Stand Up and Shout—It Is Another DLOM Put Model!

Marc Steven Katsanis, CPA/ABV, CFA

European style fixed strike, lookback, and asian put option models have been proposed and utilized by business valuation practitioners to estimate discounts for lack of marketability. Another form of put option, a shout put or shout floor option, more closely mimics marketability than do the previously mentioned forms of put option because both marketability and a shout put option give a stockholder the right to lock in a selling price (the prevailing marketable stock price) for the stock at any point in time over the term of the option. By comparison, over the term of the option the European fixed strike put gives the stockholder the right to lock in a selling price equal to the current stock price; the lookback put gives the stockholder the right to lock in a selling price equal to the highest stock price achieved; and the asian put gives the stockholder the right to lock in a selling price equal to the average of all stock prices achieved. Because the shout put option more closely mimics marketability than do the alternative put option models, it would be a valuable addition to every valuation practitioner’s toolbox.

Numerous types of put option models are used by business valuation practitioners to estimate discounts for lack of marketability.1 However, none of these put option models quite fits the situation that they are attempting to model. Marketability provides the owner of a share of stock the opportunity to lock in a selling price equal to the prevailing marketable price at any time. The only form of option that affords its holder the same opportunity is a shout put or shout floor option.

This form of shout put option is one that has no set exercise price at inception. The strike price is set at any time during the life of the option by the option holder shouting. Upon shouting, the exercise price of the option is set at the prevailing marketable stock price at the time the shout occurs.2 The shout feature mimics marketability because both give a stockholder the right to lock in a selling price (the prevailing marketable stock price) for the stock at any point in time.

The value of the shout put serves as an estimate of the marketability and liquidity value embedded within the marketable share value so that the following relationship exists:

Marketable Share Value = Shout Put Value + Nonmarketable Share Value

Similarly, the shout put value serves as an estimate of the cost of purchasing marketability and liquidity for an otherwise nonmarketable share. The shout put value is also an estimate of the discount for lack of marketability to convert a marketable share value to a nonmarketable one and an estimate of the marketability premium to convert a nonmarketable share value to a marketable one.

A shout put option also solves some of the shortcomings of fixed strike, lookback, and asian put options that have been proposed to estimate discounts for lack of marketability. A shout option is one of many types of reset strike options that financial engineers have developed in recent years. Dai, Kwok, and Wu identified numerous financial instruments with embedded shout features including a reset feature in the Geared Equity Investment offered by Macquarie Bank, executive stock options whose terms have been reset, and Canadian segregated funds with multiple reset rights.3

In 1993 my colleague and mentor D. B. H. Chaffe, III, proposed the use of a Black-Scholes put option pricing model as a proxy for a discount for lack of marketability in an article that has sparked much discussion and the use of various other put option models to estimate discounts for lack of marketability.4

Black-Scholes is designed to price a European style fixed strike put option, which gives its holder the right to lock in a selling price (for the stock) at the beginning of the holding period. As cited by Chaffe, one shortcoming of using a European style fixed strike put option model as a proxy for a discount for lack of marketability is that a European style fixed strike put option does not allow the optionholder to realize an intermediate gain (a gain caused by an increase in the underlying stock price that occurs during the holding period) quickly and efficiently. A shout option allows the optionholder to realize an intermediate gain by simply shouting.

In a critique of the Chaffe model, Damodaran states the following: ‘‘[L]iquidity does not give you the right to sell a stock at today’s market price anytime over the next 2 years. What it does give you is the right to sell at the prevailing market price anytime over the next 2 years.’’5 In addition, a shout option does not assume any particular market timing ability, as does the Longstaff lookback model, which assumes that the option holder has perfect market timing ability and can sell the underlying stock at the highest stock price achieved over the option term.6 Because an investor in corporate stock does not possess the ability to time the market perfectly, it is unlikely that an owner of a marketable corporate stock would achieve the lookback value by selling exactly at the highest price achieved. In most cases, a lookback put allows the optionholder to lock in a selling price that is above the prevailing market price.7 Because of this, the lookback optionholder possesses a right that is superior to, and more valuable than, marketability

Stockdale suggests that the Black-Scholes put option pricing model is of limited usefulness in measuring a discount for lack of marketability over long time periods because the Black-Scholes put value declines as the option term increases after reaching a peak.8 Figure 1 shows an example of this. A shout option does not share this characteristic with a European style fixed-strike put option. The shout put option increases in value until reaching a peak and then plateaus at that peak. Figure 2 demonstrates this attribute of a shout put option. This author believes that the shout option pricing model is a better proxy for lack of marketability discounts than is the Black Scholes—especially in cases in which the option term exceeds the term in which the Black-Scholes value reaches its maximum—because it is not intuitively appealing that a lack of marketability discount would decline as the restricted period increases.

Another variation of option that is used to estimate discounts for lack of marketability is an asian put option. An asian put gives its holder the right to lock in a selling price equal to the average of all prevailing marketable prices achieved during the option term. The asian put does not mimic marketability as closely as a shout put because the asian put assumes that marketability gives an investor the ability to sell an asset at an average price rather than at the then-prevailing marketable price. That is, an investor who owns 100 shares of Microsoft stock that is trading at $25 per share can sell it at $25 per share, not at the average of all of the closing daily stock prices over the holding period.

Dai, Kwok, and Wu state that the analytic price formula for a shout floor (shout put) is as follows9,10:

where SP(t) 5 the formula for an at-the-money BlackScholes put option model; S 5 the stock price and the strike price of the option; t 5 the term to expiration in years; t* 5 the term to expiration in years at the point at which the shout premium begins to have value; r 5 the risk free rate with a term equal to the remaining term of the option; and q 5 the dividend yield on the underlying stock.

