Simple Analysis is Never a Solution in Valuing Complex Derivative Securities

Executive Summary

Valuing Complex DerivativesIncreased complexity in corporate capitalization tables demands more sophisticated analysis of complex derivatives.  Cookie-cutter templates and simple rules of thumb will either overstate or understate the value of complex corporate securities and ultimately compound future valuation issues as the issuer looks to complete subsequent transactions or experiences a decline in value.  Establishing the proper methodologies & valuation constructs early allows for accurate reporting for both shareholders and issuers.

The valuation of complex derivative securities (options, warrants, and conversion elements with a variety of features) within these companies has evolved from a niche academic pursuit into a widespread practice and is often required for tax, financial accounting, reporting fair values, or for management purposes.  Sadly, many valuation analysts resort to simplified or “canned” models and assumptions that are inappropriate for valuing contemporary derivative securities: employing inappropriate volatility assumptions; using inappropriate models or formulas; and/or ignoring the interactions between the values of various outstanding securities and dilutive securities within a company’s capital structure as the value of the company changes over time.  Three important considerations demand the use of more sophisticated valuation modeling: unknown potential corporate volatility, leverage & related default risk, and multiple securities within a single corporate entity.

Special attention should be paid to both observed historical volatilities and implied future volatilities for the subject firm and for guideline securities, especially when the company’s securities are not publicly traded or have insufficient, unreliable, or inapplicable trading histories.  Adjustments may be required to the volatility assumptions to account for differences in leverage, dilution, and other developments within the company and the guideline companies considered.

When valuing derivative securities in companies with a greater risk of default, or where multiple securities must be valued, a contingent claims model of the firm is often required.  The Merton contingent claims model is generally required when valuing common shares and other junior securities issued by more highly leveraged companies, when multiple substantially dilutive derivative securities are outstanding, or when the values of multiple securities issued by the company must be jointly determined.

Valuing Complex Derivative Securities

Both investors and issuers should know and understand the value and cost of complex securities.  Unfortunately, too often the values being derived and reported are wrong, and the consequences of these errors can be significant.  Valuing complex derivative securities is one of the most difficult and challenging tasks confronted by business valuation experts.  In performing these tasks, the valuation expert must consider the following:

  1. the specific terms of the subject derivative in terms of the mix of rights and obligations of the holder and the issuer of the derivative security;
  2. the likely range of payoffs (or consequences) to the holder and issuer and how they relate to the value of the underlying asset (or assets or index price) at some future date or dates;  and,
  3. the likely distribution of probable future outcomes that would affect the realized value of the derivative security.

The right model needs to reflect a compromise between capturing the relevant complexities while being simple enough to be tractable and periodically replicated.  Specific derivative security terms and related modeling issues are often too unique for simple formulas from text books or “canned” models. Thus, some type of simulation analysis (running a large number of different scenarios and averaging the conclusions) is often required to value the derivative securities accurately and appropriately.

When a company has a greater risk of financial distress, has greater financial leverage, or has a particularly complex capital structure, a completely different model is required. Such a model is based on the Contingent Claims Model for Asset Pricing developed by Robert Merton.[1] In this approach, each outstanding security is valued as a derivative security on the underlying asset value or capital value of the company as a whole.

What is a Derivative Security?        

 A derivative security is an agreement or contract with a value contingent upon the value or price of an underlying asset or set of assets, dependent on a market or index price, or dependent on certain events (“underlying basis or bases”).  Derivative securities may be settled (if exercised or effective) by either delivery of the underlying asset or assets or by cash settlement based upon a formula with the underlying bases determined at the exercise date.  The most common underlying bases for widely traded derivative securities include: market common equity share prices; market currency exchange rates or currencies; and market prices for commodities (most often mining commodities, including metals or energy, or agricultural commodities).

