Why the Black-Scholes model overvalues hedge fund assets
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Why the Black-Scholes model overvalues hedge fund assets

By: Mark Kitchen | April 18, 2011

Espen Robak, president and founder of Pluris Valuation Advisors, argues that Black-Scholes and other theoretical valuation models have long failed to value illiquid securities properly, but their inadequacy is especially an issue today because of the prevalence of illiquid assets, regulatory insistence on fair value and the availability of more accurate methodologies.

Hedge funds frequently use valuation models that produce inaccurate results for illiquid assets, risking accounting irregularities that can result in potential litigation, increasing redemptions and
other issues.

For years it has been common practice for hedge funds to use simple theoretical approaches to valuation, of which the Black-Scholes model is a prime example.

Black-Scholes and other theoretical models fail to value illiquid securities properly, but their inadequacy is especially an issue today for a number of reasons.

Many funds are flush with illiquid assets. Since the beginning of the financial crisis, many formerly liquid stocks are much less actively traded, have become hard to borrow or show other signs of
illiquidity. Any derivatives on such shares have become increasingly misvalued as a result.

Today’s regulations demand more accurate ‘fair value’ valuations. In the past, fund managers could use Black-Scholes and still comply with US GAAP, even though Black-Scholes produces inaccurate results because it assumes perfect liquidity. But, since the 2007 passage of FAS 157 (codified in ASC 820), hedge funds have been required to value all assets based on their fair value – whether the assets are liquid or not.

Financial theory has moved beyond the old models. Newer research indicates that securities with any discontinuity in trading show large and growing deviations from model prices.

Finally, more accurate methodologies are available. Secondary markets have developed in recent years that provide data on an ongoing basis that can be used to value even the most illiquid assets.

Black-Scholes and PIPE securities

Black-Scholes is not designed to account for illiquidity. As an example, consider its use for valuing options and warrants.

In private issuances, securities are often subject to sale and transfer restrictions or, in many cases,have no public market at all. US hedge funds invested USD36.7bn in 5,667 PIPE deals between the beginning of 2007 and February 28, 2011; 3,768 of the PIPEs included warrants as sweeteners, while 1,035 issued convertible debt. The 1,262 issuers involved with the PIPE deals raised about USD410bn and 740 of them were quoted on either the OTC Bulletin Board or Pink Sheets.

Pluris Valuation Advisors has been collecting data on the sale of warrants in its proprietary LiquiStat database for several years, and has compared private market pricing with Black-Scholes values. In the worst cases, the model overvalues warrants substantially over their fair value.

The Black-Scholes overvaluation ranges from a median of 0.18 times for the bottom quintile to 2.12 times for the top quintile. Black-Scholes generates the worst overvaluations for warrants that are out of the money and issued by smaller, riskier companies. Prior research on the ‘price of illiquidity’ supports these basic relationships.

What about when convertible debt is issued instead of warrants? Convertible debt is a hybrid financial instrument comprised of two components, debt and a conversion option, but it may also include other embedded derivatives.

A range of models may be applied to such securities by both issuers and investors, ranging from simple to quite complicated. In all cases, however, if the model does not take the illiquidity of the securities into account, the resulting value estimates of the equity securities are likely off by a wide margin.

Relying exclusively on theoretical models can result in an overvaluation of the asset (conversion options for convertible debentures, for example), a misallocation between the debt and equity (conversion option) components, and a mis-statement in earnings if a conversion option is no longer deemed to have characteristics of equity – for example, if the deal is modified so that cash is paid
out at conversion instead of equity shares.

Any of these situations can lead to accounting irregularities, including undesirable earnings volatility for the issuer. For issuers, misuse of models like Black-Scholes may even entail a restatement of financial statements.

When illiquidity is a factor, companies must determine an illiquidity discount in order to estimate fair value in accordance with the measurement date and exit price concepts set forth in US GAAP. Rather than using an arbitrary portfolio haircut to determine a discount, it is best to consider relevant market transactions and whether these transactions occur in active or inactive markets.

The most representative way to estimate the fair value of a convertible debt instrument is to bifurcate the instrument into its two components: an ‘optionality’ portion that is appropriately discounted for liquidity, and a fixed income portion.

The optionality portion reflects the holder’s option to convert the note into a predetermined number of the issuer’s common shares at an agreed conversion rate. Pluris uses a model that accounts for illiquidity and adjusts fair value accordingly. Our model determines an appropriate discount by evaluating actual discounts reflected by recent secondary market transactions from the LiquiStat database.

This approach enables hedge funds to comply with fair value requirements and avoid the inaccurate results associated with Black-Scholes.