceofriskisthespotprocessandtheoptimalcontrolisthedelta-strategywhichcancelstherisk.However,undertheintroductionofstochasticvolatility,themarketmodelbecomesincomplete.Theresultingriskisniteandthedelta-strategyisno

3、toptimal.Aportfoliooptimizationproblemappearsalsonaturallyifweassumethatthemarketisilliquidandthetradingstrategyaectsthepricemovements.Inthefollowing,wewillfocusontheseoptimalcontrolproblemswhenthemarketisincompleteandthemarketisilliquid.Ourstudyinvolvestheuseofperturbationme

4、thodsfornon-linearPDEs.12.1IntroductionSincethefamouspapersofBlack-Scholesonoptionpricing[65],someprogresshasbeenmadeinordertoextendtheseresultstomorerealisticarbitrage-freemarketmodels.Asareminder,theBlack-Scholestheoryconsistsinfollowinga(hedging)strategytodecreasetheriskofl

5、ossgivenaxedamountofreturn.Thistheoryisbasedonthreeimportanthypotheseswhicharenotsatisedunderrealmarketconditions:Thetraderscanrevisetheirdecisionscontinuouslyintime.Thisrsthypothesisisnotrealisticforobviousreasons.Amajorimprovementwasrecentlyintroducedin[6]intheirtime-dis

6、cretemodel.Theyintroduceanelementarytimeafterwhichatraderisabletorevisehisdecisionsagain.Theoptimalstrategyisxedbytheminimizationoftheriskdenedbythevarianceoftheportfolio.Theresultingriskisnolongerzeroandinthecontinuous-timelimitwheregoestozero,onerecoverstheclassicalresul

7、tofBlack-Scholes:theriskvanishes.339©2009byTaylor&FrancisGroup,LLC340Analysis,Geometry,andModelinginFinanceThespotdynamicsisalog-normalprocess(withaconstantvolatility).Asaconsequence,themarketiscompleteandtheriskcancels.Thissecondhypothesisdoesn'ttruthfullyre ectthemarketasin

8、dicatedbytheexistenceofanimpliedvolatility.Inchapters5and6,we