The most commonly valued derivative securities are options.  A common, or “vanilla,” call option is a right (but not an obligation) to purchase an asset or a quantity of assets at a specified price per unit (the “strike” or exercise price) within a specified time period.  A common, or “vanilla,” put option is a right to sell an asset or a quantity of assets at a specified price per unit (the “strike” or exercise price) within a specified time period.  An American option gives the holder the right to exercise the option at any time up to a specific date (the expiration date).  A European option gives the holder the right to exercise the option at a specified date (the expiration date).  A warrant is typically a call option where the issuer/seller must issue the underlying security in the event of exercise. Technically, employee (and director) stock options are actually warrants.[2]

A futures contract is an agreement between two parties to a future exchange of an asset for a cash price at a specified date.  A forward contract is a private contract that is similar to a futures contract.

A swap is an agreement to exchange two assets at a future date at a specific exchange rate or to exchange the terms of two securities (typically to exchange interest rates on fixed income securities or currencies at some specific exchange rates) over a period of time or at certain specified dates.  A swap option is an option that provides the holder (or issuer in some instances) the right (but not the obligation) to engage in a swap transaction at a future date or over a future period of time.

A convertible security is a security that may be converted into another security (in whole or in part) upon certain events or actions of either the holder or the issuer.   Convertible securities are securities with an “embedded” or attached derivative.[3]  The most common convertible security is a fixed income security (a debt instrument that pays interest or a preferred share that pays dividends and has a liquidation preference) that is convertible into common shares at a certain exercise price with the number of shares to be received determined by the par value[4] at the date of exercise. These types of convertible securities are popular in private placement transactions and private equity investments where the investor wants additional “downside” protections and/or priority control relative to common equity investors but also seeks to share in any upside in the future.

What Makes a Derivative Security Complex?

 We refer to a complex derivative security as a derivative security that is more difficult to value because it has certain unique additional contingencies or variable features as part of the agreement, or the derivative security is, or must be, valued as a compound derivative security.  A compound derivative security exists when the underlying asset is also effectively a derivative security.  These and similar types of derivative securities are often also referred to as “exotic” derivatives.

While some more common exotic derivatives have formulas that can provide estimates of value, the characteristics of exotic derivatives are often such that the use of a formula to derive an estimate of value is not sufficiently reliable, and the valuation exercise will require some form of simulation modeling to derive a more reliable valuation estimate and range of value for the derivative.

Common examples of exotic options that often require more extensive valuation analyses, include:

