Black-Scholes Valuation
The Six Inputs to a Black-Scholes Valuation
When valuing stock options for purposes of Accounting Standards Codification 718 – Stock Compensation (“ASC 718”) various valuation methods can be applied. Some of the more common methods include the Black-Scholes formula, a lattice model, and a Monte Carlo simulation.
This article assumes use of the Black-Scholes formula (a closed-form model); as this is the method most private companies use.[2] We will explain where the typical inputs for each of these six factors are found and in certain cases, how they can be modified to fit the facts and circumstances of a specific situation.
Regardless of which method is applied, the standard stipulates that reasonable and supportable estimates must be documented for the following six factors:
The Exercise Price of the Option
This should be the easiest input to determine since the exercise price is typically stated within the option contract.
The Expected Term of the Option
The expected term of an option is the period of time for which the instrument is expected to be outstanding (that is, the period of time from the service inception date to the date of expected exercise or other expected settlement).[3] ASC 718-55-31 outlines factors that should be taken into account when determining the expected term. They are, the vesting period of the award, employees’ historical exercise and post vesting employment behavior, expected volatility of the underlying share price, blackout periods, and employees’ ages, lengths of service, and home jurisdictions.
In other words, it should not be assumed that the contractual term is equal to the expected term. When using the Black-Scholes method, the expected term may often be shorter than the contractual term because employees may choose to exercise the options early.
Here are some things to keep in mind when choosing an expected term:
- The shorter the expected term, the lower the option value.
- The term should not be shorter than the required vesting period, as it is impossible to exercise an option prior to it vesting.
- Some options have performance, service, or market conditions that must be satisfied before the option can be exercised. When estimating the expected term, it is appropriate to take these factors into account.
- It may be appropriate to combine the options into groups, if the estimated term of the groups differ.
The Current Price of the Underlying Share
For some private companies estimating the fair value of the underlying stock will be the most difficult assumption to support. Sources often used to estimate this input include: a recent arms-length transaction, an ESOP valuation, a recently completed gift or estate valuation, or a buy sell agreement that would be applicable to the underlying stock. If nothing similar to these exists, an appraisal will need to be performed to estimate the fair value of the underlying stock.
If the option contains transfer restrictions, or the stock obtained from the option contains transfer restrictions, this must be factored into the grant date fair value. The standard suggests that transfer restrictions could be accounted for by taking a discount to the fair value of the unrestricted stock (freely traded price). If this approach were taken, we would expect the applicable discount to shrink as the restriction period is shortened. This may require an upward adjustment to the selected volatility rate.
The Expected Volatility of the Price of the Underlying Share for the Expected Term of the Option.
The standard does not require any particular method be used to select the expected volatility. However, it does require the following factors be considered:
- The historical or future volatility of the share price, if it can be determined.
- The volatility of the share price as determined by the market prices for other financial instruments, if it can be determined.
- The interval period at which price movement is measured.
- The capital structure of the entity.[4]
For most private companies there will be insufficient data to determine the historical or future volatility of the share price. Because of this, we typically the volatility of publicly traded companies used as a proxy for the private company’s volatility. If public companies are being used to estimate the stock’s volatility, the historical time period analyzed will typically equal the expected term chosen for the options.
It is this author’s opinion that an index should never be used to benchmark a volatility rate. This is due to the effect diversification has on volatility. If an index is deemed as comparable, it would be more appropriate to look at each security’s volatility within that index individually, than to simply look at the volatility of the entire index.
When using comparable publicly traded companies it is also important to take into account differences in the capital structure of the public companies and the issuer of the options. All else equal, a company with more debt in its capital structure will have a higher equity volatility rate. One possible way to account for this difference is to convert the equity volatility rates of the public companies to asset volatility rates. The selected asset volatility rate can then be converted back to an equity volatility rate based on the subject company’s capital structure. See paragraph 6.36b and Table I-13 of the AICPA’s – Accounting & Valuation Guide: Valuation of Private Held Company Equity Securities Issued as Compensation, for an example.
The Expected Dividends on the Underlying Share for the Expected Term of the Option.
Use of the Black-Scholes formula requires an estimate of the expected dividend rate (as a percentage of the stock’s value). The higher the dividend rate, the lower the value of the option. The expected dividend assumption should only take into account the amount of dividends the option holders will not have a right to while holding the options. Therefore, if any dividends paid during the options existence reduce the amount of the exercise price, it may be proper to set the expected dividend rate to zero.
In addition, if the option holders are entitled to dividends as though they owned the stock, it may be appropriate to include the dividend rate in the option model and to separately calculate the present value of the expected dividend stream. The sum of the option valuation and the present value of the dividend stream would equal the option’s value.
The Risk-free Interest Rate for the Expected Term of the Option.
Per ASC 718-55-28, when a closed-form model (Black-Scholes method) is utilized, the risk-free interest rate should be the implied yield currently available on U.S. Treasury zero-coupon bonds with a remaining term equal to the expected term. [5]
Final Thoughts
When the Black-Scholes method is used to value options, ASC 718 requires that each of the six inputs be reasonable and supportable. To fulfill this requirement, it is important that the selected inputs are consistent with the facts and circumstances of the company, the option agreements, and market information (when available).
More Information
For questions, or guidance on where to begin, contact our valuation experts.
[1] ASC 718-10-55-21.
[2] There are many circumstances where a lattice model or Monte Carlo simulation would be a more appropriate method for valuing share based payments. These are not discussed in this article.
[3] ASC 718-10-55-30.
[4] ASC 718-10-55-28 and 718-10-55-37.
