THE VALUATION OF EMPLOYEE STOCK
OPTIONS
Black-Scholes-Merton
versus
Binomial Tree
Groningen, August 2006
Rutger Meiberg
S1076124
Faculty of Economics / Faculteit der Economische Wetenschappen
Programme: Economics
/
Economie
Specialization: Finance & Risk Management / Financiering & Risico Management
Supervisors:
Dr. P.P.M. Smid
University of Groningen (Rijksuniversiteit Groningen)
Dr. H. Oosterhout
Mr. F. Bollmann
Chapter 1: Abstract ... 4
Chapter 2: Introduction ... 6
2.1 Introduction... 6
2.2 Employee stock options ... 6
2.3 Problem description ... 8
2.4 Research questions... 11
Chapter 3: Theoretical Articles and Empirical Research ... 14
3.1 Introduction... 14
3.2 Option valuation models ... 14
3.2.1 Introduction ... 14
3.2.2 The Black-Scholes-Merton model (*) ... 15
3.2.3 The binomial tree (*) ... 20
3.2.4 The trinomial tree (*)... 27
3.2.5 Valuing options incorporating market-based vesting requirements ... 28
3.3 Incentive effects of employee stock options ... 29
3.4 Valuation of employee stock options... 33
3.4.1 Introduction ... 33
3.4.2 Value to the executive ... 34
3.4.3 Cost to the company ... 36
3.5 Summary ... 40
Chapter 4: Accounting Standards (*) ... 42
4.1 Introduction... 42
4.2 Accounting practices development ... 43
4.3 IFRS 2: Basic outline ... 45
4.4 Summary ... 45
Chapter 5: Standard & Poor’s Corporate Value Consulting Binomial Tree Model ... 46
5.1 Introduction... 46
5.2 The CVC binomial tree model ... 46
5.2.1 Description of the CVC binomial tree model ... 46
5.2.2 Traditional input parameters... 56
5.2.3 Other input parameters ... 57
5.3 Extension of the model: market-based vesting ... 60
5.4 Data ... 65
5.5 CVC model remarks... 66
5.6 Summary ... 68
Chapter 6: Valuing Employee Stock Option Plans... 69
6.1 Introduction... 69
6.2 Methodology ... 69
6.3 Fair value measurement company A, B and C... 70
6.3.1 Introduction ... 70
6.3.2 Contractual conditions ... 70
6.3.3 Traditional input parameters... 71
6.3.4 ESO-specific input parameters ... 72
6.3.5 Fair value measurement... 74
6.4 Fair value measurement of ESOs exhibiting market-based vesting... 76
6.4.1 Introduction ... 76
6.4.2 Fair value measurement... 76
6.5 Summary ... 82
Chapter 7: Fair Value Sensitivity Analysis ... 83
7.1 Introduction... 83
7.2.1 Introduction ... 83
7.2.2 Traditional input parameters... 86
7.2.3 ESO-specific input parameters ... 89
7.2.4 Expected life ... 92
7.3 Summary ... 94
Chapter 8: Summary ... 96
Appendix I.A: Input parameters versus option value in the BSM model (*)... 99
Appendix I.B: Put-call parity (*) ... 101
Appendix I.C: Hull & White model (2004) ... 103
Appendix II: IFRS 2 Application guidance (*) ... 106
Appendix III: Statistical analysis (*) ... 109
Literature list ... 111
Foreword
Beginning of May 2004, I started the process of writing this Masters thesis on the valuation of Employee Stock Options. Within the specialization Finance and Risk Management, I have always had a strong interest in courses relating to financial instruments in general and financial
derivatives in specific. Therefore, the possibility to write a thesis that combines theoretical aspects regarding option pricing with practical valuation considerations provided an interesting and challenging opportunity for me. Also, the fact that this process was to be conducted in a business setting, namely within the Amsterdam practice of Standard & Poor’s Corporate Value Consulting (currently Duff & Phelps), added an extra dimension to the entire process.
From this place I would like to thank all who contributed to and supported me in the process of writing this thesis. This list of people includes my family, friends and fellow students as well as university staff and colleagues at Duff & Phelps and is too extensive too list here. However, some of these people have played such an important role that I would like to explicitly mention them here.
First, I would like to thank Dr. Peter Smid of the University of Groningen for being my university supervisor and, as such, providing me with his assistance throughout the process of writing this thesis. I have drawn heavily upon his patience and would like to thank him for his understanding and advice. Second, I would like to thank Henk Oosterhout and Frank Bollmann of Duff and Phelps Corporate Value Consulting for providing me with all the advice and help I needed, as well as for the time and efforts they have spent reading and correcting the draft versions of this thesis.
In addition, I would like to thank all my friends for contributing to the very rewarding time I have had studying Economics both in Groningen and Aarhus (Denmark).
Last, but not least, my sincerest and utmost thanks go out to my parents, my girlfriend and my brothers, in no particular order, for their everlasting support.
As a final note, I would like to express my hope that this thesis has turned into an enjoyable and understandable read for anyone interested in the valuation of Employee Stock Options.
Chapter 1: Abstract
The study conducted in this thesis deals with the valuation of Employee Stock Options (ESOs). Employee stock options are non-traded options, granted by a company to its employees, that have specific restrictions. For instance, ESOs cannot be traded. Second, ESOs cannot be exercised during pre-vesting periods. Finally, ESOs are typically dependent upon employment with the firm. These characteristics make that ESOs differ from traded options. Their specific
characteristics are more likely to be adequately captured by lattice models, rather than by closed-form solution models, such as the Black-Scholes-Merton (BSM) model. The study in this thesis focuses on the fair value measurement of employee stock options under the new accounting guidelines set forth in IFRS 2, and the differences between the CVC model and a modified version of the Black-Scholes-Merton model.
The CVC model is a binomial tree model, developed in-house by Standard & Poor’s Corporate Value Consulting (currently Duff & Phelps), that explicitly takes into account the ESO-specific characteristics set forth above. This existing model was extended and applied to value a number of real-life case studies.
The case study approach, an in-depth analysis of a small number of option plans, is considered appropriate for two reasons. First, one of the research objectives is to illustrate the working of the model. This can be achieved by looking at a smaller number of option plans as well. Second, another research objective is the comparison between the CVC model and the BSM model. From the small number of option plans some indications can be derived regarding the sign of this difference. However, in order to derive any statistically significant conclusions regarding the relative amount of this difference and regarding the impact of expensing on the profit and loss statement, future research examining a larger number of option plans is needed.