  1. Warrants or options with variable exercise prices, payoffs, or quantities:
    1. Warrants with “reset” features. A reset feature will change the exercise price or the quantity of assets that may be purchased, sold or exchanged upon certain subsequent events occurring.  Often securities issued in private placements contain “anti-dilution” features.  The most common anti-dilution features provide that the exercise price will be reduced (downward ratchet), and/or the number of underlying securities that may be purchased will be increased upon certain future funding events.  If the terms of a subsequent funding event imply a lower per share value for the company’s shares (a “down” round funding event), the exercise price will be reduced and/or the number of shares that may be purchased will be increased.
    2. Look-back options are options where the exercise price, payoff, or quantities to be purchased are contingent on future events or prices. In a standard look-back option, the cash settlement is based on a minimum, average, median, or maximum observed asset price prior to maturity (determined at discrete measurement dates).  Asian options are options where the exercise price is based on the average observed price over certain dates.  A “shout” option is an option where the holder may declare (shout) at a given date the intrinsic value of the option and will receive upon exercise the greater of the intrinsic value of the option on the shout date or the intrinsic value of the option at the exercise date.
  1. Warrants and options with contingent or restricted exercise periods:
    1. A vesting period is common for stock options issued by companies to employees and directors. With a vesting option, a specified time must pass and certain conditions must be met (continued employment) in order to be able to exercise the option.
    2. Barrier options are options where the right to exercise is triggered based on the underlying asset price hitting a certain threshold (often on certain specified dates or in specified time periods). The right to exercise will cease (“knock out”) if certain future events or conditions occur, or the right to exercise is not triggered until certain conditions are met (“knock in”).  Alternatively, a barrier option may have certain terms modified in the event the underlying asset price reaches or passes a certain threshold.
    3. Forfeiture (or premature expiration) conditions are similarly a part of stock options granted by companies to employees or affiliates. A forfeiture condition exists when the right to exercise is contingent on certain continuing conditions being met (continued employment in the case of employee stock options or principal remaining in the case of a convertible debt security with allowance for prepayment of principal prior to the expiration of the conversion feature).
    4. Early exercise of a warrant or option is another contingent issue often not adequately considered. This issue often arises when the holder is more risk averse, has significant wealth in the derivative, or is impatient and the derivative has gained sufficient intrinsic value that the holder seeks to exercise the option well prior to the expiration date because there is little additional “option” value to be realized by waiting and there is greater “perceived” relative risk to the holder associated with waiting to exercise the option.
    5. Contingent exercise or expiration dates and awards are often observed particularly in private companies. These are like barrier options except that the options rights are triggered (knock in) or are lost (knock out) out upon certain events.  Typical events include the company meeting certain non-stock related performance figures (such as earnings or growth targets or asset return targets) or a major transaction event such as a public offering of shares, a merger or acquisition of the company or its shares, retirement of a debt or preferred security, or a sale of certain assets and liquidation of proceeds.
  1. Securities with multiple rights or embedded derivatives:
    1. Convertible securities with both “put” and “call” features, such as the right of the issuer to prematurely retire principal (or “call” the security) prior to maturity and the right of the holder to “put” the security back to the issuer for repayment or assets upon certain events have such multiple events.
    2. Convertible securities with various forfeiture, barrier, participation rights, and/or anti-dilution features.
    3. “Chooser” options or conversion features that entitle the holder or issuer to choose which asset (from a set of choices) to receive (in general or upon certain conditions).
  1. Compound options and unusual underlying asset distributions:
    1. Most commonly, mezzanine debt (with or without conversion features) becomes a compound option when valued in the context of highly leveraged companies (companies with significant debt or default risk) or in connection with more highly leveraged assets (such as real estate).[5]
    2. Stock options and warrants (including embedded derivatives in securities convertible into common or preferred equity) to purchase or sell equity securities should be similarly valued as compound options when the underlying company or asset is highly leveraged, has a substantial risk of reaching a value of zero (or very close to zero), or is subject to certain discrete event risks.
    3. Even a vanilla call option or put option may become a complex derivative or require more effort to model and value when the underlying asset does not have a well-defined or easily modeled distribution or characteristics. These include common circumstances such as when the underlying asset value periodically changes substantially at discrete moments in time (“jump” process), the underlying asset value can approach or reach zero with some probability, or the volatility of the underlying asset is not constant over time.[6]

Modeling Complex Derivatives for Valuation

 Modern finance and the development of more powerful computers have combined to allow for proper analysis and valuation of complex derivative securities. Analysts can often employ short cuts to avoid having to estimate certain complex issues if given certain current information or assumptions about values and prices and assumptions about the future distributions of those values and prices relevant to the specific derivative security being evaluated.  The most common short cut (used to derive the Black-Scholes option pricing model) is the risk neutral framework.  Derivative securities, by their nature, tend to have implicit leverage.  Leverage means that the volatility of the derivative security is not constant, and its risk varies with the intrinsic value of the underlying asset.  The risk neutral model allows the analyst to avoid having to calculate dynamically over time the expected appreciation in the value of the underlying basis (and by extension the appreciation of the derivative security) and to avoid calculating the risk averse discount rate required by the investor for the derivative security because the expected appreciation in the value of the derivative security at any moment of time (minus a risk free rate of return) is offset by the risk premium over the risk free interest rate in the valuation process.[7]  This risk neutral assumption allows the valuation analyst (with certain observable current price information or value assumptions) to value the derivative security without having to estimate certain parameters normally required to value the security (in the absence of the risk neutral framework) but difficult to reliably estimate.

Notwithstanding these developments, complex derivative securities typically cannot normally be valued (or solved) based on a formula or solution provided in a book.   While books of option pricing formulas exist,[8] the formulas provided are often merely approximations, require very specific assumptions about the characteristics of the underlying asset and the likely distributions over time, and often do not match the exact terms of the derivative security of interest.   When a formula (or solution) is not readily available, then one is required to employ some form of simulation modeling to solve the problem and derive a value for the derivative security.

