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A Fair Market Approach To Valuing Stock Options

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By: Espen Robak | September 30, 2007

Are estate planners shortchanging their clients by applying obsolete and inappropriate valuation rules that overvalue transferred stock options?

Regardless of how you feel about the ability of older valuation methods to produce accurate estimates of fair market value in the past, as market realities have changed, the valuation methods specified in Rev. Proc. 98-34 and Rev. Proc. 2002-45 have become obsolete.

For many clients, the old methods produce option prices far in excess of fair market value.  And, with option grant plans from 4,000 companies covering more than 9 million plan members as of 2006 1 – and with more and more plans allowing inter-family transfers – the stakes for calculating accurate valuations are high.

The relevant revenue procedures are based on the valuation rules of Statement of Financial Accounting Standards No. 123, which the Financial Accounting Standards Board’s (FASB) revised in December 2004 (SFAS 123R).  The financial accounting standard prescribes valuation methods based on standard option-pricing models, adjusted for the historical experience of the issuing entity with early-exercise behavior.  However, SFAS 123R also says that "the best evidence of fair value for employee stock options is observable market prices of identical or similar instruments in active markets."  There is a tension between the prescribed methods and the market-based context of the financial accounting standard.  However, for the relevant revenue procedures, the tension is even greater, as the definition of fair market value is much more explicitly market-based.

Because they lack the liquidity of the publicly traded options the valuation models were designed to price, employee stock options are worth significantly less.  So what’s the best methodology for valuing stock options, and how can the most-accurate valuation methods be reconciled with the rules and procedures of tax valuation practice?

Tax Valuations vs. Accounting Valuations

First, start with an understanding of what SFAS 123R allows, and the recognition that different valuation methods serve different purposes.

Statement 123R establishes acceptable valuation methods for determining the compensation expenses associated with the reporting company’s employee stock option grants, so the methods allowed are not designed for valuing options for tax purposes.   Further complicating valuations, the standard of value in financial accounting is "fair value" while the standard of value in tax cases is "fair market value"
The most important difference between an SFAS 123R valuation and the true fair market value of illiquid options is that SFAS 123R does not allow discounts for illiquidity.  Instead, SFAS 123R requires that companies estimate the time-to-exercise of options granted, based as much as possible on actual early-exercise behavior of plan participants.2

SFAS 123R allows the use of closed-form models, such as the Black-Scholes method, and methods such as binomial/trinomial trees and Monte-Carlo simulation.  Black-Scholes may provide accurate values for short-term publicly traded options, but it overvalues stock options that are not publicly traded.  The Black-Scholes formula, in its simplest form, is as follows:

graph1, where

graph2, and

graph3

The model relies on these assumptions:3

  1. The stock price follows a constant Brownian motion (with µ and ? constant).
  2. Short selling with full use of proceeds is permitted.
  3. There are no transaction costs or taxes and all securities are perfectly divisible.
  4. There are no dividends during the life of the option or warrant.
  5. There are no riskless arbitrage opportunities.
  6. Security trading is continuous for both the option and the stock.
  7. The risk-free rate of return is constant and the same for all maturities.

While none of these assumptions holds perfectly in real-world situations, they hold well enough to make Black-Scholes the most commonly used model among options traders valuing fully liquid stock options on actively traded stocks.  Known biases in the model are minor and trading software compensates for them automatically. 

With non-tradable options and warrants, though, the Black-Scholes model overstates prices, often by a wide margin.  Black-Scholes and other SFAS 123R methodologies are inaccurate, even for financial accounting purposes, because the financial accounting standard rejects the application of a liquidity discount.  The alternative method of shortening the options’ expected average lives cannot accurately account for the discount an arm’s-length investor would apply to illiquid options due to their lack of tradability.  For one, early-exercise behavior is highly individual and, further, even the "average" plan participant in very large option grant plans will tend to exhibit highly variable early exercise behavior over time.4  Estimating expected terms at grant is often almost impossible, especially for young firms with limited option plan histories.  Such estimates tend to be conservative and overestimate the expected term.  As we shall also see later in this article, even if we could shorten the term accurately, these methods would still overvalue the average illiquid stock option. 

