# Current Controversies Regarding Option Pricing Models

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

It is an interesting theoretical exercise to consider what a put option on nonmarketable shares would look like if it could be underwritten. Chances are it would be much more expensive than a put option on similar marketable shares.

First, the analyst may consider who would underwrite such an option. The potential underwriter would have to be willing and able to take an ownership interest in the underlying securities. Yet the potential underwriter cannot be so interested in owning the underlying securities that he or she would write a put option on the securities rather than simply buying the stock outright. These circumstances greatly reduce the number of potential underwriters for put options on nonmarketable stock.

Let’s assume an underwriter was identified. Let’s also assume that the subject interest—a 10 percent interest—was recently valued on a marketable basis at $100 and on a nonmarketable basis at $80, and that the goal of the put option was primarily to gain liquidity in two years. Therefore, the security owner would want a put option with a two-year term and a strike price that was close to the current market price of the stock (i.e., $100).

The put option underwriter is probably concerned with one additional type of risk that does not affect put options on marketable securities: the underlying securities are not marketable, but the OPM price is based on marketable securities. Therefore, the underwriter will need to be compensated for the fact that the securities are nonmarketable by incorporating either a lower strike price or a higher premium in the option price.

If the put option price is based on the nonmarketable stock price, and the underwriter agrees with the selected DLOM, then this risk is mitigated.

However, the OPM studies assume that the option is written based on the marketable stock price. If the underwriter writes the option with a strike price equal to the marketable stock price, then the underwriter will likely charge a premium over the formula-derived option price to reflect the fact that the underlying security is not marketable. The OPM studies assume the option price is equal to the price derived from the OPM (i.e., no premium is added).

One way or another, the underwriter will expect to be compensated for the fact that the underlying securities are not marketable. Since OPMs don’t compensate the option writer for marketability-related risk, this risk will have to be accounted for another way. Note that this compensation is above and beyond the price that results from a theoretical option-pricing model.

Based on the scenario just described, OPM studies may understate the DLOM. This is because the put option on nonmarketable shares does not include a component in the price to provide liquidity to nonmarketable shares. That is, the theoretical price of a put option on nonmarketable shares (the cost to gain liquidity) may be greater than a put option on otherwise similar marketable shares (which already have liquidity). However, some analysts take an opposite view—that the cost of put option overstates the DLOM.

This point is addressed next.

**The Cost of the Put Option May Overstate the DLOM **

We previously illustrated why some analysts believe the cost of the put option understates the DLOM. If a put option was written on a nonmarketable security, the option underwriter would likely charge a premium relative to the price indicated by an OPM in order to account for the lack of marketability inherent in the underlining security. Based on this argument, the put option cost from an OPM understates the DLOM.

In 2009, Internal Revenue Service analyst Harry Fuhrman critiqued the LEAPS studies (a type of OPM study) for *overstating* the DLOM. Fuhrman reasoned that (1) OPMs estimate the DLOM as the cost to lock in a price for the subject security; and (2) purchasing a put option overstates the cost to lock in a security’s price (the price is not locked in since the investor can still benefit from any appreciation in the price of the stock).

According to Fuhrman, “To ‘lock-in’ a security’s price today, an investor would undertake two courses of action: 1) purchase a put option to protect against any downside risk . . . or 2) have the ability to sell a call option related to any upside potential in the stock.”^{15}

By netting the cost to acquire the put option with the income from selling a call option, the investor has eliminated both the downside and upside related to changes in the price of the underlying security. His or her return is certain.

Fuhrman’s estimated DLOM using this procedure is calculated as the cost of the put option minus the income from the call option, divided by the stock price.

In the example provided in his critique of the LEAPS studies, Fuhrman illustrates how the DLOM estimated based only on the cost of the put option would range from 20 percent to 28 percent using AT&T put options in January 2009 and would be between 4 percent and 8 percent if the cost of the put option was netted against the income from a call option.

