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1、CHAPTER6partialdifferentialequationsInthisChapter...•propertiesoftheparabolicpartialdifferentialequation•themeaningoftermsintheBlack–Scholesequation•somesolutiontechniques6.1INTRODUCTIONTheanalysisandsolutionofpartialdifferentialequationsisaBIGsubject.Wecanonlyskimthesu
2、rfaceinthisbook.Ifyoudon’tfeelcomfortablewiththesubject,thenthelistofbooksattheendshouldbeofhelp.However,tounderstandfinance,andeventosolvepartialdifferentialequationsnumerically,doesnotrequireanygreatdepthofunderstanding.Theaimofthischapteristogivejustenoughbackgroundto
3、thesubjecttopermitanyreasonablynumeratepersontofollowtherestofthebook;Iwanttokeeptheentryrequirementstothesubjectaslowaspossible.6.2PUTTINGTHEBLACK–SCHOLESEQUATIONINTOHISTORICALPERSPECTIVETheBlack–Scholespartialdifferentialequationisintwodimensions,Sandt.Itisaparabolice
4、quation,meaningthatithasasecondderivativewithrespecttoonevariable,S,andafirstderivativewithrespecttotheother,t.Equationsofthisformaremorecolloquiallyknownasheatordiffusionequations.Theequation,initssimplestform,goesbacktoalmostthebeginningofthe19thcentury.Diffusionequati
5、onshavebeensuccessfullyusedtomodel•diffusionofonematerialwithinanother,suchassmokeparticlesinair•flowofheatfromonepartofanobjecttoanother102PartOnemathematicalandfinancialfoundations•chemicalreactions,suchastheBelousov–Zhabotinskyreactionwhichexhibitsfascinatingwavestruct
6、ure•electricalactivityinthemembranesoflivingorganisms,suchastheHodgkin–Huxleymodel•dispersionofpopulations,suchasindividualsmovingbothrandomlyandtoavoidover-crowding•pursuitandevasioninpredator-preysystems•patternformationinanimalcoats,suchastheformationofzebrastripes•d
7、ispersionofpollutantsinarunningstreamInmostofthesecasestheresultingequationsaremorecomplicatedthantheBlack–Scholesequation.Thesimplestheatequationforthetemperatureinabarisusuallywrittenintheform∂u∂2u=(6.1)∂t∂x2whereuisthetemperature,xisaspatialcoordinateandtistime.Thise
8、quationcomesfromaheatbalance.Considertheflowintoandoutofasmallsectionofthebar.Theflowofheatalongthebarisproporti
