This paper was presented at the Financial Management Association Europe, Southern Finance Association, and Northern Finance Association meetings, and a faculty seminar at the University of Oklahoma. The authors thank Jean Canil, Andrea Heuson, Wei-Ling Song, Cliff Stephens, John Wald, Adam Yore, David Koslowsky, and Ivan Brick for comments and suggestions.

Original Article

# THE PRICE-TAKER EFFECT ON THE VALUATION OF EXECUTIVE STOCK OPTIONS

Article first published online: 13 MAR 2014

DOI: 10.1111/jfir.12027

© 2014 The Southern Finance Association and the Southwestern Finance Association

Additional Information

#### How to Cite

Yang, T.-H. and Chance, D. M. (2014), THE PRICE-TAKER EFFECT ON THE VALUATION OF EXECUTIVE STOCK OPTIONS. Journal of Financial Research, 37: 27–54. doi: 10.1111/jfir.12027

#### Publication History

- Issue published online: 13 MAR 2014
- Article first published online: 13 MAR 2014

- Abstract
- Article
- References
- Cited By

### Abstract

- Top of page
- Abstract
- Introduction
- Previous Research
- Theoretical Model
- Numerical Results
- Conclusions
- Appendix: The Estimation of Optimal Effort and Executive Option Value
- References

Because of vesting requirements and the absence of liquidity, executive stock options are valued at less, and often far less, than Black–Scholes–Merton values. We argue that this view assumes the subtle condition that option holders are price takers and therefore cannot influence the payoffs of their options, an assumption that clearly contradicts the very reason for granting options. We build a model to incorporate the executive's effort and perception of his quality, private beliefs, and confidence to show that these options are worth considerably more than previously believed and under some conditions even more than Black–Scholes–Merton values.

### Introduction

- Top of page
- Abstract
- Introduction
- Previous Research
- Theoretical Model
- Numerical Results
- Conclusions
- Appendix: The Estimation of Optimal Effort and Executive Option Value
- References

Traditional financial models almost always assume that no investor can influence the price of an asset by trading. Investors are said to be price takers. This assumption, so common that it is rarely even mentioned, is second nature in models of competitive markets. The markets for publicly traded options are generally competitive enough to support this assumption, and as such, the Black–Scholes–Merton model and almost all of its variants assume that all market participants are price takers.

There is a history, both in research and in practice, of using the Black–Scholes–Merton model, with some adjustments, to value executive stock options. Thus, doing so implicitly assumes that executives are price takers. Because the options are awarded in part to incentivize executives, the notion that an executive is a price taker clearly belies a critical characteristic of these options.

A significant body of published papers shows that executives value their options at substantially less than Black–Scholes–Merton values. Such a conclusion is a natural result of penalizing the options for their lack of liquidity. Awarding executives financial assets over which they have no control of the payoffs and that they cannot sell will certainly cause the executives to value those assets at less than would other investors who similarly have no control over the outcomes but who are free to sell at will.

Accurate valuation of executive stock options is an important issue, as it affects our understanding of how options incentivize executives. The literature acknowledges that the executive and the company do not hold mirror images of each other's position.1 The company's cost of the option can be substantially different from the executive's value of the option. If an executive has a different opinion of the value of his options than the company believes the options are worth, it will not award the proper number of options to incentivize the executive. As noted, under traditional models of executive stock option valuation, the option value tends to lie well below the cost to the company, but as also noted, these models assume the executive is a price taker. Recognizing the fact that the executive can influence the payoff of the option, it follows that the executive is likely to have a higher opinion of the value of the option than he would if he had no influence. As such, the company's belief of how the executive values the option would be well below what the executive actually believes. Failing to recognize this fact could lead the company to award more options than necessary to incentivize the executive and therefore would raise the cost to the company. Indeed, the large numbers of options seemingly awarded to executives may well be driven by the belief that executives value these options at much less than they really do.

In this article we reexamine the traditional utility-based approach of executive stock option valuation and remove the price-taker assumption by incorporating the effect of the executive's perception of his effort, ability, and confidence on his private valuation of the options. It shows that these factors lead executives to value options at far more than suggested by traditional models and that under some circumstances, at more than Black–Scholes–Merton values. We also examine how incorporation of these effects influences the executive's decision to exercise early.

### Previous Research

- Top of page
- Abstract
- Introduction
- Previous Research
- Theoretical Model
- Numerical Results
- Conclusions
- Appendix: The Estimation of Optimal Effort and Executive Option Value
- References

A large body of literature exists on the valuation of executive stock options. Probably the most widely referenced work is Hall and Murphy (2002), who model a risk-averse executive holding a stock option and stock in the firm. They set up an expected utility maximization problem in which the executive determines the certainty equivalent value of the option as the cash that would be accepted in lieu of the option. The executive is assumed to be unable to sell the option, but may, after the vesting day, choose to exercise it early. There are numerous extensions of the Hall–Murphy model, but all are generally framed around the idea that a risk-adverse executive holds an illiquid option with limited ability to diversify within a moderate to extremely high degree of risk concentration in his portfolio.2 These models generally conclude that the factors that limit diversification substantially affect the option value.

All of these models, however, fail to incorporate how the executive can alter the behavior of the stock. These models treat the executive as a passive and poorly diversified investor, who is simply constrained a bit more than the shareholders. A few studies address the question of how executives can modify the distribution of returns of the equity (Palmon et al., 2004; Hodder and Jackwerth 2005) and others incorporate executive effort into the model (Cadenillas et al., 2004; Agliardi and Andergassen 2005; Schaefer 1998; Feltham and Wu 2001). Cadenillas et al. (2004) consider managerial effort in the optimal contract problem. They find explicit solutions for the optimal effort and volatility but do not take these factors into account in the valuation of stock options. Chang, Chen, and Fuh (2013) look at employee sentiment and find some evidence that because of overconfidence, options are valued higher than previously thought, even to the point of exceeding Black–Scholes–Merton values.

Agliardi and Andergassen's (2005) study is most closely related to our approach but differs in two significant ways. One is that they do not assume that the executive can hold the company's stock. The executive's nonoption holdings are in the form of the market portfolio and the risk-free asset. Yet it is the executive's exposure to the firm that results in such a poor degree of diversification, and much of that exposure comes from the holding of stock. A second difference from our model is that Agliardi and Andergassen do not permit optimal effort if the firm grants cash instead of options. As we show later, the solution to our model is an equation with cash equivalent on the left-hand side and options on the right-hand side, including optimal effort from shareholding on both sides. Our result is the cash equivalent that would provide the same utility and produce optimal effort as would the options in question. Moreover, we examine the effect of some additional factors on option value, such as the executive's total wealth, nonoption wealth, quality, and risk aversion. A further contribution of our article is that we analyze early exercise. The examination of these effects is important to provide a more comprehensive perspective on the factors that determine option value from the point of view of the executive.

The notion that executives might value stock options far more, rather than far less, than given by traditional models is suggested in the theories of Oyer and Schaefer (2005) and Bergman and Jenter (2007). Arya and Mittendorf (2005) argue that when employees accept options, it is an indicator of the employees' confidence. Thus, employee and executive views on the values of options tend to be optimistic and lead to more optimal hiring. Furthermore, experimental evidence by Hodge, Rajgopal, and Shevlin (2009) suggests that a group of entry- to mid-level managers are excessively optimistic, producing option values that often exceed the Black–Scholes–Merton values. Carpenter, Stanton, and Wallace (2010) also find that option values can exceed Black–Scholes–Merton values.

These concepts are consistent with our central hypothesis. The personal views of managers about their own abilities, combined with the control of key corporate decisions, can lead to substantially higher option values than previously observed. There is a substantial body of literature that suggests that most executives are very self-confident, with many rightfully so given their past successes.

### Theoretical Model

- Top of page
- Abstract
- Introduction
- Previous Research
- Theoretical Model
- Numerical Results
- Conclusions
- Appendix: The Estimation of Optimal Effort and Executive Option Value
- References

Recall that the objective of the model is to determine the executive's perception of the value of the option, not the firm's cost of the option. This perception is a function of the executive's private beliefs concerning his ability and confidence coupled with the executive's intended effort. These private beliefs need not match those of the market and indeed will not likely do so, nor is there a necessary convergence of investor beliefs and executive beliefs over time. We assume that although executives may hold pessimistic short-run views, they are optimistic in the long run. Moreover, they are highly confident in the short run as well as in the long run of their own ability to create value for shareholders. It can be argued that these assumptions imply that executives can overstate their abilities and the effects of their efforts and never learn, but we believe that executives are by nature confident and optimistic over the long run. If they perceive that the market does not share those opinions, we assume that the executives believe they are correct and that the market will eventually share those beliefs. If the market does not, executives continue to believe that they can produce positive results. Moreover, we are not assuming that if the executive's expectations do not converge to reality, the executive has failed and should be terminated. The executive can be quite successful earning normal returns and yet remain forever optimistic and confident. A number of authors have argued the notion that executives are highly confident if not overconfident (Van den Steen 2004; Gervais, Heaton, and Odean 2011; Malmendier and Tate 2005; March and Shapira 1987; Gilson 1989; Goel and Thakor 2008; Heaton 2002), so this assumption is well supported.3

#### The Determination of Optimal Effort

We start at time 0 with current stock price of *S*_{0} and expected return of *E*(*r*) and assume that the firm grants stock options with a maturity *T*. The executive immediately decides on his optimal effort over the lifetime of the options. At time *idt*, the executive believes that the effect of his effort will be reflected in the stock price and will result in an abnormal return over the period of *t = *0 to *idt*. From that point forward, he expects that investors will price his effort into the stock, so no further abnormal returns arising from this grant would be expected to occur.

