Under U.S. accounting rules, stock options and similar instruments given to employees as compensation must be recognized based on their fair value. (Similar instruments include stock appreciation rights and restricted stock units.) With few exceptions, this rule applies to both public and private companies. Some may argue that it is difficult or impossible to accurately determine fair value when these options are not actively traded; however, GAAP says that only in "rare circumstances" will it not be possible to reasonably estimate the fair value of stock options.

The fair value of options and similar instruments is calculated using option-pricing models. By far the most popular is the Black-Scholes-Merton (BSM) formula. The binomial lattice is also used, but by far fewer companies than those using BSM.

Black-Scholes-Merton

BSM is a deceptively simple formula used to value publicly traded stock options and stock options granted to employees. It uses six inputs (stock price, exercise price, and expected term, volatility, risk-free rate, and dividend rate) to calculate the current fair value of an option. IF each of the six inputs are accurate then BSM produces an accurate value. However, obtaining accurate inputs is difficult because each of the inputs other than the stock and exercise prices must reflect the expectations for the term of the option.

Expected option term. BSM values an option based on its expected term, not its contract term. The expected term is the time from the grant of the option until the date the employee exercises it. The term of an option is one of the primary drivers of fair value, so it is important to make an accurate estimate.

Expected volatility. The current value of an option incorporates the expected volatility of the underlying stock over the expected option term. The starting point for estimating future volatility may be historical volatility, but in some cases history may be secondary to the supportable expectations for the future. Volatility is another primary driver of option value, so an accurate estimate of future volatility is a must.

Expected risk-free rate. An option pricing model incorporates the risk-free interest rate appropriate to the expected term of the option. For example, if the option has an expected term of seven years then the expected risk-free rate of a seven-year U.S. Treasury bond is an acceptible risk-free rate. Notice the word "expected" again. This implies that when valuing an option we must estimate the future rate of the appropriate instrument. Admittedly, a difficult task.

Expected dividends. Finally, BSM requires an estimate of the expected dividend rate over the expected term of the option.

When looking into the future a company may conclude that input rates will vary over the expected term of the option. When this is the case a weighted average of the future rates is used as inputs into the BSM pricing model.

Binomial lattice

Although much less used than BSM, the binomial lattice has the potential for more accurately valuing stock options and similar instruments awarded to employees. The lattice is not a single formula, but is instead a software model that estimates (1) the stock price behavior over the contract term of an option, and (2) the option exercise behavior of the employees holding the options. Because it is a software model, not a single formula, it can do things that BSM cannot:

Changing assumptions. The lattice permits us to change assumptions for expected rates during the option contract. For example, we may determine that volatility should be 35% for the first five years of the option and 25% for the next five years. The lattice permits us to indicate different volatility rates for each reporting period during the term of the option. Similar changing inputs can be made for the expected risk-free rate and expected dividends.

Exercise behavior. The lattice permits us to input in to the model the future conditions that may cause employees to exercise options. For example, if the option vests over four years we can specify that exercise cannot take place during the first four years. If we expect future blackout periods, during which options cannot be exercised, we can build these into the model. Finally, we can input assumptions regarding the circumstances that motivate exercise. These include, e.g., the stock reaching a price that is a certain multiple of the option exercise. This particular assumption is referred to as the suboptimal exercise factor (SOE). For example, we might input "2" in to the model as the SOE factor, which tells the model to assume an exercise when the stock price is two times the exercise price.

Services offered

We provide option valuation services using both the Black-Scholes-Merton formula and the binomial lattice. We can assist with such things as volatility calculations, including when and when not to use historical volatility; calculation of the expected term of an option, when using BSM; and calculation of the suboptimal exercise factor when using a binomial lattice.

Contact David Hardesty for more information...