further finite-difference methods for one-factor models

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1、CHAPTER78furtherfinite-differencemethodsforone-factormodelsInthisChapter...•implicitfinite-differencemethodsincludingCrank–Nicolson•Douglasschemes•Richardsonextrapolation•American-styleexercise•exoticoptions78.1INTRODUCTIONWecontinuewithone-factornumericalmethods,discussingthe

2、moredifficulttoprogramimplicitmethods.Theextracomplexityofthemethodsisoutweighed,though,bytheirsuperiorstabilityproperties.Ialsoshowhowtoextendthefinite-differencemethodtocopewithearlyexerciseandpath-dependentcontracts.78.2IMPLICITFINITE-DIFFERENCEMETHODSThefullyimplicitmethod

3、usesthepointsasshowninFigure78.1tocalculatetheoptionvalue.Theschemeissuperficiallyjustliketheexplicitmethodusingfinite-differenceestimatesoftheoptionvalue,itsdeltaandgammabutnowatthetimestepk+1.TherelationshipbetweentheoptionvaluesonthemeshissimplyVk−Vk+1iiδt
Vk+1−2Vk+1+Vk+1+

4、ak+1i+1ii−1iδS21228PartSixnumericalmethodsandprogramsSCalculateoptionfromtheoptionvaluesatthesevalueatthispointpointstFigure78.1Therelationshipbetweenoptionvaluesinthefullyimplicitmethod.
Vk+1−Vk+1+bk+1i+1i−1i2δS+ck+1Vk+1=0.ii(Itdoesn’tmattermuchwhetherthecoefficientsa,bandc

5、areevaluatedatthetimestepk+1ork.)ThemethodisstillaccuratetoO(δt,δS2).ThiscanbewrittenasAk+1Vk+1+(1+Bk+1)Vk+1+Ck+1Vk+1=Vk(78.1)ii−1iiii+1iwhereAk+1=−νak+1−1νbk+1,i1i22iBk+1=2νak+1−δtck+1i1iiandCk+1=−νak+1+1νbk+1i1i22iwhereδtδtν1=andν2=.δS2δSAgain,Equation(78.1)doesnotholdfori

6、=0ori=I,theboundaryconditionssupplythetworemainingequations.furtherfinite-differencemethodsforone-factormodelsChapter781229Thereisaworldofdifferencebetweenthisschemeandtheexplicitfinite-differencescheme.Thetwomaindifferencesconcernthestabilityofthemethodandthesolutionprocedure

7、.Themethodnolongersuffersfromtherestrictiononthetimestep.Theassetstepcanbesmallandthetimesteplargewithoutthemethodrunningintostabilityproblems.Thesolutionofthedifferenceequationisnolongersostraightforward.TogetVk+1fromiVkmeanssolvingasetoflinearequations;eachVk+1isdirectlyli

8、nkedtoitstwoneighborsiiandthusindirectlylinkedtoeveryoptionvalueatthesameti