Human Capital & Careers

Forget Black-Scholes?

Why the traditional option-pricing model may not be the best way to value employee grants.
Craig SchneiderMay 7, 2004

Let the mudslinging begin — again. No, we’re not talking about the Presidential campaign. We’re talking about the latest battle over the Financial Accounting Standards Board’s plan to require companies to treat employee stock options as an expense. Now that the long-awaited proposal is out, corporate and political opponents are reviving their oft-heard arguments against the idea in an effort to get the board to reverse course.

But while opponents of expensing have forced FASB to back down in the past, their efforts appear less likely to succeed this time around, in large part because the unrestrained granting of stock options has been widely blamed for fueling the excesses of the 1990s’ stock-market bubble.

The critics have at least one legitimate complaint about the rule — valuation. It is difficult to estimate the actual expense of employee options. But FASB has not let this problem (which also applies to many other items used to calculate earnings) derail its proposal. The board’s view is that subtracting even a rough estimate of options expense from earnings will produce more-transparent financial results than simply footnoting the extent of dilution caused by options, as the current rule allows. Moreover, the board’s view on how to value options may take an important step forward.

At this point, the traditional method for valuing options, the Black-Scholes model, seems likely to be eclipsed for purposes of expensing employee grants by an alternative known as the binomial method (also known as the Cox, Ross, Rubinstein model). While the alternative may be easier to adopt than some experts warn, it requires more assumptions than Black-Scholes to be used effectively. Because of the potential that creates for abuse, auditors and regulators may require more documentation before they accept a company’s financial statements.

The Black-Scholes Problem

The reason the Black-Scholes formula may not be appropriate for purposes of expensing employee grants is that it is widely considered to overstate their value by an unacceptable margin. That’s because the model does not take into account the essential differences between traditional exchange-traded stock options and those granted to employees.

Unlike conventional options, employee options are subject to vesting schedules and forfeiture conditions, and cannot be transferred. As a result, they are invariably exercised before their usual 10-year term expires. These characteristics reduce the value of an option.

To properly value an option grant, a company must be able to estimate the effects these variables have on an employee’s ability and inclination to exercise the option. “You can’t do that with Black-Scholes,” explains Ron Rudkin, a vice president of Boston-based consultancy Analysis Group. “It just doesn’t have the flexibility.”

The most widely used alternative, the binomial method, is more flexible than Black-Scholes because it uses a lattice framework (see “Up the Lattice,” at the end of this article). The framework divides the time from an option’s grant date to its expiration date into small increments. That enables the model to take into account many more assumptions about a grant’s features, and better estimate employees’ likely behavior regarding investments.

That difference hasn’t been lost on FASB. As proposed, the new rule recommends use of a lattice-based model unless a company lacks sufficient historical data.

Troublesome Variables

Thanks to its additional flexibility, the binomial model tends to produce a lower estimate of option value than Black-Scholes. A study by Analysis Group found that the Black-Scholes model overstated the value of some grants by anywhere from 28 percent to 56 percent.

But some critics worry that the flexibility of the binomial model could help unscrupulous executives manipulate their financial results. The fear is that they will take advantage of the model’s flexibility to underestimate their compensation costs and overstate their earnings. “My concern is that people will use the more-powerful models in the wrong ways,” says Mark Rubinstein, a finance professor at the University of California at Berkeley who helped create the binomial model. Rubinstein says the first question he gets from managers is, “Can you tell me how I can get lower numbers?”

For better or worse, small changes in the assumptions concerning an option grant’s features can produce wildly different estimates of its value, and that’s obviously a bigger problem under the binomial method than under Black-Scholes, notes Chris Kruse, a consultant with New York­based CFX Inc. who studied under Rubinstein. “You can incorporate a lot of things in the binomial model that will significantly reduce the value of the option,” warns Kruse.

Forfeiture assumptions represent a particularly troublesome variable, he says, since they are something of “a random element.” That’s because at least some option recipients quit before their options vest, and thereby relinquish the right to exercise them. In those cases, of course, a company would not incur any expense at all. So simply by projecting a higher-than-likely rate of employee attrition, explains Kruse, a company could artificially deflate the expense it reports for its grants. It may be easy enough to convince auditors to go along, he contends, since the probability of forfeiture is related to such difficult-to-quantify factors as an employee’s opportunities elsewhere. “This is probably highly variable, especially when moving up the food chain to the executive suite,” says Kruse. “It is also highly idiosyncratic, since a firm’s outlook may affect employee attrition.” This variable alone, he cautions, gives companies tremendous leeway in estimating option values.