Estimating Volatility

 The formulas and even the simulations outlined in textbooks typically assume well-behaved distributions for prices and values that are continuous and do not allow for discrete events or discontinuities such as are present when barrier conditions or reset features are present in the terms of a derivative security.  These formulas also do not take into account the existence of multiple potentially dilutive securities and the effect that might have on volatility over the time period being modeled in the context of different scenarios.  Finally, the formulas and simulations assume that volatility will remain constant over time in the future over all possible future scenarios.  These assumptions are often unrealistic and sufficiently violated as to cause the resulting valuation models to be unreliable in practice.

The famous Black-Scholes option pricing model and its various extensions and applications, for example, assumes that the distribution of the underlying asset follows a process called geometric Brownian motion (assets provide a log-normal return distribution).  In this process, the standard deviation (volatility assumption) is constant over time as a percentage of the underlying asset or basis value, transactions may be made continuously over time at no cost to investors, and there are no discrete events.  Additionally, the price of the underlying asset or basis and its distribution is not contingent on other possible future events.

While these assumptions are generally reasonable and have proven reasonably valid for valuing short-term and even some medium options for options on well-established and relatively stable publicly traded securities, options on certain commodities, and options on currency exchange rates under certain circumstances, it is well-known that the assumptions of the Black-Scholes model are clearly violated even by publicly observed trades of such options trading on regulated exchanges.[9]  Most notably, the volatility implied by the Black-Scholes model based on observed or quoted options prices (“implied volatility”) at a point in time on a given asset will vary materially with the exercise price of the option.[10]  Even more importantly, the implied volatility will typically vary significantly as the time to expiration of an option on a given asset is significantly altered.

Additionally, for options on publicly-traded common shares, the implied volatility is also a function of the state of the development of the company and its products and the amount of leverage (debt and preferred securities) present in the company’s capital structure.  For example, for pharmaceutical and biotechnology company stocks, the volatility (as a percentage of the share price) will tend to decrease over time as the company proceeds through its development process and begins to realize significant revenues from product sales.  The volatility will also tend to decrease as the value of the company (and its share price) increases.  Companies with medical products in the early stages of clinical or pre-clinical development (with a substantial uncertainty of success and limited or no revenues) can have annualized implied volatilities of 100% and above.

Furthermore, as the issuing company’s value declines, the expected volatility of its share price will tend to increase substantially.  This increase in volatility occurs as a result of most companies having operating leverage (the presence of fixed costs and various obligations such as employment agreements and lease agreement).  This problem is exaggerated when the company has a substantial amount of financial leverage (debt securities and preferred securities with priority to common equity).  Therefore, the assumptions underlying the standard options valuation models cannot and should not be used without correction to value companies with a substantial amount of operating leverage or financial leverage.

What is seen far too often is valuation analysts taking an average of the observed historical volatilities for a relatively short and finite period of time and assuming that the volatility appropriate for valuing a given derivative security or set of derivative securities is that average.  When valuing a publicly-traded company, the historical volatility is typically based on the historical volatility of its share price over a relatively limited period of time.  There is often no adjustment made for the differences in the value of the company, its state of development, or changes in its leverage over time.

When valuing the derivative securities of public companies that have sufficiently active options markets, these issues mean that one must consider the implied volatility “surfaces”[11] on actively traded options.  A volatility surface provides some insight as to how the volatility of the underlying basis might change as the underlying basis value changes and as the time to expiration changes.  However, observed and calculated volatilities for even actively traded public companies are typically only accurate for options with a remaining life of up to one to two years at most.   Even if one has implied volatility data on options trading involving the public company’s options, one should consider the volatility surfaces as guidance as to how implied volatility might change as a function of the exercise price, asset price, and time to expiration and guidance as to how to extrapolate the short-term and medium-term implied volatilities observed into longer term volatility assumptions appropriate for the longer-term derivative securities being evaluated.   Also, implied volatilities for less actively traded options or less active exercise prices and longer time to expiration dates may have significant bid-ask spreads and vary significantly from one day to the next in the data, suggesting that some averaging or “filtering” of the data over multiple observation days may be required in order to more reliably estimate expected future volatility for valuation purposes.

