Equity Compensation in a Post-Expensing World

From Chapter 2, "How to Value an Employee Stock Option"

After mapping the tree of possible outcomes, the binomial model does a brute force calculation. It determines the value of the future gain at each node along the tree, for several consecutive branches or intervals. Then it weights each outcome according to its probability and discounts the whole tree to present value. The model's magic lies in the estimates of the likelihood and magnitude of each up or down movement. While the up/down probabilities can be spoon-fed, the more typical way is to derive them from volatility. The other assumption required is the number of intervals in the tree.

An interesting fact about the binomial model is that as we increase the number of intervals, the model converges toward the Black-Scholes model. In fact, it does not take very many node intervals to get estimates that are very close between the two models. This is not by coincidence, as the models do not conflict. The authors of the binomial were not trying to get around or go beyond Black-Scholes so much as "come at it from a different angle."

In practical terms, the binomial model often gives little advantage over Black-Scholes, as the price outputs converge. However, as earlier mentioned, the binomial model's design allows for the pricing of American options (i.e., options that can be exercised before the end of the term). For non-dividend-paying stocks, this advantage is not consequential. However, for dividend-paying stocks, an early exercise gets the option holder into dividends sooner rather than later. This makes the early exercise feature valuable. For this reason, when pricing an American-style option with a high dividend yield (e.g., an option on a REIT stock), it is in theory more accurate to use a binomial model or a special variation of the Black-Scholes model . However, as a practical matter, it may not be worth the trouble.