The inputs to a shout floor option pricing model and a brief discussion of how to estimate these inputs in the context of estimating a marketability discount follow:

  1. Stock Price: This input is equal to the marketable stock price. If no marketable stock price is available, for example, in the case of a privately held company, one must perform a valuation of the company in order to determine the marketable stock price for this input.
  2. Volatility: The volatility input is an estimate of the expected annualized standard deviation of logarithmic percentage changes in the marketable stock price over the estimated holding period. If no marketable stock or traded option price (which can be used to calculate implied volatilities) is available for the subject company shares, one can estimate the share price volatility of an appropriate publicly traded peer group to serve as the basis for the volatility input.
  3. Term to Expiration: This input variable is equal to estimated holding period.
  4. Risk-Free Rate: This input assumption is equal to the zero coupon U.S. Treasury (if in the United States) security yield with a term equivalent to the term to expiration assumption.
  5. Dividend Yield: This is equal to the annualized dividend divided by the marketable stock price.
  6. Strike price is not necessary as an input because it is always equal to the marketable stock price for an atthe-money option.

The value of a shout floor option consists of two components: (1) an at-the-money European style put option and (2) a shout premium. When the risk-free rate is less than or equal to the dividend yield, then the shout premium equals 0. When the risk-free rate is greater than the dividend yield, the shout premium is 0, except when the option term (or restriction period) exceeds the critical term (t*).11 The critical term is determined by the riskfree rate, dividend yield, and volatility inputs.

Figure 3 shows a graphical comparison of the difference between a shout and Black-Scholes put (the shout premium) for selected volatility inputs.

The shout floor value generally exceeds the BlackScholes put option value the most (1) at high and low levels of volatility, (2) at high risk-free rates, and (3) for options with long terms to expiration.

Marketability allows the owner of a share the opportunity to lock in a selling price equal to the prevailing marketable price at any time of his choosing. The more closely a put option provides its owner with the same opportunity, the more effective that option will be at estimating a discount for lack of marketability. The shout put option model mimics this opportunity better than other forms of put models because the optionholder can shout and set the strike price equal to the prevailing marketable price at any point in time. This matches exactly the marketable security owner’s ability to lock in the prevailing marketable price. By comparison, the other forms of put options mentioned in this article offer the optionholder a set of selling price (or strike price) assumptions that are different from what marketability provides. For this reason, it should be a part of every valuation practitioner’s toolbox.

Marc Steven Katsanis is Senior Vice President of Valuation Services at Chaffe & Associates, Inc., New Orleans, Louisiana.

1In the context of this article, the term marketability is used to describe both marketability and liquidity.
2Subsequent to the occurrence of the shout, the option becomes an ordinary fixed strike put for the remainder of the option term. 3See M. Dai, Y. Kwok, and L. Wu, ‘‘Optimal Shouting Policies of Options With Strike Reset Right,’’ Mathematical Finance 14 (2004):383–384. 4David B. H. Chaffe, III, ‘‘Option Pricing as a Proxy for Discount for Lack of Marketability in Private Company Valuations,’’ Business Valuation Review 12 (1993):182–185. 5Aswath Damodaran, ‘‘Marketability and Value: Measuring the Illiquidity Discount,’’ Stern School of Business (July 2005):42. This quote is also included in the AICPA Working Draft ‘‘Practice Aid Valuation of Privately Held Company Equity Securities Issued As Compensation,’’ 2011, 86–87.
6Francis A. Longstaff, ‘‘How Much Can Marketability Affect Security Values?,’’ The Journal of Finance, volume 50, issue 5 (December 1995):1768. 7That is, at all times except when the stock is currently trading at its highest point over the option term, the optionholder has the right to sell at a price higher than the prevailing market price. 8See John J. Stockdale, ‘‘A Test of DLOM Computational Models,’’ Business Valuation Review 27 (2008):137. 9For the pricing formula see M. Dai, Y. Kwok, and L. Wu, ‘‘Optimal Shouting Policies of Options With Strike Reset Right,’’ Mathematical Finance 14 (2004):387–388. 10A trinomial or binomial lattice model can also be used to price a shout put option as described in Cheuck, T. and T. Vorst. ‘‘Shout Floors.’’ Financial Engineering Review 1 (2) (October 2003):24–27. 11The critical term (t*) is also the term that results in the peak BlackScholes value referenced by Chaffe and Stockdale.

References
AICPA Equity Securities Task Force. Working draft practice aid valuation of privately held company equity securities issued as compensation. New York: American Institute of Certified Public Accountants, Inc., 2011.
Chaffe, David B. H., III. 1993. ‘‘Option Pricing as a Proxy for Discount for Lack of Marketability in Private Company Valuations.’’ Business Valuation Review 12 (4):182–185.
Cheuck, T., and T. Vorst. 2003. ‘‘Shout Floors.’’ Financial Engineering Review 1 (2):15–35.
Dai, M., Y. Kwok, and L. Wu. 2004. ‘‘Optimal Shouting Policies of Options With Strike Reset Right.’’ Mathematical Finance 14 (3):383–401.
Damodaran, Aswath. 2005. ‘‘Marketability and Value: Measuring the Illiquidity Discount.’’ Stern School of Business. Published on the internet only at http://pages. stern.nyu.edu/,adamodar/pdfiles/papers/liquidity.pdf.
Longstaff, Francis A. 1995. ‘‘How Much Can Marketability Affect Security Values?’’ The Journal of Finance 50 (5):1767–1774.
Stockdale, John J. 2008. ‘‘A Test of DLOM Computational Models.’’ Business Valuation Review 27 (3):131–137.