When valuing a derivative security issued by a private company or issued by a public company with a limited or unrepresentative prior history, often valuation analysts use the average historical volatility observed for some finite period of time from some selected “comparable” public companies.  However, the averages are often unreliable without adjustments for the differences in operating and financial leverage between the subject company and the public companies and without adjustments for the changes in volatility in the market and industry over the observed period of time.  Additionally, where implied volatility information is available, it should be considered as additional information for adjusting the future industry volatilities to account for investor’s expectations for future volatility as compared with historical industry volatility.

In valuation assignments, the derivative securities being evaluated are often long-term in nature (two to ten years to expiration).  Volatility within an industry or company can vary significantly as a result of numerous changes over time such that the observed historical volatilities may not be reflective of expected future volatility and the implied volatilities observed may be limited to options expiring in the near term.  Thus, a valuation analyst needs to use considerable judgment and experience and may need to understand how volatility has changed over time in the past or is likely to change in the future within the industry as a function of leverage, general market volatility, a company’s development, and other factors.

Additional dilution & volatility considerations

 Once an analyst has examined and estimated how volatility is likely to change over time in general, or as a function of the underlying asset price, or as related factors such as the state of development or enterprise value are taken into consideration, the next step is to model the valuation of the subject derivative security.  A variety of models may be constructed given the tradeoff being capturing the essential elements that affect valuation and creating a model that is tractable and can be replicated in practice.

Beyond the issues of non-constant volatility, two additional issues tend to be neglected by valuation analysts in the modeling process when valuing derivatives securities involving companies.  First, many derivative securities outstanding may be dilutive in nature relative to the underlying asset.  The existence of multiple dilutive securities with different maturity dates invalidates the constant volatility assumption and creates potential “kinks” or “bends” in the future value of the company’s shares as a function of its future enterprise value as a whole.  Second, the existence of securities with greater priority claims (“financial leverage”-fixed costs and commitments, debt securities and preferred securities relative to common equity) means that the company’s share price may plausibly be close to or equal to zero for medium-term and especially for longer-term derivative securities in reasonably probable scenarios.  The risk of default or bankruptcy (financial distress) is often material to the valuation of the company’s junior securities and equity securities. Two consequences follow from the existence of significant financial leverage: the log-normal (constant percentage volatility) assumption is no longer valid for the stock price because a log-normal distribution implies zero probability of the asset having zero value or price in the future; and the distribution the future asset value or price may need to be modeled in the context of to the risk of failure (a risk of failure is like a barrier option with a knock out feature).

When a company has multiple different derivative securities outstanding, and the combined dilutive effect of such securities is potentially significant, the distribution of returns on the company’s share price will not be consistent with the standard log-normal distribution assumption due to skewness and the volatility will not be constant over time in the future.  It is common for analysts evaluating derivative securities of public companies to not account for the effect of dilution associated with dilutive derivative securities.  However, the ability to ignore dilution effects is based on two critical implicit assumptions: that the market price per share already reflects the effect of the dilution;[12] and the total amount of dilution is not significant enough (or likely significant enough) to materially alter the volatility process prior to the expiration date of the derivative being evaluated.    Unfortunately, when valuing private companies or public companies with non-public debt securities and/or multiple derivative securities that are dilutive in nature, these assumptions are often violated.[13]

Asset-Based Simulation: The Merton Contingent Claims Model

 One natural solution to address both leverage and the problem of multiple dilutive derivative securities is to employ an asset-based simulation model for a company.  The initial development of this model in finance is attributed to Professor Robert Merton.[14]  It is called the contingent claims model of the company.  Extensions of this work have now led to the development and refinement of practical models for estimating default probabilities for pricing and rating corporate debt and preferred securities.  These simulation models are increasingly used by credit ratings agencies, by securities analysts, and by asset managers.[15]  Additionally, this model is recommended (potentially required) by the AICPA when valuing certain private company issued equity and derivative securities.[16]

