Options can be valued: it's time for generally accepted option valuation principles. (Special Section: Stock Options).
To say that options can't be valued flies in the face of economic reality. The can be valued. They are valued. Money changes hands every day between willing buyers and willing sellers, and both sides of each transaction are acting in their own self-interest. There is no compulsion on either party. While it's difficult to prove, there is no evidence in the options market of the types of shenanigans that periodically have spoiled other markets.
Black-Scholes--Just What Is It?
The Black-Scholes option-pricing formula is the solution to the Black-Scholes-Merton differential equation under specific boundary conditions. Unfortunately, the result is not directly intuitive to nonmathemeticians, but there are numerous textbooks and articles that derive the formula if you want to look further. You only need to obtain a flavor of what Black-Scholes is about to understand the issues, and you don't have to be able to derive the formula mathematically in order to be able to use it effectively.
(see sidebar, p. 53).
Here we're only looking at what an option is and the variables that influence its value. You can take the formula on trust. If you want to see a straight application of the formula to any option valuation, you can go to http://www.blobek.com/black-scholes.html, which has a calculator you can use for free. Enter the variables, and the computer will give you the value. For a history of the Black-Scholes model and the chance to test your skill at managing a pretend portfolio without risking any money, visit www.pbs.org. Search by the subject "Business & Finance," and then click on NOVA: Trillion Dollar Bet.
Here are the key points about option valuation that you as a finance manager need to know.
* An option is a contract that gives the holder the right to acquire an underlying security, at a fixed price, for a period of time.
* An option is itself a derivative security, and its value depends on the price of the underlying stock.
The holder can exercise the option if it's favorable to do so but can let the option expire unexercised if it isn't beneficial.
* The option is marketable. That is, the holder has the choice of exercising it or selling the option to someone else.
* If the option is exercised, then the underlying stock can be sold immediately.
* The more volatile the underlying stock, the more valuable the option. This is because a more volatile stock is more likely to fluctuate and at some point be above the exercise or strike price.
* The longer the duration of the option contract, the more opportunity there is to realize value from exercising the option, so the option is more valuable. An option that expires next week to buy stock at $40 when the stock today is $35 is not worth much. An option with the same strike or exercise price and market price that still has 10 years to run will be quite valuable.
* Dividends paid on the common stock reduce the value of options. Option holders don't own the underlying security directly, so they don't receive dividends.
When the Financial Accounting Standards Board (FASB) issued its guidelines for stock options (Statement of Accounting Standards (SFAS) No. 23, "Accounting for Stock-Based Compensation"), it suggested that the valuation of stock options be performed by utilizing Black-Scholes or some other option-pricing model. There are, in fact, several different option-pricing models, but they usually provide very similar answers. We focus on the popular and oft-relied-upon Black-Scholes model. (For an example of the model, see the article "Accounting's Phoenix" on p. 45.) Essentially, the FASB requirement (SFAS No. 123 ** 19) is that whatever option-pricing model is used, the following variables must be included:
* Exercise price,
* Expected life of option,
* Current price of the underlying stock,
* Expected volatility of the stock,
* Expected dividends on the stock, and
* Risk-free interest rate.
Finally, the FASB says that, "The fair value of an option estimated at the grant date shall not be subsequently adjusted for changes in the price of the underlying stock or its volatility, the life of the option, dividends on the stock, or the risk-free interest rate." Thus, once the option is valued on the grant date, any subsequent changes--either up or down--are disregarded.
Overstatement of Black-Scholes Option Values Must Be Corrected
Now here's the clincher: The Black-Scholes model overstates the value of stock options granted to executives. It's this overstatement, not the inability to value the options, that has caused certain CEOs and CFOs to state that their options "cannot be valued" What they said would, in practice, be true if it were enlarged to say, "Executive options cannot be fairly valued by Black-Scholes without making several important adjustments."
Why does the model overprice executive stock options?
First, and foremost, stock options granted to employees are not marketable. If you owned an October 2002 call option on IBM at $80, when the stock was at $76.50 you could have called your broker and sold the option in one minute at $3.10 [quote from WSJ 8/16/02]. Depending on what you had paid to buy the option initially, you would have been able to lock in a gain or loss, whenever you wanted, with a single phone call.
