Valuation Techniques: Discounted Cash Flow, Earnings Quality, Measures of Value Added, and Real Options

Lay out the binomial tree of stock prices to the period when the option expires.

For the period when the option expires, insert the value of the option corresponding to the value of the stock. The value of a call option is the greater of zero or the stock price less the exercise price. The value of a put option is the greater of zero or the exercise price less the stock price.

Step back one period prior to expiration. Look ahead at the next two possible option values, multiply them by p and 1 − p, respectively, add the results, and divide by 1 plus the risk-free rate.

Step back one period and repeat Step 3. Continue the process, stepping back one period and repeating Step 3 until Time 0 is reached, at which point the option value is obtained.

Understanding what was done here is not difficult, but accepting that this type of model is realistic may be more difficult. There are, after all, more than two prices to which a stock can move. To make the model more like the real world, we can add a large number of time periods. But if we do so, we may find that having the stock continue to go up or down 50 percent and the interest rate be 5 percent per period is unrealistic. For an option with a fixed life, we can shrink the up and down factors and the risk-free rate so that over the life of the option, the stock price movement and the interest rate are more realistic for that length of time.

We shall not get into the details of this procedure, but one can simply compare this approach ...