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1、IntroductionandPrefaceAnoptiongivesonetheright,butnottheobligation,tobuyorsellasecurityunderspecifiedterms.Acalloptionisonethatgivestherighttobuy,andaputoptionisonethatgivestherighttosellthesecurity.Bothtypesofoptionswillhaveanexercisepriceandanexercise
2、time.Inaddition,therearetwostandardconditionsunderwhichoptionsoper-ate:Europeanoptionscanbeutilizedonlyattheexercisetime,whereasAmericanoptionscanbeutilizedatanytimeuptoexercisetime.Thus,forinstance,aEuropeancalloptionwithexercisepriceKandexercisetimet
3、givesitsholdertherighttopurchaseattimetoneshareoftheunderlyingsecurityforthepriceK,whereasanAmericancalloptiongivesitsholdertherighttomakethepurchaseatanytimebeforeorattimet.Aprerequisiteforastrongmarketinoptionsisacomputationallyeffi-cientwayofevaluati
4、ng,atleastapproximately,theirworth;thiswasaccomplishedforcalloptions(ofeitherAmericanorEuropeantype)bythefamousBlack–Scholesformula.TheformulaassumesthatpricesoftheunderlyingsecurityfollowageometricBrownianmotion.ThismeansthatifS(y)isthepriceofthesecur
5、ityattimeythen,foranypricehistoryuptotimey,theratioofthepriceataspecifiedfuturetimet+ytothepriceattimeyhasalognormaldistributionwithmeanandvarianceparameterstμandtσ2,respectively.Thatis,S(t+y)logS(y)willbeanormalrandomvariablewithmeantμandvariancetσ2.
6、BlackandScholesshowed,undertheassumptionthatthepricesfollowageo-metricBrownianmotion,thatthereisasinglepriceforacalloptionthatdoesnotallowanidealizedtrader–onewhocaninstantaneouslymaketradeswithoutanytransactioncosts–tofollowastrategythatwillre-sultina
7、sureprofitinallcases.Thatis,therewillbenocertainprofit(i.e.,noarbitrage)ifandonlyifthepriceoftheoptionisasgivenbytheBlack–Scholesformula.Inaddition,thispricedependsonlyonthexiiIntroductionandPrefacevarianceparameterσofthegeometricBrownianmotion(aswellaso
8、ntheprevailinginterestrate,theunderlyingpriceofthesecurity,andtheconditionsoftheoption)andnotontheparameterμ.Becausethepa-rameterσisameasureofthevolatilityofthesecurity,itisoftencalledthevolatilityparameter.Arisk-neutralinvestorisonewho
