The Black-Scholes Model

By November 12, 2018Financing
stock price computer

Both public companies and nonpublic companies use the Black-Scholes Model (or BSM) to value their options because of its relative simplicity and widespread acceptance. Though the Black-Scholes Model is relatively simple, the formulas and calculations used to value an option can still be daunting without a basic understanding of the model. This article explains the purpose of the Black-Scholes Model, its inputs, its limitations, and its alternatives. Additionally, this article points out complexities that companies may need to work through as they implement the Black-Scholes Model.

Purpose

Many companies compensate employees using stock options. These options have value, and determining the correct value is an important part of accounting for stock compensation. However, determining the value of stock options can be difficult because a company must estimate the value.

Ideally, your company can value its options using observable market prices of identical or similar options. However, options in observable markets are rarely identical or similar to those of your company. The Black-Scholes Model helps to estimate the value of your company’s options when comparable options are not available in an observable market and is the most common method of valuing options.

The Black-Scholes Model calculates the fair value of option-like financial instruments, such as the following:

  • Employee stock options: a form of compensation that gives an employee the right to buy company stock at a predetermined price after vesting conditions have been satisfied.
  • Put options: options that give the owner the right to sell stock at a predetermined price.
  • Call options: options that gives the owner the right to buy stock at a predetermined price.
  • Warrants: financial instruments that are similar to stock options and allow company shares to be purchased at a set price. Follow the article link to our article on warrants to see how warrants and stock options differ.

Inputs

When using the Black-Scholes Model, an option’s fair value is estimated by predicting potential future outcomes, probability weighting those outcomes, and discounting the outcomes back to the present value. The more likely it is that an option will expire in the money (i.e., the fair value at the time of exercise will be greater than the exercise price), the more valuable that option is. Six inputs are used in the Black-Scholes Model—the current stock price, the exercise price, the risk-free rate, the expected term, the volatility, and the dividend yield.

BSM Summary Chart

Input Relationship to Fair Value Difficulty to Determine Impact on Fair Value
Current Stock Price Positive

As the stock price increases relative to the exercise price, the fair value of the option increases.

Low (public entities) to High (nonpublic entities)

Determining the current stock price is more difficult for nonpublic companies because nonpublic companies require a 409A valuation provided by valuation experts.

Low ­to Moderate
Exercise Price Negative

As the exercise price increases relative to the stock price, the fair value of the option decreases.

Low

The exercise price is stated in the option grant agreement.

Low
Expected Term Positive

An option that has a longer expected term is more valuable because the option is more likely to expire in the money.

Moderate to High

Because U.S. options can generally be exercised anytime between the vesting date and the expiration date, estimating the expected term requires analysis and judgement.

High
Risk-free Rate Positive

The exercise price is discounted by the risk-free rate, so a larger risk-free rate causes the exercise price to be smaller and, thus, the fair value of the option to be larger.

Low

Risk-free rates are easy to find online at the Treasury’s website. Picking a relevant risk-free rate requires first developing the expected term estimate.

Low
Volatility Positive

A more volatile stock is more likely to expire in the money, so higher volatility causes the value of an option to increase.

High

Volatility can be difficult to measure, especially for nonpublic companies. Measuring volatility requires a high degree of judgement in analyzing relevant data and reaching a meaningful conclusion.

High
Dividend Yield Negative

Because an option holder does not have rights to the dividends paid out by a company, a higher dividend yield decreases the value of an option.

Low

Most companies will use their current dividend yield under the assumption that the dividend yield will remain constant over time, which for many nonpublic companies is 0%.

Low

Current Stock Price

The fair value of a stock option increases as the current stock price increases relative to the exercise price.

The current stock price is the value of the stock on the grant date of the option. For publicly-traded companies, the current stock price is readily available on the company’s listing. Valuing the stock price of nonpublic companies is a bit more complex. A more thorough analysis of the valuation process for nonpublic companies can be found in our Valuation article. 409A valuations, specifically, are frequently used by nonpublic companies when issuing stock compensation to employees. 409A valuations help ensure that the company is compliant with Section 409A of the tax code (and ensure the company avoids a 20 percent tax penalty) by supporting the value of a company’s underlying equity and the resulting stock price. Outside experts are often hired to perform a 409A valuation for nonpublic companies.

Exercise Price

The fair value of a stock option increases as the exercise price decreases relative to the stock price.

