# What's the right Black-Scholes value?

As the SEC and FASB stock-option initiatives build momentum,
it's clear that the special characteristics of executive options,
especially vesting restrictions, must be incorporated into the valuation
model.

The collective outcry from the government, public and investors over executives' huge option gains has fueled the perception that executive pay in Corporate America is dangerously out of control. As a result of these and other pressures, the Financial Accounting Standards Board and Securities and Exchange Commission have introduced initiatives on accounting for and disclosing stock options, making the need for a credible valuation technique an urgent issue that demands resolution.

The Black-Scholes model is increasingly viewed as the model of choice, but it may not merit this status. Outside the investment community, where it has been used for some time, the model isn't well understood, and crucial considerations like the impact of vesting on option values remain unresolved. In particular, the FASB's current position on recognizing option vesting restrictions is flawed. It recognizes the impact on the value of stock-option forfeitures created by vesting restrictions, but not the loss of the right to exercise the option for the vesting term.

The model was originally developed to value market stock options, but the executive stock option is an entirely different animal. The terms to expiration are typically much longer than those of market stock options. And unlike market stock options, executive stock options aren't transferable and are often subject to restrictions on option exercise.

Proponents of using the model to value executive options think it's the best-available approach because it offers easily obtained, objective inputs, permits the user to calculate values with relative ease and can be applied consistently from company to company without excessive subjectivity. Therefore, proponents skirt some of the problems by adjusting the model's values to recognize the differences between executive stock options and market stock options.

This has ironed out some of the kinks, but other important questions remain. Although the model is easy to manipulate, it is difficult to understand how it really works and what its outputs mean. Other problems are the model's highly restrictive assumptions when applied to executive options and its inability to deal convincingly with the differences between executive and market stock options.

The original model's inputs are the volatility of the underlying stock's returns, the option strike price, the stock's market price, a risk-free interest rate and the option's term. Other modifications permit the use of less-restrictive assumptions in valuing stock options. These include adjustments for dividends, as well as variable-interest-rate and non-constant-volatility assumptions.

Although many users employ modified versions of the model that permit liberalizing the underlying assumptions, generally the resulting values do not differ dramatically from the original model adjusted for dividends. When valuing executive stock options, this version seems to appropriately balance the trade-offs between ease of use and theoretical precision.

One of the crucial model inputs is the volatility of the underlying stock's returns. The model implies that the value of the stock option is positively related to the stock return's volatility. Therefore, the higher the volatility, the greater the stock-option value, assuming all other inputs remain the same. This suggests a stock with a volatile price has a greater probability of being "in the money" than a less-volatile one.

As the model's inputs demonstrate, the values derived by the model don't depend on growth in the underlying stock's value. Some users like this because it doesn't force an assumption about the future value of the stock, although it does implicitly require several assumptions about the future. But this seems counter-intuitive to many, since it suggests that the factors that contribute to a stock's value are irrelevant to the value of the stock option, even over the long terms to expiration typical of executive options.

THE FASB TAKES A LOOK

In its exposure draft on stock-compensation accounting, the FASB has addressed the need to modify the option values to adjust for the differences between executive stock options and market stock options. Typically, if an option isn't vested, the executive has no right to exercise it and will forfeit it if he or she leaves the company. Vesting criteria clearly limit the executive's option rights, making an option with vesting restrictions less valuable than one that can be exercised at any time. The FASB has responded to vesting restrictions by recommending adjustments for expected forfeitures and allowing companies to spread the expense attributable to the option over the vesting period.

Most executives exercise their stock options before the options expire, because of retirement, personal considerations, job changes or other factors. As a result, the FASB has suggested that the expected term of the option, rather than the full term, be used in determining the value of the executive stock option.

Presumably, these adjustments mean that if option holders are expected to forfeit 5 percent of the options granted and executives are expected to exercise the options an average of six years after the grant date, then only 95 percent of the options granted would be used to determine the expense amount and the term used to value the option would be six years instead of the full option term.

This technique for adjusting option values to reflect experience is a practical solution to recognizing some of the differences between executive stock options and market options. Nevertheless, recognizing expected forfeitures only addresses part of the impact of vesting restrictions on the Black-Scholes value of an option.