We make the assumption that there is no stock price reaction at the grant date. Although Dittman and Maug (2007) assume that the market anticipates executive effort, the empirical evidence on this issue is mixed. Clearly there must be uncertainty about whether option grants lead to positive results.4 Our assumption reflects that uncertainty and is consistent with Schaefer (1998) and Hall and Murphy (2002). We further assume that differences of opinion between the executive and investors cannot be arbitraged away by the executive. In other words, if the executive perceives that the stock is worth far more to him than its value in the market, he cannot exert sufficient demand pressure through personal stock purchases or signal to the market with such credibility that it causes the market price to rise to his personal valuation. The assumption that the executive can perceive the value of the stock differently from the shareholders and that there is no automatic corrective mechanism is an important and accepted paradigm in finance.

Although *S*_{0} is the stock price at the grant date, we are interested in determining the executive's private assessment of the stock price that would exist if the option is awarded, the executive puts forth additional effort as a result of receiving the options, and the executive is confident that his effort will be successful. As noted, the executive expects that the market will determine the results of his effort at time *idt* and the stock price will change to

- (1)

where is the executive's belief of the true value of the stock after taking the effort into account at time *idt*.5 *S*_{idt} is the stock price on date *idt* with minimum effort equal to one, *q*_{idt} is the measure of executive effort from period *t = *0 to *idt*, and *δ* is the measure of the executive's perception of his quality, which is the elasticity of the stock price with respect to his effort, *δ* ≥ 0.6 This quality measure also incorporates the executive's confidence and his private beliefs about his ability to take actions that will increase the price of the stock.7 Under the same effort, the higher the *δ*, the higher the executive believes will be the after-effort stock price.8

We need not specify when the effort is expended during the interval *t = *0 to *idt*. It could occur early, late, or spaced evenly throughout. Also, the executive expects to continue the effort after *t = idt*, but he expects no further abnormal return beyond that point because investors are expected to become aware of his effort, which will then be reflected in the stock price.

We let the market price, *S*_{t}, follow the well-known geometric Brownian motion process

where *α* is the drift, *σ* is the standard deviation of the raw stock return, and *w*_{t} is a standard Brownian motion. We assume the continuous-time capital asset pricing model (CAPM) holds so that *α* = *r*_{f} + *β*(*E*(*r*_{m}) – *r*_{f}), where *r*_{f} is the risk-free rate, *β* is the measure of systematic risk, and *E*(*r*_{m}) is the expected market return. We assume that early exercise decisions have no effect on the executive's choice of optimal effort.9 Therefore, executives determine their optimal effort immediately after accepting the option. We can view equation (1) in terms of the expected return by dividing by *S*_{0} and taking the expectation of the log returns:

- (2)

Under the geometric Brownian motion assumption, the observed market return follows a normal distribution with mean *µ* and standard deviation *σ*. We define *µ** as the after-effort expected log return and *η* as the incremental expected return resulting from executive effort. These parameters reflect the executive's private assessment of the value of the stock, based on what actions he expects to take and his expected outcomes of those actions:

- (3)

Therefore, we relate executive effort to incremental expected return as follows:

- (4)

From equation (4), we know that the executive's effort is exponentially related to incremental expected return, the length of time, and the perceived quality of the executive.10 As noted, the executive expects that the market price will converge to his perception of the true value that reflects effort at *t = idt*. Any price before that time, such as *S*_{dt}, does not reflect effort. Nonetheless, the executive will have this private opinion of the price, , which by recursive evaluation, will equal, .11 Because the executive has a private awareness of his effort and its impact on the value of the stock, the executive will be more optimistic about the future prospects for the stock and the option than would other investors.

The fundamental idea behind the model is that while we hold the risk constant the executive believes his effort shifts the expected return upward, increasing the probabilities of states in which higher returns are received and decreasing the probabilities of states in which lower returns are received. Because the executive believes the results of his effort will show up in the abnormal return, however, we must be cautious in our definition of the expected return. The executive believes he shifts the distribution by *η*. If we incorporate *η* into the true expected return, there would naturally be no abnormal return. Thus, before *t = idt*, we need to clearly distinguish the expected return without effort from the expected return with effort, the latter of which is not observed by investors.

Because this model is characterized by information asymmetry and private beliefs, we must make a careful distinction between the expected return as perceived by investors, denoted as *E*(*r*), and the expected return as perceived by the executive, denoted as *E*(*r**) and defined as the sum of *E*(*r*) and *η*. This measure is the executive's private assessment of the expected return, which will differ from *E*(*r*) by the incremental expected return. The executive believes that at time *idt*, *E*(*r*) converges to *E*(*r**) and *η* is zero. The executive still exerts the same level of optimal effort but he expects no further abnormal return.

Upon receipt of the grant, the executive decides on the optimal effort. To reduce some of the complexity of the solution, we specify the executive's wealth to consist of three components: $*c* in cash, *m* shares of the firm's stock, and *n* stock options.12 The terminal wealth is *W*_{T} = *c*(1 + *r*_{f})^{T} +* mS*_{T}* + *nMax*(*S*_{T}* – *K*,0), where *T* is the maturity of the options, *r*_{f} is the risk-free rate, is the terminal after-effort stock price, and *K* is the exercise price of *n* options. As described, we assume that the executive believes he can affect firm value by choosing his level of effort and therefore increasing the expected return of the stock, which increases the return on the executive's stock-based compensation. If effort were costless, however, the executive would clearly work to increase the stock price without limit. Therefore, we must impose a cost to effort, which we do with a disutility function as a quadratic with respect to effort.13 To analyze the trade-off relation mentioned previously in the expected utility model, we represent executive effort and disutility in terms of incremental expected return. Therefore, the disutility function of effort is

- (5)

Recall that *q*_{idt} = *e*^{ηidt/δ}, where *idt* is the period over which the effort is expected to convert into the abnormal return. As noted before, the executive expects investors to become aware of the results of his effort at time *idt* and adjust the expected return in the market to the expected return with effort. After that, there is no longer an abnormal return from executive effort but the executive expects to maintain the same level of effort. Otherwise, investors can identify the reduction of effort at the next observation point and there will be a negative abnormal return. To compute the disutility of effort over the entire option life, we assume that the total effort from time 0 to time *T* is the product of effort in each interval of *idt*, that is, *T* = *idt*. Thus, from equation (4), total effort can be expressed as *q*_{T} = *e*^{ηT/δ}.

To examine the impact that effort and the executive's private beliefs have on option valuation, we must compare the results of this model and standard models that do not take such factors into account. Thus, we need to consider only the disutility of the effort beyond *q*_{T} = 1. Therefore, we can write the disutility function (equation (5)) as

This implies that the cost of minimum effort is zero. Following Feltham and Wu (2001) and Agliardi and Andergassen (2005), we assume the executive has negative exponential utility with coefficient of absolute risk aversion *ρ*:14

The executive determines the optimal effort by maximizing the expected utility with respect to terminal wealth net of the disutility of effort, which is

- (6)

We assume additively separable utility for terminal wealth and the disutility of effort.

Finally, we again emphasize that success and abnormal returns are not guaranteed simply by granting stock options. The executive's private assessment of the value of the stock need not be the true value. It need only reflect the executive's own assessment of the value, which will be driven by his beliefs of how he feels he will be able to take actions that increase the value of the stock. It is these private beliefs that cause the executive to value that stock at a level higher than that of the stockholders. There is no built-in or necessary convergence of the executive's private belief with the market price of the stock. Again, we rely on the assumption that the executive is long-term optimistic about his ability to produce positive results, as are most high-level decision makers in business and in other professions.

#### Executive Option Values

Some models of stock option pricing focus on the cost of the options to the company and the differential between that cost and the value as perceived by the executive. We do not address the question of firm cost. Our attention is paid to how the executive views the option and how that view differs when the executive's position as a price taker is considered.

