simple generalizations of the black–scholes world

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1、CHAPTER8simplegeneralizationsoftheBlack–ScholesworldInthisChapter...•complexdividendstructures•jumpconditions•time-dependentvolatility,interestrateanddividendyield8.1INTRODUCTIONThischapterisanintroductiontosomeofthepossiblegeneralizationsofthe‘Black–Scholesworld.’Inparticular,Iwilldiscussthe

2、effectofdividendpaymentsontheunderlyingassetandhowtoincorporatetime-dependentparametersintotheframework.Thesesubjectsleadtosomeinterestingandimportantmathematicalandfinancialconclusions.Thegeneralizationsareverystraightforward.However,later,inPartFive,Idescribeothermodelsofthefinancialworldthat

3、takeusalongwayfromBlack–Scholes.8.2DIVIDENDS,FOREIGNINTERESTANDCOSTOFCARRYInChapter5IshowedhowtoincorporatecertaintypesofdividendstructuresintotheBlack–Scholesoptionpricingframework,andtheninChapter7Igavesomeformulaeforthevaluesofsomecommonvanillacontracts,againwithdividendsontheunderlying.Th

4、edividendstructurethatIdealtwithwastheverysimplestfromamathematicalpointofview.Iassumedthatanamountwaspaidtotheholderoftheassetthatwasproportionaltothevalueoftheassetandthatitwaspaidcontinuously.Inotherwords,theownerofoneassetreceivedadividendofDSdtinatimestepdt.Thisdividendstructureisrealist

5、iciftheunderlyingisanindexonalargenumberofindividualassetseachreceivingalumpsumdividendbutwithallthesedividendsspreadoutthroughtheyear.Itisalsoagoodmodeliftheunderlyingisacurrencyinwhichcasewesimplytakethe‘dividendyield’tobetheforeigninterestrate.Similarly,iftheunderlyingisacommoditywithacost

6、ofcarrythatisproportionaltoitsvalue,140PartOnemathematicalandfinancialfoundationsthenthe‘dividendyield’isjustthecostofcarry(withaminussign,webenefitfromdividendsbutmustpayoutthecostofcarry).Torecap,iftheunderlyingreceivesadividendofDSdtinatimestepdtwhentheassetpriceisSthen∂V∂2V∂V+1σ2S2+(r−D)S−r

7、V=0.∂t2∂S2∂SHowever,iftheunderlyingisastock,thentheassumptionofconstantandcontinuously-paiddividendyieldisnotagoodone.8.3DIVIDENDSTRUCTURESTypically,dividendsarepaidoutquarterlyintheUSandsemi-annuallyorquarterlyintheUK.Thedividendissetbythebo