The exercise price or strike price of an option is the fixed price at which the stock can be purchased or sold after the option has vested. The exercise price is stated in the option grant agreement. As defined by the agreement, the exercise price is almost always equal to the current stock price on the grant date.

Expected Term

The expected term is positively correlated with the value of an option. As estimated time frame of an option increases there is more time for the volatility of the option to create in-the-money value.

The expected term refers to the time period between when the options have vested and when they are either exercised or forfeited. Like the other inputs in the Black-Scholes Model, this input is a single number. Determining the correct estimate for this input is difficult because U.S. options may be exercised anytime between the vesting date and the expiration date—a gap that may extend over many years. Some options are even exercisable before they vest. This estimate also has a relatively large impact on the output of the Black-Scholes Model.

The lowest possible value for the expected term is the vesting term, and the maximum value is the end of the contractual term (i.e., when the options expire). If the options qualify as “plain vanilla1” options, a company may estimate the option using the “simplified” method. In the simplified method, the expected term is calculated by taking the average of the vesting term and the contractual term:

Expected term = (vesting term + contractual term)/2

Often, a company will have an award with multiple vesting tranches (i.e., vesting periods), such as when the award is subject to graded vesting. A more detailed explanation of graded vesting can be found in our article Stock Options 101. When a company has multiple vesting tranches, the company will typically estimate the vesting term as the weighted-average vesting term of the tranches. For example, a company with two vesting tranches may use a formula like the following:

Expected term = [(vesting term A + vesting term B)/2 + contractual term]/2

Under US GAAP, a company may only use the simplified method if it does not have sufficient historical data or relevant, public information to develop a supportable estimate for the expected term. However, in practice many companies use the simplified method when other methods are not practicable. ASC 718-10-55-32 states that a company may use “whatever relevant and supportable information is available” in developing its expected term, such as industry averages, public company benchmarks, or public academic research. In developing an expected term, your company should also consider its stock price history, blackout periods2, future strategic plans, and expected volatility. These factors may warrant an adjustment to your expected term.

Risk-free Rate

The risk-free rate of return is positively correlated with the value of an option. One component of the Black-Scholes Model is a calculation of the present value of the exercise price, and the risk-free rate is the rate used to discount the exercise price in the present value calculation. A larger risk-free rate lowers the present value of the exercise price, which increases the value of an option.

A risk-free rate is the rate of return of an investment with zero risk. The risk-free rate is developed by using the rate of a secure government bond that is in the same currency as the option. For example, a company with an option in USD should determine the risk-free rate using U.S. Treasury Bonds, which can be found on the Treasury’s website. The time horizon of the risk-free rate corresponds to the expected term input discussed above.

Volatility

Volatility is positively correlated with the value of an option. A larger volatility indicates a greater potential for the stock price to be higher, so the Black-Scholes Model assigns greater value to options with a higher volatility, holding all other inputs constant.

Volatility measures the magnitude of estimated stock price fluctuations during a given period. If available, your company should assess volatility using your historic stock prices. For publicly-traded companies, this information is readily available on active markets; however, the process is more complicated for pre-IPO companies.

A nonpublic company may determine the historic volatility of its stock based on transactions where the company used stock as consideration or issued new equity or convertible debt instruments. However, nonpublic companies rarely have enough historical information to support these estimates. Absent sufficient historical information, nonpublic companies should look at the historic volatility of comparable publicly-traded companies or the volatility of an appropriate “industry sector index” (ASC 718-10-30-20). In picking an appropriate industry sector index, the size and the industry of the nonpublic company should be considered. After picking an appropriate index, the nonpublic company should analyze the historic volatility over a time horizon corresponding with its expected term.

Nonpublic companies may calculate the historic volatility of a comparable public company or a peer group of companies from actual stock prices. Firms that provide 409A valuations will often calculate volatility for the nonpublic company, and nonpublic companies may be able to pull the calculated volatility from the 409A report into the Black-Scholes Model.  A nonpublic company may also find data disclosed by comparable public companies useful in calculating volatility. At times, a publicly-traded company may simply disclose its expected price volatility.

Other times, volatility can be solved with other publicly-available information. For example, a company with publicly traded options will have (a) a market option price, (b) a remaining contractual term, (c) a stated exercise price, and (d) readily available marketing information for the risk-free rate and dividend yield. These inputs can be put into the Black-Scholes formula to solve for volatility. This is referred to as implied volatility. In practice, historic volatility is used much more than implied volatility.