To operationalize the model, we need to be able to identify the effort that an executive would expend and his perceived value of the option, taking that effort and his confidence into account. In a utility-based model, the typical method of finding the value of an executive stock option is the certainty equivalent approach. The option value is viewed as the cash amount, *CE*, at the valuation date that has the same expected utility as the option. Therefore, without considering effort, the option value would be determined by solving for *CE* below,

- (7)

The value of one option is, therefore, *CE*/*n*. This is the standard approach taken in models that value the option as a certainly equivalent without the effect of effort. Were we focused on the cost to the firm, this amount could be viewed as the freely investable cash that the company would have to pay to induce the executive to work.

There are, however, some differences between our approach and that of others. Note that on the left-hand side, the executive receives a cash award, *CE*, while on the right-hand side, he receives options. The cash award is merely a mechanism for identifying the amount of cash that the executive would take that would have the same expected utility as the option and would therefore have the same value as the option. But if cash were actually granted in lieu of options, the cash would clearly imply lower incentives. The executive would have some incentives with a cash grant, however, because of stock ownership, but clearly stock and options would have more incentives than stock and cash. This problem, which is common to all models of this type, is recognized by Hall and Murphy (2002) by acknowledging that the stock price distribution does not change when options are granted.

Our approach deals with this problem by incorporating a differential between the stock price in the market and the executive's perception of the value of the stock. Our left-hand side reflects cash and stock, with stock providing some incentive and, therefore, some effect of executive effort. Our right-hand side reflects options and stock, with both stock and options providing an effect of executive effort. The differences in incentives are accounted for in our model. Because the effort that maximizes expected utility determines the stock return distribution, which then determines the certainty equivalent value of the option, we first need to determine the optimal effort. Accounting for both the utility benefits of effort and the utility cost, our option value is *CE* in the following:

- (8)

In equation (8), is the stock price after taking into account the optimal effort from stock ownership, and is the stock price after taking into account the optimal effort from both stock and option ownership. Then *η*_{1} and *η*_{2} are the corresponding incremental expected returns on the left-hand and right-hand sides, respectively. In the Appendix, we show how to find a numerical solution. When the options are exercisable early, we take a similar but slightly different approach.

#### Early Exercise

Other factors that can affect the executive option value are early exercise and the vesting schedule. Early exercise and effort combine to pose an interesting question about which comes first. Does the ability to exercise early affect effort or does effort affect the early exercise decision? It is impossible to definitively state that one is cause and one is effect. We assume that effort is chosen independently of the early exercise feature, which means that optimal effort would be no different for American options than for European options. But we then assume that effort can affect the decision of when to exercise early. If early exercise occurs, we invest the proceeds in the risk-free asset until expiration. The executive will exercise early when the expected utility of early exercise is higher than that from holding the options. The assumption that the effort decision precedes the early exercise decision is reasonable in that effort is determined when the options are awarded, while early exercise is decided later.

As is commonly required when analyzing the early exercise of standard options, a numerical method is used to capture the early exercise decision. Following the simulation method in Hall and Murphy (2002), we also use a binomial tree to decide when to exercise early. The expected utility at each node in a binomial tree after time *t* is

where *U*(*W*_{t}^{u}) and *U*(*W*_{t}^{d}) are the utilities at time *t* with up and down moves, respectively, and *U*(*W*_{t–1}^{E}) is the utility from early exercise. Following this rule, we can find the expected utility considering early exercise at time 0. Then, the value of the options is the cash amount received at time 0 and invested in the risk-free asset that provides the same expected utility.

Our objective is to determine how option values differ when we account for the executive's perception of his ability to influence the payoff of the option. We will do that by assuming a reasonable range of values of the inputs required by the model, computing values under the model and comparing those with values computed by traditional models of executive stock option valuation.

#### Parameter Assumptions and Estimates

We examine a range of reasonable values for the model inputs. There are 12 parameters in the model, which can be grouped into Black–Scholes–Merton variables, CAPM variables, and executive properties.

The Black–Scholes–Merton variables are the current stock price, *S*_{0}; the exercise price, *K*; the risk-free rate, *r*_{f}; the volatility of the stock return, *σ*; and the time to maturity, *T*.15 Using Standard & Poor's ExecuComp, we find that between 2000 and 2005 more than 99% of stock options were granted at the money. The mean exercise price in 2005 was $32.47 and the median was around $29.20. We use $30 as the exercise price. We use at-the-money options as the benchmark but also consider out-of-the-money and in-the-money options by examining stock prices of $20 and $40, respectively. These options can reflect the effects of issuing premium and discount options. For the risk-free rate, the three-month T-bill rate is 4.95% and 10-year Treasury maturity rate is 5.09% in July 2006. We use 5% as the risk-free rate in our estimates.

The average volatility reported in ExecuComp from 2000 through 2005 to compute the Black–Scholes–Merton value is 47%. We use 50% as the benchmark and 30% and 70% to represent lower and higher volatility companies, respectively. The most common time to maturity for original-issue executive stock options is 10 years. To examine how option values change over their lives, we examine 10-, 7-, and 5-year options. Although companies do not typically issue 7- and 5-year options, examining these maturities can give us an idea of the effect of issuing options shorter than the standard period. In addition, the common but simplistic view that options are exercised with certainty after a fixed number of years can be somewhat applicable to this assumption, though we will address early exercise directly.

We use the value-weighted return on all NYSE, AMEX, and NASDAQ stocks as a proxy for the expected market return. The average market return from 1992 through 2005 is 11.88%. We use 12% as the benchmark for the expected market return. We also run estimates with 10% and 14% expected market returns. We use a beta of 1.0 as the benchmark and betas of 0.5 and 1.5 to show the results for firms with different levels of systematic risk. To conserve space and focus on the most important variables, we do not report or discuss the results with high and low values of beta and the expected market return as the variation is not sufficiently large.

There are three relevant components of the executive's personal wealth: cash, the firm's stock, and stock options. As noted, we assume the executive has negative exponential utility, which has the characteristic of constant absolute risk aversion. Moreover, the executive's perception of the elasticity of the stock price with respect to his effort is also a crucial component in our model in relation to others. Therefore, we establish benchmark values for this perceived elasticity of the stock price, the amount of nonoption wealth, the number of shares of stock and options, and the coefficient of absolute risk aversion.

As we have emphasized, the elasticity coefficient represents a factor that captures the executive's perception of his personal impact on the value of the stock. He chooses a level of effort and this level converts, through the elasticity, into a higher stock price. As a basis for estimating this value, we turn to the literature on the relation of effort, as proxied by work hours, to sales and sales to stock price. First, we note that Kim, Lim, and Park (2009) estimate that the change in stock return divided by the change in sales deflated by market value of equity is 0.8599. Multiplying this figure by their mean market value of equity of 1,419.41 and dividing by their mean sales figure of 1,997.53 gives 0.611 as an estimate of the elasticity of stock price with respect to sales. For convenience, we round off to 0.625. Bitler, Moskowitz, and Vissing-Jorgensen (2005) estimate the elasticity of sales with respect to working hours as 0.40. We then multiply these two figures to get 0.25 as a benchmark estimate of the elasticity of stock price with respect to working hours, which is our proxy for the elasticity of stock price with respect to effort. We also use 0.1 and 0.5 to represent low and high values, respectively.16

Because of vesting requirements and possibly a negative signaling effect, most executives hold more than the optimal level of their firms' stock. Therefore, the stock component of nonoption wealth should be higher than the optimal level in the benchmark. Based on the optimal holding of risky assets from Merton (1969), the optimal holding in the benchmark is 24%.17 We assume the executive invests 40% of his wealth in the firm's stock as a benchmark to show that the executive bears higher than optimal firm-specific risk. In addition, we extend the stock–wealth ratio to 30% and 50% for low and high stock holdings. From ExecuComp, we find that the average stock wealth in year-end 2005 is $39.2 million, but the median is $1.6 million. Using the median stock wealth and 40% stock–wealth ratio assumption, we use $4 million as a benchmark for total nonoption wealth. The less and more wealthy executives are assumed to have $2 and $6 million in their nonoption wealth, respectively. The number of shares of stock is equal to the stock wealth divided by the current stock price.

From ExecuComp, we find that the median number of options granted is 21,000 and the mean is 78,970. Therefore, the distribution of options granted is highly skewed. When we use only the CEO in the database, the median and mean are 60,000 and 191,000, respectively. We use the median of these grants and set the number of granted options equal to 40,000, and use 20,000 and 60,000 options to observe the effect of low and high option grants.18

The last executive parameter is the coefficient of risk aversion. From Pratt (1964), the relation between absolute risk aversion, *ARA*, and relative risk aversion, *RRA*, is *ARA*_{t} = *RRA*_{t}/*W*_{t}. The commonly used value of *RRA* is from 2.0 to 4.0. We use *RRA* = 2.0 as the benchmark and *RRA* = 1.0 and 3.0 to represent lower and higher relative risk aversions. Because negative exponential utility has the characteristic of constant absolute, rather than relative, risk aversion, the coefficient of *ARA* is 0.0000005 in the benchmark, and 0.00000025 and 0.00000075 are for *RRA* = 1.0 and *RRA* = 3.0, respectively.

