Under Financial Accounting Standards Board Statement no. 123, Accounting for Stock-Based Compensation, companies are required to provide new note disclosures about employee stock options based on their fair value at the date of the grant. Companies also may base the recognition of compensation cost for new and modified options on these fair values. Since options granted to employees generally are not traded on an exchange, Statement no. 123 requires companies to use recognized option pricing models to estimate the fair values. The new disclosures are required in financial statements for fiscal years beginning after December 15, 1995; earlier application is permitted. However, the disclosures must include the pro forma effects of options and other awards granted in fiscal years beginning after December 15, 1994.

Many CPAs have heard of the option pricing model first published by Fischer Black and Myron Scholes in 1973. But only those involved with sophisticated investment activities are likely to have a familiarity that goes beyond a college finance course. The October 1995 issuance of Statement no. 123 means CPAs who compile and audit the financial statements of many public companies will need to develop new skills or refresh ones learned years ago.

The use of other models, such as the more complicated binomial model, also is permitted. However, most employee stock options don't have the unusual features that make use of these models necessary. The Black-Scholes model can be programmed into computer spreadsheets and even some pocket calculators. The same features that make Black-Scholes the primary tool of option dealers and other sophisticated investors to value a wide variety of options make it attractive to those needing to value employee stock options under Statement no. 23.

The Black-Scholes model assumes that stock prices follow the skewed lognormal distribution above in blue. This assumption means the percentage changes in stock prices follow the bell curve normal distribution shown in red. By expressing the problem in terms of the stock's rate of return, Black and Scholes were able to use common statistics formulas for normal probability distributions to solve stock option pricing problems.

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STARTING AT THE END

Before an option's current value can be calculated, its value at expiration must be considered. A call option gives the holder the right to buy a share of stock from the issuer at a predetermined exercise price. Exhibit 1, page 74, shows in purple the value at expiration of a call option with a $50 exercise price graphed as a function of the price of the related stock. At stock prices below $50, the option is worthless because the holder would buy the stock at the lower market price rather than exercise the option. At stock prices above $50, the option is worth the difference between the stock price and the $50 option price.

Exhibit 1 also shows a related bullish portfolio consisting of stock bought on margin overlayed in red and blue. This comparison shows that the value of the option at expiration is equal to the value of the bullish portfolio when the stock price is greater than $50. That is, at expiration the call option is equivalent to a portfolio of two hypothetical instruments:

* A contingent liability that requires the payment of $50 only when the stock price is $50 or greater. * A security whose value is equal to the price ofthe stock when the stock price is $50 or greater and zero otherwise.

All modern option pricing models say the present value of a so-called European-style option--one exercisable only at expiration--is the sum of the present values of each of the two hypothetical instruments in the equivalent portfolio. So far, this does not appear to be much of a step forward, since neither hypothetical instrument is traded. However, Black and Scholes made a breakthrough by formulating this problem in a way that made it possible to calculate the present value of each of the hypothetical instruments.

THE BLACK-SCHOLES MODEL

In developing their model, Black and Scholes made several assumptions about the stock and money markets. For purposes of this article, the most important assumptions are:

* Stock options are freely traded.

* The total rate of return-price changes plus dividends-on the stock in question, measured on the basis of continuous compounding over the option's life, is a random variable drawn from a normal bell curve distribution.

Exhibit 1: Call Option and Bullish Portfolio

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Since employee stock options generally do not trade, the first assumption obviously is not appropriate. Statement no. 123 explicitly addresses this, as discussed below. The statement also implicitly makes a substitute assumption that option issuers and holders are risk neutral-they would accept $1 for taking a 1 in 2 chance of losing $2 or a 1 in 3 chance of losing $3 and so forth. Practitioners will use this assumption when a practical answer is required. In this case, it also makes understanding the model easier.

Using the Black-Scholes model, what is the fair value of the option in exhibit 1, assuming it expires in one year! If the risk-free discount rate is 8.5% on a bond basis, the present value of $50 to be paid unconditionally in one year is $46. According to the risk-neutral assumption, the present value of the hypothetical contingent liability is $46 multiplied by the probability the liability will be paid. Black and Scholes reformulated the problem based on the probability the total return on the stock would be large enough so the stock price would grow to $50 or more on the expiration date, making exercise desirable and causing the contingent liability to be paid. This formulation and the second assumption allow common statistics formulas for normal probability distributions to be used to solve the problem.

To see how this works, assume today's stock price is $46.50 and the stock pays no dividends. First compute (on the basis of continuous compounding) the total return that would be realized if the stock grew from $46.50 to $50 in one year. One way to express the continuous compound interest formula is

future value = present value x e (rate of return x time) where e is a constant number equal to approximately 2.71828. Rearranging this formula yields In(future value t present value) = rate of return x time

Note that "ln" is the natural logarithm function, which uses the number e as its base, as opposed to the common logarithm function that uses the number 10 as its base.

Solving for the threshold rate of return, the result is

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The question then becomes: What is the probability the actual return on the stock for the next year will be at or above 7.26%?

This probability can be computed easily if it is known how many standard deviations the threshold rate is above or below the average rate of return. The letter Z often is used to name the threshold when it is expressed in terms of the number of standard deviations above or below the average. Without explaining the mathematical derivation, the Black-Scholes formula for Z is:

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"Volatility" is the name given to the standard deviation of the stock's rate of return. For a stock that pays no dividends, the present value is its current price.