In a response to Fuhrman’s critique, Seaman notes that a DLOM “does not attempt to ‘lock-in a security’s price today,’ which Fuhrman states as the objective in his example. A DLOM simply attempts to measure the investor’s risk, a major part of which is the risk of loss in value over time. That is precisely the risk measured in an analysis using LEAPS put options.”^{16}

Fuhrman is not the only one to critique the OPM studies for producing discounts that are too large. In an April/May 2013 article, Jay Fishman and Lester Barenbaum wrote that, “it is our view that use of the cost of put options overstate [sic] the discount for lack of marketability.”^{17}

Fishman and Barenbaum suggested that a better way to use options to estimate the DLOM would be the cost of a prepaid variable forward (PVF) contract, which involves buying a put option, selling a call option, and borrowing money.

Since the theory of using a prepaid variable forward contract is relatively new, it is unclear how this method to estimate the DLOM will be received by practitioners, courts, and academics. However, the DLOM using prepaid variable forward contracts may be similar to a company’s cost of debt.

As noted in the Fishman and Barenbaum article, a PVF contract is “a constructive sale that fully monetizes an asset position with *borrowing cost representing the discount*”^{18} [emphasis added]. This result seems unreasonably low.

**The Cost of a Put Option May Not Reflect Marketability **

A prior section discussed the fact that an underwriter of a put option on nonmarketable stock may charge a premium based

on the lack of marketability in the underlying securities. This suggests that option pricing models measure factors other than liquidity.

A distinction between put options and marketability is that a put option becomes more expensive (valuable) as the time to option expiration increases while the nonmarketable stock become less valuable as the expected holding period increases. So, although both options and the DLOM are affected by time horizon and volatility, they are affected in different ways. In this way, stock options aren’t analogous to the DLOM; they are more like opposite sides of the same coin.

We next examine the relationship between the cost of a put option and the magnitude of the DLOM by comparing the factors that affect each. This relationship is analyzed to determine if/how the cost of a put option is related to marketability.

**Factors that Affect the Option Price **

An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price at or before the date of expiration. Since it is a right and not an obligation, the holder can choose to not exercise the right and allow the option to expire.

Stock options are bought and sold by three types of investors: (1) hedgers, (2) speculators, and (3) arbitrageurs. Hedgers use options to protect themselves against negative price movements (i.e., an investor that owns shares in Apple, Inc., common stock may want to protect against a near-term decline in the stock price).

Speculators use options to make bets about future price movements (i.e., an investor that wants to bet that shares of Apple common stock will increase can magnify his or her losses and gains using options compared to owning the stock outright).

Arbitrageurs use options to lock in a known profit by taking multiple positions in the stock, options, and futures markets. Arbitrageurs are important to the financial markets because they help establish the prices of stock options. Note that none of the investors that use stock options do so to gain liquidity. Stock options are purchased and sold for reasons completely unrelated to liquidity.

To illustrate how arbitrageurs establish option prices, let’s consider a simple portfolio that is (1) long one stock traded at $10/share and (2) short in two call options with a strike price of $11/share. Let’s assume there are two possible outcomes for the price of the stock in three months: either $9 or $11.

If the stock ends up at $11/share, then the portfolio is worth $9/share ($11/share × 1 share−2 options). If the stock ends up at $9/share, then the portfolio is also worth $9 ($9/share × 1 share).

Regardless of what happens to the stock price, the value of the portfolio will always be $9/share. Since a riskless portfolio earns the risk-free rate, one can derive the option price based on (1) the risk-free rate (let’s assume it is 3 percent), and (2) the facts in the example above. The present value of the portfolio, based on the above, is $8.93 (i.e., the present value of $9/share).

One can also determine the price of the call option in this scenario. Based on the information outlined above, the price of the call option such that no arbitrage opportunities exist is $1.07.

In this example, if the price of the call option was greater than $1.07, then the portfolio would cost less than $8.93 to set up and would earn more than the risk-free rate of return.

If the price of the call option cost less than $1.07, then shorting the portfolio would provide a way of borrowing money at less than the risk-free rate of return. If the price of the call option was anything other than $1.07, arbitrageurs would invest in such a way to earn a riskless profit and, eventually, supply and demand factors should force the call option price back to $1.07.