The extensions made to the CVC model include the ability to deal with market-based vesting requirements and the impact of dividends on exercise decisions. The outcomes of applying the CVC model to the case studies were compared to the outcomes resulting from a modified version of the BSM model, in which expected life is used rather than contractual life. In addition, this modified model (BSM EL model) adjusts the option price for the effect of forfeiture during the pre-vesting period of the option.
Based on the valuation of the case studies performed in this thesis, there seems to be an indication that the CVC model is likely to result in a lower option value as compared to the BSM EL model. This was the case for eight out of nine observations. However, given the small number of
sample. In addition, the input parameters describing employee behavior in the CVC model were based on analysis of historical data. The BSM EL value was calculated using the expected life estimate as determined by the companies included in the sample. As a result, the ESO-specific input parameters to the CVC model, and the resulting implied expected life, may not be
consistent with the expected life estimate used in the BSM EL model. This is a potential source of value differences in the case studies. Therefore, whether or not the CVC model results in lower values as compared to the BSM EL model, when observing real-life cases, depends on the consistency between the two models with regard to expectations of employee behavior. In order to rule out any value differences between the two models arising due to differences in expected life estimates, the CVC model and the BSM EL model were calibrated to expected life using a hypothetical ESO plan. When calibrating the models to expected life, the CVC value resulted in lower option values as compared to the BSM EL model providing another indication that the CVC model is likely to result in lower values as compared to the BSM model. When expected life is the same, any value differences are expected to be in the range of negative 11.2% to negative 2.6%. For the H&W model this range is negative 6.0% to 2.6%. It should be noted that these results might depend on the set-up of the hypothetical ESO plan. The differences are observed over an expected life interval ranging from 6 to 8.5 years. This is the interval resulting from using a wide range of input parameters to the CVC model.
In general, there seem to be two main explanations for differences between the BSM EL model and the CVC model:
a) a difference in the expected life used in the BSM EL model and the implied expected life in the CVC model; and
Chapter 2: Introduction
2.1 Introduction
The subject of this thesis is the valuation of employee stock options (ESOs). This chapter describes the research methodology of the study conducted in this thesis. An introduction to employee stock options and some commonly used option terminology is provided in section two. Section three turns to the problem at hand, the valuation of ESOs. The fourth section lists and discusses the research questions addressed in the study.
2.2 Employee stock options
Employee stock options are call options, granted by a company to its employees or management, with the company’s stock as the underlying asset. A call option on a stock is the right, but not the obligation, to buy the underlying stock, at a pre-determined price. This price is called the exercise price or strike price. Options can only be exercised during or at the end of a pre-determined period. This period is referred to as the time to maturity or the life of an option. Options that can be exercised at any point in time during the option’s life are named American-style or simply American options. European-style options are options that can only be exercised at the end of the option’s life. The final day on which the option can be exercised is also referred to as the
expiration date.
The value of an option can be divided into two components: time value and intrinsic value. The latter component is also referred to as exercise value. This is the maximum of: a) the difference between the stock price and the exercise price and b) zero. Since an option is the right to buy, a rational investor will never exercise a call option if the stock price is lower than the exercise price and therefore the minimum value of an option is zero. From this it can be seen that options have an a-symmetric pay-off profile: the pay-off of an option is truncated on the downside at zero while the up-side potential is unlimited. When the intrinsic value of an option is positive, i.e. the stock price exceeds the exercise price, an option is said to be in-the-money. The terms at-the-money option and out-of-the-at-the-money option are used to refer to options for which the price of the underlying stock is equal to or below the strike price, respectively.
price, while they are protected from decreases in the stock price below the exercise price the uncertainty regarding future stock price movements holds value. This can be illustrated using the following example. Consider an out-of-the-money call option. By definition, the intrinsic value is zero. However, this does not mean the option is worthless. Since there is a chance that the option will become in-the-money before maturity, the option still holds value.
Publicly traded American companies generally disclose some information regarding the employee stock options they grant to their employees. When looking at these disclosures, a few high-level observations can be made that hold for almost all ESOs. First, ESOs are American options. Second, ESOs have long maturities, typically ranging from 5 years up to as long as 10 years. In comparison, most exchange-traded options have maturities of two years or less. Third, the underlying shares of ESOs can be either existing shares or new shares. In the latter case, the existing share capital is diluted. Tradable options on new shares are called warrants. Nevertheless, most companies refer to employee stock options on new shares as “options” rather than as
“warrants”. Finally, it is important to note that in practice ESOs are granted to employees at all levels of an organization, from rank-and-file workers up to the CEO and to employees at all organizational levels in between. The percentage of option value to total pay differs drastically between organizational levels. Throughout the remainder of this thesis, the term “executive” is used to refer to the holder of an ESO. The reason for this is that most academic literature focuses on stock options held by executives rather than by employees in general. However, the term “executive” can be read more generally as “employee” or “option holder”. Only with regard to the discussion of incentives in section 3.3, some distinctions are made between executives on the one hand and lower-level employees on the other.
Employee stock option contracts hold some restrictions that make ESOs different from exchange-traded options. A very important restriction of an ESO is its non-tradability. This is the main difference between ESOs and exchange-traded call options. Companies issuing ESOs do not allow the option holders to sell or transfer their options to other persons. In addition, companies do not allow their employees to trade the company’s stock or to trade in options on the
company’s stock. As a result, the holders of ESOs cannot hedge their exposure to the company’s stock price.
period. Vesting conditions can also be linked to performance measures. For instance, the percentage of options that will vest may depend on the total shareholder return (TSR), the company’s share price development or the growth in earnings per share (EPS) over a certain period.
Another difference between ESOs and tradable options is that holding ESOs is very often conditional on employment by the company, even after the pre-vesting period. For instance, the contractual terms of ESOs may disallow an option holder to continue to hold ESOs after his retirement from the company. Typically, when an option holder’s employment is terminated, he is typically forced to exercise any vested options. If the options are out-of-the-money, they are typically forfeited. Summarizing, ESOs are subject to three restrictions:
1. ESOs cannot be traded;
2. ESOs cannot be exercised during pre-vesting periods; and 3. ESOs are typically dependent upon employment with the firm.
Finalizing this section, the following should be noted with regard to option terminology. The term “grant date” refers to the date at which an ESO is granted. Similarly, the “vesting date” refers to the date at which an ESO vests. Only options that have vested can be exercised. The time period during the grant date and the vesting date is often referred to as the “vesting period” or “pre-vesting period”. In this thesis, the latter term is used. Similarly, the term “post-“pre-vesting period” refers to the time between the vesting date and the maturity of the option. In this thesis, the term “option grant” is used to refer to either one or more options granted to a single employee. All options that fall under the same grant are similar with respect to for instance their exercise price, time-to-maturity and grand date. However, the options within an option grant may have different vesting dates. For instance, the contractual conditions of an option grant may state that 50% of the options vest after a year, while the remainder vests after two years. The term “option plan” is used to refer to a collection of option grants.