Though rare, companies may make adjustments to the estimated volatility to account for factors that may affect volatility. For example, if future performance is expected to differ from historic experience, a company might adjust the volatility calculation accordingly. This is because volatility is an estimate of future volatility.

Dividend Yield

Dividends are negatively correlated with an option’s value. This is because option holders do not have rights to the dividends issued by a company. The dividend yield is built into the Black-Scholes Model by subtracting the expected dividend rate from the current stock price.

Typically, determining the dividend yield does not require extensive analysis. Many companies will assume that current dividend yields are expected to remain constant over time and will use the current dividend yield at the grant date. Companies may also estimate the dividend yield by taking the average of several recent dividend payments. High growth companies (including pre-IPO companies) will frequently assume a dividend yield of zero percent.

Limitations

While the Black-Scholes Model is relatively simple and is widely used in the financial community, it does have limitations. The model is rigid in its inputs and makes significant assumptions.

Originally, the model was created to value European call options. As mentioned above, U.S. options can be exercised anytime between the vest date and the expiration date, whereas European options are only exercisable at the expiration date. Thus, U.S. options are more valuable than comparable European options because the options are more likely to be exercised in the money. Additionally, the Black-Scholes Model doesn’t allow any one activity to have multiple inputs. For example, though your company may identify consistent exercise patterns at various points in time after the options vest, your company must narrow this input estimate down to a single number. Having only a single input for each activity leads to a less precise measurement of the fair value of an option.

Further, the Black-Scholes Model has limitations in its assumptions. The Black-Scholes Model assumes that some factors remain constant over time, including dividends, risk-free rates, and volatility. It also assumes no transaction costs or taxes exist when purchasing options and that markets are perfectly efficient (i.e., large unexpected changes will not occur).

To account for these limitations, a company may choose to use a modified version of the Black-Scholes Model or to use a more complex model, such as a lattice model. In choosing between the Black-Scholes Model and a lattice model, your company should consider the trade-off between the precision of the estimate and the additional cost and effort required to create that estimate.

Lattice Models

The Black-Scholes Model is an example of a closed-form model—a model that uses an equation to solve for the fair value of an option. Lattice models, on the other hand, are more flexible and complex. Some examples of lattice models include binomial, trinomial, and finite-difference models. A lattice model assumes that price changes will happen over multiple time periods. Each period used in the model, therefore, has at least two price movements.

A lattice model can accommodate more detailed assumptions in its inputs, such as employee exercise patterns, volatility, dividends, and interest rates. For example, a lattice model forecast can include multiple estimates for exercise dates within each sub-period of an option’s life. By using multiple inputs rather than a single input, the fair value produced by the lattice model is able to be more precise.

Although lattice models can be more precise, the fair value that results from the lattice model may not be significantly different from the fair value produced by the Black-Scholes Model. PwC notes that “simpler types of lattice models do not always produce results that are comparable to results from more complex lattice models . . . In fact, very simple lattice models may be less reliable than the Black-Scholes Model with simple but well-supported assumptions.” A lattice model may not be helpful, for instance, if changing circumstances cause historical data to become irrelevant to future exercise patterns. Thus, companies should ensure that the model they use is well supported, weighing the advantages and disadvantages of each model before applying any one model.

Conclusion

Determining a supportable estimate of a stock option’s value will help a company correctly account for its stock compensation. Absent relevant market data, a stock option’s value can be estimated using either a closed-form model, such as the Black-Scholes model, or a lattice model. Understanding and weighing the benefits and costs of each model will help your company choose a model that will best develop an accurate estimate of an option’s value.

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Resources Consulted

 


 

Footnote

  1. SAB Topic 14 defines a “plain-vanilla” option as an option that is granted at-the-money, is conditional only on the employee completing service through the vesting date, has a time limit to exercise if an employee leaves the company, and is not transferable or hedgeable.
  2. Blackout periods are periods of time where stock options cannot be exercised.
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Author Brett Bartholomew

Brett has a passion for technical accounting and hopes to be involved in the cutting technologies that are disrupting the accounting industry. Outside of accounting, Brett loves playing ultimate frisbee, golf, and basketball. Lehi, Utah, is Brett’s home town, but he has traveled throughout the U.S., Thailand, Europe, and Central America.

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