# Current Controversies Regarding Option Pricing Models

## A view of the use and limits of option models (Part 1 of 2)

Option pricing models (OPMs) are increasingly used to estimate the discount for lack of marketability (DLOM) in the business valuation profession. Some analysts disagree about whether OPMs are applicable for estimating the DLOM. Since OPMs were originally derived to determine option prices for publicly traded securities, many analysts question the merits of applying them to closely held securities. This discussion explores the controversies of applying OPMs to estimate the DLOM for nonmarketable securities.

Business valuations prepared for gift and estate purposes usually involve the valuation of a closely held company. That is, the subject company shares are nonmarketable. In these valuations, the valuation analyst typically first estimates the value of the company as if the underlying shares were marketable. Then, the analyst incorporates a discount for lack of marketability (DLOM) to reflect the fact that the underlying shares are nonmarketable.

According to the American Institute of Certified Public Accountants (AICPA) Statement on Standards for Valuation Services (SSVS), the discount for lack of marketability is defined as “an amount or percentage deducted from the value of an ownership interest to reflect the relative absence of marketability.”^{1}

For purposes of this discussion, the concept of marketability relates to how quickly and with what degree of certainty the investment can be converted into cash at the owner’s discretion.

A very liquid security is an ownership interest of an actively traded stock. This security can typically be converted into cash within three business days of the sell decision. This is the typical investment benchmark for a fully marketable security. At the other end of the investment marketability spectrum is an ownership interest in a privately owned company that pays no dividends or other distributions, requires capital contributions, and limits ownership of the company to certain individuals. Of course, there exists a myriad of positions in between these two extremes in the investment marketability spectrum.

Since the 1980s, valuation analysts have used empirical DLOM studies to estimate the DLOM related to a noncontrolling interest in a closely held company. Such studies include (1) studies of price discounts on the sale of restricted shares of publicly traded companies (i.e., the restricted stock studies); and (2) studies of price discounts on private stock sale transactions prior to an initial public offering (i.e., the pre-IPO studies).

In 1993, David Chaffe introduced the concept of using an option pricing model (OPM) to estimate the DLOM. Chaffe wrote that by purchasing a put option to sell nonmarketable securities at the current stock price, the securities’ owner has effectively purchased marketability for the shares. And, therefore, the cost of the put option represents the DLOM, with the put option cost divided by the stock price representing the percentage DLOM.

Since Chaffe first introduced the concept of using put options to estimate the DLOM, other option pricing models to estimate the DLOM have been created and promulgated. A large majority of the subsequent literature about OPMs, including new OPM studies and critiques, has focused more on the mechanics of using OPM studies to estimate the DLOM than on critiquing whether or not a hedging strategy based on stock options is a legitimate way to estimate the DLOM for nonmarketable securities.

This discussion (1) introduces and summarizes commonly used OPMs to estimate the DLOM; and (2) describes the current controversies in regard to using OPMs to estimate the DLOM.

**OVERVIEW OF OPTION PRICING MODELS**

**Chaffe European Put Option Model**

As mentioned above, David Chaffe authored a 1993 DLOM option pricing study in which he related the cost to purchase a European put option to the DLOM.^{2} Chaffe theorized that “if one holds restricted or non-marketable stock and purchases an option to sell those shares at the free market price, the holder has, in effect, purchased marketability for those shares. The price of that put is the discount for lack of marketability.”^{3}

In other words, let’s assume there are two securities and the only difference between them is that one is marketable with a freely traded price and the other is nonmarketable. Let’s suppose someone tried to sell you the security that is not marketable, but also gave you the option to sell it back at its freely traded price at any time in the future.

Under these circumstances, the nonmarketable security is assumed to be equivalent to the fully marketable security. Therefore, the value of the option is equal to the DLOM, and the question becomes how should this hypothetical option be valued?

Chaffe relied on the Black-Scholes-Merton option pricing model (BSM model) to estimate the price of the option in his model. The inputs in the BSM model are:

- Stock price
- Strike price
- Time to expiration
- Interest rate
- Volatility

In the Chaffe model, the stock price and strike price equal the marketable value of the private company stock as of the valuation date; the time to expiration equals the time the securities are expected to remain nonmarketable; the interest rate is the cost of capital; and, volatility is a judgmental factor that is often estimated by reference to the volatility of guideline publicly traded stocks.

According to Chaffe, volatility for small privately owned companies is likely to be 60 percent or greater. Chaffe reached this conclusion based on the volatility for small public companies that are traded in the over-the-counter market.