For the binomial version, we examine both weekly and monthly time steps and find no notable differences. We elect to go with a monthly time step, so *h* = 12 in the binomial model. The optimal effort and executive option values are lower than those in the continuous-time model, but the difference does not change our qualitative results. When we examine early exercise, we specify vesting periods of 0, 2, and 4 years, and focus only on 10-year options.

### Numerical Results

- Top of page
- Abstract
- Introduction
- Previous Research
- Theoretical Model
- Numerical Results
- Conclusions
- Appendix: The Estimation of Optimal Effort and Executive Option Value
- References

To identify the implications of our model, we produce numerical estimates using the various ranges of input parameters. These estimates are based on the condition that upon receipt of options, the executive maximizes his expected utility by choosing the optimal level of effort. Given the chosen effort along with the other input variables, the value of the option can be derived as the certainty equivalent. A description of the full estimation procedure is available from the authors on request.

#### Optimal Executive Effort

The optimal effort for the various parameter values is shown in Table 1. The columns labeled “Lower Value,” “Benchmark,” and “Higher Value” refer to the assumption of the lower of the three values of the input we vary, as shown in the first column. Within each column of the three main columns, we show results for three maturities: 10, 7, and 5 years. Within each cell, three moneyness levels are shown vertically with the top value representing the lowest stock price (out of the money) and the bottom value representing the highest stock price (in the money). The optimal effort can be translated into the stock return by equation (4).19 Six variables are indicated in the row heads in Table 1. Combined with variation in maturity and moneyness, we can observe the effects of eight variables on effort.

Lower Value | Benchmark | Higher Value | |||||||
---|---|---|---|---|---|---|---|---|---|

T = 10 | T = 7 | T = 5 | T = 10 | T = 7 | T = 5 | T = 10 | T = 7 | T = 5 | |

Note ^{}The numbers in each cell are the optimal effort when the current stock prices are $20, $30, and $40, respectively. When changing one parameter at a time, we keep all other parameters as their benchmark values. The bold values are the optimal effort of the at-the-money options. Benchmark values are shown only once because they do not change when quality, nonoption wealth, stock–wealth ratio, number of options, and absolute risk aversion are their benchmark values. Benchmark values for these variables are the middle of the three vertical values for each corresponding variable.
| |||||||||

Volatility (30%, 50%, 70%) | |||||||||

$20 | 1.557 | 1.931 | 2.570 | 1.598 | 1.997 | 2.681 | 1.595 | 2.002 | 2.702 |

$30 | 1.547 | 1.911 | 2.524 | 1.598 | 1.994 | 2.671 | 1.596 | 2.003 | 2.703 |

$40 | 1.536 | 1.888 | 2.476 | 1.596 | 1.989 | 2.655 | 1.596 | 2.003 | 2.701 |

Executive quality (0.1, 0.25, 0.5) | |||||||||

$20 | 1.541 | 1.905 | 2.527 | 1.602 | 1.973 | 2.577 | |||

$30 | 1.542 | 1.906 | 2.530 | 1.598 | 1.962 | 2.547 | |||

$40 | 1.542 | 1.906 | 2.529 | 1.593 | 1.949 | 2.517 | |||

Nonoption wealth (2,000,000, 4,000,000, 6,000,000) | |||||||||

$20 | 1.691 | 2.152 | 2.966 | 1.515 | 1.862 | 2.440 | |||

$30 | 1.688 | 2.145 | 2.946 | 1.514 | 1.861 | 2.434 | |||

$40 | 1.685 | 2.135 | 2.920 | 1.513 | 1.858 | 2.426 | |||

Stock–wealth ratio (30%, 40%, 50%) | |||||||||

$20 | 1.563 | 1.975 | 2.629 | 1.636 | 2.020 | 2.739 | |||

$30 | 1.561 | 1.972 | 2.615 | 1.635 | 2.017 | 2.732 | |||

$40 | 1.559 | 1.965 | 2.596 | 1.633 | 2.012 | 2.719 | |||

Number of options (20,000, 40,000, 60,000) | |||||||||

$20 | 1.599 | 1.998 | 2.682 | 1.598 | 1.997 | 2.680 | |||

$30 | 1.598 | 1.996 | 2.676 | 1.597 | 1.992 | 2.665 | |||

$40 | 1.597 | 1.993 | 2.667 | 1.595 | 1.985 | 2.645 | |||

Absolute risk aversion (0.00000025, 0.00000050, 0.00000075) | |||||||||

$20 | 1.748 | 2.256 | 3.170 | 1.486 | 1.813 | 2.351 | |||

$30 | 1.747 | 2.252 | 3.156 | 1.486 | 1.811 | 2.345 | |||

$40 | 1.745 | 2.246 | 3.139 | 1.484 | 1.807 | 2.333 |

In all cases, optimal effort is greater than one, which means that it is always optimal for the executive to exert effort beyond that required to maintain the current stock price. Thus, at that minimum level of effort, the marginal benefit of effort exceeds the marginal cost. As expected, the optimal level of effort varies widely for different inputs, being in some cases three times as high as in others. Based on these results, we conclude that the executive's effort from option compensation can improve the firm's performance, which is consistent with the empirical finding in Core and Larcker (2002). After firms adopt target ownership plans, Core and Larcker find that the mean and median one-year excess stock return is around 5.7%. In addition, Hanlon, Rajgopal, and Shevlin (2003) show that the future operating income associated with a dollar of Black-Scholes-Merton value of an ESO grant is $3.71, which supports the positive impact of option grants on firm performance.

We find that optimal effort decreases with moneyness, which means that out-of-the-money options induce more effort than in-the-money options. With out-of-the-money options, there is a lower probability of the options expiring with value, so executives recognize that additional effort is required. But in all cases the differences in effort by moneyness are small. Therefore, moneyness does not appear to be a strong factor in motivating executives, but we will return to this issue later.

We also see that optimal effort decreases with maturity, and the effect is strong. That is, longer term options result in lower effort. When the maturity is shorter, there is less time for the option to expire in the money. The executive is therefore motivated to work harder. In contrast, if the maturity is longer, such as 10 years, there is more time for the options to expire in the money. The executive's odds of success are much more favorable for a given level of effort with a longer term option. Moreover, the executive benefits from the mere passage of time and the positive drift of the stock price. In contrast, with a shorter time to expiration, the executive perceives that additional effort on his part may be necessary to boost the option's payoff. In addition, disutility of effort is greater the longer the period over which the effort is expended. In short, the executive is far more willing to work harder over a short period than over a longer one, and therefore, companies might wish to consider shortening the original issue terms of their options.20 Effort changes in a slightly positive manner with the stock–wealth ratio and in a slightly negative manner with nonoption wealth, which make sense. If the executive's options represent only a small portion of his overall wealth, he will not be motivated to work hard. Effort also varies sharply and inversely with risk aversion. This result is also logical as more risk-averse executives are less inclined to make the effort. The number of options exerts a moderately negative effect on effort. Hence, the more options awarded, the less effort exerted, though the effect is not strong.

Recall that the elasticity of the stock price reflects the executive's perception of his own quality. We refer to this variable as quality, but let us not forget that it is the executive's opinion of his quality. When we refer to executives with high, medium, or low quality, we are describing the executive's perception of his ability, which can be a combination of his true ability, his perception of his ability, and his general optimism. We see that executive effort has a concave relationship with quality, with the greatest effect occurring at medium quality.21 This result implies that medium-quality executives work harder than low- or high-quality executives. This result has different explanations for different quality executives. From equation (4), executive effort is negatively related to *δ*, which means executives with high quality exert less effort. That is what we observe from benchmark quality to high quality. For executives with low quality, one unit of their effort has the same disutility as others but less influence on stock price. It is optimal for them to exert less effort.

Optimal effort in general has a moderately positive relation with volatility. Volatility is well known to have counterintuitive effects on executive stock option values. Higher volatility makes options more valuable, but these options are valued by expected utility, not by arbitrage. Therefore, higher volatility also has a negative effect on option value, as well as on stock value. Therefore, the effect of volatility on effort reflects the combined effects of the stock and the options.

In Table 1 we observed that optimal effort is moderately negatively related to moneyness. That is, out-of-the-money options lead to greater effort. We explore this result further here by estimating the optimal effort with respect to different exercise prices according to volatility and show this relation in Figure I. For 30% volatility, we see a positive asymptotic relation between effort and exercise price, and see that out-of-the-money options induce more effort than at- or in-the-money options. This effect weakens with 50% volatility, and virtually disappears with 70% volatility. Figure I also shows that options with positive exercise prices induce more effort than those with an exercise price of zero. This result implies that restricted stock, which is equivalent to an option with zero exercise price, induces less effort than ordinary stock options, which of course have positive exercise prices.