The illiquidity discount for non-tradable options and warrants varies significantly, depending on how far in the money the option or warrant is.  That’s partly because the holder of the option or warrant can often immediately realize its intrinsic value by exercising it early.  However, liquidity has value in and of itself, and illiquidity discounts apply to all non-tradable securities, separate from the possibility (in the case of options) of early exercise.

The "safe harbor" revenue procedures of the IRS – Rev. Proc. 98-34 and Rev. Proc. 2002-45 – are equally problematic.  They generally require use of the Black-Scholes or binomial models, do not allow illiquidity discounts, and place limitations on specification of the inputs to the models.

Rev. Proc. 98-34, which applies to valuation of stock options for calculating gift, estate and generation-skipping transfer taxes, allows the use of any valuation method that takes into account each of the following:

  • Exercise price of the option
  • Expected life of the option
  • Current price of the underlying stock
  • Expected volatility of the underlying stock
  • Expected dividends on the underlying stock
  • Risk-free interest rate for the expected term of the option

The volatility of the underlying stock must be based on the volatility disclosed in the financial statement for the fiscal year in which the valuation is made.  Likewise for dividends used in the valuation.  In determining the factor for the risk-free interest rate, you must use the yield to maturity of zero-coupon U.S. Treasury bonds as of the valuation date with a remaining term nearest to the expected life of the option.
To calculate the expected life of the option, you can use either the "maximum remaining term" of the option, which is the number of years remaining from the valuation date rounded to the nearest tenth of a year, or the "computed expected life," which is calculated by multiplying the maximum remaining term by the quotient of the weighted-average expected life, divided by the number of years from the date the option was granted.  However, the maximum remaining term must be used in many cases, including when:

  • The transferor of the option (or the decedent) is not the person initially granted the option.
  • The person transferring the option is not an employee or director of the company that granted the option as of the valuation date (except when the transfer takes place at death).
  • The option being valued does not terminate within six months after employment or service as a director ends (except when the transfer takes place upon the owner’s death or disability).
  • The terms of the option being valued permit the option to be transferred to a charity or to someone other than "the natural objects of the transferor’s bounty."
  • The option being valued has an exercise price that is not fixed on the valuation date (except when the transfer takes place at death). 
  • The option being valued has terms and conditions such that if all of the options granted in the fiscal year of the company that includes the valuation date had the same terms and conditions, the weighted-average expected life for the year would be more than 120% of the weighted average expected life reported for the year.
  • The company is not required by FAS 123 to disclose an expected life of the options granted in the fiscal year of the company that includes the valuation date.

    2. The FASB’s position is based on empirical research of early exercise patterns.

    3. Hull, J. (2006) Options, Futures, and Other Derivatives, 6th ed. Pearson Prentice Hall. pp 290-291.

    4. For example, early-exercise behavior changes with changing employee demographics and with something as unpredictable as the changing fortunes of the company’s stock in the market.

A second regulation, Rev. Proc. 2002-45, covers valuation of options for tax returns, and claims for refunds, credit or abatement when an option is granted as compensation or becomes fully vested contingent on a change in ownership or control.  Rev. Proc. 2002-13 refers to both Rev. Proc. 98-34 and Rev. Proc. 2002-13, which says a taxpayer may value a stock option "using any valuation method that is consistent with generally accepted accounting principles," as well as other factors from the tax regulations.

Rev. Proc. 2002-45 also notes that "a stock option will not be considered properly valued if the option is valued solely by reference to the spread between the exercise price of the option and the value of the stock at the time of the change in ownership in control or without regard to the other factors" included in the regulations.

So how can you obtain a value for stock options that is defensible for tax purposes, while also serving your clients’ needs?