[5] Treasury yields can be found at: https://www.federalreserve.gov/releases/h15/
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The Six Inputs to a Black-Scholes Valuation
When valuing stock options for purposes of Accounting Standards Codification 718 – Stock Compensation (“ASC 718”) various valuation methods can be applied. Some of the more common methods include the Black-Scholes formula, a lattice model, and a Monte Carlo simulation.
This article assumes use of the Black-Scholes formula (a closed-form model); as this is the method most private companies use.[2] We will explain where the typical inputs for each of these six factors are found and in certain cases, how they can be modified to fit the facts and circumstances of a specific situation.
Regardless of which method is applied, the standard stipulates that reasonable and supportable estimates must be documented for the following six factors:
The Exercise Price of the Option
This should be the easiest input to determine since the exercise price is typically stated within the option contract.
The Expected Term of the Option
The expected term of an option is the period of time for which the instrument is expected to be outstanding (that is, the period of time from the service inception date to the date of expected exercise or other expected settlement).[3] ASC 718-55-31 outlines factors that should be taken into account when determining the expected term. They are, the vesting period of the award, employees’ historical exercise and post vesting employment behavior, expected volatility of the underlying share price, blackout periods, and employees’ ages, lengths of service, and home jurisdictions.
In other words, it should not be assumed that the contractual term is equal to the expected term. When using the Black-Scholes method, the expected term may often be shorter than the contractual term because employees may choose to exercise the options early.
Here are some things to keep in mind when choosing an expected term:
- The shorter the expected term, the lower the option value.
- The term should not be shorter than the required vesting period, as it is impossible to exercise an option prior to it vesting.
- Some options have performance, service, or market conditions that must be satisfied before the option can be exercised. When estimating the expected term, it is appropriate to take these factors into account.
- It may be appropriate to combine the options into groups, if the estimated term of the groups differ.
The Current Price of the Underlying Share
For some private companies estimating the fair value of the underlying stock will be the most difficult assumption to support. Sources often used to estimate this input include: a recent arms-length transaction, an ESOP valuation, a recently completed gift or estate valuation, or a buy sell agreement that would be applicable to the underlying stock. If nothing similar to these exists, an appraisal will need to be performed to estimate the fair value of the underlying stock.
If the option contains transfer restrictions, or the stock obtained from the option contains transfer restrictions, this must be factored into the grant date fair value. The standard suggests that transfer restrictions could be accounted for by taking a discount to the fair value of the unrestricted stock (freely traded price). If this approach were taken, we would expect the applicable discount to shrink as the restriction period is shortened. This may require an upward adjustment to the selected volatility rate.
The Expected Volatility of the Price of the Underlying Share for the Expected Term of the Option.
The standard does not require any particular method be used to select the expected volatility. However, it does require the following factors be considered:
- The historical or future volatility of the share price, if it can be determined.
- The volatility of the share price as determined by the market prices for other financial instruments, if it can be determined.
- The interval period at which price movement is measured.
- The capital structure of the entity.[4]
For most private companies there will be insufficient data to determine the historical or future volatility of the share price. Because of this, we typically the volatility of publicly traded companies used as a proxy for the private company’s volatility. If public companies are being used to estimate the stock’s volatility, the historical time period analyzed will typically equal the expected term chosen for the options.
It is this author’s opinion that an index should never be used to benchmark a volatility rate. This is due to the effect diversification has on volatility. If an index is deemed as comparable, it would be more appropriate to look at each security’s volatility within that index individually, than to simply look at the volatility of the entire index.
When using comparable publicly traded companies it is also important to take into account differences in the capital structure of the public companies and the issuer of the options. All else equal, a company with more debt in its capital structure will have a higher equity volatility rate. One possible way to account for this difference is to convert the equity volatility rates of the public companies to asset volatility rates. The selected asset volatility rate can then be converted back to an equity volatility rate based on the subject company’s capital structure. See paragraph 6.36b and Table I-13 of the AICPA’s – Accounting & Valuation Guide: Valuation of Private Held Company Equity Securities Issued as Compensation, for an example.
The Expected Dividends on the Underlying Share for the Expected Term of the Option.
Use of the Black-Scholes formula requires an estimate of the expected dividend rate (as a percentage of the stock’s value). The higher the dividend rate, the lower the value of the option. The expected dividend assumption should only take into account the amount of dividends the option holders will not have a right to while holding the options. Therefore, if any dividends paid during the options existence reduce the amount of the exercise price, it may be proper to set the expected dividend rate to zero.
In addition, if the option holders are entitled to dividends as though they owned the stock, it may be appropriate to include the dividend rate in the option model and to separately calculate the present value of the expected dividend stream. The sum of the option valuation and the present value of the dividend stream would equal the option’s value.
The Risk-free Interest Rate for the Expected Term of the Option.
Per ASC 718-55-28, when a closed-form model (Black-Scholes method) is utilized, the risk-free interest rate should be the implied yield currently available on U.S. Treasury zero-coupon bonds with a remaining term equal to the expected term. [5]
Final Thoughts
When the Black-Scholes method is used to value options, ASC 718 requires that each of the six inputs be reasonable and supportable. To fulfill this requirement, it is important that the selected inputs are consistent with the facts and circumstances of the company, the option agreements, and market information (when available).
More Information
For questions, or guidance on where to begin, contact our valuation experts.
[1] ASC 718-10-55-21.
[2] There are many circumstances where a lattice model or Monte Carlo simulation would be a more appropriate method for valuing share based payments. These are not discussed in this article.
[3] ASC 718-10-55-30.
[4] ASC 718-10-55-28 and 718-10-55-37.
[5] Treasury yields can be found at: https://www.federalreserve.gov/releases/h15/
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