If the stock's volatility is assumed to be 30%, the Z threshold can be computed as follows

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The probability for this threshold can be looked up in traditional normal distribution statistical tables found in many textbooks. As shown in the sidebar on page 77, the built-in function in a computer spreadsheet also can be used to find that 45.45% of the time, the stock's rate of return will be higher than the 7.26% threshold rate or .114 standard deviations above average. This means the present value of the hypothetical contingent liability is $20.91, which is ($46 X 45.45%).

Computing the value of the hypothetical security is slightly more complicated because there are many possible future stock prices greater than $50, each with its own probability of occurrence. Fortunately, a single computation using the stock's present value and an adjusted probability solves this problem. For a stock that pays no dividends, the current stock rice can be thought of as the sum of the present values of all future possible stock prices zero or greater multiplied by their individual probabilities of occurrence. The adjusted probability can be thought of as the percentage of that total present value resulting from future stock prices at or above $50. The threshold for the adjusted probability--called Z prime--is equal to Z minus an amount equal to the volatility level. This is equivalent to placing a minus sign before the first term of the prior Z formula, as follows:

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The probability is 57.37% that the stock's rate of return will be higher than the Z' threshold of -.186 or .186 standard deviations below average. This means the present value of the hypothetical security is $26.68, which is ($46.50 X 57.37%).

Exhibit 2: Put Option and Bearish Portfolio

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To put the two pieces of the model together, subtract the $20.91 value computed for the hypothetical contingent liability from the $26.68 value computed for the hypothetical security to get a net option value of$5.77.

COMPLICATIONS

A number of factors may complicate the situation and make option values more difficult to calculate.

Early exercise. The Black-Scholes model was developed for European-style options exercisable only at expiration. Most employee stock options are American-style and can be exercised at any point during their life, after a vesting period. If options can be freely traded, this generally doesn't matter because the holder would be better off using the option as margin collateral or selling it at its higher fair value rather than exercising it early. However, this is not true for most employee stock options, which cannot be pledged or sold and often are exercised early.

Statement no. 123 addresses this problem by stipulating that the option life is the expected time until the option is exercised, instead of its contractual term. Shortening the option's life reduces its value. The FASB concluded this was a reasonable way to recognize that restricted options were less valuable than freely traded ones. The statement also implicitly treats the option as if it were a European-style option-- thereby conforming it to the Black-Scholes model. Statement no. 123 also permits a different type of adjustment that assumes the option is exercised when the stock price reaches a particular level. Implementing this method requires the more complicated binomial option pricing model.

Dividends. The original Black-Scholes model dealt with stocks that paid no dividends. Adjusting BlackScholes to reflect dividends requires reducing the current stock price by the value of dividends to be paid during the life of the option, since most options do not give holders the benefit of dividends paid before exercise. This is commonly done by assuming dividends will be a known percentage of the stock price and discounting the current price by that amount.

For example, if the stock in question paid dividends of 0.825% of its price every quarter-which is a 3.3% annual rate- they could be reinvested so one share of stock could grow, with compounding, into 1.0334 shares in a year. Using the standard interest formula, the present value of one share of stock in one year, without dividends, is ($46.50 1.0334), or $45. Because part of the total return is now paid in dividends instead of being represented by stock price growth, this adjustment reduces both probabilities while increasing the total return threshold. By assuming the dividend payout rate is constant, the stock growth rate distribution has the same shape and standard deviation as the total return, so no adjustment to volatility is necessary. Using this additional assumption about dividends, the example option would have a value of$4.94, which is [($45 X 53.04%) - ($46 X 41.14%)].

Volatility, The Black-Scholes model uses the volatility expected over the life of the option--the coming year in the previous example. This prediction of future volatility is a forecast and is the variable to which option values typically are most sensitive. Volatility varies over time according to changes in conditions affecting markets generally and the stock in particular. Actual historical volatility can be measured, as shown in the computer spreadsheet in the sidebar, and used as a guide for estimating future volatility. Market-traded options can be analyzed to determine the volatility their prices imply. In any event, careful judgment is required to choose an appropriate volatility estimate over the typically long life of employee stock options.

Volatility generally is expressed on an annualized basis. However, the probabilities in the Black-Scholes model are based on the cumulative volatility over the option life. Cumulative volatility is commonly computed by multiplying the annual volatility by the square root of the option term measured in years. Therefore, the cumulative volatilities for a three-month option on the stock in the example would be 15%, which is (30% X sqrt(1/4)) and for a four-year option, 60%, which is (30% X sqrt(4)).

Put options. The Black-Scholes model can value put options--which give the holder the right to sell stock at a predetermined price--as easily as call options. As exhibit 2, page 75, suggests, all that needs to be done is to use the reciprocal probabilities-which is one minus the call probability-and change the signs to make the hypothetical stock position short instead of long and the hypothetical borrowing a cash investment instead.

TIME TO START IS NOW

Statement no. 123 adds option pricing modeling to the basic tools CPAs use to compile and audit the financial statements of companies that grant employee stock options. Honing these skills will take practice and gathering the necessary information to use them will take time. Depending on the volume of option activity, a company may need new data processing applications to implement Statement no. 123. For calendar-year companies, options granted in 1.995 must be included in the pro forma disclosures. Management also may wish to revisit option grant practices and related accounting policies in light of Statement no. 123. Together, these concerns mean CPAs must start now to assemble the pieces of the option pricing models.