In the contingent claims model, the observed equity volatility rates must be adjusted to enterprise value (asset) volatility rates.  This model is particularly useful when the value of the either the company as a whole is known or estimable, or the value of a specific security or bundle of securities is known or observed as of, or relatively close to, the valuation date.[17]

In many instances, one may be able to estimate the company value or to observe the price of a given unit of investment that represents a bundle of securities including derivative securities, but may be unable to determine individually the values of most of the individual securities issued by the firm without simulation analyses.  Using the current or estimated market value of the firm (or a reasonable guess as a starting point), one can simulate the future scenarios on a current value basis by estimating the evolution of the current market value of the firm.  While simulating the market value of the firm over time, one can track the payments required to more senior securities (principal payments, interest and dividend payments, and scheduled repurchases) and estimate and account for such payments in terms of reducing the remaining enterprise value at each step.  Then, one can use the current value (intrinsic value) basis as the enterprise value in order to determine the intrinsic values of both the common equity and derivative securities at each step.  Once the intrinsic value of common equity and each derivative security is determined, then one can determine three things:

  • whether a specific derivative security would be exercised at its expiration date (or some other date when relevant);
  • the value realized from the exercise of the derivative security at its expiration date,
  • and the effect of that exercise of that derivative security on the intrinsic values of the common equity and other remaining derivative securities in the simulation model.

These models can be considerably more complex and may require multiple simulation runs in order to approximate (“calibrate”) the positive return (“positive drift” assumption) on current value of the firm required to offset the effects of dilution and to estimate certain the starting parameters and the starting value assumptions.  However, these models provide a more powerful and potentially accurate means of estimating the values of the outstanding securities and derivative securities of a company, especially when companies have multiple sets of securities, are highly leveraged, and/or have multiple dilutive securities with material amounts of dilution. [18]

The Merton contingent claims model is particularly useful when analyzing the securities of private equity backed companies where most of the value of the company is in the form of senior and mezzanine debt and preferred securities and/or substantial value is held in the form of embedded derivative securities, warrants, and options.  The valuation of common shares in such companies is often required for IRC Section 409A purposes, potential employee compensation purposes, and for financial accounting purposes (allocating values to the specific securities and embedded derivatives and determining the cost of incentive compensation in the company).  In order to both value the underlying shares and the incentive awards based on the value of underlying shares, a contingent claims model is required because the principal amounts of debt and preferred securities will often not be accurate estimates of their respective fair values. Contingent claims models are also particularly useful when estimating fair values for warrants and options that are significantly dilutive and/or subject to forfeiture or early exercise risk, as is commonly the case for employee stock options.

A simplistic version of the Merton contingent claims model is often used in private equity investing to determine the likely internal rate of return on investment that will be realized by the respective parties investing in a company (including management and shareholders retaining interests after closing) and the carried interest payments that the private equity manager will receive upon sale of the acquired firm assuming a finite set of (or range of) plausible realized future enterprise values.  When these finite scenarios are probability weighted and used to value the securities and interests of different parties, this is referred to as a PWERM analysis (Probability Weighted Expected Return Method).[19]  The problems with such methods are that the correct discount rate is typically unknown because the risk-neutral assumption is violated and the correct risk premium cannot be easily calculated.  Also, the scenarios may be insufficient in number or range to capture fully all possible outcomes realistically, and the probability weights for the scenarios are simply educated guesses (and often biased toward successful scenarios).[20]

In many circumstances, a contingent claims approach is overkill and unnecessary.  However, often it is not considered or employed when clearly required.  Many valuation analysts are unfortunately not familiar with such contingent claims models or capable of developing and applying such models to value a derivative security or a common equity security that is effectively a derivative security on the residual value of the firm.  We have observed instances recently where the failure to use such contingent claims models led to significant potentially adverse tax consequences,[21] potential allegations of breaches of fiduciary duties against management or directors,[22] and/or substantial errors and misstatements in audited financial statements.[23]

Summary and Conclusion              

The valuation of derivative securities has evolved from a niche academic innovation into widespread practice and is often required for tax or financial accounting purposes.  It may also be advisable that derivatives be evaluated in order for issuers or investors to understand what they are getting or giving up in an exchange.  The developments in our understanding of securities markets and the empirical pricing of publicly-traded options and other derivatives has led to a number of developments that allow us to value complex derivative securities using more conceptually correct and reliable models.

