Discounts for illiquidity: where are we standing now? Part 3

We continue this series on DLOM – the discount for illiquidity applied to the valuation of restricted assets. As you recall, in the first post we saw the key concepts for the discounts, and in the second we showed the earliest theoretical approach made by Chaffe in 1993.

The second earliest valuation estimate of the DLOM was proposed by Longstaff in 1995[1]. He followed Chaffe’s idea of modelling DLOM as a protective European put option on the restricted asset, with a change: while the exercise price (the strike price, or strike) in Chaffe’s model is the current price of the asset, Longstaff’s strike price is the maximum trading price that the asset will reach during the time of illiquidity. That is, Chaffe’s approach hedges the holder of the illiquid asset against the fall of its price in relation to the current price; Longstaff’s model adds an extra layer of protection by choosing as strike the maximum achievable price of the asset.

Admittedly, this new assumption for the strike price generates a very high DLOM, possibly a ceiling, as it maximizes the odds of making a profit with the put option. At first sight, it would seem a bold assumption, but it is not too far from real life if one takes in mind that some people with superior information may be able to choose the perfect time to sell the corresponding unrestricted asset[2]. To be clear, the perfect timing hypothesis means that the holder of the restricted asset is able to foresee when the maximum price of the corresponding unrestricted asset is to be achieved, that is, he or she can see the future.

In summary, Chaffe’s model is applicable to investors with average knowledge and Longstaff’s to people with perfect knowledge (and foresight). We will show in the next posts that other models try estimate DLOM with a less-than-perfect-timing assumption.

Just like Chaffe’s, Longstaff′s calculation is also based on the closed-end formulas of the Black, Scholes & Merton model - BSM:

where:

DLOM: Discount for illiquidity as a percentage of the price of the unrestricted asset.

D: Duration of the restriction on the asset.

s: Constant expected volatility – the standard deviation of the asset’s rate of return, for the duration (D).

N(): Cumulative standard normal probability distribution, mean equal to zero, standard deviation equal to 1.

e: Euler’s constant.

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For the sake of brevity, we will not discuss the technical assumption of BSM, but check the key hypotheses and parameters that make the size of DLOM:

-??????? The size of the DLOM is positively correlated to the volatility and to the restriction time. Unlike Chaffe’s model, the interest rate has no influence on DLOM, as the model assumes that the proceeds from selling the corresponding unrestricted asset will be reinvested at the risk-free interest rate from the time of the sale to the end of the restriction time.

-??????? The European version of the put option allows one single time to exercise, while the buyer’s opportunity cost include the exercise in several points in time, like in an American put. However, unlike Chaffe’s model, the upper bound of DLOM calculated by Longstaff with the BSM will not underestimate the true upper bound of DLOM as the put option is exercised at the maximum price, regardless of the time when that maximum is achieved.

Longstaff points out one key weakness in Chaffe’s model: the latter only captures the opportunity cost of selling the asset immediately for the current price, or the premium of an insurance against the decrease of the asset price from the current level. Longstaff shows that are more valuable missed opportunities in the illiquidity period, and DLOM should reflect them.

On the other hand, Longstaff’s formula has a critical flaw: it is so sensitive to high volatilities and longer illiquidity periods that it can produce discounts greater than 100%. That implies negative prices for restricted asset, which goes against common economic sense.

Under less extreme hypothesis, we will discuss in the last posts of this series how far Longstaff’s upper bound of DLOM is from empirical market data.

The following posts will show alternative models that assume a less-than-perfect-timing hypothesis for exercise of the protective put.

I hope you are enjoying this posts series so far, and look forward to read your comments.

I kindly thanks Wulaia Consultoria 's team for reviewing, commenting and sharing this article series.

[1] LONGSTAFF, Francis A. How much can marketability affect security values? Journal of Finance, Volume 50, Issue 5, December 1995.

[2] The option markets trade floating-strike lookback European puts, an exotic type of option that allows for optimal-timing by retrospectively choosing the strike price as the maximum price the asset achieved during the exercise period.