According to the Chaffe study, the appropriate DLOM for a privately held stock with a two-year required holding period and volatility between 60 percent and 90 percent is between 28 percent and 41 percent.

According to Chaffe, “considering that volatility for shares of most smaller, privately held companies fit the ‘VOL 60%-70%-80%-90%’ curves, a range of put prices of approximately 28% to 41% of the marketable price is shown at the two-year intercept. At the four-year intercept, these ranges are 32% to 49%, after which time increases do not substantially change the put price.”

Chaffe noted that his findings are downward biased due to the reliance on European options in his model. Therefore, Chaffe concluded that his findings should be viewed as a minimum applicable DLOM. Exhibit 1 presents representative DLOMs based on the Chaffe model.

According to the Chaffe model, the implied DLOM is between 14.5 percent and 70.4 percent for stocks with (1) volatility between 25 percent and 125 percent and (2) holding periods between 1 year and 4 years, as shown in Exhibit 1.

Although this is a large range for these DLOMs, the results are consistent with other DLOM studies. In order to analyze the reasonableness of the Chaffe model output, this discussion compares the implied DLOM under various scenarios to the results of the FMV Opinions DLOM Study.^{5} These results are presented in Exhibit 2.

The FMV Opinions Study breaks down the DLOM by quintile and shows various financial data associated with each quintile. As shown in Exhibit 2, the DLOM increases as volatility increases. According to the Chaffe model, under a one-year holding period and assuming 75 percent volatility, the implied DLOM is 27.7 percent. This DLOM is similar to the fourth quintile DLOM of 27.9 percent in the FMV Opinions study, which reports 80 percent volatility.

In general, it appears from Exhibit 1 and Exhibit 2 that the DLOMs reported in the Chaffe model are reasonable for moderate volatilities, but at higher volatilities the Chaffe model generates high DLOMs.

**Longstaff Model**

Francis A. Longstaff authored a study that relies on stock options to estimate the DLOM for a private company.^{6} Whereas Chaffe based his study on avoiding losses, Longstaff based his study on unrealized gains. Another difference is that the Longstaff study allegedly provides an estimate for the upper limit on the value for marketability.

The Longstaff study is based on the price of a hypothetical “lookback” option. A “lookback” option differs from most other options in that the holder can look back at the end of the option’s life and retroactively exercise the option at either the lowest stock price (for a call option) during the holding period or the highest stock price (for a put option) during the holding period.

The Longstaff study assumes an investor has a single-security portfolio, perfect market timing, and trading restrictions that prevent the security from being sold at the optimal time. The value of marketability, based on these assumptions, is the payoff from an option on the maximum value of the security, where the strike price of the option is stochastic.

Longstaff analyzed securities with volatility between 10 percent and 30 percent because, “[t]his range of volatility is consistent with typical stock return volatilities.”^{7} However, small stocks (such as those traded over-the-counter and analyzed by Chaffe) typically have greater volatility, all else equal.

When comparing the Longstaff model results (Exhibit 3) to the FMV Opinions Study results (Exhibit 2), the Longstaff model reports DLOMs that (1) are far greater than the observed discounts from restricted stock transactions and (2) exceed 100 percent at reasonable levels of volatility.

**Transformed Longstaff Discount**

The Longstaff model DLOMs exceed 100 percent under certain assumptions—an illogical conclusion.

However, there is disagreement about whether the Longstaff model, and OPMs in general, conclude a DLOM or a liquidity premium (which needs to be converted to a discount).

Ashok Abbot believes that the Longstaff model results in a premium and not a discount. Abbot suggests the following transformation of the Longstaff model in order to convert the Longstaff model results into a DLOM:

DLOM=(Longstaff Discount)/(1+Longstaff Discount)

After applying this formula, the transformed Longstaff model returns lower DLOMs, as presented in Exhibit 4.

Compared to the FMV Opinions Study, the DLOMs under the transformed Longstaff model remain relatively high. For example, a one-year holding period with 75 percent volatility returns a DLOM of 43 percent from the transformed Longstaff model, which is much higher than the FMV Opinion Study’s third and fourth quintiles which use 72 percent volatility and 80 percent volatility, respectively.

The DLOMs from the transformed Longstaff model appear more reasonable than the DLOMs that result from the Longstaff model prior to the transformation. The issue of whether the OPM studies conclude a discount or a premium is explored in greater detail later in this discussion.