2.3 Problem description
On February 17, 2004, the International Accounting Standards Board (IASB) released
date. Second, it provides guidelines for estimating the fair value of these payments. In the absence of market prices, the IASB requires companies to use option valuation models to estimate the fair value of Employee Stock Options. Typically, market prices for these options are not available. Since they cannot be traded there are no secondary markets for ESOs. The valuation models have to meet minimum requirements in the form of a number of option characteristics that have to be included in the model. Current business practice is to disclose information about share-based payments in the footnotes to the financial statements. The valuation of these payments is often based on the Black-Scholes-Merton (BSM) model (see section 3.2.2).
Beginning January 1, 2005, all publicly quoted companies in Europe were required to apply IFRS as their primary generally accepted accounting principles (GAAP) and therefore compliance with IFRS 2 is mandatory. Chapter 4 provides an overview of the main features of IFRS 2 and
discusses the rules for measuring the fair value in more detail. Before, no formal accounting standards regarding ESOs were in place.
Following the issuance of IFRS 2, the Financial Accounting Standards Board (FASB) published the exposure draft ”Share-Based Payment”. The guidelines in this draft are very similar to those in IFRS 2. The FASB states that it will require companies to record the fair value of ESOs and other forms of share-based payment as an expense in their income statement as of the grant date. In addition, the FASB recommends the use of so-called lattice models to estimate the fair value of ESOs. Examples of lattice models are the binomial tree and the trinomial tree model discussed in chapter three. The terms lattice and tree are used interchangeably throughout this thesis. The exposure draft was finalized by the issuance of FAS 123R in December 2004. FAS 123R is a revision of FAS 123. These accounting standards are discussed more in-depth in chapter four, with a focus on European accounting principles in the form of IFRS 2.
Expensing of ESOs is quite a controversial issue. Announcements made by the FASB in the early nineties that the FASB would require the expensing of ESOs revoked heavy opposition,
Also in Europe there has been quite some discussion on the impact and desirability of the new accounting standards. Many opponents argue that employee stock options are very hard, if not impossible, to value accurately and therefore should not be expensed. However, proponents argue that recognizing the value of ESOs by expensing them increases financial transparency, which will benefit investors and other stakeholders.
The issue whether or not ESOs should be expensed is strongly related to another issue at stake: how should the value of ESOs be measured? Until now, one of the most widely used methods to value employee stock options for accounting purposes is the Black-Scholes-Merton model. However, this model has a number of shortcomings in valuing ESOs. This is acknowledged by both the IASB and the FASB. IFRS 2 states that ESOs have certain characteristics that may preclude the use of the BSM model, while the FASB initially took it one step further by explicitly recommending the use of lattice models in its exposure draft. However, in the subsequent
statement FAS 123R this statement was dropped.
The main reason for recommending companies to apply lattice models lies in the fact that these models allow for much more flexibility in valuing options than the BSM model does. Employee stock options have some specific features that require more modeling flexibility than the BSM model allows for. These features include the vesting requirements, forfeiture rules and non-transferability identified earlier.
Non-transferability is a characteristic that does not apply to exchange-traded option contracts. As mentioned before, employee stock options cannot be sold or transferred. Therefore, when option holders, want to liquidate their option positions for instance for liquidity or diversification reasons, they have to exercise their options. The Black-Scholes-Merton model is not able to take this early exercise into account. In an attempt to include the value impact of early exercise when using the BSM model, the contractual life of the option is typically replaced by an estimate of the expected life of the option. In contrast, lattice models value the model over the full contractual life of the option, while also taking into account early exercise.
The two adjustments to the BSM value mentioned above, using expected life and adjusting for pre-vesting forfeiture, are both part of the approach described in FAS 123.
Furthermore, the BSM model uses some other strict assumptions to value options. For instance, the BSM model assumes constant interest rates and volatility. Lattice models are able to include the term structures of interest rates and volatility.
Summarizing, employee stock options are non-traded options that have specific characteristics that differ from traded options. Their specific characteristics are more likely to be adequately captured by lattice models, rather than by closed-form solution models, such as the BSM model. The study in this thesis focuses on the fair value measurement of employee stock options under the new accounting guidelines set forth in IFRS 2, and the differences with other valuation models.
2.4 Research questions
Following from the general description of the problem, the research questions addressed in this thesis are:
1. How does the CVC binomial tree model value ESOs?
The mechanics of the model in general and, more specifically, the way in which it deals with ESO-specific features such as vesting, forfeiture, early exercise etc. are discussed. Also, some shortcomings of the model are listed.
2. How can the CVC model be extended to value ESOs that include market-based vesting requirements?
Three real-life case studies are discussed. The fair values using the CVC model, the values resulting from the BSM model and the subsequent impact on the profit and loss statements are disclosed.
4. Is the fair value, measured by the CVC model, significantly over- or underestimated by the Black-Scholes-Merton option value?
The BSM values of the ESO plans under consideration is calculated. Furthermore, any differences between the results from the CVC model and the results from the BSM model are discussed. Although only a small sample of ESO plans is considered, these results may shed some light on the question whether or not BSM values are higher than the fair values based on the CVC model.
5. What is the sensitivity of option value to factors such as pre-vesting period, exercise behavior, dividend yield, termination probabilities etc.?
For example, questions that will be answered are: what is the effect on the option’s fair value if the volatility estimate changes? What effect does this change have on the BSM option value? It is important to note that there are interdependencies between option features. An increase in the pre-vesting period reduces the decreasing early exercise effect, but increases the
value-decreasing forfeiture effect. The aim of this research question is to illustrate and explain these interdependencies. Investigating these effects will help companies gain insight into the dynamics that drive option values and how these effects impact company earnings.
The remainder of this thesis is divided as follows. In order to put this subject into an academic and regulatory context, previous academic work on the valuation of ESOs is discussed in chapter three while the accounting guidelines regarding the accounting treatment of ESOs are the subject of chapter four. It should be noted that chapter three start with a summary of two standard option pricing models: the BSM model and binomial tree model. These sections are marked with an asterisk and can be skipped by readers with a basic understanding of these models.