While the FASB has addressed the forfeiture provision attributable to vesting by valuing only the options expected to vest, it hasn't considered the most interesting limitation, which says the executive can't exercise the option during the vesting period. For example, if a stock option with a 10-year term is granted with a three-year vesting term, the executive is not permitted to exercise it for three years. If the option has value, it derives from the right conferred during the last seven years of the 10-year term.

A DIFFERENT PERSPECTIVE

From a different perspective, suppose an investor buys a 10-year option. If the investor only wants the right to exercise the option between the beginning of the fourth year to the end of the 10th year of the option term, he or she could sell the rights to the first three years of the option -- an option with a three-year term, the same strike price, and the same date of grant as the 10-year option. The net cost of the transaction -- buying a 10-year option and selling a three-year option -- is equivalent to the value of a 10-year option with three-year cliff-vesting restrictions on exercise, because it puts the investor in an equivalent position.

This means that under a Black-Scholes framework, the value of an option with vesting restrictions can be determined simply by subtracting the value of the option from the value of the option determined using its expected term. Except for the option term to expiration, both stock-option values would be generated using the same Black-Scholes inputs -- the same dividend yield, volatility, risk-free interest rate, strike price and market-price assumptions. This calculation is the arithmetical equivalent of buying the full-term stock option and selling the corresponding stock option covering the vesting period. Therefore, it's a simple, rational approach to recognizing the impact of vesting restrictions on stock-option exercise.

The bottom line is that recognizing vesting restrictions on option exercise substantially reduces the calculated value of an executive stock option. Assume we have a 10-year option that cliff vests three years after the date of the grant. Also assume that its market and strike price are $25, the risk-free interest rate is 7 percent, the volatility of the underlying stock's returns is 0.30 and, for simplicity's sake, no dividends are paid on the stock. The Black-Scholes model generates a value for the 10-year option of roughly $14.70, or 59 percent of the value of the underlying stock on the grant date. The value of a three-year stock option, with the same input parameters, is about $7.40. Consequently, a 10-year option with a three-year restriction would be worth $7.30 ($14.70 - $7.40).

If we modify the assumptions to include a 3 percent dividend yield, the impact is even more pronounced, with the value of the option shrinking to $3.32, or about 13.3 percent of the market value of the underlying stock. The table on page 58 shows the impact of vesting restrictions on exercise for options granted at market, assuming a 7 percent interest rate, a 3 percent dividend yield, a 0.30 volatility assumption and a $25 stock price. Option-term and vesting assumptions are shown in the table.

As the table shows, losing the right to exercise the stock option early in the term creates stock-option values substantially less than the ones many companies are reporting. Note that applying the proposed FASB methodology to the option definition used in the table, assuming a five-year expected term and no forfeiture risk, generates a value of $7.31, regardless of vesting restrictions. If we assume that the option cannot be exercised for one year, recognizing those restrictions yields a value of slightly more than half the FASB value.

More generally, a stock option with no adjustment for vesting restrictions on exercise would typically be valued at 30 percent to 40 percent of the market price of the underlying stock at grant. But options with vesting requirements can yield values that are only 10 percent to 20 percent, or less, of the value of the underlying stock.

CONSIDER THE OPTIONS

The arithmetical approach to recognizing vesting restrictions to value options is rational and theoretically correct, but it runs counter to the FASB's current position, and it does suggest a need to resolve some significant model outcomes. In particular, if we accept the model and its corresponding framework, we must also accept that an option with vesting restrictions is worth much less than an option without them and that much of an option's value derives from the right to exercise it in the first years after the grant.

Whether the implied reduction in value is consistent with expectations is as much a question about the model's validity for valuing executive stock options as it is about determining the option's value. If the model is to be useful, it must account for not only practical considerations, such as the forfeiture potential and shorter terms to exercise, but also the effect of vesting restrictions on the value of the exercise right. Finally, since the question of incorporating vesting into the valuation remains unresolved, the Black-Scholes model should be considered one of several techniques, not the sole yardstick, to value executive stock options.

Mr. Young is a consultant in the Pittsburgh office of Buck Consultants.

The collective outcry from the government, public and investors over executives' huge option gains has fueled the perception that executive pay in Corporate America is dangerously out of control. As a result of these and other pressures, the Financial Accounting Standards Board and Securities and Exchange Commission have introduced initiatives on accounting for and disclosing stock options, making the need for a credible valuation technique an urgent issue that demands resolution.

