19 F. Black and M. Scholes, “The Pricing of Options and
Corporate Liabilities,” Journal of Political Economy 81
(1973): 637–654.
20 Toillustrate,ifσ 2 isthevarianceinln(stockprices),theup
and downmovements in thebinomialcanbe estimated as
follows:
whereuanddaretheupanddownmovementsperunitof
timeforthebinomial,Tisthelifeoftheoption,andmisthe
number of periods within that lifetime.
21 Warrantsarecalloptionsissuedbyfirms,eitheraspartof
management compensationcontracts orto raiseequity.We
discussed them in Chapter 11.