# The Black-Scholes Model

What is the Black-Scholes Model (BSM), how is it developed, and how does it compare to lattice models? The answers can be found in this article.

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****

### 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 beft develop an accurate estimate of an option’s value.

### Related Articles

### Resources Consulted

**Footnotes**

**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.- Blackout periods are periods of time where stock options cannot be exercised.