Now look at an option on IBM stock held by its CFO. No matter what the strike price is, and no matter what the current price of IBM stock is, the option itself can't be sold. The CFO can exercise the option or continue to hold the option, but he can't cash in his "gain" simply by selling the option to someone else. Executive stock options aren't traded on an exchange; in fact, they are not marketable in any way whatsoever.
There's a tremendous amount of published research, supported by the IRS and court decisions, which states that securities that aren't marketable are worth less than those that are marketable. Common sense supports this observation. If you had a choice between two otherwise identical securities, one of which was fully marketable through a broker and the other that wasn't marketable at all, which would you choose? Obviously the marketable security is more valuable. The only question is, "How much would an illiquid security need to be discounted in order to induce people to buy it rather than its marketable counterpart?"
Research suggests that a 25% to 35% or even higher discount is necessary to induce people to hold a nonmarketable security. Black-Scholes, without adjustment, assumes that the option is marketable. Since executive options are not marketable, the model overstates the value without an adjustment for marketability.
Second, the longer the term of an option, the more valuable the option is. Most corporate stock option grants are for 10 years, with the proviso that the options don't vest for a period of years and the employee still must be employed in order to exercise the option. While illness, retirement, or death of the employee may permit immediate exercise, if the employee leaves the company, say to work for a competitor, the options expire unexercised.
There's a significant probability that of, say, 100 employees granted options in 1999, only a fraction of them will still be working for the same company in 2009. In fact, there's some evidence to suggest that, on balance, employee stock options with a 10-year horizon are exercised or expire within perhaps five years. Keep in mind that a five-year option is worth less than a 10-year option. Option theory suggests that early exercise of an option always costs the holder because they immediately lose the time value of the option. Financial theory suggests that for ordinary options that are marketable, it never pays to exercise; rather, you should sell the option to someone else who can use the remaining period of the option. This is great theory, but it doesn't apply to executive options because they can't be sold.
The third problem with Black-Scholes is volatility. All other things being equal, an option on a more volatile stock is worth more than an option on a stock that barely moves from day to day. Volatility provides value to an option holder.
But the same volatility that adds value to the derivative security tends to reduce value in the underlying security. If you have a choice of owning a stock with a history of slow and steady growth, compared to a highly volatile stock, many financial planners would recommend the former. Only gamblers and short-term traders like their holdings to be volatile. Well-known statistics of 10%-11% compound growth for equities over a 20- or 30-year time horizon assume that the company survives and grows. Highly volatile stocks, like those in the Internet and telecom sectors, may not even be around in 10 years!
The point is that while volatility increases the value of an option, it tends to decrease the value of the underlying stock. Now look at the situation of a corporate executive treated as an "insider" by the SEC. Stock held by insiders, whether previously acquired or currently available from existing stock options, can only be sold in certain tightly controlled windows. In short, even the insider's stock to be received in exercise of the option may not be marketable for a period of time, during which heightened volatility will certainly have an adverse impact on overall value.
A NEW METHOD
We have performed a number of valuations of options, primarily for determining fair market value for gift- and estate-tax purposes. In every case, the factors we've discussed were incorporated in the final valuation of the options. Each valuation assignment is fact-specific. But the fair value (the FASB's term) typically is less than 50% of the nominal value derived from an unadjusted application of Black-Scholes.
This gap between true fair market value and the unadjusted Black-Scholes value approach--greater than 50%--is supported by other valuation practitioners. In fact, the burden of proof should lie with the FASB to justify why they didn't explicitly allow for these value-affecting factors to be included in their GAAP requirements.
At the present time there's no definitive "generally accepted" approach to valuing executive stock options. In effect, each practitioner, each auditor, is on his or her own. But that the Black-Scholes valuation model overstates values by some significant amount is without question. We just don't know by how much.
The SEC wants to enhance comparability among companies, which requires similar approaches to valuing options. It's necessary that companies, appraisers, and auditors adopt a uniform approach to valuing executive options, but such an agreed-upon methodology doesn't yet exist.
Peer pressure is currently forcing many companies, so far at least on a voluntary basis, to account for option grants as an expense--as we're seeing daily in the business press--either because of future International Accounting Standards Board (IASB) or FASB mandates or because of voluntary compliance with the alternative provided in SFAS No. 123. Companies today can expense option grants, although up to now most have chosen to show only footnote disclosure.
