Behavioral Factors in the Valuation ofEmployee Stock Options∗

Jennifer N. CarpenterNew York University

Richard StantonU.C. Berkeley

Nancy WallaceU.C. Berkeley

November 16, 2012

∗Financial support from the Fisher Center for Real Estate and Urban Economics and the Societyof Actuaries is gratefully acknowledged. For helpful comments and suggestions, we thank WenxuanHou, Ulrike Malmendier, Kevin Murphy, Terry Odean, Jun Yang, David Yermack, and seminar par-ticipants at Cheung Kong Graduate School of Business, Erasmus University, Lancaster University,New York University, Oxford University, Peking University, Securities and Exchange Commission,Shanghai Advanced Institute of Finance, Temple University, Tilburg University, and Tsinghua Uni-versity. We thank Terrence Adamson at AON Consulting for providing some of the data used in thisstudy. We also thank Xing Huang for valuable research assistance. Please direct correspondenceto carpenter@stern.nyu.edu, stanton@haas.berkeley.edu, or wallace@haas.berkeley.edu.

Behavioral Factors in the Valuation ofEmployee Stock Options

Abstract

Options are a major component of corporate compensation and accounting methods signif-icantly affect allocations, yet valuation remains a challenge. We develop the first empiricalmodel of employee exercise suitable for valuation, and estimate the exercise rate as a functionof stock price and employee characteristics in the first sample of all employee exercises atover 100 firms. These characteristics significantly affect exercise behavior and cost, in waysthat Black-Scholes-based approximations used in practice fail to capture. For example, exer-cises are slower, and option cost greater, at firms with higher market correlation; top-decileoption holders within a firm exercise faster than lower-decile holders; and men exercise fasterthan women, resulting in 2–3% lower cost.

JEL classification: G14.

Employee stock options (ESOs) are a major component of corporate compensation and

a material cost to firms. Frydman and Jenter (2010) find that options represented 25% of

CEO pay in 2008, and the 2010 General Social Survey of the National Opinion Research

Center found that 9.3 million employees hold options. As a fraction of outstanding shares of

common stock, the median number of shares underlying outstanding options at S&P 1500

firms was 4.4% in 2010, with almost 1% of outstanding shares allocated to new option grants

annually, according to Equilar (2011). At the same time, accounting valuation methods for

ESOs significantly affect corporate decisions about compensation structure and allocation,

and thus affect both employee incentives and firm cost. For example, the 2005 change in

accounting valuation precipitated a decline in the granting of options from previous levels

and an increase in the allocation to top managers.1 Valuation methods for employee stock

options are therefore an important issue for investors.

ESO valuation remains a challenge in practice. The value of these long-maturity Ameri-

can options depends crucially on how employees exercise them, but because employees face

hedging constraints, standard option exercise theory does not apply. For example, employees

systematically exercise options on non-dividend-paying stocks well before expiration (see, for

example, Huddart and Lang, 1996; Bettis et al., 2005), and this substantially reduces their

cost to firms. Modifications to Black Scholes used for valuation in practice are inadequate

because they ignore key features of employee exercise patterns. At the same time, exist-

ing empirical descriptions of option exercise behavior based on OLS or hazard rate models,

such as Heath et al. (1999), Armstrong et al. (2007), and Klein and Maug (2011), are not

appropriate as an input for option valuation, as we shall show.

In this paper we develop the first empirical model of employee option exercise suitable for

valuation and apply it in a sample of all employee exercises at over 100 publicly traded firms

from 1981 to 2009 to show empirically how employee exercise patterns affect option cost to

firms. We develop a GMM-based methodology that is robust to both heteroskedasticity and

correlation across option exercises to estimate the rate of voluntary exercise as a function

of the stock price path and of firm, contract, and option-holder characteristics. We show

that these characteristics significantly affect exercise behavior, in a manner consistent with

both portfolio theory and results from the behavioral literature, such as those in Barber

and Odean (2001). We use the estimated exercise function to value options and show that

these characteristics also materially affect option value. For example, men exercise options

significantly faster than women, resulting in 2–3% lower option value, and exercise rates are

1Equilar (2011) reports that the median number of shares underlying outstanding options at S&P 1500firms fell from 6.1% in 2006 and to 4.4% in 2010, while the fraction of options allocated to CEOs and namedexecutives rose from 21% to 26%.