The above example provides a simplified method to estimate the price of a call option. In that example, there were only two possible outcomes for the price of the stock and each possibility was just as likely to occur. A more complex and commonly used method to value stock options is the BSM Model.

The BSM Model is analogous to the no-arbitrage example discussed above. However, in the BSM Model, the position that is set up is riskless only for an instantaneously short period of time.

The value of an option using the BSM Model is determined primarily by the six variables listed below.^{19}

*Current value of the underlying asset*: Options derive their value from an underlying asset.*Change in value of the underlying asset (i.e., volatility)*: The buyer of an option has the right to buy or sell the underlying asset at a fixed price. The higher the variance in the expected value of the underlying asset, the greater the value of the option.*Dividends paid on the underlying asset*: The value of the asset can be expected to decrease if dividends are paid on the asset during the life of the option. The value of a call on the asset decreases with the size of the expected dividend payments. The value of a put increases with expected dividend payments due to the cost of delaying exercising options that are in-the-money.^{20}*Strike price of option*: The value of a call will decline as the strike price increases, and the value of a put will increase as the strike price increases.*Time to expiration on option*: Both calls and puts are more valuable as the time to expiration increases. The long time to expiration provides more time for the value of the underlying asset to change, which increases the value of both types of options.*Risk-free interest rate*: Since a buyer of an option must pay the price up front, purchasing options involves an opportunity cost which depends on the level of interest rates and the time to expiration. The risk-free interest rate also is part of calculating the present value of the exercise price. Increases in the interest rate will increase the value of calls and reduce the value of puts.

Every variable above except volatility is directly observable in the market. When pricing a stock option, the volatility variable in the BSM Model can be estimated one of two ways. It can be estimated based on the historical stock price of the underlying security. Or, it can be estimated by calculating the implied volatility from an actively traded stock option on a similar security.

Changing the volatility assumption in the BSM Model leads to significant movements in the price of the stock option. It may be appropriate to view the BSM Model as a procedure to estimate volatility rather than a model to estimate the price of a stock option. This may be true if (1) the market price of the stock option is determined based on a no-arbitrage condition and not an empirical financial model (that is, supply/demand of arbitrageurs determine the option price); and (2) every variable in the BSM model except volatility is known.

In this scenario, (1) the analyst can solve for volatility; and (2) the BSM Model is more useful to estimate the market’s opinion about the volatility of a stock instead of to determine the price of an option.

The above examples provide a basic theoretical and mathematical understanding of how options are priced. The analyst should understand how a stock option price is determined if he or she will use the option price as a proxy for the DLOM.

**Factors That Affect the DLOM**

When looking at the relationship between option prices and DLOMs, what are the common elements to suggest an option price is related to the DLOM? When comparing the six option pricing variables listed above to the factors that affect the DLOM, there appear to be some common elements.

In *Mandelbaum v. Commissioner*,^{21} the Tax Court listed nine factors to consider when determining the DLOM:

- Financial statement analysis
- Dividend policy
- Nature of the company, its history, its position in the industry, and its economic outlook
- Management
- Amount of control in the transferred shares
- Restrictions on transferability
- Holding period of the stock
- Company redemption policy
- Costs associated with a public offering

For purposes of this discussion, let’s assume that these nine Mandelbaum factors will affect the magnitude of the DLOM and that no other factor has a material impact on the magnitude of the DLOM.

Some of the factors that affect the magnitude of the DLOM are similar to the variables that affect the price of stock option. For example, both are affected by (1) holding period, (2) dividends, and (3) volatility of the underlying securities. In the BSM model, these three factors are considered directly. Holding period is considered in the DLOM in Mandelbaum factors 6 through 9 (based on the numbering in the above list); dividends are considered in factor 2; and volatility of the underlying securities is considered in factors 1, 3, and 4.

When comparing the variables that affect the price of stock options and the magnitude of the DLOM, it is apparent that there are similarities. This perhaps suggests that OPMs have some relevance for determining DLOMs.

