finite-difference methods for one-factor models

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1、CHAPTER77finite-differencemethodsforone-factormodelsInthisChapter...•finite-differencegrids•howtoapproximatederivativesofafunction•howtogofromtheBlack–Scholespartialdifferentialequationtoadifferenceequation•theexplicitfinite-differencemethod,ageneralizationofthebinomialmethod77.

2、1INTRODUCTIONRarelycanwefindclosed-formsolutionsforthevaluesofoptions.Unlesstheproblemisverysimpleindeedwearegoingtohavetosolveaproblemnumerically.InanearlierchapterIdescribedthebinomialmethodforpricingoptions.Thisusedtheideaofafinitetreestructurebranchingoutfromthecurrentasset

3、priceandthecurrenttimerightuptotheexpirydate.Onewayofthinkingofthebinomialmethodisasamethodforsolvingapartialdifferentialequation.Finite-differencemethodsarenomorethanageneralizationofthisconcept,althoughwetendtotalkaboutgridsandmeshesratherthan‘trees.’Oncewehavefoundanequati

4、ontosolvenumericallythenitismucheasiertouseafinite-differencegridthanabinomialtree,simplybecausethetransformationfromadifferentialequation(Black–Scholes)toadifferenceequationiseasierwhenthegrid/mesh/treeisniceandregular.Moreover,therearemany,manywaysthefinite-differencemethodca

5、nbeimprovedupon,makingitfasterandmoreaccurate.Thebinomialmethodisnotsoflexible.Andfinally,thereisagreatdealinthemathematical/numericalanalysisliteratureontheseandothermethods;itwouldbesuchashametoignorethis.Themaindifferencebetweenthebinomialmethodandthefinite-differencemethodsi

6、sthattheformercontainsthediffusion,thevolatility,inthetreestructure.Inthefinite-differencemethodsthe‘tree’isfixedbutparameterschangetoreflectachangingdiffusion.Tothoseofyouwhoarenewtonumericalmethodsletmestartbysayingthatyouwillfindtheparabolicpartialdifferentialequationveryeasyt

7、osolvenumerically.Ifyouarenotnewtotheseideas,buthavebeenbroughtuponbinomialmethods,nowisthetimetowean1200PartSixnumericalmethodsandprogramsyourselfoffthem.Onapersonalnote,forsolvingproblemsinpracticeIwouldsaythatIusefinite-differencemethodsabout75%ofthetime,MonteCarlosimulatio

8、ns20%,andtherestwouldbeexplicitformulae.Thoseexplicitformulaearealmo