Too often valuation analysts resort to simplified models and assumptions that fail to reflect the developments in finance and empirical research over the past thirty years.  Thus, they employ inappropriate models of the volatility of the underlying assets, overly simplistic models, or formulas that fail to capture the salient terms of the subject derivative securities.  They also often ignore or fail to consider the interaction of the values of different securities and the existence of multiple dilutive securities on the volatility of the underlying security on which the derivative security is based.  In this context, care must be taken to ensure that the assumed volatility process for a given underlying asset is sufficiently understood and modeled to properly value the subject derivative securities.  This may require a better understanding of both observed historical volatilities and implied volatilities observed in the market for the subject firm or for guideline securities when the company’s securities are not publicly traded or have insufficient, unreliable, or inapplicable trading histories.  It also may require adjustments in the volatility assumptions and model for changes in leverage and developments within the company over time.

When valuing derivative securities in companies with a greater degree of risk of default or failure, or where multiple securities must be jointly valued, a more complete contingent claims model of the firm may be required.  This broader model is particularly important when valuing common shares or junior securities within more highly leveraged companies, when multiple dilutive derivative securities are outstanding, or when the values of multiple securities issued by the company must be jointly determined.



[1] See Merton, “A General Dynamic General Equilibrium Model of the Asset Market and its Application to the Pricing of the Capital Structure of the Firm” Sloan School of Management Working Paper #497-070, M.I.T. (December 1970) and Merton, “Theory of Rational Option Pricing,” The Bell Journal of Economics and Management Science 4, Spring 1973, pp. 141-183
[2] Employee (and director) stock options are actually warrants because the underlying common shares are not issued and outstanding and the underlying security must be issued by the company or a cash settlement must be paid by the company upon exercise.
[3] The conversion feature is a form of call or put option, warrant, or mix of derivatives.
[4] The par value may include or exclude earned but unpaid dividends or interest and may be different than the amount paid or invested by the holder for the security.
[5] See footnote 1.
[6] There are a variety of observed circumstances where the assumption of constant volatility is not observed.  Most assets tend to have periods of greater and lesser volatility.  For example, a “jump” or change in the value of an asset following a discrete event is often followed by a period of increased volatility due to greater investor uncertainty and heterogeneity of investor beliefs as to the future value of the asset.  Similarly, periods of greater general economic uncertainty may lead to periods of increased volatility in asset prices which well then tend to revert to a more normal level of volatility over time.  See, for example, Taylor, Asset Price Dynamics, Volatility, and Prediction, 2005, pp. 69 and 82 and more extensive discussions in Chapters 8 to 11 and 13 concerning these issues.
[7] This will be exactly true in a perfectly efficient market, but is only approximately true in the real world for very actively traded commodities and securities with active options and futures markets.  On average, and in general, the assumption is still valid generally for fair value reporting purposes.
[8] See, for example, Nelken, The Handbook of Exotic Options: Instruments, Analysis, and Applications, 1996.
[9] See, for example, Jackwerth, Jens Carsten, “Option-Implied Risk-Neutral Distributions and Implied Binomial Trees: a Literature Review,” Journal of Derivatives 7, Winter 1999, pp. 66-82.
[10] See, for example, Taylor, Asset Price Dynamics, Volatility, and Prediction, 2005, pp. 378-383.
[11] A volatility surface is a map estimated on observed options trades. The surface models how the implied volatility changes as a function of the time to expiration and/or the exercise price.  It might also be studied over time to understand how the implied volatility changes with changes in the underlying asset price.  It can help identify how volatility assumptions might be altered as the asset price changes or the time to expiration changes.
[12] See, for example, Handley, “On the Valuation of Warrants,” The Journal of Futures Markets, 22, 2002, pp. 765-782.
