ESOs: Using the Black-Scholes Model
Companies need to use an options-pricing model in order to "expense" the fair value of their employee stock options (ESOs). Here we show how companies produce these estimates under the rules in effect as of April 2004.
An Option Has a Minimum Value
When granted, a typical ESO has time value but no intrinsic value. But the option is worth more than nothing. Minimum value is the minimum price someone would be willing to pay for the option. It is the value advocated by two proposed pieces of legislation (the Enzi-Reid and Baker-Eshoo congressional bills). It is also the value that private companies can use to value their grants.
If you use zero as the volatility input into the Black-Scholes model, you get the minimum value. Private companies can use the minimum value because they lack a trading history, which makes it difficult to measure volatility. Legislators like the minimum value because it removes volatility - a source of great controversy - from the equation. The high-tech community in particular tries to undermine the Black-Scholes by arguing that volatility is unreliable. Unfortunately, removing volatility creates unfair comparisons because it removes all risk. For example, a $50 option on Wal-Mart stock has the same minimum value as a $50 option on a high-tech stock.
Minimum value assumes that the stock must grow by at least the risk-less rate (for example, the five or 10-year Treasury yield). We illustrate the idea below, by examining a $30 option with a 10-year term and a 5% risk-less rate (and no dividends):
You can see that the minimum-value model does three things: (1) grows the stock at the risk-free rate for the full term, (2) assumes an exercise and (3) discounts the future gain to the present value with the same risk-free rate.
Calculating the Minimum Value
If we expect a stock to achieve at least a risk-less return under the minimum-value method, dividends reduce the value of the option (as the options holder forgoes dividends). Put another way, if we assume a risk-less rate for the total return, but some of the return "leaks" to dividends, the expected price appreciation will be lower. The model reflects this lower appreciation by reducing the stock price.
In the two exhibits below we derive the minimum-value formula. The first shows how we get to a minimum value for a non-dividend-paying stock; the second substitutes a reduced stock price into the same equation to reflect the reducing effect of dividends.
Here is the minimum value formula for a dividend-paying stock:
s = stock price
e = Euler's constant (2.718…)
d = dividend yield
t = option term
k = exercise (strike) price
r = risk-less rate
Don't worry about the constant e (2.718…); it is just a way to compound and discount continuously instead of compounding at annual intervals.
Black-Scholes = Minimum Value + Volatility
We can understand the Black-Scholes as being equal to the option's minimum value plus additional value for the option's volatility: the greater the volatility, the greater the additional value. Graphically, we can see minimum value as an upward-sloping function of the option term. Volatility is a "plus-up" on the minimum value line.
Those who are mathematically inclined may prefer to understand the Black-Scholes as taking the minimum-value formula we have already reviewed and adding two volatility factors (N1 and N2). Together, these increase the value depending on the degree of volatility.
Black-Scholes Must Be Adjusted for ESOs
Black-Scholes estimates the fair value of an option. It is a theoretical model that makes several assumptions, including the full trade-ability of the option (that is, the extent to which the option can be exercised or sold at the options holder's will) and a constant volatility throughout the option's life. If the assumptions are correct, the model is a mathematical proof and its price output must be correct.
But strictly speaking, the assumptions are probably not correct. For example, it requires stock prices to move in a path called the Brownian motion - a fascinating random walk that is actually observed in microscopic particles. Many studies dispute that stocks move only this way. Others think Brownian motion gets close enough, and consider the Black-Scholes an imprecise but usable estimate. For short-term traded options, the Black-Scholes has been extremely successful in many empirical tests that compare its price output to observed market prices.
There are three key differences between ESOs and short-term traded options (which are summarized in the table below). Technically, each of these differences violates a Black-Scholes assumption - a fact contemplated by the accounting rules in FAS 123. These included two adjustments or "fixes" to the model's natural output, but the third difference - that volatility cannot hold constant over the unusually long life of an ESO - was not addressed. Here are the three differences and the proposed valuation fixes proposed in FAS 123 that are still in effect as of March 2004.
The most significant fix under current rules is that companies can use "expected life" in the model instead of the actual full term. It is typical for a company to use an expected life of four to six years to value options with 10-year terms. This is an awkward fix - a band-aid, really - since Black-Scholes requires the actual term. But FASB was looking for a quasi-objective way to reduce the ESO's value since it is not traded (that is, to discount the ESO's value for its lack of liquidity).
Conclusion - Practical Effects
The Black-Scholes is sensitive to several variables, but if we assume a 10-year option on a 1% dividend-paying stock and a risk-less rate of 5%, the minimum value (assumes no volatility) gives us 30% of the stock price. If we add expected volatility of, say, 50%, the option value roughly doubles to almost 60% of stock price.
So, for this particular option, Black-Scholes gives us 60% of stock price. But when applied to an ESO, a company can reduce the actual 10-year term input to a shorter expected life. For the example above, reducing the 10-year term to a five-year expected life brings the value down to about 45% of face value (and a reduction of at least 10-20% is typical when reducing the term to the expected life). Finally, the company gets to take a haircut reduction in anticipation of forfeitures due to employee turnover. In this regard, a further haircut of 5-15% would be common. So, in our example, the 45% would be further reduced to an expense charge of about 30-40% of stock price. After adding volatility and then subtracting for a reduced expected-life term and expected forfeitures, we are almost back to the minimum value!
ESOs: Using the Binomial Model
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