Thebinomialmodelprovidesinsightintothedeterminantsof
optionvalue.Thevalueofanoptionisnotdeterminedbythe
expectedpriceoftheassetbutbyitscurrentprice,which,of
course,reflectsexpectationsaboutthefuture.Thisisadirect
consequenceofarbitrage.Iftheoptionvaluedeviatesfrom
thevalueofthereplicatingportfolio,investorscancreatean
arbitrage position (i.e., one that requires no investment,
involvesnorisk,anddeliverspositivereturns).Toillustrate,
iftheportfoliothatreplicatesthecallcostsmorethanthecall
doesin themarket, aninvestorcouldbuythecall,sellthe
replicatingportfolio,andguaranteethedifferenceasaprofit.
Thecashflowsonthetwopositionswilloffseteachother,
leadingto nocashflowsin subsequentperiods.Theoption
valuealsoincreasesasthetimetoexpirationisextended,as
thepricemovements(uandd)increase,andwithincreasesin
the interest rate.
Whilethebinomialmodelprovidesanintuitivefeelforthe
determinantsofoptionvalue,itrequiresa largenumberof
inputs,intermsofexpectedfuturepricesateachnode.Aswe
make time periods shorter in thebinomial model, we can
make one of two assumptions aboutasset prices.We can
assume that price changes become smaller as periods get
shorter;thisleadstopricechangesbecominginfinitesimally
smallastimeperiodsapproachzero,leadingtoacontinuous
price process. Alternatively, we can assume that price
changesstaylargeevenastheperiodgetsshorter;thisleads
toajumppriceprocess,wherepricescanjumpinanyperiod.
Inthis section, weconsidertheoptionpricingmodels that
emerge with each of these assumptions.
Black-Scholes Model