1

significantly lower when a firm’s stock return correlation with the market is higher, resulting

in greater option cost.

Our proprietary dataset is the first in the academic literature to contain all employee

exercises at a large number of firms.2 Since estimation of exercise behavior requires a large

sample of stock price paths, and thus a large number of firms, our paper is the first that can

investigate differential option exercise patterns and option cost across top and lower-ranked

employees. To the extent that options are granted where their performance incentives have

greatest benefit, large option holders within a firm are likely to be the key decision makers.

We find that top-decile option holders exercise faster than lower-decile holders within a given

firm, and consequently their options are typically worth 2–3% less. Similarly, in a subsample

with data on employee titles, senior executives exercise faster than lower-ranked executives.

The propensity of top employees to exercise faster may stem from their greater exposure

to undiversifiable firm risk, which can have a theoretically important effect on the exercise

decision (see, for example, Hall and Murphy, 2002; Ingersoll, 2006; Carpenter et al., 2010).

Wealth differences may also play a role. Option compensation delivers both wealth, which

would reduce the propensity of an employee (with decreasing absolute risk aversion) to

exercise options early, and risk, which would increase the early-exercise rate. We attempt to

disentangle these effects by including in our estimation proxies for both wealth and stock-

based risk. Consistent with portfolio theory, we find that portfolio risk increases the exercise

rate and wealth decreases it. At the same time, the employee rank effect remains even when

we control for the risk of the employee’s option portfolio, perhaps reflecting top employees’

use of preplanned exercise policies under SEC rule 10b5-1, which shields against prosecution

for insider trading. It may also reflect their greater exposure of human capital to firm

risk, interactions of option value with corporate decision making, more volatile corporate

information flow, overconfidence, or other psychological factors.

In our estimation we also find that the rate of voluntary option exercise is positively

related to the level of the stock price and the imminence of a dividend, and negatively

related to stock-return volatility and option time to expiration, consistent with standard

option theory. In addition, the exercise rate is higher when the stock price is in the 90th

percentile of its distribution over the past year or in the two weeks after a vesting date,

consistent with employees’ use of cognitive benchmarks in decision making.

To assess the economic impact of the sensitivity of employee exercise rates to volatility

and dividends, we compare option values calculated using the estimated exercise function

with option values from Black Scholes approximations, which account for volatility and div-

2For example, the samples of Heath et al. (1999) and Armstrong et al. (2007) contain only seven and tenfirms, respectively, while the sample of Klein and Maug (2011) contains only top executives.

2

idend effects in the stock price process, but not in exercise behavior. We consider both

traditional Black Scholes and Black Scholes with option expiration set equal to the average

of the vesting date and stated expiration date, which is an approximation permitted by the

Financial Accounting Standards Board in SAB 110 for firms with limited option exercise

data. The pricing errors are large and vary systematically with firm and option charac-

teristics, highlighting the importance of properly accounting for the dependence of exercise

behavior on these characteristics.

In practice, most firms use the Modified Black Scholes method, permitted by FAS 123R,

in which the option expiration date is set equal to the option’s expected term. This expected

term has to be estimated, which entails significant error since option outcomes depend on a

firm’s stock price path and a given firm has option data based only on its own realized stock

price path. We show, moreover, that even without error in the estimation of the option’s

expected term, this method creates systematic biases. In particular, we calculate the option’s

expected term exactly from the estimated exercise function to get the Modified Black Scholes

value and show that the approximation error still varies widely and systematically with firm,

option, and option-holder characteristics. Firms frequently need to revalue old options in

the event of a corporate or compensation plan restructuring. We find that the Modified

Black Scholes approximation errors are even larger for out-of-the-money options than for

at-the-money options, underscoring the need for a more robust valuation methodology.

Our data are provided by corporate participants in a sponsored research project funded

by the Society of Actuaries in response to regulatory calls for improved ESO valuation meth-

ods. The Financial Accounting Standards Board’s SAB 110, p.6, states a clear expectation

that “more detailed external information about exercise behavior will, over time, become

readily available to companies.” This paper responds to FASB’s call by providing the first

comprehensive analysis of exercise behavior and its implications for valuation.