Chapter 3: Theoretical Articles and Empirical Research
3.1 Introduction
This chapter provides an overview of relevant theoretical articles and empirical research on employee stock options (ESOs) and is divided as follows. Section two briefly discusses the basics of traditional option pricing theory by discussing the two most well-known and most widely used option pricing models: the Black-Scholes-Merton (BSM) model and the binomial tree model. An understanding of these models is required, since these models or extensions of these models will be applied in this study. For further reference, see Hull (2005) or Wilmott (2006). They provide extensive coverage of both models and of the underlying economic foundations. Section three summarizes academic work with regard to the incentive effects of ESOs. These incentive effects form the economic rationale for granting ESOs and provide insight into executive behavior and companies’ practices regarding the setting of option terms. The fourth section presents an overview of previous academic work on ESO valuation. This overview discusses how previous academic studies attempt to incorporate ESO-specific characteristics in valuing employee stock options. Hence, this section illustrates the most important caveats in valuing ESOs. Finally, the chapter concludes with a summary.
This chapter provides the necessary theoretical background regarding option valuation in general and the valuation of ESOs in specific and provides an important building block for the remainder of this thesis.
3.2 Option valuation models
3.2.1 Introduction
3.2.2 The Black-Scholes-Merton model (*)
In their famous article “The Pricing of Options and Corporate Liabilities”, Fischer Black and Myron Scholes (1973) construct an option pricing formula that prices European-style options on non-dividend paying stocks. This formula values options based on five parameters:
1. the stock price; 2. the exercise price;
3. the volatility of the returns of the underlying asset; 4. the time to maturity; and
5. the risk-free interest rate.
Table 3.1 lists the relationships between these parameters and the value of a call option.
Note that the expected dividend yield is included in table 3.1. This parameter is added, since later in this section an extension to the model that includes the effect of dividends is introduced. The signs in the second and third column indicate whether the option value increases or decreases with an increase in the associated parameter, all other things equal.
Table 3.1: Effect on call option value from an increase in an input parameter.
Input parameter Call option value
American European
Stock price + +
Exercise price - -
Volatility + +
Time to maturity + ?
Risk-free interest rate + +
Expected dividend yield - -
A detailed explanation of these relationships is provided in appendix 1.A.
Having established the relationships between input parameters and option value, the question of how the option value is determined still remains. The Black and Scholes formula is based on two important economic foundations.
provide the same future pay-offs. The price of instrument A is higher than that of instrument B. An investor who sells short instrument A and uses the proceeds to buy instrument B will earn an immediate profit, without any future cash-flow consequences. This trading activity is an example of arbitrage. Simply put, the no-arbitrage principle means that all financial instruments are correctly priced relative to each other. Hence, assuming no-arbitrage, instruments A and B introduced earlier should have the same price. The no-arbitrage principle excludes the possibility of investors earning riskless profits. Therefore, in the absence of arbitrage opportunities, a riskless portfolio would earn the risk-free interest rate.
Black and Scholes construct a riskless portfolio consisting of a short position in a call option and a long position in the shares underlying the call option. They show that a short position in an option can be perfectly hedged with a long position in the underlying asset. Hedging involves taking a position with an opposite pay-off profile compared to the initial portfolio in order to reduce or eliminate risk.
However, this riskless portfolio is only riskless for an infinitesimally short time interval. The price of the option changes due to the passage of time and due to changes in the price of the underlying asset. As a result, the amount of shares needed to perfectly hedge, or offset, the short option position changes. Therefore, the riskless portfolio must be continuously rebalanced in order to remain riskless. The process of continuously rebalancing in order to remain riskless is also referred to as dynamic hedging or delta hedging.
Following from the no-arbitrage principle, the return on this risk-free portfolio must equal the risk-free rate, as stated earlier. As a result, the BSM model does not include any assumptions regarding the expected return of the underlying stock. Expected returns are based on investor’s risk preferences. Since expected returns do not enter the equation, no assumptions regarding investor’s risk preferences are included in the model. This leads to the other keystone following from the BSM model: the principle of risk-neutral valuation.
compensation for the risk they bear, only for the time value of their investment. The concept of risk-neutral valuation states that valuing a derivative assuming that investors are risk-neutral gives the correct price for a derivative in all situations with different risk attitudes, not just in a risk-neutral world. This can be explained in the following way. Moving from a risk-neutral world to a risk-averse world has two effects. First, the expected growth rate changes to reflect the compensation for risk. Second, the discount rate changes also as a result of the increase in risk. These two opposing effects always offset each other exactly. Summarizing, the concept of risk-neutral valuation states that it is correct to assume investors are risk-risk-neutral when valuing an option or other derivative.
The Black and Scholes formula values options under some strict assumptions. Hull (2005) lists these assumptions:
1. the returns on the stock price follow a process called geometric Brownian motion; 2. the short selling of securities with the full use of proceeds is permitted;
3. there are no transaction costs or taxes and all securities are perfectly divisible; 4. there are no dividends during the life of the derivative;
5. there are no riskless arbitrage opportunities; 6. security trading is continuous; and
7. the risk-free rate of interest is constant and the same for all maturities.
with:
c = the value of the call option;
S0 = the stock price at time t = 0;
X = the exercise price;
r = the continuously compounded risk-free rate on an annual basis;
σ2 = the volatility of the return of the underlying asset on an annual basis;
T = the time to maturity of the option measured in years; ln(x) = the natural logarithm of x; and
N(x) = the cumulative probability distribution for a variable that is normally distributed with a mean of zero and a variance of one.
This formula can be explained at a more intuitive level in the following way. The N(d) terms, given by equations 3.2 and 3.3, can be loosely interpreted as the risk-adjusted probabilities that the option will expire in-the-money. First, assume the N(d) terms are close to one and the option is highly likely to expire in-the-money. Based on equation 3.1 this would result the option to be worth approximately
S
−
Xe
−rT0 , which is the difference between the stock price and the present
value of the exercise price. If the probability of expiring in-the-money is close to 1, the option will almost certainly return its intrinsic value. However, the exercise price has to be paid at exercise. Therefore, the exercise price is discounted to its present value.
Next, consider the N(d) terms to be close to zero. If the probability of expiring in-the-money is zero, the option is worthless. As expected, the right-hand side of the BSM formula in equation 3.1 approximately reduces to zero when setting the N(d) terms close to zero. As a final remark, note that the N(d) terms cannot be exactly equal to zero or equal to one. Uncertainty regarding the future stock price causes the probability that the option expires in-the-money to be larger than zero and smaller than one at any time before maturity.
in-the-money, the smaller the volatility of the stock price and the smaller the time to maturity. As a result, the N(d) terms increase with the probability that an option expires in-the-money.
The value of a put option can be derived from the price of a call option. The relationship between call option and put option prices is called put-call parity and is elaborated on in appendix 1.B. Mathematically, the BSM for call options can be used to value the put options by inputting a negative volatility. The absolute value of the resulting option value is the value of a put option. Merton (1973) sets forth an alternative derivation of the Black-Scholes model. Furthermore, he extended the model. His extended version of the formula provides the value of an option on a dividend-paying stock. The extended formula is shown in equation 3.4.