**Finnerty Model**

John D. Finnerty conducted an option-pricing study that “tests the relative importance of transfer restrictions on the one hand and information and equity ownership concentration effects on the other in explaining private placement discounts.”^{8}

The Finnerty Option-Pricing Study is an extension of the Longstaff study. H owever, unlike Longstaff, Finnerty did not assume that investors have perfect market timing ability. Instead, Finnerty modeled the DLOM as the value of an average strike put option.

As shown in Exhibit 5, the Finnerty model generates DLOMs that are relatively close to the average DLOMs reported in the FMV Opinions study. Assuming 75 percent volatility and a one-year holding period, the Finnerty model returns a DLOM of 16.3 percent. The FMV opinions study shows a DLOM of 16.7 percent using 72 percent volatility.

It appears the Finnerty model works reasonably well at lower volatilities, but yields low DLOMs at higher volatilities when compared to the restricted stock transactions presented in Exhibit 2.

Some OPMs are suitable at certain levels of volatility and produce results that appear reasonable. However, no OPM appears to line up well with the restricted stock transactions at all levels of volatility.

**LONG-TERM EQUITY ANTICIPATION SECURITIES (LEAPS) STUDIES**

In September 2003, Robert Trout published a LEAPS study that analyzed LEAPS and marketability discounts.^{9} Ronald Seaman updated the Trout LEAPS study in March 2010.^{10} Each of these LEAPS studies were conducted using a similar research logic and research design.

A long-term equity anticipation security is essentially a long-term stock option that offers price protection for up to two years into the future. Therefore, an investor who desires protection against stock price declines can purchase a LEAPS put option.

The LEAPS studies examined the cost of buying LEAPS put options and concluded that the cost of the LEAPS put option divided by the stock price is the DLOM.

The authors of the LEAPS studies concluded that the observed DLOMs are appropriately viewed as benchmark minimum price discounts when applied to privately held companies. This is because (1) the underlying securities on which the LEAPS are based are often much larger than the privately held subject company; (2) the underlying securities on which the LEAPS are based are marketable; (3) the LEAPS themselves can be sold at any time during the holding period; and (4) there is a known liquidity event (i.e., the sale of the underlying security) for the LEAPS.

**HOW COMMON ARE OPMS?**

It is difficult to determine to what extent OPMs are currently used in the valuation profession. Valuation Products and Services, LLC, presented a webinar on the DLOM. During that webinar, a poll was given that asked participants how they determined the DLOM.^{11}

According to the poll, approximately 85 percent of participants used restricted stock benchmark data and 53 percent used IPO benchmark data to help determine a DLOM as of September 2011. Similarly, participants were asked how many people use LEAPS,^{12} the Chaffe model, the Longstaff model, or the Quantitative Marketability Discount Model (QMDM).^{13}

Fourteen percent of participants said they used LEAPS, 8 percent said they used the Chaffe model, 9 percent said they used the Longstaff model, and 25 percent said they used the QMDM. Sixty-one percent said they used none of the above models. Therefore, according to the webinar poll, a meaningful number of listeners were using an option pricing model to help determine a DLOM.^{14}

In addition, according to Robert Duffy, most Big Four accounting firms incorporate the Finnerty put option model into their DLOM analyses.

**CURRENT CONTROVERSIES REGARDING OPTION PRICING MODELS**

There are some controversies regarding the use of OPM studies to estimate the DLOM for nonmarketable securities, including the following:

- Stock options do not exist for nonmarketable securities.
- The cost of put options may understate the DLOM.
- The cost of put options may overstate the DLOM.
- The cost of put options may be unrelated to the marketability of closely held company stock.
- DLOMs from OPM studies may result in a premium, and not a discount.

These five controversies are discussed below.

**Options Are Not Available for Owners of Nonmarketable Stock**

Given that there is no available option market for closely held shares, an owner of nonmarketable stock cannot purchase a put option to sell his or her shares at a later date. Therefore, an investor cannot really purchase liquidity for his or her interest in nonmarketable stock the way Chaffe theorized in 1993. And, since this strategy is not practically possible, many valuation analysts dismiss the use of OPMs to measure the DLOM for nonmarketable stock.

In addition to being unavailable in the market, an option-based strategy will be unavailable to the owner of closely held company stock if he or she may be contractually restricted from selling the stock. Operating agreements, partnership agreements, or stock transfer agreements often contain provisions that serve to restrict the marketability of the underlying shares of stock. When this is the case, even if the owner of a nonmarketable stock could purchase a put option to sell his or her shares at a later date, provisions in the shareholder agreements might prevent that exercise of the put option.

