a monte carlo approach to value exchange options using a single stochastic factor

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1、AMonteCarloapproachtovalueexchangeoptionsusingasinglestochasticfactorGiovanniVillaniAbstract.ThisarticledescribesanimportantsamplingregardingmodificationoftheMonteCarlomethodinordertominimisethevarianceofsimulations.Inaparticularway,weproposeageneralisationoftheantitheticmethodandanewa

2、-samplingofstratifiedprocedurewitha!=1tovalueexchangeoptionsusingasinglestochasticfactor.Asiswellknown,exchange2optionsgivetheholdertherighttoexchangeoneriskyassetVforanotherriskyassetDandtherefore,whenanexchangeoptionisvalued,wegenerallyareexposedtotwosourcesofuncertainity.Thereductio

3、nofthebi-dimensionalityofvaluationproblemtoasinglestochasticfactorimpliesanewstratificationproceduretoimprovetheMonteCarlomethod.Wealsoprovideasetofnumericalexperimentstoverifytheaccuracyderivedbya-sampling.Keywords:exchangeoptions,MonteCarlosimulations,variancereduction1IntroductionSi

4、mulationsarewidelyusedtosolveoptionpricing.Withthearrivalofeverfastercomputerscoupledwiththedevelopmentofnewnumericalmethods,weareabletonumericallysolveanincreasingnumberofimportantsecuritypricingmodels.Evenwhereweappeartohaveanalyticalsolutionsitisoftendesirabletohaveanalternativeimp

5、lementationthatissupposedtogivethesameanswer.SimulationmethodsforassetpricingwereintroducedinfinancebyBoyle[3].Sincethattimesimulationhasbeensuccessfullyappliedtoawiderangeofpricingproblems,particularlytovalueAmericanoptions,aswitnessedbythecontributionsofTilley[10],BarraquandandMartin

6、eau[2],BroadieandGlasserman[4],RaymarandZwecher[9].TheaimofthispaperistoimprovetheMonteCarloprocedureinordertoevaluateexchangeoptionsgeneralizingtheantitheticvariatemethodologyandproposinganewstratificationprocedure.Torealisethisobjective,wepricethemostimportantexchangeoptionsusingasin

7、glestochasticfactorPthatistheratiobetweentheunderlyingassetVandthedeliveryoneD.Forthisreason,weneedaparticularsamplingtoconcentratethesimulationsintherangeinwhichthefunctionPismoresensitive.M.Corazzaetal.(eds.),MathematicalandStatisticalMethodsforActuarialSciencesandFinance©Springer-V

8、erlag