Obtaining "Fair Market Value"

Rev. Proc. 98-34 emphasizes the importance of the market-based context.  As is common in almost all tax valuation guidelines, Rev. Proc. 98-34 says, "the value of property is the price at which the property would change hands between a willing buyer and a willing seller."  Indeed, the overriding goal of any tax valuation exercise is to arrive at fair market value, regardless of what safe harbor techniques are provided by the revenue procedures.  If the safe harbors calculate values that are significantly higher than fair market value, prudent advisors can, and in many cases should, advise clients to file their returns based on the fair market value of transferred assets, rather than the safe harbor value. 

The most accurate way to obtain the "fair market value" of a security, of course, is to sell it.  What it sells for is its fair market value.

When there is no public market for an asset, the next best thing is an estimate of what it would sell for privately.

To obtain the most accurate possible fair market values for illiquid assets, Pluris Valuation Advisors LLC has developed the LiquiStat™ database, which includes transactions involving illiquid securities sold through the Restricted Securities Trading Network (www.RestrictedSecurities.net).  While the number is growing, at the time of this writing the LiquiStat™ database included more than100 sales of non-traded warrants, exercisable for shares of publicly traded companies.

Pluris uses real-world transactions from LiquiStat to determine "fair market value" for illiquid assets.  Unlike prices derived from Black-Scholes and other methods, whether with early exercise or not, prices derived using LiquiStat are based directly on "observable market prices."

Table 5 below provides a description of the warrant sample.  For each transaction in the database, we computed the theoretical model value of the warrant, which is the value it would hold if both the warrant and the underlying stock were fully liquid and all assumptions of the Black-Scholes model held.5 

Table 5 – LiquiStat Warrant Trades

 

 

 

 

 

 

 

 

Intrinsic
Value

Moneyness

Time to Expiration (Years)

Market Price

Volatility

Black-Scholes
Discount

Mean

$1.10

0.15

3.3

$5.39

75.6%

41.5%

Standard Deviation

1.47

0.42

1.2

4.99

25.4

18.0

Minimum

0.00

-0.99

0.1

0.40

43.9

1.8

Maximum

6.76

1.72

5.0

21.12

178.5

71.4

             

Medians – Data Sorted by Magnitude of Discount from Black-Scholes Value

1st Quintile

$1.68

0.33

2.2

$4.75

54.9%

16.1%

2nd Quintile

1.30

0.24

3.6

4.66

72.5

33.5

3rd Quintile

0.26

0.00

3.5

5.50

68.8

44.0

4th Quintile

0.25

0.04

4.0

5.75

82.8

54.4

5th Quintile

0.00

-0.10

3.3

1.95

93.5

61.5

The "intrinsic value" of an option or warrant is the price it would yield if exercised (i.e., the stock price is greater than the strike price).  The "moneyness" of an option or warrant is its stock price divided by its strike price (S/K). 

Note that the average intrinsic value is significantly higher for warrants in the low-discount quintiles, while the average moneyness is significantly lower for warrants in the high-discount quintiles.

These real-world transactions show that non-traded options or warrants would never sell at full Black-Scholes value, using volatility inputs from the market and the full time to expiration.  The size of these discounts is evidence that Rev. Proc. 98-34 can dramatically overvalue non-traded options, especially if any of the exceptions requiring use of the maximum remaining term are operative.

In the LiquiStat database, illiquidity discounts are calculated from both (a) the theoretical option value and (b) the theoretical time value of each warrant (the full theoretical value minus the intrinsic value).  In the LiquiStat database, time-value discounts range from 20% to more than 100%, with a median of 61%.  Such discounts are far greater than discounts typically derived by adjusting the Black-Scholes inputs for shorter expected terms.

The LiquiStat database represents the first-ever study of real-world empirical data on transactions in non-traded options and warrants.  We believe there are two reasons why the discounts for options and warrants are so high:

  • Options and warrants are more leveraged than shares, since a certain percentage appreciation in the price of shares will lead to a greater percentage appreciation in the price of the option.
  • Holding periods for options and warrants, i.e., the period required to get the full benefit of their time values, are long.  In comparison, LiquiStat found lower discounts for restricted stock, which has a one-year holding period under Rule 144.

