Replicating portfolio theory is the foundation of the ground-breaking Black-Scholes equation originally developed to value financial options. This theory is also the basis for the binomial method developed for the same purpose by John Cox, Stephen Ross, and Mark Rubinstein. ^[*] The advantage of the binomial method is that it is a transparent, efficient, numerical method that uses elementary mathematics to value not only the simple European options but also the more complex American options, where premature exercise may be optimal. In its special limiting case, the binomial method reduces to the Black-Scholes equation. This appendix briefly presents the theory behind the replicating portfolio approach and the derivation of equations for the binomial method to solve financial as well as real options problems.

A call option provides the owner of the option the right to buy the underlying asset (normally a stock) at a predetermined price (strike price) on or before the expiration date. The replicating portfolio method uses a portfolio that consists of a certain number of underlying stocks and risk-free bonds that correlates perfectly with the option value irrespective of whether the stock prices go up or go down in the future. Since the portfolio correlates perfectly with the value of the option, the current price of the call option is calculated as the current value of the replicating portfolio.

To better illustrate the principle, let us assume that the current price of a stock is S _o. Also assume that in the next time...