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A cornerstone of modern financial theory, the Black-Scholes model was originally a formula for valuing options on stocks that do not pay dividends. It was quickly adapted to cover options on dividend-paying stocks. Over the years, the model has been adapted to value more complex options and derivatives. For example, a modified Black-Scholes model could be used to value an option with an exercise price that moves in relation to a stock index.
To estimate an option's fair value using the Black-Scholes model, it is first necessary to develop assumptions at the measurement date (generally the grant date). See SC 2.6.1 and SC 9 for information about the grant date and developing assumptions, respectively). The six key variables are:
  • Per share market price of the underlying stock
  • Exercise price of the option
  • Expected term of the option
  • Risk-free interest rate for the duration of the option's expected term
  • Expected annual dividend yield on the underlying stock
  • Expected stock price volatility over the option's expected term

The stock price is simply the quoted market price for publicly-traded securities or the estimated fair value of a share of stock for a private company on the measurement date. The exercise price is generally defined by the terms of the award. Developing the valuation model inputs (assumptions) for the remaining four variables requires judgment.
As described in ASC 718-10-55-27, the assumptions used to estimate an award’s fair value should be determined in a consistent manner. For example, if an entity uses the closing share price or the share price at another specified time as the current share price on the grant date in estimating fair value, that technique should be employed consistently from period to period for all awards.
Figure SC 8-1 summarizes the manner in which each assumption impacts the value of an option.
Figure SC 8-1
Impact of Black-Scholes assumptions on fair value
Assumptions
Impact on option's fair value as assumption/input increases
Impact on option's fair value
Less significant
More significant
Stock price
Increase
X
Exercise price
Decrease*
X
Expected term
Increase
X
Expected volatility
Increase
X
Expected dividend yield
Decrease
X**
Risk-free interest rate
Increase
X
* Assuming an at-the-money option, a higher exercise price (and stock price) would drive a higher option fair value, purely as a result of the higher time value component of the option value. For an in-the-money option, holding the stock price constant, the exercise price will have an inverse relationship on the intrinsic value of the option—i.e., a higher strike price would reduce the option's fair value.
**For a large change in dividend yield (e.g., a change from 3% to 6%) this assumption can become more significant.
We note that that Figure SC 8-1 represents high-level general trends that ignore the potential interactions between assumptions. For example, in certain cases, a longer expected term assumption may decrease the fair value of an award that is significantly in the money if a high dividend yield is assumed.

8.4.1 Expected term of an option

The Black-Scholes model uses a single input for an option's expected term (the weighted average expected term)—the anticipated time period between the measurement date (typically the grant date) and the exercise date or post-vesting cancellation date—to estimate the fair value of an employee stock option. The expected term falls between the option's vesting and contractual expiration dates. It can never be less than the period from the grant date to the vesting date. However, as employees may exercise options at widely varying times, developing the expected term assumption is highly judgmental.
SAB Topic 14 provides SEC registrants with a simplified method to calculate the expected term assumption for "plain vanilla" options when the company has no relevant exercise experience on which to develop their assumption. ASC 718-10-30-20A through ASC 718-10-30-20B provide a similar simplified method for nonpublic companies. If a company cannot apply this simplified method, it should develop its expected term assumption by analyzing its employees' past exercise patterns for similar options. See SC 9.3.1 for information on the simplified method for developing the expected term assumption and the factors to be considered by companies that do not use the simplified method.
An option's expected term can have a significant effect on its fair value. Figure SC 8-2 shows how varying expected term assumptions affect the fair value of options issued by a typical emerging company and by a mature company. A change in the expected term assumption will have a greater impact on an option's fair value if the option has a shorter expected term. In contrast, the impact tends to flatten out for longer expected terms. When there is less volatility in the price of the underlying stock (as is the case for the mature company), the fair value of options is lower for all possible expected terms as compared to options for a stock with higher volatility. The fair value is also more linear in relation to expected term.
Figure SC 8-2
Sensitivity of sample fair value to expected term input