Issues related to the inability of nonmarketable stockholders to use options to purchase liquidity are criticisms with using OPMs to estimate the DLOM.

Therefore, in order to apply OPMs to closely held shares, two primary assumptions are needed. First, closely held shares combined with a put option are equivalent to marketable securities. Second, the price of the put option measures the DLOM.

While some may question the validity of these assumptions, OPMs may still provide a useful way of thinking conceptually about the DLOM for privately held shares. Just because a theory may initially appear implausible does not mean it is not useful or cannot provide further insight to the issue at hand. There are examples in other fields where some assumptions are used that seem unrealistic or oversimplified in order to achieve a better understanding of a given topic.

For example, some macroeconomic models, such as dynamic stochastic general equilibrium (DSGE) models, use simplified microeconomic assumptions to forecast economic growth, business cycles, and the effects of monetary and fiscal policy.

By analyzing the interaction of agents making microeconomic decisions, the models attempt to better forecast macroeconomic variables. The assumptions are relatively simplified, yet DSGE models have been improved upon and have allowed economists to think about how the economy will evolve over time.

Some economic theories of consumption also use oversimplified assumptions yet offer insight into spending patterns. An example is the permanent income hypothesis where a person’s spending reflects both permanent and transitory income. According to the theory, the average propensity to consume depends on the ratio of permanent income to current income.

When current income temporarily rises above permanent income, the average propensity to consume temporarily falls. When current income temporarily falls below permanent income, the average propensity to consume temporarily rises. This model uses very simple assumptions, but over long periods of time one should observe a relatively constant average propensity to consume, which is what the model suggests and the data support.

While these model assumptions are oversimplified approximations of consumer behavior, they still make useful contributions. The general acceptance of these models—in spite of their oversimplified assumptions—suggests that it may be premature to dismiss a model just because the model’s assumptions appear simple or unrealistic.

In the case of using OPMs to value a DLOM, the fact that there exists no option market for private shares does not necessarily mean OPMs should not be used. Rather, they may still provide insight or an approximation that is useful.

This article first appeared in the Autumn 2013 issue of *Insights*, a publication of Willamette Management Associates.

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Notes:

*Statement on Standards for Valuation Services*(SSVS1) (New York: AICPA, 2007): 43.- “European” options have a single exercise date. In contrast, the holder of an “American” option can exercise the option at any time during the existence of the option.
- David B.H. Chaffe III, “Option Pricing as a Proxy for Discount for Lack of Marketability in Private Company Valuations,”
*Business Valuation Review*(December 1993): 182–6. - Ibid.: 184.
- Lance S. Hall, “Responding to the IRS DLOM Job Aid,” Business Valuation Resources, LLC Webinar (October 12, 2011): slide 42.
- Francis A. Longstaff, “How Much Can Marketability Affect Security Values?”
*The Journal of Finance*(December 1995): 1767–74. - Ibid.: 1771.
- John D. Finnerty, “The Impact of Transfer Restrictions on Stock Prices.”
*Analysis Group/Economics*(October 2002). - Robert R. Trout, “Minimum Marketability Discounts,”
*Business Valuation Review*(September 2003): 124–6. - Ronald M. Seaman, “Minimum Marketability Discounts—5th Edition” (March 2010), http://www.dlom-info.com/pdf/Full_Report_2009_Study.pdf
- James Hitchner and Michael Gregory, “Navigating the IRS and the IRS DLOM Job/Practice Aid,” ASA Advanced Business Valuation Conference Presentation (October 8, 2012).
- LEAPS are identical in all respects to short-term options except they have a longer expiration date.
- The QMDM uses the discounted cash flow method to determine an appropriate DLOM.
- See Robert Duffy,“Why Finnerty’s Put Option Model Is the DLOM Model of Choice,”
*Financial Valuation Litigation Expert*(August/September 2011): 40.

*Aaron Rotkowski is a manager with Willamette Management Associates based out of Portland, OR. Mr. Rotkowski has performed the following types of valuation and economic analyses: business and stock valuations, fairness opinions, solvency and insolvency analyses, acquisition purchase price allocations, reorganization and restructuring analyses, tangible/intangible asset transfer price analyses, and lost profits/economic damages analyses. He can be reached at amrotkowski@willamette.com*

*Michael A. Harter is a senior associate with Willamette management Associates and also based out of the Portland, OR office. Mr. Harter can be reached at mahrter@willamette.com.*