Master the Six Inputs of Black Scholes Model

Written by: Caitlin Keefe, CPA, Kathryn E. Ryan, CPA

Is your business a start-up? Are you planning to issue – or have you recently issued – stock options or warrants? Then chances are you will need to know how to fair value those options or warrants.

Stock options are often used to attract, motivate, and retain employees. They are particularly popular with start-up businesses, as they do not have any up-front cash costs, but they do have potential future financial benefit to employees if the business continues to be successful.

As a founder or CFO of a start-up business, you will need to determine the fair value of stock options issued. If you do not have cap table software to rely on for that calculation, you will need to use a fair-value pricing model such as the Black Scholes Model yourself. Although it is not entirely necessary to understand the underlying mathematics of the calculation, it is imperative that you understand the basic inputs, as incorrect inputs can lead to drastic changes in stock compensation expense. Read more below to learn about the six key inputs used in every Black Scholes Model.

Underlying Stock Price

Definition: The fair value of the stock, generally common stock, on the day the option is issued.

Where can you find the underlying stock price or fair value of the stock? For a start-up business, the most common place to find this input is in the 409A valuation. A 409A valuation is usually prepared by a third-party and is used to determine the fair value of the company’s common stock per share as of a certain date. This value is typically good for one year following the valuation date, however, any significant change in the company may warrant the completion of a new 409A.

Exercise Price/ Strike Price

Definition: The price that the option holder will pay when they exercise their stock options.

The exercise (strike) price is what an employee will pay per share when they “cash in” their stock options. This price is fixed over the term of the option and should be clearly stated in the employee’s option agreement.

Example: If an employee has 1,000 options granted with an exercise price of $5 per share, they will pay $5 for each of the 1,000 options, or a total exercise price of $5,000.


Definition: The estimated remaining contractual term the individual has to exercise their stock option at the time of pricing.

The term used in the Black Scholes formula is different for stock options issued to employees and non-employees. Options issued to non-employees, such as consultants or advisors, should use the contractual term stated in the stock option agreement. Whereas options issued to employees should use the best estimate of the actual term of the option. In other words, the term should be estimated based on how long the company expects the option to be outstanding. This would incorporate estimates of when the option would be exercised or settled through forfeiture or otherwise.

Also, the company should be able to support their estimate of the remaining term for employees. If the company has granted options before, historical exercise data showing how long those options were actually outstanding can be used to determine the estimated term. If historical data is not yet available or is insufficient, the company should use the “simplified method.” This method calculates the expected term by taking the average of the sum of the vesting term plus the original contractual term.

Expected Term = (Vesting Term + Original Contractual term)/2

When calculating the vesting term, both the vesting period and the amount that vests each year should be factored into the equation above.

Example: If stock options are granted with a 4 year vesting period and 25% of the shares vest each year, and the original contractual term is 10 years, the expected term would be 6.25 years.

Expected Term = ((1*25% + 2*25% + 3*25% + 4*25%) + 10)/2 = 6.25 years


Definition: How much the security price is expected to move during the life of the option. This is expressed as a percentage per year.

This is another input for which a 409A valuation will come in handy. Non-public companies typically use the volatility stated in their most recent 409A valuation, which is derived from the volatilities of comparable public companies. If you do not have a 409A, you will need to do your own volatility calculation typically by evaluating volatilities of similar public companies. Volatility calculations get a little technical so you may need to dust off your college Statistics textbook!

To calculate the volatility of a similar public company, take the standard deviation of the daily change in the company’s closing stock price over the applicable period, for example, three years of daily price history from the grant date. Then, multiply by the square root of the number of trading days in a year. Excel is very useful here.

Annual Rate of Dividends

Definition: The annual rate of dividends that are expected to be declared on outstanding stock. This is expressed as a percentage.

As a company in its early stages, it is not likely that you will be declaring dividends. If this is the case, then this input is simple: your annual rate of dividends will be 0%. If your company regularly or historically declares dividends, refer to the following example showing how to calculate the dividend yield percentage.

Example: If a company has an annual dividend of $0.25 per share and the price per share is $5, the dividend yield would be $0.25 divided by $5, or 5%.

Risk-Free Interest Rate

Definition: The interest earned on cash with zero risk. This is expressed as a percentage.

Typically, this input is represented by the interest rate on U.S. Treasury securities, as they are considered to have zero credit risk. You should find the yield on a U.S. Treasury security with a term that most closely matches the expected (or contractual) term of the option being valued. Historical interest rate data on U.S. Treasury securities is posted by the U.S. Department of the Treasury on a daily basis and can be found on their website.

Example: If the term of the option granted on 3/31/17, is seven years, then the risk-free rate for a seven-year Treasury security would be 2.22%.

Once you have an understanding of these six inputs and where to find them, you will be in good shape to determine the fair value of stock options using this Black Scholes Valuation Model. There are various free versions of the Black Scholes equation calculator or spreadsheet online that can be used to calculate the fair value of your options and or warrants.