Time value discounts in the LiquiStat sample average about 1.5 times the full-value discounts obtained using Black-Scholes.  The longer the time to expiration, the higher the volatility, and the further out of the money an option is, the larger the discount.

Other Studies

That investors will not pay full Black-Scholes values for non-traded options and warrants has been predicted and analyzed in several theoretical papers.  These papers from the academic literature are generally confirmed by the analysis of the LiquiStat database.  Together, the empirical data and theoretical papers provide strong evidence that market-based valuations are better than the rules-based approaches of the relevant revenue procedures and financial accounting standards.

Most published work focuses on the behavior of holders of illiquid stock options (mostly, employee stock options) and provides proof both that Black-Scholes and other methods that treat options as if they are fully liquid produce valuations that are too high and that even adjusting for early exercise is unlikely to fully account for the illiquidity discount.  Kulatilaka and Marcus note that a holder who wants to reduce his option position would sell part of the position.6  "Because employee stock options are not transferable, however, the only way to cash them in is to exercise them … "  Such early exercise reduces the market value of the options, but do prior exercise patterns provide sufficient guidance for estimating current option values?  The evidence is that they do not.

Results from the Kulatilaka and Marcus study imply that historical exercise patterns, since they are driven by past stock price performance, are a poor guide for future exercise patterns.

For example, Kulatilaka and Marcus derive a model where early exercise is driven by the need for diversification.  While the value of traded options always increases with volatility, Kulatilaka and Marcus found that the value of illiquid options may sometimes decrease with increasing volatility, depending on the level of investor risk aversion, because higher volatility may lead to earlier exercise.

A study by Hall and Murphy found that executives demand large premiums for accepting stock options in lieu of cash compensation, because options are worth less to executives than they cost to the issuing firm.7  Applying a certainty-equivalent approach, they find that the Black-Scholes model always overvalues non-traded stock options, that far in-the-money executive options are routinely exercised at vesting or fairly shortly thereafter because the expected utility from locking in their gains exceeds the utility from holding the options.  Their model indicates that executives with low levels of risk aversion and a high concentration of wealth tied up in the company’s equity assign values to stock options between 25% and 70% of the Black-Scholes value.  In fact, in this model, assigned values in some cases are below intrinsic value.

Finally, there is also specific evidence that the FASB and IRS methods of shortening the time to exercise cannot fully account for the effect of limited liquidity.  A study by Finnerty shows that assuming early exercise alone provides discounts for lack of marketability for employee stock options that are too low and tends to overstate fair market values.8  Finnerty notes that since options are leveraged investments, the "impact of any transfer restrictions will be magnified, and the discount for lack of marketability should be greater" for options than for restricted stock.  Finnerty finds that employee stock options, at grant, are worth approximately half their Black-Scholes values.  Finnerty concludes that extrapolating exercise behavior from past stock-price patterns that may not be repeated produces inaccurate values.

Professionals who rely on accurate values to comply with regulatory requirements, and who want to ensure that their clients are paying no unnecessary taxes, would do well to consider a valuation methodology that is based on true market values, rather than hypothetical exercise periods.

5. For simplicity, the differences between actual and model prices are referred to herein as "illiquidity" or "marketability" discounts. The discounts may, in fact, represent any number of divergences from the theoretical model values in addition to just liquidity issues (for example, the Black-Scholes formula may consistently overvalue warrants with long time to expiration). However, we believe that the lack of marketability for these securities is the cause of the majority of the discount. The goal, of course, is not to separate various elements of the warrant discount, but to arrive at workable valuation models for non-traded warrants. This we can do without differentiating between the various causes of the discount.

6. Kulatilaka and Marcus (1994) "Valuing Employee Stock Options" Financial Analysts Journal 50 (Nov/Dec) p. 46-56.

7. Hall and Murphy (2002) "Stock Options for Undiversified Executives" Journal of Accounting and Economics 33 (Feb) p. 3-42.

8. Finnerty (2005) "Extending the Black-Scholes-Merton Model to Value Employee Stock Options" Fordham University working paper, January 2005.