The Black-Scholes model is increasingly viewed as the model of choice, but it may not merit this status. Outside the investment community, where it has been used for some time, the model isn't well understood, and crucial considerations like the impact of vesting on option values remain unresolved. In particular, the FASB's current position on recognizing option vesting restrictions is flawed. It recognizes the impact on the value of stock-option forfeitures created by vesting restrictions, but not the loss of the right to exercise the option for the vesting term.

The model was originally developed to value market stock options, but the executive stock option is an entirely different animal. The terms to expiration are typically much longer than those of market stock options. And unlike market stock options, executive stock options aren't transferable and are often subject to restrictions on option exercise.

Proponents of using the model to value executive options think it's the best-available approach because it offers easily obtained, objective inputs, permits the user to calculate values with relative ease and can be applied consistently from company to company without excessive subjectivity. Therefore, proponents skirt some of the problems by adjusting the model's values to recognize the differences between executive stock options and market stock options.

This has ironed out some of the kinks, but other important questions remain. Although the model is easy to manipulate, it is difficult to understand how it really works and what its outputs mean. Other problems are the model's highly restrictive assumptions when applied to executive options and its inability to deal convincingly with the differences between executive and market stock options.

The original model's inputs are the volatility of the underlying stock's returns, the option strike price, the stock's market price, a risk-free interest rate and the option's term. Other modifications permit the use of less-restrictive assumptions in valuing stock options. These include adjustments for dividends, as well as variable-interest-rate and non-constant-volatility assumptions.

Although many users employ modified versions of the model that permit liberalizing the underlying assumptions, generally the resulting values do not differ dramatically from the original model adjusted for dividends. When valuing executive stock options, this version seems to appropriately balance the trade-offs between ease of use and theoretical precision.

One of the crucial model inputs is the volatility of the underlying stock's returns. The model implies that the value of the stock option is positively related to the stock return's volatility. Therefore, the higher the volatility, the greater the stock-option value, assuming all other inputs remain the same. This suggests a stock with a volatile price has a greater probability of being "in the money" than a less-volatile one.

STOCK-OPTION VALUES WITH VESTING CRITERIA UNDER BLACK-SCHOLES Note: This example assumes an underlying stock price of $25. 10-Year Option Term Term That Option Option Option Value As % of Stays Unvested Value Stock Price at Grant 0 years $9.15 36.6% 1 year $5.81 23.2% 3 years $3.32 13.3% 5 years $1.84 7.4% 5-Year Option Term Term That Option Option Option Value As % of Stays Unvested Value Stock Price at Grant 0 years $7.31 29.2% 1 year $3.97 15.9% 2 years $2.53 10.1% 3 years $1.48 5.9%

As the model's inputs demonstrate, the values derived by the model don't depend on growth in the underlying stock's value. Some users like this because it doesn't force an assumption about the future value of the stock, although it does implicitly require several assumptions about the future. But this seems counter-intuitive to many, since it suggests that the factors that contribute to a stock's value are irrelevant to the value of the stock option, even over the long terms to expiration typical of executive options.

THE FASB TAKES A LOOK

In its exposure draft on stock-compensation accounting, the FASB has addressed the need to modify the option values to adjust for the differences between executive stock options and market stock options. Typically, if an option isn't vested, the executive has no right to exercise it and will forfeit it if he or she leaves the company. Vesting criteria clearly limit the executive's option rights, making an option with vesting restrictions less valuable than one that can be exercised at any time. The FASB has responded to vesting restrictions by recommending adjustments for expected forfeitures and allowing companies to spread the expense attributable to the option over the vesting period.

Most executives exercise their stock options before the options expire, because of retirement, personal considerations, job changes or other factors. As a result, the FASB has suggested that the expected term of the option, rather than the full term, be used in determining the value of the executive stock option.

Presumably, these adjustments mean that if option holders are expected to forfeit 5 percent of the options granted and executives are expected to exercise the options an average of six years after the grant date, then only 95 percent of the options granted would be used to determine the expense amount and the term used to value the option would be six years instead of the full option term.

This technique for adjusting option values to reflect experience is a practical solution to recognizing some of the differences between executive stock options and market options. Nevertheless, recognizing expected forfeitures only addresses part of the impact of vesting restrictions on the Black-Scholes value of an option.

