Note that the author is pricing European options, not American options. The standard BSM formula is for European options, not American. If anyone wants to dive deep into a lot of the different models out there, check out The Complete Guide to Option Pricing Formulas by Espen Haug(1).

His site has some wonderful material too(2). I especially like his ‘Exotic Option Fantasy Land’(3)

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(1) https://www.amazon.com/Complete-Guide-Option-Pricing-Formula...

(2) http://www.espenhaug.com/articles.html

(3) http://www.espenhaug.com/manhat.html