[13] See, for example, Galai and Schneller, “Pricing of Warrants and the Value of the Firm,” Journal of Finance 33, 1978, pp. 1333-1342; Crouchy and Galai, “The Interaction Between the Financial and Investment Decisions of the Firm: The Case of Issuing Warrants in a Levered Firm,” Journal of Banking and Finance 18, 1994, pp. 861-880; Galai, “A Note on ‘Equilibrium Warrant Pricing Models and Accounting for Executive Stock Options,” Journal of Accounting Research 27, 1989, pp. 313-315; Darsino and Stachell, “On the Valuation of Warrants and Executive Stock Options: Pricing Fomulae for Firms with Multiple Warrants/Executive Options,” Cambridge Working Papers in Economics 0218, 2002; Chang and Liao, “Warrant Introduction Effects on Stock Return Processes,” Applied Financial Economics 20, 2010, pp. 1377-1395; Yagi and Swaki, “The Pricing and Optimal Strategies of Callable Warrants,” European Journal of Operational Research 206, 2010, pp. 123-130; and Bajo and Barbi, “The Risk-Shifting Effect and the Value of a Warrant,” Quantitative Finance 10, 2010, pp. 1203-1213.
[14] Initially written as a working paper “A General Dynamic General Equilibrium Model of the Asset Market and its Application to the Pricing of the Capital Structure of the Firm” Sloan School of Management Working Paper #497-070, M.I.T. (December 1970) and later developed in a series of published papers and texts by Merton, including:  “Theory of Rational Option Pricing,” The Bell Journal of Economics and Management Science 4, Spring 1973, pp. 141-183; “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance 29, 1974, pp. 449-470; “Contingent Claims Pricing and the Modliani-Miller Theorem,” Journal of Financial Economics 5, 1977, pp. 241-249.
[15] See, for example, Loffler and Posch, Credit Risk Modeling Using Excel and VBA, Second Edition, 2011.
[16] See AICPA’s Accounting and Valuation Guide: Valuation of Privately-Held-Company Equity, 2013, Chapter 6, particularly 6.01 and 6.30 to 6.41 for a brief discussion of the benefits and issues with the methodology.  However, the written text is very brief and does not consider the full range of applications and practice experiences now available.
[17] This is often the result of an observed merger or acquisition transaction or a funding event where the price per unit or the price of the company as a whole is known but the values of individual company securities are not known or easily estimable.
[18] Enterprise value distributions are more consistent with the log-normal distribution assumption than common share value distributions, and the constant volatility assumption is closer to being accurate for enterprise values than for common share values. These facts, combined with the nature of financial leverage, mean that simulation models based on simulating enterprise value will often be more robust in estimating junior debt and preferred security values, common share values, and derivative values in companies with significant financial leverage or with relatively high common share value volatilities (in excess of 100%).
[19] Sometimes, the private equity firm will use an inflated internal return of require requirement (a “hurdle rate”) to discount the value of the future value that will be realized by an investor or the private equity fund to determine the value of the investment or to determine whether to invest.  Unfortunately, this can result in highly misleading and inaccurate values and return expectations even though commonly done.  See, for example, Smith, Smith, and Bliss, Entrepreneurial Finance: Structure, Valuation and Deal Structure, 2011, pp. 344-351.
[20] Ibid.
[21] In some cases, significant taxes, interest and penalties were determined to be owed by employees that received derivative securities that were illiquid due to “in-the-money” derivative securities or the undervaluation of securities awarded at the time of grant or upon vesting by the recipients of incentive compensation awards.  Similarly, taxes, interest, and penalties were be owed by an estate that made gifts or for estate taxes due to substantial undervaluation.
[22] The officers and directors awarded themselves compensation that proved to be excessive by setting artificially low exercise prices or awarding excessive amounts of derivative or common equity securities due to undervaluation.  An exchange of common equity for preferred over-valued the preferred and undervalued the common equity securities such that the persons receiving preferred interests received less-than-equivalent value for their prior interests in a reorganization or restructuring transaction.
[23] Misstatements of the values of embedded derivatives or employee stock options are surprisingly common even in public companies due to the issues previously discussed.
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