paul wilmott on quantitative finance multi-asset options

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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.ˆAlthoughthemodelingandmathematicsiseasy,thefinalstepofthepricingandhedging,the‘solution,’canbeextremelyhardindeed.Iexplainwhatmakesaproblemeasy,andwhatmakesithard,fromthenumericalanalysispointofview.

4、11.2MULTI-DIMENSIONALLOGNORMALRANDOMWALKSThebasicbuildingblockforoptionpricingwithoneunderlyingisthelognormalrandomwalkdS=µSdt+σSdX.Thisisreadilyextendedtoaworldcontainingmany,sayd,assetsviamodelsforeachunderlyingdSi=µiSidt+σiSidXi.184PartOnemathematicalandfinancialfound

5、ationsHereSiisthepriceoftheithasset,i=1,...,d,andµiandσiarethedriftandvolatilityofthatassetrespectivelyanddXiistheincrementofaWienerprocess.WecanstillcontinuetothinkofdXiasarandomnumberdrawnfromaNormaldistributionwithmeanzeroandstandarddeviationdt1/2sothatE[dX]=0andE[dX

6、2]=dtiibuttherandomnumbersdXianddXjarecorrelated:E[dXidXj]=ρijdt.Hereρijisthecorrelationcoefficientbetweentheithandjthrandomwalks.Thesymmetricmatrixwithρijastheentryintheithrowandjthcolumniscalledthecorrelationmatrix.Forexample,ifwehavesevenunderlyingsd=7thecorrelationma

7、trixwilllooklikethis:1ρ12ρ13ρ14ρ15ρ16ρ17ρ211ρ23ρ24ρ25ρ26ρ27ρ31ρ321ρ34ρ35ρ36ρ37
=ρ41ρ42ρ431ρ45ρ46ρ47ρ51ρ52ρ53ρ541ρ56ρ57ρ61ρ62ρ63ρ64ρ651ρ67ρ71ρ72ρ73ρ74ρ75ρ761Notethatρii=1andρij=ρji.Thecorrelationmatrixispositivedefinite,sothatyT
y≥0.Thecovariancematrixis

8、simplyM
M,whereMisthematrixwiththeσialongthediagonalandzeroseverywhereelse.Tobeabletomanipulatefunctionsofmany