While the FASB has addressed the forfeiture provision attributable to vesting by valuing only the options expected to vest, it hasn't considered the most interesting limitation, which says the executive can't exercise the option during the vesting period. For example, if a stock option with a 10-year term is granted with a three-year vesting term, the executive is not permitted to exercise it for three years. If the option has value, it derives from the right conferred during the last seven years of the 10-year term.

A DIFFERENT PERSPECTIVE

From a different perspective, suppose an investor buys a 10-year option. If the investor only wants the right to exercise the option between the beginning of the fourth year to the end of the 10th year of the option term, he or she could sell the rights to the first three years of the option -- an option with a three-year term, the same strike price, and the same date of grant as the 10-year option. The net cost of the transaction -- buying a 10-year option and selling a three-year option -- is equivalent to the value of a 10-year option with three-year cliff-vesting restrictions on exercise, because it puts the investor in an equivalent position.

This means that under a Black-Scholes framework, the value of an option with vesting restrictions can be determined simply by subtracting the value of the option from the value of the option determined using its expected term. Except for the option term to expiration, both stock-option values would be generated using the same Black-Scholes inputs -- the same dividend yield, volatility, risk-free interest rate, strike price and market-price assumptions. This calculation is the arithmetical equivalent of buying the full-term stock option and selling the corresponding stock option covering the vesting period. Therefore, it's a simple, rational approach to recognizing the impact of vesting restrictions on stock-option exercise.

The bottom line is that recognizing vesting restrictions on option exercise substantially reduces the calculated value of an executive stock option. Assume we have a 10-year option that cliff vests three years after the date of the grant. Also assume that its market and strike price are $25, the risk-free interest rate is 7 percent, the volatility of the underlying stock's returns is 0.30 and, for simplicity's sake, no dividends are paid on the stock. The Black-Scholes model generates a value for the 10-year option of roughly $14.70, or 59 percent of the value of the underlying stock on the grant date. The value of a three-year stock option, with the same input parameters, is about $7.40. Consequently, a 10-year option with a three-year restriction would be worth $7.30 ($14.70 - $7.40).

If we modify the assumptions to include a 3 percent dividend yield, the impact is even more pronounced, with the value of the option shrinking to $3.32, or about 13.3 percent of the market value of the underlying stock. The table on page 58 shows the impact of vesting restrictions on exercise for options granted at market, assuming a 7 percent interest rate, a 3 percent dividend yield, a 0.30 volatility assumption and a $25 stock price. Option-term and vesting assumptions are shown in the table.

As the table shows, losing the right to exercise the stock option early in the term creates stock-option values substantially less than the ones many companies are reporting. Note that applying the proposed FASB methodology to the option definition used in the table, assuming a five-year expected term and no forfeiture risk, generates a value of $7.31, regardless of vesting restrictions. If we assume that the option cannot be exercised for one year, recognizing those restrictions yields a value of slightly more than half the FASB value.

More generally, a stock option with no adjustment for vesting restrictions on exercise would typically be valued at 30 percent to 40 percent of the market price of the underlying stock at grant. But options with vesting requirements can yield values that are only 10 percent to 20 percent, or less, of the value of the underlying stock.

CONSIDER THE OPTIONS

The arithmetical approach to recognizing vesting restrictions to value options is rational and theoretically correct, but it runs counter to the FASB's current position, and it does suggest a need to resolve some significant model outcomes. In particular, if we accept the model and its corresponding framework, we must also accept that an option with vesting restrictions is worth much less than an option without them and that much of an option's value derives from the right to exercise it in the first years after the grant.

Whether the implied reduction in value is consistent with expectations is as much a question about the model's validity for valuing executive stock options as it is about determining the option's value. If the model is to be useful, it must account for not only practical considerations, such as the forfeiture potential and shorter terms to exercise, but also the effect of vesting restrictions on the value of the exercise right. Finally, since the question of incorporating vesting into the valuation remains unresolved, the Black-Scholes model should be considered one of several techniques, not the sole yardstick, to value executive stock options.

Mr. Young is a consultant in the Pittsburgh office of Buck Consultants.

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Title Annotation: | Corporate Reporting; valuation of executive stock options |
---|---|

Author: | Young, Christopher K. |

Publication: | Financial Executive |

Date: | Sep 1, 1993 |

Words: | 1892 |

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