Do European Put Options Measure Marketability?
Proponents of the option pricing methodologies commonly argue that an At-the-Money (ATM) European Put Option provides a lower bound estimate of the discount for lack of marketability. According to the argument, if an investor purchases an ATM European Put Option with a strike price equal to the marketable price, the investor has, in effect, purchased the ability to sell the restricted shares at the marketable price. In the words of Chaffe, the investor has purchased marketability. If this is the case, then we should expect a portfolio of a ATM European Put Option and a restricted security to provide the same economic payoffs at expiration as that of owning liquid stock directly. However, working through the math, one will discover that a portfolio comprised of a European put option and a restricted security actually provides better economic payoffs at expiration than owning liquid stock directly. To illustrate this important economic concept, consider the following hypothetical example:
Suppose the marketable value of the common stock is currently $10 per share and the underlying stock price volatility is 40%. The stock pays no dividends, and the risk free rate is currently 5%. The illiquid stock is subject to a 1 year holding period restriction. Therefore, under these assumptions, the cost of a 1 year European ATM Put Option (strike of $10) is $1.32 per share (computed using the Black Scholes Option Pricing Model) and the theoretical value of the illiquid stock is $8.68 per share (i.e. $10 – $1.32 = $8.68). Now, assume an investor purchases the illiquid stock at its theoretical value of $8.68 and simultaneously purchases the ATM Put Option at its theoretical value of $1.32. In effect, this investor has just purchased marketability.[1] Therefore, we should expect the portfolio to have the same economic payoffs as that of owning the liquid stock directly. However, consider the economic payoffs at expiration in all states of the world (i.e. when the liquid stock expires in the money, at the money, and out of the money). For this illustration we will assume the liquid stock trades at $8, $10, and $12 at expiration. Keep in mind that at expiration the illiquid stock is now liquid and, therefore, trades at its liquid value. The table below illustrates the payoffs at expiration for all three scenarios:
Price of Liquid Stock at Expiration |
Profit on European Put Option At Expiration[2] |
Profit on Illiquid Stock at Expiration[3] |
Profit on Portfolio of Option and Illiquid Stock[4] |
Profit on Liquid Stock at Expiration[5] |
$8 |
$0.68 |
-$0.68 |
$0 |
-$2 |
$10 |
-$1.32 |
$1.32 |
$0 |
$0 |
$12 |
-$1.32 |
$3.32 |
$2 |
$2 |
[1] This assumption is presumed to be true according to Chaffe’s theory.
[2] European put profit = MAX(K – S, 0) – $1.32 (i.e. cost of option); where K = $10 and S = Price of liquid stock
[3] Profit on illiquid stock = S – $8.68 (cost of illiquid stock); where S = price of liquid stock (recall that the illiquid stock is liquid at expiration of the option)
[4] Profit on portfolio = sum of profit on option and profit on illiquid stock.
[5] Profit on liquid stock = S – $10; where S = price of liquid stock at expiration.
As one can see, the portfolio provides the holder with superior economic payoffs when the put options expires in-the-money. Accordingly, a portfolio comprised of restricted stock and a put is more valuable than liquid stock and, therefore, we should expect a European Put Option to actually overstate the cost of lack of marketability.
Does this Mean that European Put Options are Not Useful?
No, not necessarily. Although a European Put Option does not directly measure (and likely overstates) the cost of illiquidity, the option does eliminate a large portion of pricing fluctuation over the restriction period. A rationale market participant can, therefore, use a European Put Option to eliminate price risk and guarantee a specific economic payoff, or price, at expiration of the option. For example, suppose an investor owns a restricted security and simultaneously purchases a 1 year ATM put option at $10 per share. This technique will guarantee $10 per share for the investor at expiration of the option. And, while this economic payoff is slightly different than the real option of liquidity (i.e. liquidity allows the purchaser to receive $10 today), and can actually be constructed in a less costly fashion (i.e. with a costless collar), the strategy, nevertheless, is a viable hedging option for an investor owning illiquid securities.
Excellent. I have been making this argument for some time now.
Bob Dohmeyer, ASA