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1、CHAPTER11multi-assetoptionsInthisChapter...?howtomodelthebehaviorofmanyassetssimultaneously?estimatingcorrelationbetweenassetpricemovements?howtovalueandhedgeoptionsonmanyunderlyingassetsintheBlack–Scholesframework?thepricingformulaforEuropeannon-path-dependentoptionson
2、dividend-payingassets?howtopriceandhedgequantosandtheroleofcorrelation11.1INTRODUCTIONInthischapterIintroducetheideaofhigherdimensionalitybydescribingtheBlack–Scholestheoryforoptionsonmorethanoneunderlyingasset.Thistheoryisperfectlystraightforward;theonlynewideaisthatof
3、correlatedrandomwalksandthecorrespondingmultifactorversionofIto’slemma.?Althoughthemodelingandmathematicsiseasy,the?nalstepofthepricingandhedging,the‘solution,’canbeextremelyhardindeed.Iexplainwhatmakesaproblemeasy,andwhatmakesithard,fromthenumericalanalysispointofview.
4、11.2MULTI-DIMENSIONALLOGNORMALRANDOMWALKSThebasicbuildingblockforoptionpricingwithoneunderlyingisthelognormalrandomwalkdS=μSdt+σSdX.Thisisreadilyextendedtoaworldcontainingmany,sayd,assetsviamodelsforeachunderlyingdSi=μiSidt+σiSidXi.184PartOnemathematicaland?nancialfound
5、ationsHereSiisthepriceoftheithasset,i=1,...,d,andμiandσiarethedriftandvolatilityofthatassetrespectivelyanddXiistheincrementofaWienerprocess.WecanstillcontinuetothinkofdXiasarandomnumberdrawnfromaNormaldistributionwithmeanzeroandstandarddeviationdt1/2sothatE[dX]=0andE[dX
6、2]=dtiibuttherandomnumbersdXianddXjarecorrelated:E[dXidXj]=ρijdt.Hereρijisthecorrelationcoef?cientbetweentheithandjthrandomwalks.Thesymmetricmatrixwithρijastheentryintheithrowandjthcolumniscalledthecorrelationmatrix.Forexample,ifwehavesevenunderlyingsd=7thecorrelationma
7、trixwilllooklikethis:??1ρ12ρ13ρ14ρ15ρ16ρ17?ρ211ρ23ρ24ρ25ρ26ρ27????ρ31ρ321ρ34ρ35ρ36ρ37???=?ρ41ρ42ρ431ρ45ρ46ρ47????ρ51ρ52ρ53ρ541ρ56ρ57???ρ61ρ62ρ63ρ64ρ651ρ67ρ71ρ72ρ73ρ74ρ75ρ761Notethatρii=1andρij=ρji.Thecorrelationmatrixispositivede?nite,sothatyTy≥0.Thecovariancematrixis
8、simplyMM,whereMisthematrixwiththeσialongthediagonalandzeroseverywhereelse.Tobeabletomanipulatefunctionsofmany