8.4.2 Expected volatility of an option

Stock price volatility is another key input in all option-pricing models. ASC 718-10-20 defines volatility as "a measure of the amount by which a … price has fluctuated … or is expected to fluctuate … during a period," and also as "a probability-weighted measure of the dispersion of returns about the mean." In mathematical terms, in the context of the Black-Scholes model, volatility is the annualized standard deviation of the natural logarithms of periodic stock price changes over the option's expected term. In other words, volatility is a statistical measurement of a stock's relative propensity towards wide price movements over a given time and reflects the expected variability of the returns on a company's stock. The price of a less volatile stock fluctuates over a smaller range than does the price of a more volatile stock.
Volatility has a significant impact on the fair value of a stock option. Because a more volatile stock has greater upside potential (and greater downside risk) as a percentage of the stock price than a less volatile one, an option on a stock with high volatility has greater value than an option on a stock with low volatility, assuming all other assumptions are equal. The volatility assumption reflects the benefit of an option holder's right to participate in the upside potential (i.e., stock price increases) with less exposure to downside risk (i.e., stock price decreases). While a number of factors can affect a stock's expected volatility, in general terms, a more mature company is likely to exhibit lower share price volatility than an emerging or high growth company.
Option values are sensitive to changes in volatility assumptions. Figure SC 8-3 illustrates the sensitivity of fair value to stock price volatility for an emerging and a mature company with different expected term assumptions. The fair values for the mature company are higher than for the emerging company because the mature company has a longer expected term. However, the effect of the longer expected term would typically be offset to some degree by a lower volatility assumption for the mature company. For example, the fair values of options for the two companies shown in Figure SC 8-3 would be equivalent (about $50) if the expected volatilities of the emerging company and the mature company were approximately 73% and 53%, respectively.
Figure SC 8-3
Sensitivity of sample fair value to volatility input

8.4.3 Risk-free interest rate for options

The use of an interest rate in valuing an option reflects the time value of the exercise price for the period (the expected term) over which the option holder is able to defer the cash outlay of the exercise price. Management must determine the expected term of an option before it can select the risk-free interest rate because the interest rate must correspond to the duration of the option. ASC 718 requires that the assumed risk-free interest rate be based on the yield on the measurement date of a zero-coupon instrument, such as US Treasury STRIPS, with a remaining duration to maturity equal to the award's expected term. The higher the interest rate, the higher the fair value of the option.

8.4.4 Dividend yield of an option

Since the market price of a stock reflects, in theory, the value of all future dividends expected to be paid, the dividend yield assumption serves to reduce the value of an option for the dividends that will be paid prior to the point at which the option holder becomes a shareholder entitled to participate in dividends. Under ASC 718, the dividend yield assumption usually reflects a company's historical dividend yield (i.e., average annualized dividend payments divided by the stock price on the dates recent dividends were declared) adjusted for management's expectations that future dividend yields might differ from recent ones. The dividend yield assumption represents the expected average annual dividend payment over the life of the award. Because option or other award holders typically do not receive dividend payments prior to exercise or vesting, a higher dividend yield assumption will reduce the fair value of an award if all other assumptions and conditions of the award are equal. For awards when the holder is entitled to receive dividends prior to exercise or vesting, a 0% dividend yield is generally appropriate. See SC 9.6.3 for more details.

8.4.5 Black-Scholes model: Underlying theory

As noted earlier, the Black-Scholes model is based on the theory that a replicating portfolio can be built that exactly reproduces the payoff of an option based on certain assumptions. The replicating portfolio does this through a combination of shares of stock and risk-free bonds. The fair value of an option can be computed in terms of (1) the price of the underlying stock (or short positions in the stock) and (2) the price of a zero-coupon bond of the appropriate maturity (or short position on the bond), so long as the balance of long and short positions can continually be adjusted to exactly match the option's payoffs upon expiration.
Describing how the Black-Scholes model allocates the components of the replicating portfolio involves advanced financial theory and mathematics that are beyond the scope of this guide. Because some knowledge of the underlying theory may be helpful in understanding what drives an option's fair value, the following pages present an overview of two basic components of an option's fair value: intrinsic value and time value. Time value is itself subdivided into two further sub-components: minimum value and volatility value.