To address this problem, FASB’s proposed rule would require companies to disclose all the assumptions they have used to calculate the expense of their options. Those disclosures would not be limited to a grant’s standard assumptions — regarding such elements as stock price, option strike price, interest rate, dividend yield, stock volatility, and term — but would also include those concerning vesting restrictions and forfeiture requirements.

Even so, some experts say FASB should either move away from option-price models entirely or try to minimize their room for error. Rubinstein himself recommends that the board simply require companies to calculate the present value of options based on the stock’s price at the time of the grant, and then adjust that as the price changes. That way, a company would have to “true up” any difference between the value of the option estimated at the time it was granted and the actual value when exercised.

Another alternative, proposed by Stanford University professors Jeremy Bulow and John Shoven, would instead greatly shorten the time frame involved in modeling a grant’s value. Since most programs require employees who leave the company to exercise their options within 90 days, the Bulow-Shoven approach would assume that a new grant is made every quarter, requiring a company to subtract the value of extending its existing options another three months as an expense, along with the three-month cost of any newly granted options. The authors contend such an ongoing method would give a more-accurate picture of the cost of outstanding grants than any attempt to provide a once-and-for-all valuation.

FASB says it will take these issues into account in its deliberations over the final rule, due out by the end of the year. “We expect that over time, companies will move to the binomial method, because it provides a better measure of the option value,” says board member G. Michael Crooch. And he notes that the use of estimates is a necessary evil. “FASB must rely on management and auditors to properly apply our standards,” he says.

For Better and Worse

Meanwhile, however, some finance executives worry that the different results produced by the Black-Scholes and binomial models will make it harder for investors to compare companies’ performance. “Flexibility is nice,” says John Cox, chief accounting officer at Houston-based BMC Software Inc., “but it brings with it other issues.”

And, some experts predict that finance executives will find the binomial model difficult to apply, at least initially. “Black-Scholes is so widely used that there are lots of software packages to run the model,” says Rebecca McEnally, vice president of the Charlottesville, Virginia-based Association for Investment Management and Research — an analyst group that advocates expensing options — and an adviser to the International Accounting Standards Board (IASB), FASB’s counterpart outside the United States. In contrast, says McEnally, the binomial method “is not quite as easily adopted.”

None of these obstacles has stopped some companies that are already expensing options from using the binomial model, including such insurers as American International Group and Washington Mutual. Those raising capital abroad may have another reason to adopt the alternative: IASB has effectively endorsed the model in its own, already finalized rule requiring that employee option grants be expensed starting next year. In essence, the international standards-setter requires that companies decide which valuation model to use based on what “a knowledgeable, willing market participant,” such as a derivatives trader, would use. The rule also notes, “This may preclude the use of the Black-Scholes model.”

If it’s any solace, Kruse contends that reports of the binomial model’s relative difficulty of use are exaggerated. While he concedes that the method takes time to get used to, those who adopt it, he insists, will soon find it “quite simple.” What’s more, he says, “you can find an out-of-the-box [binomial] calculator for $100.”

Too bad the gap between the estimated and actual value of a company’s option grants is unlikely to always be that small.

Up the Lattice

In its proposed rule for expensing stock options, the Financial Accounting Standards Board has acknowledged that the most popular option-pricing model, known as Black-Scholes, may be less appropriate than a lattice-based method for valuing employee grants. That’s simply because a lattice-based method can take into account assumptions that reflect the conditions under which employee options are typically granted. The binomial model is the most commonly used lattice-based method, but other methods may be better suited to compensation programs that link vesting to specific performance objectives, as this summary provided by consulting firm Analysis Group illustrates:

Binomial. This model can calculate option values when the price of the underlying stock moves either up or down over a short interval. It is considered better than Black-Scholes at taking into account how factors such as forfeiture and exercise before expiration, as well as an employee’s risk aversion and lack of diversification, affect an option’s value at the time it is granted.

Trinomial. The trinomial model goes a step further by allowing for the underlying stock price to either remain unchanged or move up or down. That’s useful for valuing performance-based options that vest only if the stock price exceeds a certain level over time.

Multinomial. This model can take many more factors into account than either the binomial or trinomial framework. Such additional flexibility may be required to value options that cannot be exercised unless the underlying stock price exceeds the performance of one or more indices. But when there are more than two such sources of uncertainty, a Monte Carlo simulation may be preferable, since it is easier to apply than lattice models.