When the price process is continuous (i.e., price changes
become smaller as time periods get shorter), the binomial
model for pricing options converges on theBlack-Scholes
model.Themodel,namedafteritsco-creators,FischerBlack
andMyronScholes,allows usto estimatethevalueofany
optionusingasmallnumberofinputsandhasbeenshownto
be remarkably robust in valuing many listed options.
The Model
WhilethederivationoftheBlack-Scholesmodel isfartoo
complicatedtopresenthere,itisalsobased ontheideaof
creatingaportfoliooftheunderlyingassetand theriskless
assetwiththesamecashflowsandhencethesamecostasthe
option being valued. The value of a call option in the
Black-Scholesmodelcanbewrittenasafunctionofthefive
variables:
S= Current value of underlying asset
K= Strike price of option
t= Life to expiration of option
r= Riskless interest rate corresponding to life of option
σ 2 = Variance in ln(value) of underlying asset
The value of a call is then: