Black-Scholes Overvalues Stock Options

A study released by Financial Executives Research Foundation (FERF), the nonprofit research affiliate of Financial Executives International (FEI), analyzed the impact of six valuation models on employee stock options.   The report, entitled Valuing Employee Stock Options: A Comparison of Alternative Models, covers 27 separate stock option grants totaling more than 403 million options, which were granted by nine major US companies.

The report provided a description of their methods and the pros and cons of the six models included in the comparison:

• Black-Scholes-Merton model – assumes both the option and the stock trade in liquid markets.   It also assumes that the risk-free interest rate and the stock’s price volatility remain constant during the option’s life.   This premise tends to not hold true with the long durations typical of stock options.
• Black-Scholes-Merton model with the FAS 123 modification – substitutes the average time to exercise for the contractual life of the option in the Black Scholes model to calculate the fair value of stock options.   However, this model does not account for the stock option’s lack of transferability during the vesting period.   This method is currently the most widely used.
• Analysis Group/Economics Binomial model – designed to both meet the requirements of FAS 123 and address unique features of stock options.   It is calibrated to observed measures of exercise and forfeiture behavior.   While more complex than some of the other evaluated models, it is the one capable of incorporating more complex, non-traditional features, such as, indexed and performance-vested options.
• Analysis Group Enhanced Black-Scholes-Merton model – extends the Black-Scholes model to reflect vesting requirements, transfer restrictions, early exercise and forfeiture features of stock options.   This model measures a stock option grant’s fair value based on the hypothetical price a fully diversified outside investor would pay in an arm’s-length transaction to purchase the right to receive the stock option’s cash flow.
• Minimum Value model – based on one’s willingness to buy an at-the-money call option on a share of stock with the right to defer payment of the exercise price until the end of the option’s term.   The model has the advantage of simplicity but does not capture the effect of share price volatility.

Also evaluated was the Intrinsic Value method. However, this method was not included in the empirical analysis of the grants because this method results in a value of zero at the grant date when the strike price equals the current market price, as it did for most of the 27 grants evaluated.

Study Results

The comparison demonstrates that, in most cases, the Minimum Value model produced the lowest valuations, and the unadjusted Black-Scholes-Merton model produced the highest valuations.

In an example of one grant, the Minimum Value model valued the grant at $8.76 per share, while the unadjusted Black-Scholes-Merton model produced a value of $25.27 per share. The Analysis Group models produced values substantially below Black-Scholes-Merton but usually higher than those produced with the Minimum Value model.

The study compared how each of the six models treated key features of stock options including

• Volatility
• Early exercise
• Forfeiture before vesting
• Forfeiture after vesting
• Non-transferability
• Vesting schedules.

Each model was also analyzed in terms of accuracy, comparability, consistency, simplicity and reproducibility.