To summarize our findings so far, these results confirm that options do serve as incentives for executives to exert more effort than they would otherwise. The variables that have the greatest impact on effort are maturity, nonoption wealth, and risk aversion with greater effort found with shorter maturity options, executives with lower nonoption wealth, and less risk-averse executives. Slightly greater effort is found for medium-quality executives, executives of more volatile firms, executives with higher stock–wealth ratios, and executives with fewer options. Out-of-the-money options are slightly more effective at motivating executives to work harder.

#### Option Values

To determine how effort affects the executive's private valuation of his options, we estimate option values with optimal effort, *q* > 1, and minimum effort, *q* = 1. Results are shown in Table 2 with optimal effort (Panel A), without reflecting effort (Panel B), and the Black–Scholes–Merton values (Panel C) under different parameters. The middle case, without reflecting effort, can be viewed somewhat as the conventional model of executive stock option valuation, reflecting illiquidity as well as the executive's risk aversion.

Lower Value | Benchmark | Higher Value | |||||||
---|---|---|---|---|---|---|---|---|---|

T = 10 | T = 7 | T = 5 | T = 10 | T = 7 | T = 5 | T = 10 | T = 7 | T = 5 | |

Note ^{}The numbers in each cell are the option values when the current stock price is $30, which is an at-the-money option. When changing one parameter at a time, we keep all other parameters as their benchmark values. We use the optimal effort in the previous table to compute the option values in Panel A. When we change a parameter other than volatility, the Black–Scholes–Merton values are the same as those in the benchmark case in Panel C. Therefore, we report these values only once.
| |||||||||

Panel A. Executive Option Value with Optimal Effort | |||||||||

Volatility | $12.52 | $17.89 | $23.94 | $2.97 | $4.85 | $7.32 | $0.99 | $1.71 | $2.77 |

Executive quality | $1.88 | $2.74 | $3.71 | $5.71 | $10.07 | $16.35 | |||

Nonoption wealth | $7.01 | $10.63 | $15.04 | $1.49 | $2.49 | $3.90 | |||

Stock ratio | $4.21 | $6.61 | $9.70 | $2.20 | $3.66 | $5.66 | |||

Number of options | $3.52 | $5.71 | $8.56 | $2.57 | $4.22 | $6.40 | |||

Absolute risk aversion | $9.18 | $13.85 | $19.47 | $1.28 | $2.16 | $3.38 | |||

Panel B. Executive Option Value without Optimal Effort | |||||||||

Volatility | $4.65 | $5.11 | $5.25 | $1.31 | $1.76 | $2.17 | $0.48 | $0.74 | $1.02 |

Executive quality | $1.31 | $1.76 | $2.17 | $1.31 | $1.76 | $2.17 | |||

Nonoption wealth | $2.91 | $3.66 | $4.24 | $0.67 | $0.93 | $1.20 | |||

Stock ratio | $1.91 | $2.49 | $2.99 | $0.93 | $1.27 | $1.60 | |||

Number of options | $1.54 | $2.05 | $2.51 | $1.14 | $1.54 | $1.91 | |||

Absolute risk aversion | $3.61 | $4.48 | $5.11 | $0.59 | $0.83 | $1.08 | |||

Panel C. Black–Scholes–Merton Value | |||||||||

Volatility | $15.64 | $12.94 | $10.69 | $20.14 | $17.32 | $14.83 | $23.75 | $21.09 | $18.55 |

First, we note an interesting and counterintuitive result. Executive option values are inversely related to time to expiration, which is opposite that of standard option valuation intuition.22 Holders of standard options, of course, benefit from longer time to expiration, and if they need to liquidate before expiration, they can sell their options. Executives in need of liquidity, however, cannot sell their options. Hence, the liquidity penalty is a heavy one that dominates the time value benefit. As expiration approaches, however, the option value increases because the liquidity penalty is smaller. We will see later that this effect vanishes when the options can be exercised early.

Second, comparing Panels A and B, we find as expected that option values increase after taking optimal effort into account. Thus, the executive values the option higher when taking into account his effort, quality, and confidence. In some cases, the differences are large. For example, our benchmark 10-year option is worth $1.31 without accounting for effort and $2.97 after accounting for effort. The 7-year option is worth $1.76 without effort and $4.85 with effort, and the 5-year option is worth $2.17 without effort and $7.32 with effort. These results cast a new light on the standard view of executive option valuation whereby executives would not value their options so highly because of illiquidity, vesting, and the poorly diversified nature of their wealth. But those models ignore the fact that executives can influence the outcomes of the options and have confidence in their ability to produce positive results. They know that with additional effort exerted to the point where the value of the marginal effort equals the marginal cost, these options can be made more valuable. Thus, we should not attempt to determine executive option values by treating executives as though they were ordinary investors merely constrained by poor diversification and illiquid holdings.

Third, the effect of effort is such that we even observe cases in which executive option values are perceived by executives as greater than Black–Scholes–Merton option values. For example, we find in Panel A that executive option values are higher than Black–Scholes–Merton option values when the volatility or risk aversion is low and the maturity is 7 years or less. Thus, in spite of the options being illiquid, executives like some options more than do ordinary investors. When the executive's ability to affect the payoff along with his confidence is considered, it should not be surprising that executives favor options more than other models suggest.

Table 3 shows the ratio of option value with effort to value without effort, or the ratio of Panels A and B of Table 2. Of course, these values also reflect the executive's perception of his quality. We see that option values with optimal effort and quality are consistently higher than those with minimum effort, in most cases more than twice as high. This result holds for both in-the-money and out-of-the-money options for almost all variations of parameters, the exception primarily being low-quality executives. Interestingly, the ratios decrease with maturity, which is consistent with the pattern we observed for optimal effort in Table 1. We know that the executive exerts more effort when the maturity is shorter. This additional factor magnifies the difference between option values with and without optimal effort. Note also the not surprising result that the ratio is highest when the executive's quality is high. Even though high-quality executives exert slightly less effort, they still value their options higher than other executives. Recall, of course, that the quality variable is the executive's belief in his own quality and could be excessive. The ratios are also higher the lower the volatility, the lower the nonoption wealth, the higher the stock–wealth ratio, the lower the number of options, and the lower the risk aversion, but quality and volatility are the only variables that have a strong effect. The effect of moneyness is inconsistent but in most cases, it has an inverse effect.

Lower Value | Benchmark | Higher Value | |||||||
---|---|---|---|---|---|---|---|---|---|

T = 10 | T = 7 | T = 5 | T = 10 | T = 7 | T = 5 | T = 10 | T = 7 | T = 5 | |

Note ^{}The numbers in each cell are the ratios of executive option values when the current stock prices are $20, $30, and $40, respectively. When changing one parameter at a time, we keep all other parameters as their benchmark values. The bold values are ratios of the at-the-money options. The ratios are computed as executive option value with effort divided by executive option value without effort. When we change a parameter other than volatility, the ratios do not change in the benchmark case. Therefore, we report these ratios only once.
| |||||||||

Volatility (30%, 50%, 70%) | |||||||||

$20 | 3.23 | 4.61 | 6.74 | 2.48 | 3.09 | 3.94 | 2.20 | 2.52 | 3.08 |

$30 | 2.69 | 3.50 | 4.56 | 2.26 | 2.76 | 3.38 | 2.05 | 2.31 | 2.71 |

$40 | 2.42 | 3.00 | 3.72 | 2.11 | 2.50 | 2.95 | 1.90 | 2.15 | 2.44 |

Executive quality (0.1, 0.25, 0.5) | |||||||||

$20 | 1.39 | 1.56 | 1.76 | 5.12 | 7.14 | 10.11 | |||

$30 | 1.43 | 1.56 | 1.71 | 4.34 | 5.74 | 7.54 | |||

$40 | 1.38 | 1.48 | 1.60 | 3.86 | 4.91 | 6.28 | |||

Nonoption wealth (2,000,000, 4,000,000, 6,000,000) | |||||||||

$20 | 2.58 | 3.25 | 4.19 | 2.29 | 2.97 | 3.83 | |||

$30 | 2.41 | 2.91 | 3.55 | 2.23 | 2.68 | 3.24 | |||

$40 | 2.26 | 2.66 | 3.15 | 2.06 | 2.41 | 2.82 | |||

Stock ratio (30%, 40%, 50%) | |||||||||

$20 | 2.39 | 2.98 | 3.80 | 2.56 | 3.19 | 4.08 | |||

$30 | 2.20 | 2.65 | 3.25 | 2.37 | 2.89 | 3.53 | |||

$40 | 2.07 | 2.43 | 2.88 | 2.19 | 2.59 | 3.05 | |||

Number of options (20,000, 40,000, 60,000) | |||||||||

$20 | 2.50 | 3.13 | 4.01 | 2.46 | 3.06 | 3.89 | |||

$30 | 2.28 | 2.79 | 3.41 | 2.25 | 2.75 | 3.35 | |||

$40 | 2.13 | 2.52 | 2.98 | 2.10 | 2.48 | 2.94 | |||

Absolute risk aversion (0.00000025, 0.0000005, 0.00000075) | |||||||||

$20 | 2.79 | 3.54 | 4.60 | 2.08 | 2.76 | 3.58 | |||

$30 | 2.54 | 3.10 | 3.81 | 2.17 | 2.61 | 3.14 | |||

$40 | 2.38 | 2.82 | 3.36 | 1.99 | 2.34 | 2.73 |

Table 4 presents the ratio of the executive's perception of option value to the Black–Scholes–Merton value or, effectively, the ratio of Panels A and C in Table 2. These ratios are low in the case of 10-year maturities, but they increase with shorter maturity options, which is consistent with our previous analysis of the negative time value effect.23