8.4.6 Intrinsic value of an option

The first component of the fair value of an employee stock option is intrinsic value. It is the value, if any, at any given date that an employee could realize if the option were exercised (i.e., the amount by which the underlying stock's market price is greater than the option's exercise price). The intrinsic value for a vested and unvested option is the same, even though an unvested option cannot actually be exercised until it is vested.
On the grant date, the intrinsic value of most employee stock options issued by US companies is zero because the exercise price typically equals the price of the underlying stock. Such options are said to be issued at-the-money. An option with a positive intrinsic value is said to be in-the-money, while one where the exercise price exceeds the underlying stock price has no intrinsic value and is said to be underwater or out-of-the-money.
Options have different risks from those of the shares underlying them. The risk of loss is always lower for an option-holder than a shareholder because an option-holder cannot sustain a loss greater than the value of the option—which is always worth less than the value of the underlying stock—while a stockholder can lose the entire price paid for or current fair value of the shares. As a result, option-holders enjoy the same opportunities for gain as a shareholder, but with less risk of loss.

8.4.7 Time value of an option

The second component of the fair value of an employee stock option is time value. This component is comprised of two sub-components: minimum value and volatility value.

8.4.7.1 Minimum value of an option

Minimum value is dependent upon the underlying stock price at grant date, the exercise price, the time to expected exercise, the expected dividend payments on the underlying stock during the option's life, and the risk-free interest rate.
Computing an option's minimum value does not require any particular assumptions about the movement of the underlying stock (i.e., expected volatility); in fact, the only significant judgment required is an estimate of the option's expected term. Additionally, judgments regarding the appropriate risk-free interest rate and dividend yield should be made, but these assumptions usually have a much smaller impact on the estimate of minimum value. Minimum value at grant date is the current value of company stock minus the net present value of funds that will be used in exercising the option, and is calculated by subtracting from the current stock price, the present value (using the risk-free interest rate) of both the exercise price and any dividend payments expected during the option's expected term. In essence, minimum value—which is usually substantially lower than fair value—represents that portion of an option's fair value that is not contingent on volatility, but rather just reflects the benefit of the time value of not having to pay the exercise price until a later date while still enjoying any appreciation of the stock price that may occur. Figure SC 8-4 illustrates the calculation of minimum value.
Figure SC 8-4
Illustration of minimum-value calculation
Assumptions:
  • Expected term—6 years
  • Exercise price—$50
  • Stock price on grant date—$50
  • Expected annual dividend yield—1% (annually compounded)
  • Risk-free interest rate—3% (continuously compounded)

Minimum value computation:
Current stock price
$50.00
Less:
  • Present value of exercise price ($50 discounted at 3% over 6 years)
41.76
  • Present value of expected dividends (at 1% over 6 years)
2.90
Minimum value
$5.34

8.4.7.2 Volatility value of an option

Under ASC 718, stock price volatility is considered when calculating an option's fair value.
In the Black-Scholes model, an option’s fair value will equal its minimum value when volatility is assumed to be zero, or a number very close to zero. Many software versions of the Black-Scholes model will not allow an input of zero volatility, so a very small value (e.g., 0.001%) may be used as the volatility input to demonstrate this equivalence. The volatility assumption should reflect the degree of uncertainty about possible future returns (price changes) on the underlying stock. Volatility value relates to an option’s unlimited upside potential and the limited downside risk of principal loss compared with the risk of holding the underlying stock.
Under the mathematical formula underlying the Black-Scholes model, as the value of the volatility assumption increases, the fair value of the option increases since a higher volatility raises the potential payoff. For example, if volatility was assumed to be 20%, 50%, and 80% for the option illustrated in Figure SC 8-4, the estimated fair value under the Black-Scholes model would be $11.52, $23.17, and $32.59, respectively.
Due to the time value and volatility value of an option, the fair value of an option is always higher than the option's intrinsic value. Even an out-of-the-money option (which has $0 intrinsic value) generally has some amount of fair value as there is a possibility of upside if the stock price appreciates without the risk of further downside loss if the stock price declines. However, fair value begins to converge with intrinsic value as the option approaches its expiration date, as well as for deep-in-the-money options.
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