

ESOs: Using the BlackScholes Model
Companies need to use an optionspricing 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 EnziReid and BakerEshoo 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 BlackScholes 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 hightech community in particular tries to undermine the BlackScholes by arguing that volatility is unreliable. Unfortunately, removing volatility creates unfair comparisons because it removes all risk. For example, a $50 option on WalMart stock has the same minimum value as a $50 option on a hightech stock.
Minimum value assumes that the stock must grow by at least the riskless rate (for example, the five or 10year Treasury yield). We illustrate the idea below, by examining a $30 option with a 10year term and a 5% riskless rate (and no dividends):
You can see that the minimumvalue model does three things: (1) grows the stock at the riskfree rate for the full term, (2) assumes an exercise and (3) discounts the future gain to the present value with the same riskfree rate.
Calculating the Minimum Value
If we expect a stock to achieve at least a riskless return under the minimumvalue method, dividends reduce the value of the option (as the options holder forgoes dividends). Put another way, if we assume a riskless 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 minimumvalue formula. The first shows how we get to a minimum value for a nondividendpaying 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 dividendpaying stock:
s = stock price
e = Euler's constant (2.718â€¦)
d = dividend yield
t = option term
k = exercise (strike) price
r = riskless rate
Don't worry about the constant e (2.718â€¦); it is just a way to compound and discount continuously instead of compounding at annual intervals.
BlackScholes = Minimum Value + Volatility
We can understand the BlackScholes 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 upwardsloping function of the option term. Volatility is a "plusup" on the minimum value line.
Those who are mathematically inclined may prefer to understand the BlackScholes as taking the minimumvalue formula we have already reviewed and adding two volatility factors (N1 and N2). Together, these increase the value depending on the degree of volatility.
BlackScholes Must Be Adjusted for ESOs
BlackScholes estimates the fair value of an option. It is a theoretical model that makes several assumptions, including the full tradeability 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 BlackScholes an imprecise but usable estimate. For shortterm traded options, the BlackScholes 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 shortterm traded options (which are summarized in the table below). Technically, each of these differences violates a BlackScholes 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 10year terms. This is an awkward fix  a bandaid, really  since BlackScholes requires the actual term. But FASB was looking for a quasiobjective 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 BlackScholes is sensitive to several variables, but if we assume a 10year option on a 1% dividendpaying stock and a riskless 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, BlackScholes gives us 60% of stock price. But when applied to an ESO, a company can reduce the actual 10year term input to a shorter expected life. For the example above, reducing the 10year term to a fiveyear expected life brings the value down to about 45% of face value (and a reduction of at least 1020% 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 515% would be common. So, in our example, the 45% would be further reduced to an expense charge of about 3040% of stock price. After adding volatility and then subtracting for a reduced expectedlife term and expected forfeitures, we are almost back to the minimum value!
ESOs: Using the Binomial Model



Black's Model
A variation of the popular BlackScholes options pricing model ... 
LatticeBased Model
An option pricing model that involves the construction of a binomial ... 
Option Pricing Theory
Any model or theorybased approach for calculating the fair ... 
Boundary Conditions
The maximum and minimum values used to indicate where the price ... 
Optionable Stock
A stock that has options trading on a market exchange. Not all ... 
Implied Volatility  IV
The estimated volatility of a security's price.

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