Lower Value | Benchmark | Higher Value | |||||||
---|---|---|---|---|---|---|---|---|---|

T = 10 | T = 7 | T = 5 | T = 10 | T = 7 | T = 5 | T = 10 | T = 7 | T = 5 | |

Note ^{}The numbers in each cell are the ratios of executive option values to Black–Scholes–Merton option values when the current stock prices are $20, $30, and $40, respectively. When changing one parameter at a time, we keep all other parameters as their benchmark values. The bold values are ratios that are greater than one. When we change a parameter other than volatility, the ratios do not change in the benchmark case. Therefore, we report these values only once.
| |||||||||

Volatility (30%, 50%, 70%) | |||||||||

$20 | 0.73 | 1.43 | 2.67 | 0.04 | 0.07 | 0.40 | 0.03 | 0.05 | 0.10 |

$30 | 0.80 | 1.38 | 2.24 | 0.07 | 0.10 | 0.49 | 0.04 | 0.08 | 0.15 |

$40 | 0.81 | 1.32 | 2.00 | 0.08 | 0.13 | 0.53 | 0.05 | 0.10 | 0.18 |

Executive quality (0.1, 0.25, 0.5) | |||||||||

$20 | 0.06 | 0.11 | 0.18 | 0.22 | 0.48 | 1.02 | |||

$30 | 0.09 | 0.16 | 0.25 | 0.28 | 0.58 | 1.10 | |||

$40 | 0.11 | 0.19 | 0.29 | 0.32 | 0.62 | 1.12 | |||

Nonoption wealth (2,000,000, 4,000,000, 6,000,000) | |||||||||

$20 | 0.31 | 0.58 | 1.04 | 0.04 | 0.09 | 0.17 | |||

$30 | 0.35 | 0.61 | 1.01 | 0.07 | 0.14 | 0.26 | |||

$40 | 0.35 | 0.61 | 0.96 | 0.10 | 0.18 | 0.32 | |||

Stock ratio (30%, 40%, 50%) | |||||||||

$20 | 0.17 | 0.32 | 0.59 | 0.07 | 0.14 | 0.27 | |||

$30 | 0.21 | 0.38 | 0.65 | 0.11 | 0.21 | 0.38 | |||

$40 | 0.23 | 0.40 | 0.66 | 0.14 | 0.25 | 0.43 | |||

Number of options (20,000, 40,000, 60,000) | |||||||||

$20 | 0.12 | 0.24 | 0.45 | 0.09 | 0.19 | 0.35 | |||

$30 | 0.18 | 0.33 | 0.58 | 0.13 | 0.24 | 0.43 | |||

$40 | 0.21 | 0.38 | 0.63 | 0.15 | 0.27 | 0.46 | |||

Absolute risk aversion (0.00000025, 0.0000005, 0.00000075) | |||||||||

$20 | 0.40 | 0.75 | 1.32 | 0.03 | 0.07 | 0.15 | |||

$30 | 0.46 | 0.80 | 1.31 | 0.06 | 0.12 | 0.23 | |||

$40 | 0.48 | 0.81 | 1.27 | 0.08 | 0.16 | 0.27 |

The Black–Scholes–Merton values are European option values. We know there is no liquidity discount for the nontradable constraint in the Black–Scholes–Merton value. Therefore, the time value is positive and increases with maturity in the Black–Scholes–Merton formula. In contrast, the liquidity discount can dominate the positive time value in executive options. Hence, these cases are more likely to occur in options with longer maturities. The effect of effort could dominate the liquidity discount so that the values of executive stock options would be higher than Black–Scholes–Merton values. Hence, the commonly stated argument that that the Black–Scholes–Merton model gives too high a value for executive stock options is not generally correct. As we observed, the Black–Scholes–Merton model can even understate the values of these instruments. This means that the executive would place a higher value on this illiquid instrument than an outside investor would place on a perfectly liquid version of the same option. This seemingly irrational result arises because outside investors are price takers while executives have the ability to influence option payoffs and confidence that they can produce positive results.

#### Early Exercise

It is well known that executives exercise their options early for a number of reasons. Perhaps the most important is to better diversify their personal portfolios. In addition, they exercise early to obtain cash for large purchases, to capture dividends, to shift taxes, and in some cases, when they leave the company.24

We now introduce the right to exercise early. As with other utility maximization models, early exercise is motivated by the combined effect of all factors, but with the no-dividend assumption and without the incorporation of departure or private cash needs, early exercise will be driven by the desire to achieve a more diversified portfolio. Our model allows such exercises to occur, conditional on the executive's chosen effort. We focus on 10-year options that vest in 0, 2, and 4 years. Using the binomial model, we summarize the values of the options after taking early exercise into account and the ratios of option values with effort to those without effort in Table 5.

Lower Value | Benchmark | Higher Value | |||||||
---|---|---|---|---|---|---|---|---|---|

0 Years | 2 Years | 4 Years | 0 Years | 2 Years | 4 Years | 0 Years | 2 Years | 4 Years | |

Note ^{}The numbers in each cell are the option values when the current stock price is $30, which is an at-the-money option, and the maturity is 10 years. When changing one parameter at a time, we keep all other parameters as their benchmark values. We use the optimal effort in the binomial model without consideration of early exercise to compute the option values in Panel A. We assume the options are exercisable after vesting periods of 0, 2, and 4 years and use a monthly time step in the binomial model. When we change a parameter other than volatility, the results do not change in the benchmark case. Therefore, we report these values only once.
| |||||||||

Panel A. Executive Option Values with Optimal Effort and Early Exercise (1) | |||||||||

Volatility | $16.00 | $15.97 | $15.67 | $11.59 | $10.91 | $9.26 | $11.86 | $10.04 | $7.46 |

Executive quality | $9.75 | $8.94 | $7.23 | $15.30 | $14.79 | $13.26 | |||

Nonoption wealth | $15.13 | $14.69 | $13.30 | $9.89 | $9.04 | $7.25 | |||

Stock ratio | $12.71 | $12.11 | $10.54 | $10.84 | $10.09 | $8.37 | |||

Number of options | $13.41 | $12.95 | $11.30 | $10.34 | $9.50 | $7.93 | |||

Absolute risk aversion | $18.57 | $18.33 | $17.08 | $8.84 | $7.82 | $6.16 | |||

Panel B. Executive Option Values without Optimal Effort and with Early Exercise (2) | |||||||||

Volatility | $8.62 | $8.49 | $7.89 | $8.78 | $7.85 | $6.13 | $9.88 | $7.89 | $5.41 |

Executive quality | $8.78 | $7.85 | $6.13 | $8.78 | $7.85 | $6.13 | |||

Nonoption wealth | $10.25 | $9.51 | $7.94 | $7.98 | $6.90 | $5.10 | |||

Stock ratio | $9.35 | $8.52 | $6.88 | $8.34 | $7.33 | $5.56 | |||

Number of options | $9.93 | $9.23 | $7.44 | $7.96 | $6.87 | $5.26 | |||

Absolute risk aversion | $11.82 | $11.34 | $8.87 | $7.22 | $6.05 | $4.41 | |||

Panel C. Ratio of Option Value with Effort to that without Effort (1)/(2) | |||||||||

Volatility | 1.86 | 1.88 | 1.99 | 1.32 | 1.39 | 1.51 | 1.20 | 1.27 | 1.38 |

Executive quality | 1.11 | 1.14 | 1.18 | 1.74 | 1.88 | 2.16 | |||

Non-option wealth | 1.48 | 1.54 | 1.67 | 1.24 | 1.31 | 1.42 | |||

Stock ratio | 1.36 | 1.42 | 1.53 | 1.30 | 1.38 | 1.51 | |||

Number of options | 1.35 | 1.40 | 1.52 | 1.30 | 1.38 | 1.51 | |||

Absolute risk aversion | 1.57 | 1.62 | 1.93 | 1.22 | 1.29 | 1.40 |

First, note that as expected, we find that the longer the vesting period, the lower the value of the option. In comparing the European option values in Table 1 with the American option values in Table 5, we see that, as expected, the ability to exercise early increases the values of the options regardless of whether effort and quality are considered. Some of the absolute differences are large. For example, for the high-volatility case, the 10-year European-style option is worth $0.48 with no effort and $0.99 with effort (Table 2). Even with a 4-year vesting period, that option is worth $5.41 with no effort and $7.46 with effort. In Table 5 we again see that all numbers in Panel A, with effort, are greater than the corresponding values in Panel B, without effort. Panel C shows the ratio of values with effort to without. Even though the addition of early exercise reduces the difference between option value with effort and option value without effort, we still find many option values with consideration of the executive's effort and his quality perception much higher than those without.