) ( ) ( 1 2 0e N d Xe N d S c= −qT − −rT (3.4) with:
q = the continuously compounded annual dividend yield.
This argument does not hold for American options on stocks that pay dividends. For these options, early exercise can be beneficial. Recall from earlier observations that distributing dividends to shareholders leads to a drop in the share price. Therefore, exercising immediately prior to the ex-dividend date might be optimal to holding the option until maturity. Whether this is optimal depends on the amount of the dividend and the remaining time to maturity. Hull (2005) derives a lower bound of the amount of the dividend. If the dividend is larger than this lower bound, early exercise is optimal for a sufficiently high share price. This condition is depicted in equation 3.5: ( )
[
rT tn]
nX
e
D
>
1
−
− − (3.5) with: nD
= the dividend at time n;T
= time to maturity; andn
t
= time n.Black (1975) suggests an approximation procedure to take into account early exercise for American options on dividend-paying stocks. This procedure is based on the BSM model. However, to obtain a more accurate price of an American-style option, option theory suggests the use of lattice models. These models incorporate the optimal early exercise decision by testing at discrete points in time if the intrinsic value, the amount an investor receives when he exercises the option, is larger than the holding value, the value of the option if the option is not exercised. The option value is set equal to the maximum of these two values. The binomial tree, discussed in the next section, is an example of a lattice model.
3.2.3 The binomial tree (*)
binomial tree approach assumes that the returns on the underlying asset follow a random geometric Brownian motion. However, the binomial tree approach assumes that the stock price follows a discrete-time process, whereas the BSM model assumes a continuous-time stock price process. The binomial tree approach divides the stock price process in a large number of small time intervals in which the stock price can either go up or down. As the time interval between two time steps becomes infinitesimally small, the discrete-time stock price process approximates a continuous-time stock price process. In the limit, the BSM assumptions of continuous trading and a log-normally distributed stock price are satisfied when using specific parameter values as proposed by Cox, Ross and Rubinstein (1979). In that case, their binomial option value converges to the BSM value.
The main advantage of the binomial tree approach is that it allows for more flexibility than the BSM model. For instance, the binomial tree approach can easily be extended to include early exercise. Since the stock price process is divided in small intervals, at each time step, the exercise decision can be considered.
As mentioned before, the binomial model set forth by Cox, Ross and Rubinstein (1979) is based on the construction of a riskless portfolio. Similar to the approach taken by Black and Scholes (1973), this riskless portfolio consists of a short position in a call option and a long position in the shares underlying the option. The long position in the shares exactly hedges the short option position. The amount of shares needed to accomplish this is defined as Δ.
During each time interval the stock price is assumed to either go up or down. If an up move occurs the stock price is multiplied by u (u >1), while a down move leads to multiplication of the stock price by d (d <1). The initial value of the stock price is S0. During the first period, the
stock price either goes up to S0u or down to S0d. To have a riskless portfolio, the portfolio value
in the up state and down state must be the same. This equality is stated in equation 3.6.
d u
S
d
c
c
u
S
−
=
Δ
−
Δ
0 0 (3.6) with:Δ = the amount of shares in the portfolio; S0 = the stock price at time t = 0;
cu = the value of the call option after an up move; and
Equation 3.6 can be re-arranged as: d S u S c cu d 0 0 − − = Δ (3.7)
This equation shows the amount of shares the long position should consist of in order to perfectly hedge the short call option position. Equation 3.6 describes the situation faced after one period. Since the portfolio is riskless, the value of the portfolio after one period can be discounted at the risk-free rate. The present value equals the cost of setting up the portfolio to preclude arbitrage. Therefore, at time t = 0:
(
S
u
c
)
S
c
e
−rTΔ
−
u=
Δ
−
0
0 (3.8)
Substitution of equation 3.7 into equation 3.8 and simplifying yields:
(
)
[
u d]
rTpc
p
c
e
c
=
−+
1
−
(3.9) with:d
u
d
e
p
r t−
−
=
Δ (3.10)Note that
p
is not the probability that the stock moves up. One would expect the probabilities that the stock moves up or down to influence the price of an option. However, this is not the case. These probabilities are already incorporated into the stock price. Therefore, there is no need to take them into account when valuing the option. However, the termp
can be regarded as a specific probability. It is the probability of an up move in a risk-neutral world. Similarly,(
1− p)
is the risk-neutral probability of a down movement. This can be shown in the following way. The expected value of the stock at time t = 0 is given in equation 3.11:
( )
S
pS
u
(
p
)
S
d
Substituting equation 3.7 into equation 3.11 and simplifying results in:
( )
S
S
e
rTE
=
0 (3.12)This equation shows that the expected return on the stock is the risk-free rate. Therefore, setting probability
p
as described in equation 3.10 effectively assumes that the stock earns the risk-free rate. Only this probabilityp
makes the expected return of the uncertain pay-offs in the different states of the world equal to the certain return of the risk-free interest rate.Having established the economic intuition of the model, figure 3.1 provides a graphical presentation of a two-step binomial tree.
Cox, Ross and Rubinstein (1979) proposed the following equations for u en d:
t
e
u
=
σ Δ (3.13) te
d
=
−σ Δ (3.14)This ensures that these model parameters match the volatility of the underlying stock. Equation 3.13 and 3.14 are subject to the requirement:
u
r
d
<
(
1
+
)
<
(3.15)Equation 3.15 must always hold. If u and d would both be smaller than
(
1+r)
, the risk-free asset dominates the risky asset and investors would require a negative compensation for bearing risk. If u and d would both be greater than(
1+r)
, the underlying, risky asset has a higher return than the risk-free rate in each state of the world. By definition this cannot hold, because now the risky asset is no longer riskier.Now, recall the binomial distribution of the stock price development. During each time step the stock price can either go up or down. As a result, after two time steps, one would expect the tree to have four nodes. However, as can be seen in figure 3.1, the tree only consists of three nodes at the final time step. This is caused by the fact that the middle node of the tree can be reached both by an initial up move followed by a down move and by an initial down move followed by an up move. Since the product of u and d is equal to the product of d and u the tree recombines. This property of the binomial tree model is referred to as recombination or reconnectivity. Note also that, as can be seen from figure 3.1, S0ud is equal to S. This property results from the fact that d is
set as the reciprocal of u.