If expensing of options is coming, and we fully believe it will, it's incumbent on the financial community to develop a single method for valuing the options. Such an approach must accurately capture all the factors that distinguish publicly traded options from executive options. Black-Scholes works fine for the former. It fails the latter.
IMA, FEI, AICPA, American Society of Appraisers (ASA), and Association for Investment Management Research (AIMR) should jointly sponsor research that will arrive at generally accepted option valuation principles. The acronym GAOVP doesn't exactly roll off the tongue. Nevertheless, it is vital that the financial community start developing GAOVP soon.
RELATED ARTICLE: The Black and Scholes Model
BY KEVIN R. RUBASH, BRADLEY UNIVERSITY
(adapted from A Study of Option Pricing Models, http://bradley.bradley.edu/~arr/bsm/model.html)
THE BLACK AND SCHOLES OPTION PRICING MODEL didn't appear overnight; in fact, Fischer Black started out working to create a valuation model for stock warrants. This work involved calculating a derivative to measure how the discount rate of a warrant varies with time and stock price. The result of this calculation held a striking resemblance to a well-known heat transfer equation. Soon after this discovery, Myron Scholes joined Black and the result of their work is a startlingly accurate option-pricing model. Black and Scholes can't take all credit for their work, in fact their model is actually an improved version of a previous model developed by A. James Boness in his Ph.D. dissertation at the University of Chicago. Black and Scholes' improvements on the Boness model come in the form of a proof that the risk-free interest rate is the correct discount factor, and with the absence of assumptions regarding investor's risk preferences.
C = SN([d.sub.1]) - [Ke.sup.(-rt)]N[(d.sub.2)]
C = Theoretical call premium
S = Current stock price
t = time until option expiration
K = option striking price
r = risk-free interest rate
N = cumulative standard normal distribution
e = exponential term (2.7183)
[d.sub.1] = In(S/K) + (r + [s.sup.2]/2)t/ s [square root of(t)]
[d.sub.2] = [d.sub.1] - s [square root of(t)]
s = standard deviation of stock returns
In = natural logarithm
In order to understand the model itself, we divide it into two parts. The first part, SN[(d.sub.1]), derives the expected benefit from acquiring a stock outright. This is found by multiplying stock price [s] by the change in the call premium with respect to a change in the underlying stock price [N([d.sub.1])]. The second part of the model, [Ke.sup.(-rt)]N([d.sub.2]), gives the present value of paying the exercise price on the expiration day. The fair market value of the call option is then calculated by taking the difference between these two parts.
ASSUMPTIONS OF THE BLACK AND SCHOLES MODEL:
1) The stock pays no dividends during the option's life
Most companies pay dividends to their shareholders, so this might seem a serious limitation to the model considering the observation that higher dividend yields elicit lower call premiums. A common way of adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock price.
2) European exercise terms are used
European exercise terms dictate that the option can only be exercised on the expiration date. American exercise terms allow the option to be exercised at any time during the life of the option, making American options more valuable due to their greater flexibility. This limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end of the life of a call, the remaining time value is very small, but the intrinsic value is the same.
3) Markets are efficient
This assumption suggests that people cannot consistently predict the direction of the market or an individual stock. The market operates continuously with share prices following a continuous Ito process. To understand what a continuous Ito process is, you must first know that a Markov process is "one where the observation in time period t depends only on the preceding observation." An Ito process is simply a Markov process in continuous time. If you were to draw a continuous process you would do so without picking the pen up from the piece of paper.
4) No commissions are charged
Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay some kind of fee, but it is usually very small. The fees that individual investors pay are more substantial and can often distort the output of the model.
5) Interest rates remain constant and known
The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In reality there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30 days left until maturity is usually used to represent it. During periods of rapidly changing interest rates, these 30-day rates are often subject to change, thereby violating one of the assumptions of the model.
6) Returns are lognormally distributed
This assumption suggests returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options.
Alfred M King, CMA, CFM, is vice chairman of Valuation Research Corporation, an appraisal company based in Princeton, N.J. You can reach him at (609) 243-7013 or firstname.lastname@example.org.
Summer Parrish, CPA, is senior vice president of Valuation Research Corporation. You can reach her at (609) 452-0900 or email@example.com.
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|Title Annotation:||Black-Scholes-Merton option pricing model|
|Author:||King, Alfred M.; Parrish, Summer|
|Date:||Oct 1, 2002|
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