**Do OPMs Result in a Price Premium or a Price Discount?**

Some analysts theorize that the percentage result from the OPM studies is actually a price premium, and not a price discount. Ashok Abbott wrote that, “Often, however, the value of a put option premium, estimating the cost of liquidity, is presented incorrectly as the discount for lack of liquidity. This is similar to the merger premium being treated as a discount for lack of control. Neglecting to convert the option premium to the applicable discount creates the illusion that the estimated discounts are greater than 100%, an impossible solution.”^{22}

Margin Greene concurred with this sentiment, saying, “Frequently, appraisers compute the option and assume their result is a discount. In reality, the models produce a premium, which must then be converted to a discount.”^{23}

There is not universal agreement about whether the OPM studies produce a premium or a discount. Therefore, practitioners who rely on these studies should choose how to use the studies to estimate the DLOM.

**Conclusion**

There are many issues surrounding the use and applicability of OPMs in determining DLOMs. In some instances, the OPM output can be relatively similar to the ranges found in both restricted stock studies and IPO studies. However, the reasonableness of the output does not necessarily imply the OPM is measuring a DLOM.

Some of the OPMs work relatively well when certain holding periods and volatilities are used as inputs. What needs to be kept in mind is whether output that appears reasonable may be capturing something unrelated to marketability issues. This is because OPMs theoretically have little to do with marketability of the underlying security.

Instead, option prices are determined based on the concept of a no arbitrage condition. The lack of arbitrage opportunities in the market ensures that investors cannot earn more than the risk-free rate of return using a combination of leverage, stock positions, and option positions.

Another issue with using OPMs to estimate the DLOM is that in OPMs, the underlying security is assumed to be marketable. Therefore, some analysts challenge the applicability of OPMs to nonmarketable securities.

A better analogy to the DLOM for private company stock may be from the pre-IPO studies and the restricted stock studies. In both cases, the value of nonmarketable stock is compared to the value of marketable stock. Therefore, the difference in value from the two ownership interests in the pre-IPO studies and restricted stock studies is exclusively related to marketability.

In spite of the differences between how option prices are determined and the factors that affect marketability, some analysts assert that the option price is a useful proxy for the DLOM. In the context of option pricing, the support for this opinion is that both the option price and the DLOM are affected by the following:

- Volatility of the underlying security
- Time horizon

Since OPMs incorporate the relationship between volatility and time horizon, they may provide insight in determining an appropriate DLOM.

^{15.} *BVWire *#76-4 (January 28, 2009).

^{16.} Ibid.

^{17.} Jay E. Fishman and Lester Barenbaum, “Do Put Option Models Overstate Discounts for Lack of Marketability?” *Financial Valuation and Litigation Expert* (April/May 2013): 9.

^{18.} Ibid, 11.

^{19.} Aswath Damodaran, “The Promise and Peril of Real Options,” Stern School of Business, New York, New York, http://pages.stern.nyu.edu/~adamodar/

^{20.} This is because once a call option is in the money and the holder exercises the option the holder receives the stock and dividends in subsequent periods.

^{21.} *Mandelbaum v. Commissioner*, T.C. Memo 1995-255 (June 13, 1995).

^{22.} Ashok Abbott, “Discounts for Lack of Liquidity: Understanding and Interpreting Option Models.” *Business Valuation Review* 28, No. 3 (Fall 2009): 145.

^{23.} Martin Green, “Do Maximum Strike Price Lookback (Longstaff) and Other Put Option Models Produce a Marketability Premium or a Discount?” *Business Valuation Update* (October 2010): 26.

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

* *

*Aaron Rotkowski is a manager in the Portland, Oregon, practice office of Willamette Management Associates. He can be reached at (503) 243-7522 or at **amrotkowski@willamette.com**.*

*Michael Harter is a senior associate in the Portland, Oregon, practice office of Willamette Management Associates. He can be reached at (503) 243-7501 or at **maharter@willamette.com**. *