Now, the mechanics of the binomial tree are explained and illustrated using a simple, two-step numerical example. Consider a European call option on a non-dividend paying stock. The option has an exercise price of $ 30. The current stock price is $ 35. The risk-free rate is 5 percent, the volatility of the returns on the underlying stock is 30 percent and the time to maturity is six months. The life of the option is divided into two time steps. In this case the time interval per time step, ΔT, is three months.
At the final node, the option is either exercised or expires worthless. Therefore, the option value is equal to the maximum of the difference between the stock price and the exercise price and zero. Next, the option value is calculated at all nodes at the previous time step. At each node, the option value is equal to the expected value derived from the possible option values one time step later. This expected value is the weighted-average of these possible outcomes, discounted back one period. The possible outcomes are weighted by the risk-neutral probabilities of an up move,
p
, and the down move,(
1− p)
.To illustrate this process, consider the up-node at time t = 1, node A in figure 3.2. The option value at this node is derived from the possible option values at the upper and middle nodes at time t = 2, labeled B and C respectively.
At the upper node at time t = 2, the stock price is $ 47.24. The option is in-the-money and is exercised, capturing a gain of $ 47.24 – $ 30 = $ 17.24. At the middle node, the stock price is $ 35. Therefore, at this node, the option is worth $ 35 - $ 30 = $ 5. Since the option value is calculated at time t = 1, the option values at the upper and middle nodes at t = 2 need to be discounted back one time period. Equation 3.19 shows the value of the call option at the up-node at time t = 1:
(
)
{
}
r T ud uu up
c
p
c
e
c
=
*
+
1
−
*
−Δ (3.16) with:cuu = the value of the call option at the upper node at time t = 2; and
cud = the value of the call option at the middle node at time t = 2.
Equation 3.17 presents the value at this node for the option in the numerical example:
(
0
.
5043
*
$
17
.
24
+
.
4957
*
$
5
)
*
e
−0.05*.(3/12)=
$
11
.
04
(3.17)
This procedure is continued for each of the intermediate time steps, working all the way back to time t = 0, and is also referred to as ‘rolling back the tree’. Figure 3.2 shows the binomial
Since this example only has three possible option values at expiration, the option value at time t = 0 can also be expressed mathematically as shown by equation (3.18):
(
)
(
(
)
) ( )
(
)
( )
(
2 2)
0 0 2 2 0,
0
*
2
max
,
0
*
1
max
,
0
*
1
max
S
u
X
p
S
ud
X
p
p
S
d
X
p
e
c
=
−rT−
+
−
−
+
−
−
(3.18)Figure 3.2: Valuation process two-step binomial tree
A general formula for the price of a call option using the binomial tree can also be expressed mathematically:
(
p
)
(
u
d
S
X
)
p
j
n
j
n
e
c
n j n j j n j j rT−
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
− − = −∑
1
max
0
,
)!
(
!
!
0 (3.19) with:This equation can be explained more intuitively. Consider a single node at maturity T. The option pay-off at that final node,
max
(
0
,
u
jd
n−jS
−
X
)
, is multiplied by the probability of reaching this node:p
j(
1
−
p
)
n−j. This probability depends on the number of up moves needed to arrive at this node as well as on the number of total time steps. However, almost every node can be reached by more than one price path. Therefore, the product of the option pay-off and the associated chance has to be multiplied by the number of price paths that lead to this node. This number of price paths can be determined using binomial probability calculation and is represented by the factor)!
(
!
!
j
n
j
n
−
. Finally, since the pay-off takes place in the future, the product of these factors is discounted back to the present. Repeating this procedure for all nodes at maturity and summing the resulting values leads to the option value.Note that the call option considered in the example is a European call option. For American options, the optimal exercise policy has to be taken into account. When valuing American options, at each node, the value of holding the option for one more period, the holding value, is compared to the value of exercising the option, the exercise value. As mentioned before, the holding value is equal to the expected value derived from the possible option values one time step later. The option value is set equal to the greater of the holding value and the exercise value.
3.2.4 The trinomial tree (*)
Some authors turn to the trinomial tree approach rather than the binomial tree approach when calculating the fair value of employee stock options. Hull and White (2004) set forth a binomial model, but extend this model to a trinomial model when actually calculating ESO values. This model is discussed more in-depth in section 3.4.3.
The basic rationale behind the two approaches is the same. The trinomial tree can be regarded as an extended version of the binomial tree. The trinomial tree has three branches arising from each node: one corresponding with an up move, one with a down move and one corresponding to an unchanged value of the underlying asset. The trinomial tree method can be used to value barrier options. Barrier options are options that are activated or triggered when the price of the
options that include barrier features. The model developed by Hull and White (2004) assumes early exercise to take place when the stock price reaches a certain barrier. In such a case,
the property to adapt the model so that the barrier and nodes coincide at each point in time can be convenient.
3.2.5 Valuing options incorporating market-based vesting requirements
A commonly observed ESO characteristic is market-based vesting. ESOs are said to exhibit market-based vesting when the vesting of these ESOs is dependent on a market variable. In case these vesting requirements depend on a second underlying asset, the presence of such vesting requirements effectively turns an ESO into a special case of a rainbow option. A rainbow option is an option with more than one underlying asset. This section describes some methodologies to value such options. Market-based vesting requirements are discussed more in-depth in later chapters.
The difficulty when valuing rainbow options is the fact that typically the underlying assets are not independent. Hence, when valuing the option the correlation between the assets has to be taken into account. Wilmott (2006) provides closed-form solutions for a number of plain vanilla rainbow option contracts. Unfortunately, the features of ESOs make closed-form solutions impossible to derive.
One method to value an option depending on two (correlated) underlying assets is the binomial pyramid method as described by Rubinstein (1991). This method is based on an alternative version of the binomial tree model, for which
p
is set equal to 0.5. Rubinstein constructs a three-dimensional binomial tree, or pyramid as he refers to it. The correlation between the assets is reflected in the fact that the size of an up move (or a down move) of the second asset depends on the state of the first asset. This method is used to value options with market-based vesting conditions and is described in more detail in chapter five.stock options whose value depends on more than one underlying asset and introduces an extension to the CVC binomial lattice model that incorporates market-based vesting requirements.
3.3 Incentive effects of employee stock options
The agency theory set forth by Jensen and Meckling (1976) provides the basic rationale for granting ESOs. The relationship between shareholder and executive is a classic example of the principal-agent relationship they describe. The idea is that the agency problem can be overcome by the creation of incentives that align the interests of the shareholders (‘principal’) and the executive (‘agent’). ESOs tie part of the executive’s salary to stock price performance. As a result, both executives and shareholders benefit from an increase in the share price. The executive benefits through an increase in the share price-dependent component of his salary, while a
shareholder benefits from the increase in the value of his shares. Therefore, it provides the executive with an incentive to act in the shareholder’s interest. In agency theory, the incentive for an agent to expand his effort is referred to as incentive compatibility. The term incentive
alignment can also be used to refer to this motive.