From the literature, two important factors that should substantially affect the decision to exercise early are the executive's stock holdings and his risk aversion. When executives are more risk averse or have high stock holdings, they may value the benefits of exercising now to reduce exposure to the stock at more than the value of future volatility. Thus, they may choose to exercise at somewhat lower stock prices. In the previous results with effort, however, we saw that elasticity, which proxies for the executive's perception of his quality, can play an important role in the determination of option values and this effect can dominate these two factors. To examine this question, we estimate the early exercise premiums by comparing option values with effort but without early exercise to those with effort and with early exercise. These results are shown for different elasticities, maturities, and vesting periods in Table 6.

δ = 0.1 | δ = 0.25 | δ = 0.5 | |||||||
---|---|---|---|---|---|---|---|---|---|

T = 10 | T = 7 | T = 5 | T = 10 | T = 7 | T = 5 | T = 10 | T = 7 | T = 5 | |

Note ^{}The numbers in each cell in Panels A and B are the option values when the current stock price is $30, which is an at-the-money option. The options in Panel A are European-type options with different time to maturity. Those in Panel B are American-type options, and we assume different vesting periods of 0, 2, and 4 years. The numbers in Panel C are the early exercise premiums, which are the values with early exercise (Panel B) minus those without early exercise (Panel A). Except the quality (elasticity), all other parameters are maintained at their benchmark values.
| |||||||||

Panel A. Option Value with Effort but No Early Exercise (1) | |||||||||

Option value | $1.88 | $2.74 | $3.71 | $2.97 | $4.85 | $7.32 | $5.71 | $10.07 | $16.35 |

Panel B. Option Value with Effort and Early Exercise Consideration (2) | |||||||||

0 year vesting | $9.75 | $9.34 | $8.99 | $11.59 | $11.99 | $12.71 | $15.30 | $17.74 | $21.57 |

2 years vesting | $8.94 | $8.41 | $7.91 | $10.91 | $11.28 | $11.96 | $14.79 | $17.27 | $21.13 |

4 years vesting | $7.23 | $6.38 | $5.37 | $9.26 | $9.29 | $9.38 | $13.26 | $15.39 | $18.68 |

Panel C. Early Exercise Premium (B) – (A) | |||||||||

0 year vesting | $7.88 | $6.61 | $5.28 | $8.62 | $7.14 | $5.39 | $9.59 | $7.66 | $5.22 |

2 years vesting | $7.06 | $5.68 | $4.20 | $7.94 | $6.43 | $4.64 | $9.09 | $7.20 | $4.78 |

4 years vesting | $5.35 | $3.64 | $1.66 | $6.29 | $4.44 | $2.06 | $7.56 | $5.32 | $2.33 |

We see in Panel C that as expected, the early exercise premium increases with maturity and is inversely related to the vesting period. The early exercise premium can be viewed as either the cost of being unable to exercise a European option early or the benefit of being able to exercise an American option early. With a longer vesting period, this cost or benefit is lower because exercise is deferred. The early exercise premium is also directly related to quality, meaning that there is a greater advantage to exercising early for executives who believe they are of high quality.25 These executives believe they have the ability to make the stock perform well, so the ability to exercise early and capture that effect over having to wait until expiration is beneficial.

To verify the relation between quality, as measured by the elasticity of stock price, and the decision to exercise early, we compute the threshold price, which is the critical stock price for the decision to exercise early. We summarize these threshold prices with respect to the elasticity of stock price that implies different executive quality in Figure II for 10-year options. In Panel A we see that the threshold prices are positively related to the elasticity of stock price and, hence, executive quality. For example, in year 3, the executive with the lowest quality, *δ* = 0.1, will exercise the options when the stock price is above $58.58, but the executive with the highest quality, *δ* = 0.5, will wait until the stock price is over $73.73. This result is consistent with expectations. Given the executive's opinion of his ability and the effort he will exert, holding the options longer can increase the expected wealth, which also increases the expected utility. Based on the higher expected utility of holding the option, the threshold prices should be higher for more confident executives. The behavior, however, is not significantly different in the last year. From this result, we expect the executive who exerts more effort will exercise at a higher stock price. Therefore, the effect of the effort would interact with that of risk aversion or stock holdings on the behavior of early exercise, which is examined in the following analysis.

From Table 1, we know the executive would exert slightly more effort when his stock holdings constitute a higher proportion of his wealth but less effort with higher risk aversion. Hence, the decision to exercise early with respect to different risk aversion should be similar to the finding in the literature without consideration of executive effort and quality. The more risk-averse executive would exercise stock options at lower prices, after taking effort into account. In Panel B, we show the threshold stock prices for two executive qualities (elasticities) of 0.25 and 0.50 and risk aversions of 0.0000005 and 0.00000075. First we see that the more risk-averse executive exercises at a lower stock price regardless of his belief in his quality. We also see that the lower quality executive with the same risk aversion exercises at a lower stock price regardless of risk aversion. The highest threshold stock price is for the less risk-averse and higher quality executive, and the lowest is for the more risk-averse and lower quality executive. Thus, higher quality executives will hold out for a higher stock price, and this effect is even greater the less risk averse they are. But lower quality executives may hold out for a higher stock price if they are less risk averse.

A result in the literature states that less diversified executives would exercise at a lower stock price compared to ordinary diversified investors, who are of course price takers. Because an executive can reduce his personal firm-specific risk by early exercise, he would do it as soon as the options are vested. This result does not, however, consider the executive's effort and his perception of his ability. From the previous analysis, we know that there is a counteracting effect between stock wealth and executive effort resulting from the stock–wealth ratio on the behavior of early exercise. The change in the threshold stock price with respect to the stock–wealth ratio is shown in Panel C. We consider two stock–wealth ratios, 40% and 50%, and two elasticities, 0.25 and 0.5, our measure of the executive's perception of his quality. Think of these four cases as low exposure (40%) and low quality (0.25), low exposure (40%) and high quality (0.5), high exposure (50%) and low quality (0.25), and high exposure (50%) and high quality (0.5). For any maturity level and exposure, executives who believe they are highest quality will exercise at higher prices than those who believe they are of the lowest quality. For any quality perception, low-exposure executives will exercise at higher prices than will high-exposure executives. The perception of quality seems to dominate exposure, however, as high-quality and high-exposure executives exercise at higher prices than low-quality and low-exposure executives.

### Conclusions

- Top of page
- Abstract
- Introduction
- Previous Research
- Theoretical Model
- Numerical Results
- Conclusions
- Appendix: The Estimation of Optimal Effort and Executive Option Value
- References

It has become widely accepted that the values of executive stock options are less, and often far less, than the values of analogous traded options. Although it is true that executives cannot sell their options and liquidity is limited by the lack of a market as well as vesting requirements, it is also true that, unlike traded options, executives have the ability to influence the payoffs of their options. Indeed the principal reason executives are granted options is to motivate them to take actions that will increase the stock price. Thus, we should not value their options as though executives are merely passive claimants on corporate performance. Executives are not price takers, as would be other holders of options in most valuation models. In addition to their presumed ability to influence the outcomes of the options, they have their own perceptions of their ability to produce positive results. These private beliefs, whether accurate or not, should certainly play an important role in their personal valuation of their stock and options. We argue that executives have the means, confidence, and optimism to produce personal valuations of the stock, and therefore their options, that are much higher than previously believed and potentially higher than would be those of outside investors holding traded options on the stock. Although one could argue that an executive's valuation of the option is not a cost to the company, it is important to understand that the executive's valuation of the option is a critical factor in determining how well the option motivates the executive.

In our model we assume that executives hold the risk-free asset, as well as options and stock of the company. The stock-holding assumption is particularly important in that valuation of the options should be greatly influenced by exposure to the stock in whatever form, options or stock. Our model determines the option value as the equivalent amount of cash the executive would accept upon maximization of utility, given the effort the executive chooses to make. We find that the value of the option exceeds the value that would be assigned under traditional executive stock option valuation models that assume executives are price takers. In fact, in some cases, the executive's perceived values of stock options can even exceed their corresponding Black–Scholes–Merton values. In addition we find that consideration of executive effort can influence early exercise behavior, as executives who have a combination of high ability and confidence will exercise at a higher stock price, thereby postponing their decisions to exercise their options.

