While early exercise is not optimal generally, there are at least
two exceptions to this rule. One is a case where the
underlyingassetpayslargedividends,thusreducingthevalue
oftheassetandanycalloptionsonthatasset.Inthiscase,call
optionsmaybeexercisedjustbeforeanex-dividenddateif
thetime premium ontheoptions isless thantheexpected
decline in asset value as a consequence of the dividend
payment.Theotherexceptionariseswhenaninvestorholds
boththeunderlyingassetanddeepin-the-moneyputsonthat
assetatatimewheninterestratesarehigh.Inthiscase,the
timepremiumontheputmaybelessthanthepotentialgain
from exercising the put early and earning interest on the
exercise price.
Option Pricing Models
Optionpricingtheoryhasmadevaststridessince1973,when
Fischer Black and Myron Scholes published their
pathbreaking paper
19 providing a model for valuing dividend-protected
European options. Black and Scholes used a “replicating
portfolio”—aportfoliocomposedoftheunderlyingassetand
therisk-freeassetthathadthesamecashflowsastheoption
beingvalued—tocomeupwiththeirfinalformulation.While
their derivation is mathematically complicated, there is a
simplerbinomialmodelforvaluingoptionsthatdrawsonthe
same logic.
Binomial Model
The binomial option pricing model is based on a simple
formulationfortheassetpriceprocessinwhichtheasset,in
anytimeperiod,canmovetooneoftwopossibleprices.The