For granting ESOs to non-executive employees, the rationale is the same. However, the incentive effect may be different, since lower-level employees may feel they are not able to influence company performance. Furthermore, free-rider problems may limit the incentive effect of group option schemes.
ESOs may also be granted for retention reasons. ESOs provide employees with an incentive to remain with the company until their ESOs have vested.
The incentive compatibility and retention motives are not the only motives for granting ESOs. Couwenberg and Smid (1999) acknowledge these motives and add the following motives for granting employee stock options:
1. attract scarce personnel;
2. increase employee ownership; and 3. reduce the potential for hostile takeovers.
the widespread use of ESO remuneration. Furthermore, most academic research focuses on this motive.
The incentive compatibility motive is based on the notion that the maximization of shareholder value is a company’s main objective. There has been much debate between advocates of this notion and proponents of the maximization of stakeholder value as the company’s main objective. However, discussing the debate regarding shareholder value and stakeholder value is beyond the scope of this thesis. Couwenberg and Smid (1999) state that even if the maximization of
shareholder value is the correct main objective of a company, it still remains doubtful whether options are effective instruments for providing the desired incentive.
Numerous studies related to the incentive alignment effects of ESOs have been conducted. Some of the most influential are discussed below. For a discussion of other granting motives, see Hall and Murphy (2003).
Jensen and Murphy (1990) have investigated the relationship between performance pay and top-management incentives. Performance pay includes ESOs, but also other forms of executive compensation, such as performance-based bonuses, salary revisions, and restricted shares. The change in CEO wealth due to stock options is 15¢ for every $ 1,000 change in shareholder wealth compared to a change of 30¢ due to cash compensation in the form of salary and bonuses. Stock ownership creates the largest incentive: CEO wealth changes by $2.50 for every $ 1,000 change in shareholder wealth. Despite this large incentive effect from share ownership, most CEO’s hold trivial fractions of their companies’ stock. Concluding, Jensen and Murphy find that total CEO wealth changes $ 3.25 for every $ 1,000 change in shareholder wealth and they find this observed pay-performance sensitivity to be small. It should be noted that this wealth change relates to all pay-components related to executive performance.
Hall and Liebman (1998) find that the sensitivity of CEO wealth to an increase in shareholder wealth increases when more detailed information regarding stock option plans is incorporated into the study. However, the overall level still remains relatively low.
Following from the incentive compatibility argument, the issuance of ESOs should reduce agency costs. Yermack (1995) tests whether stock options’ performance incentives have significant associations with explanatory variables related to agency cost reduction. His results indicate that few agency or financial contracting theories have explanatory power for patterns of CEO stock option awards.
Core and Guay (1999) also studied the use of equity grants and the associated effect on
this optimal level. Core and Guay model the optimal incentive level and use the residuals from this model to measure deviations between CEO’s equity incentives and optimal incentive levels. They find that equity grants are negatively related to these deviations. Therefore, Core and Guay conclude that equity grants are used to manage incentive levels in an efficient way, consistent with economic theory.
A number of the studies mentioned before, measure incentives using option deltas. The delta is the sensitivity of the price of an option to a change in the price of the underlying asset. In other words, the incentive effect is measured as the extent to which executive option holdings increase in value as a result from an increase in the share price. Jensen and Murphy (1990), Yermack (1995), Hall & Liebman (1998), Core and Guay (1999) and Hall and Murphy (2000, 2002) all apply this methodology.
An example of this ‘delta approach’ can also be found in Meulbroek (2001). She makes a distinction between what she calls the ‘market value option delta’ and the ‘private option delta’. The former is based on the market value of the option, while the latter is based on the value of the option as perceived by the executive. By making this distinction she acknowledges the fact that the market value and the private value may differ. This distinction is addressed more in-depth in section four of this chapter when academic work on the valuation of ESOs is discussed. For now, it is important to note that the proper incentive measure is the private option delta. An executive’s decisions and actions are, amongst other factors, based on his level of risk aversion, non-option wealth and on his utility function. These factors are not taken into account using market valuation models and are therefore not incorporated in the measurement of the market option delta.
However, private option values include such variables.
Jin and Meulbroek (2002) applied the ‘delta approach’ to investigate whether underwater (out-of-the-money) ESOs still align incentives or not. They find that the incentives of out-of-the-money options remain remarkably intact. This finding is contradictory to the results obtained by Hall and Murphy (2000). They find a substantial decline in incentive power when options are underwater. As a result, their evidence supports ESO repricing, while Jin and Meulbroek’s results are
non-repricable options. Their analysis is based on some simplifying assumptions regarding the timing and the extent of repricing.
Another issue regarding the effectiveness of granting ESOs is the question whether ESOs lead executives to engage in risk-seeking behavior. Since the value of a tradable option increases with an increase in volatility of the underlying asset (all else equal), several authors predict that executives have an incentive to increase the volatility of their companies’ shares. However, the findings of Lambert, Larcker and Verrecchia (1991) contradict these results. They argue that an increase in variance has two opposing effects for a risk-averse executive who cannot diversify the risk associated with the option’s pay-off. The convexity of the option’s pay-off function has a positive effect, while a negative effect arises from the concavity of the expected utility function. Their analysis shows that for risk-averse executives the second effect dominates the first over certain intervals. This means that executives become more risk-averse rather than what standard option pricing models suggest.
The study by Johnson and Tian (2000), mentioned earlier, also investigates whether repricable options may increase risk-seeking executive behavior. They find evidence that repricable options provide executives with an incentive to increase the volatility of the underlying stock. For a more extensive discussion of the effect of ESOs on the risk behaviour of executives, see Carpenter (1999).
Some researchers have investigated the incentive-providing power of ESOs by comparing the incentives provided by ESOs to their cost. Hall and Murphy (2002), as well as Meulbroek (2001), have studied the efficiency of ESOs by measuring the ratio between the value of the option as perceived by the executive and the cost of the option to the company. It should be noted that this analysis does not include any ESO benefits other than incentive alignment. For instance, the positive retention effect of ESOs is not taken into account even though the full cost of granting ESOs is included in the analysis. However, as identified earlier, incentive alignment is probably the most important reason for granting ESOs.
Meulbroek (2001) reports that undiversified managers from a sample of NYSE firms value their options at an average of 70% of their cost to the firm. For a sample of internet-based companies, the average is even less, at 53%. In line with the Meulbroek study, Hall and Murphy (2002) find a significant difference between the cost to the company and the executive value.