In summary, we argue that executives cannot be treated as price takers, as is commonly the case in the research to date on executive stock option valuation. This subtle assumption is present in nearly all financial models and is rarely questioned. If executives are merely passive price takers, however, the entire notion of granting options as incentives is a *reductio ad absurdum*.

### Appendix: The Estimation of Optimal Effort and Executive Option Value

- Top of page
- Abstract
- Introduction
- Previous Research
- Theoretical Model
- Numerical Results
- Conclusions
- Appendix: The Estimation of Optimal Effort and Executive Option Value
- References

To determine optimal effort, *q*, we simulate the terminal stock price with incremental expected return, *η*, assuming lognormality, volatility *σ*, and expected return [*r*_{f} + *β*(*r*_{m} − *r*_{f}) + *η* − *σ*^{2}/2]*T*. Starting with an initial value of *η* of 0.01, we have the terminal stock price with effort, *η*, of . We then construct the distribution of terminal stock prices with effort by generating Weiner values, *z*_{T}, within +/– 6 standard deviations and convert to the stock price. Inserting this terminal stock price into terminal wealth, *W*_{T}, we have the distribution of terminal wealth reflecting cash of *c*, *m* shares, and *n* options. We then maximize the expected utility net of cost of effort in equation (6) by using the first-order condition, which is

where *f*(*S*_{T}*) is the density of *S*_{T}*. With no closed-form solution for *η*, we simulate with different values until the first-order condition is less than 0.0000001. The first *η* that meets this criterion can be converted into the optimal executive effort reported in Table 1 under given parameters.

To find the option value using the certainty equivalent approach in equation (8),

we need optimal effort with and without stock options, which are functions of *η*_{2} and *η*_{1} respectively, in the first step. Based on the optimal effort with stock and options, we find the terminal after-effort stock prices, , and the expected utility net of the disutility of effort. The certainty equivalent, *CE*, is the total value of *n* stock options. Applying the same method to find the optimal effort with *m* shares of stock but zero options, we find the optimal effort without options. Based on the optimal effort, we can simulate the possible terminal after-effort stock prices, . Finally, we compute the expected utility on the left-hand side of equation (8) by changing *CE* until

The first *CE* that satisfies this criterion is the total value of *n* options. The value of one option is *CE/n*.

### References

- Top of page
- Abstract
- Introduction
- Previous Research
- Theoretical Model
- Numerical Results
- Conclusions
- Appendix: The Estimation of Optimal Effort and Executive Option Value
- References

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- 1
For an excellent discussion of this distinction, see Carpenter (1998, pp. 131–32).

- 2
- 3
This characteristic of perpetual confidence seems to be common in virtually all positions involving high-level decision making. Coaches of poor teams seem to be perpetually optimistic that they can turn their teams around. Politicians always seem optimistic that they can win upcoming elections. Presidents exude optimism that they can adopt the right policies to improve life for citizens. Actors and producers are optimistic that their next movie will be highly successful, and if not, that the one after that will. Others may not share those beliefs, but these professionals seem to almost never lose their optimism and confidence, even in light of negative information. There is no necessary convergence of reality to expectations, nor does this lack of convergence imply failure and loss of job. It simply means that they remain confident of their abilities even when expectations do not pan out.

- 4
The empirical evidence on this issue is complicated by several factors. Grant awards are rarely discovered by investors until a later date, grant dates can be manipulated around the announcement of good news (Yermack 1997), and grand dates could at one time be backdated (Lie 2005, among others). The length of the period before the executive expects that the information is revealed is not critical to our model. There is evidence that executives may not reveal such information as soon as they possibly can (Hope and Thomas 2008), and logically, they may withhold this kind of information for competitive purposes.

- 5
- 6
By assuming a minimum level of effort of

*q*_{idt}*=*1, we mean that the executive believes that he can at a minimum take actions that will maintain the stock price. We rule out any possibility that the executive believes he will take actions that will cause the stock price to go down. While his actions may cause the stock price to decline, he does not believe that will happen. - 7
The quality measure may also reflect the overconfidence of a manager, which is consistent with Goel and Thakor (2008). Because an overconfident manager overestimates the precision of his private information and overreacts to it, he can increase firm value by alleviating the underinvestment problem. Hilary, Hsu, and Segal (2003) also present some empirical evidence that overoptimism can result in greater managerial effort.

- 8
This interpretation is the same as that in Cadenillas, Cvitanic, and Zapatero (2004). They mention that δ is an indicator of the quality of the executive. The only difference with our interpretation is that δ is the executive's opinion of his quality.

- 9
In Section III, we analyze the effect of optimal effort on the early exercise decision and discuss the assumption more thoroughly.

- 10
If

*δ*= 0, then*η*= 0, and the stock price process becomes the original process with minimum effort of*q*= 1. Therefore, the case of*δ*= 0 has the same effect on expected return as*q*= 1, even though the interpretations of these two cases are different. In either case, however, the executive would then perceive that his effort and confidence lead him to a private assessment of the stock price that is the same as the market's. - 11
At time

*idt*, . One period before time*idt*, . Continuing back to any time*jdt*gives , which is the executive's private assessment of the value of the stock and reflects the additional information he knows about his effort, as well as his confidence. - 12
Obviously the executive's wealth can consist of other asset classes and multiple tranches of options. A review of the literature reveals that virtually all models maintain fairly simple executive portfolios, some even simpler than what we assume. The effects of previously granted options as well as the expectation of future grants would be an interesting topic for future research.

- 13
- 14
The negative exponential utility function is one choice from among many possible utility functions and is preferred because it greatly facilitates the computations required in this model.

- 15
To simplify the analysis and focus on the central issue of this article, we assume no dividends, which is a common assumption in the literature.

- 16
These variations allow for the likely differences in elasticity across economies, industries, executives, and time. Cadenillas, Civitanic, and Zapatero (2004) assume low- and high-type managers arbitrarily with numbers of five and six, respectively. In the recent literature, Demerjian, Lev, and McVay (2012) quantify managerial ability, covering various industries, with a fitted value between –0.37 and 0.93, and our proxy of managerial ability is within the range. In addition, our estimation of the elasticity coefficient in essence is similar to their estimation of managerial ability.

- 17
The optimal holding of risky assets is the expected return divided by the product of relative risk aversion and the variance of the stock returns. In our benchmark case, the expected return is 12% and volatility is 50%. Assuming the coefficient of relative risk aversion is 2, the optimal holding of the firm's stock is 24%.

- 18
Lambert and Larcker (2004) identify a technical issue concerning the validity of the first-order condition. Because the option-based compensation is an increasing convex function of executive's effort, they show that the number of options granted to implement a given level of effort can become so large that the convexity of the option payoff can overwhelm the convexity of the agent's disutility. In that case, the expected utility would not be a well-behaved concave function of executive effort. We verify that this problem does not occur in our implementation of the model.

- 19
As an example, for

*T*= 7 and the low-executive-quality case, optimal effort of 1.906 translates into an abnormal stock return of ln(1.906) × 0.1 = 6.45% per annum. For the high-quality case, 1.962 translates into an abnormal stock return of ln(1.962) × 0.5 = 33.698% per annum. In the benchmark case, the abnormal stock return is 17.254% under the optimal effort of 1.994. These are the abnormal returns the executive believes he can generate. The magnitudes of these returns are not at all out of line with the returns observed in event studies of earnings, dividends, and share repurchase announcements. They are smaller, as they should be, but are clearly not unreasonable annualized abnormal returns. - 20
Recall that we do not allow executives to reset their effort as the option approaches maturity. Effort is determined at the grant date. But this suggests that if we allowed executives to reset their effort through time, effort would probably increase if moneyness and all other factors were held constant.

- 21
This result is consistent with Gervais, Heaton, and Odean (2003), who find that moderately optimistic managers are more aligned with well-diversified shareholders. Because they overstate the value of their effort, these managers can benefit the firm by exerting more effort than rational managers.

- 22
This result is not unique to our model. When we use 50% volatility in the model of Hall and Murphy (2002) and keep other parameters the same as their assumptions, the option value also decreases when we change the maturity from 10 years to 15 years.

- 23
As a practical matter, most firms do not grant short-maturity options but we see that such options should induce greater effort and would be more appreciated by executives.

- 24
Brooks, Chance, and Cline (2012) indicate another reason, the exploitation of private information. The database used in that study reports that about 94% of options are exercised early, about 3.6% are exercised on the vest date, about 13.5% appear to be exercised to capture dividends, and about 7.2% are associated with the departure of the option holder.

- 25
There is only one exception that higher quality executives have lower early exercise premiums when they can exercise early. It is when the quality changes from 0.25 to 0.5 in the case of

*T*= 5-year maturity and 0-year vesting.