现代金融中数学问题.ppt

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1、Chapter3:AmericanoptionsandearlyexerciseTwoquestions:(1)optionprice(2)what’stheoptimalexercisepolicyProbabilisticapproachBTMForEuropeanvanillaoptionsForAmericanvanillaoptionsContinuous-timepricingmodelforAmericanputAmericancallWhat’sthesolution?ContinuousdividendpaymentsAmericanputAmericancall

2、AfreeboundaryproblemPerpetualAmericanoptionsCanbeexercisedatanytime,noexpirycontinuedOptimalexerciseboundary(asafunctionoftimetoexpiry(tau))Put-CallparityTheparitydoesn’tholdforAmericanoptionsPut-callsymmetryThesymmetryholdsalsoforAmericanoptionsContinuedCalloptionC(S1,t;X1,r,q)max(S1-X1,0)toe

3、xchangeoneassetXwithdividendyieldrforanotherassetS1withdividendyieldqPutoptionP(S2,t;X2,r2,q2)max(X2-S2,0)toexchangeoneassetS2withdividendyieldq2foranotherassetX2withdividendyieldr2Ifweassumer2=q,q2=r,S2=X,X2=S(otherparameters,suchasmaturityandvolatility,aresame),thenC(S,t;X,r,q)=P(S2,t;X2,r2,

4、q2)=P(X,t;S,q,r)Earlyexercisedoesn’taffectthisresultContinuedBermudanoptionsEarlyexerciseisallowedonlyatcertaindates.BTMModel-independentandmodel-dependentModel-dependentBlack-ScholesequationsBlack-ScholespricesPut-callsymmetryBasedontheassumptionofgeometricBrownianmotionofunderlyingassetaswel

5、lastheno-arbitrageprincipleAllmodel-independentresultsmustbevalidintheBlack-ScholesframeworkModel-independentPriceofforwardcontractput-callparity,…AmericancallshouldneverbeexercisedearlyifnodividendpaymentsBasedontheno-arbitrageprincipleonly