From the discussion above, it becomes clear that evidence on the effectiveness of ESOs in providing incentives to align interests is mixed.
based on the incentive compatibility motive. He departs from this perspective by focusing on employee retention as a motive for the granting of ESOs. He proposes that the participation constraint of agency theory can help explain many common compensation schemes. The participation constraint states that both the principal and the agent must each receive utility that exceeds their individual reservation utilities in order to be willing to participate. This constraint is used to explain the decision of an executive to remain with the company.Chen (2004) shows that firms that do not reprice option grants are more vulnerable to voluntary executive turnover than firms that reprice grants. This result indicates the retention power of (in-the-money) ESOs. These studies are two examples of studies with a focus on other motives for granting ESOs besides the incentive compatibility motive.
3.4 Valuation of employee stock options
3.4.1 Introduction
This section discusses the valuation of employee stock options. It is important to note that there are two perspectives that can be taken when valuing ESOs. The previous section already briefly touched upon this distinction when discussing the ‘delta approach’. Carpenter (1998), Hall and Murphy (2002) and Kadam, Lakner and Srinivasan (2003), as well as various other authors, emphasize that the value of employee stock options can be expressed as either:
• the economic cost to the company; and
• the economic value to the (executive) recipient.
The economic cost to the company is often measured using market valuation models. The cost of the ESO is defined as the price that an outside investor would be willing to pay for the option. Traditional market valuation models of option values, such as for instance the BSM model or the binomial tree model are used as the starting point of the valuation process.
In the remainder of this thesis the value of an ESO as perceived by an executive is referred to as the ‘executive value’ of an option, in line with Hall and Murphy (2002, 2003).
From an accounting perspective, only the costs to the company are relevant. IFRS 2 acknowledges this and turns to market valuation models to estimate the fair value of ESOs. However, as Hall and Murphy (2002) and Kulatilaka and Marcus (1994) note, the value of ESOs as perceived by executives provides crucial insight into all relevant aspects of the valuation of ESOs. Therefore, the discussion of this valuation methodology is highly relevant and provides an understanding of the caveats in extending traditional market valuation models in order to value ESOs.
The remainder of this section is divided as follows. Subsection 3.4.2 discusses the valuation of ESOs from the perspective of the executive, while ESO valuation from the company’s
perspective is the subject of subsection 3.4.3. In the latter section, the Hull & White model addressed earlier is elaborated on.
3.4.2 Value to the executive
Market valuation models such as the Black-Scholes-Merton model and the binomial tree model value options assuming that investors are unconstrained and there are no arbitrage opportunities. However, executives are not unconstrained. An executive’s human capital is entirely tied up in the company. Therefore, maintaining an asset portfolio that includes shares of that company and/or options on shares of that company might be sub-optimal from a diversification perspective. In order to diversify his portfolio, an executive might want to sell his ESOs or hedge his position by shorting the underlying stock. However, employee stock options are non-tradable.
Furthermore, executives are often not permitted to sell short the shares of their companies. As a result, executives cannot sell or hedge their option position. Therefore, exercising the option position is the only way for an executive to diversify his position.
Also, executives might want to liquidate part of their asset portfolios because of consumption motives. Again, this may cause executives to exercise their options.
importance of inside information, but ignore the effects in their analysis of ESOs. For a
discussion of inside information and the exercise of ESOs, see Carpenter and Remmers (2001). Recognizing that executives are neither unconstrained nor fully diversified, the executive value of ESOs is typically measured using expected-utility models rather than market valuation models. Numerous studies develop and apply expected-utility models. These studies include Lambert, Larcker and Verrecchia (1991), Kulatilaka and Marcus (1996), Detemple and Sudaresan (1999), Hall and Murphy (2000, 2002) and Kadam, Lakner and Srinivasan (2003).
The ‘certainty equivalence’ approach that Lambert, Larcker and Verrecchia (1991) set forth, was among the first studies to depart from market valuation models by explicitly incorporating executives’ trading restrictions. Their approach defines the value of a non-tradable option to an undiversified and risk-averse executive as the amount of riskless cash compensation the executive would exchange for the option.This amount of riskless cash compensation depends on:
1. the portfolio composition of the executive; 2. the form of the expected utility function; and 3. the executive’s risk preferences.
The portfolio consists of options and of non-option wealth. The expected utility derived from this portfolio depends on the form of the expected utility function and on the executive’s risk
preferences. Lambert, Larcker and Verrecchia report a significant difference between this executive value and the corresponding Black-Scholes-Merton value.
Hall and Murphy (2002) extend the model by Lambert, Larcker and Verrecchia (1991). They utilize a binomial tree framework. At each time step the option is exercised if the expected utility from exercising the option is higher than the expected utility from holding the option for another period. They show that early exercise can be a rational and optimal decision for executives, even if the underlying stock does not pay dividend. Kulatilaka and Marcus (1994) derive the same result also using a binomial framework to model early exercise.
In contrast, as shown in section 3.2, it is never optimal to exercise tradable options on
non-dividend paying stocks before maturity. However, for tradable options on non-dividend-paying stocks, it can be optimal to exercise early.
option always increases with an increase in volatility. However, volatility has two contradicting effects on the executive value of ESOs. First, and analogous to tradable options, the executive value increases due to the increased probability that the option expires in-the-money. Second, the executive value of the option decreases, because risk-averse employees tend to exercise options with a higher volatility earlier than options with a lower volatility. Kulatilaka and Marcus show that there are intervals for which the second effect dominates the first. Over these intervals, executive value decreases with an increase in volatility. This result is analogous to the result derived by Lambert, Larcker and Verrechia (1991) described in the last section when discussing the incentive effects of ESOs. Also, Detemple and Sundaresan (1999) observe the same
relationship.
Another paradox mentioned by Kulatilaka and Marcus (1994) is that European-style ESOs are worth more than otherwise identical American-style ESOs. For tradable options, the condition that an American option is worth more or at least the same as an otherwise identical European option always holds. This can be seen in table 3.1. The cause of this paradox is, again, early exercise behavior. European employee stock options are worth more, because, by definition, they exclude the possibility of early exercise.
Most studies mentioned before utilize a binomial tree to incorporate early exercise behavior. Kadam, Lakner and Srinivasan (2003) construct a continuous-time model that computes the optimal exercise policy of ESOs. By establishing some simplifying assumptions, they are able to derive closed-form solutions for both the value to the executive and the cost to the company.
3.4.3 Cost to the company
The cost to the company of granting ESOs is defined in academic literature as the amount of money the company would have received by selling the options to an outside investor rather than granting them to the executive.