The paper proceeds as follows. Section 1 reviews the related literature. Section 2 first

describes the theoretical foundation and then develops an empirical model of employee ex-

ercise that is both flexible enough to capture observed exercise behavior and suitable as a

basis for option valuation. Section 3 describes the sample of proprietary data on employee

options. Section 4 contains estimation results for the empirical exercise function. Section 5

presents estimates of option cost and approximation errors of current valuation methods,

and Section 6 concludes.

3

1 Previous Literature

The principles of employee option valuation and the need to study exercise behavior are

well understood. One approach taken in the literature is to model the exercise decision

theoretically. The employee chooses an option exercise policy as part of a greater utility-

maximization problem that includes other decisions, such as portfolio, consumption, and

effort choice, and this typically leads to some early exercise for the purpose of diversification.

Papers that develop utility-maximizing models and then calculate the implied cost of options

to shareholders include Huddart (1994), Detemple and Sundaresan (1999), Ingersoll (2006),

Leung and Sircar (2009), Grasselli and Henderson (2009), and Carpenter et al. (2010).

Combining theory and data, papers such as Carpenter (1998) and Bettis et al. (2005)

calibrate utility-maximizing models to mean exercise times and stock prices in the data,

and then infer option value. However, these papers provide no formal estimation and the

approach relies on the validity of the utility-maximizing models used. Huddart and Lang

(1996), Heath et al. (1999), and Klein and Maug (2011) provide more flexible empirical

descriptions of option exercise patterns, but do not go as far as option valuation. Armstrong

et al. (2007) perform a valuation based on a hazard model of the exercise of option grants, but

this specification is inappropriate for valuation because employees exercise random fractions

of outstanding option grants.

A number of analytic methods for approximating employee option value have also been

proposed. FAS 123R permits using the Black Scholes formula with the expiration date re-

placed by the option’s expected life and SAB 110 permits using Black Scholes with expiration

replaced by the average of the contractual vesting date and expiration date. Jennergren and

Naslund (1993), Carr and Linetsky (2000), and Cvitanic et al. (2008) derive analytic formulas

for option value assuming exogenously specified exercise boundaries and stopping rates. Hull

and White (2004) propose a model in which exercise occurs when the stock price reaches an

exogenously specified multiple of the stock price and forfeiture occurs at an exogenous rate.

However, until the accuracy of these methods can be determined, their usefulness cannot be

assessed.

2 Modeling Exercise Behavior

This section first describes the theoretical foundation for modeling employee-exercise behav-

ior and then develops an empirical model that is both flexible enough to capture observed

exercise behavior and suitable as a basis for option valuation.

4

2.1 Theoretical Foundation

In the standard theory of American-option exercise, the option holder chooses a policy to

maximize option value, and for an ordinary American call on a stock in a Black Scholes

framework, the value-maximizing policy is described by a critical stock price, above which

it is optimal to exercise and below which it is optimal to continue holding the option. The

critical stock price is increasing in the time to expiration, the stock return volatility, and the

interest rate, and decreasing in the dividend rate (see, for example, Kim, 1990). However,

employee stock options are nontransferable and employees face hedging constraints, so in

order to diversify away from excessive stock price exposure, employees may exercise options

early. A number of papers rationalize this behavior in models of employee-option exercise in

which the option holder chooses a policy to maximize expected utility, subject to constraints

on selling the option and the underlying stock. These include Huddart (1994), Detemple

and Sundaresan (1999), Ingersoll (2006), Leung and Sircar (2009), Grasselli and Henderson

(2009), and Carpenter et al. (2010). For example, Carpenter et al. (2010) model a risk-averse

employee who chooses both an exercise policy for his options and a dynamic trading strategy

in the market and riskless asset for his outside wealth to maximize expected utility. They

show in particular that the optimal exercise policy need not be characterized by a single

critical stock-price boundary (though it is for certain utility functions, including constant

relative risk aversion). They also show either analytically or numerically how the critical

stock price (or, more generally, the employee’s continuation region) varies with employee risk

aversion and wealth and with the stock-return volatility, dividend rate, and the correlation

between the stock return and the market return. For example, the continuation region is

smaller if the employee has greater absolute risk aversion and thus is larger with greater

employee wealth if he has decreasing absolute risk aversion. In addition, the continuation

region is smaller the greater the dividend rate. In numerical examples with constant-relative-

risk-averse utility, the critical stock price is increasing in the correlation between the stock

return and the market return (in the region of positive correlation) because the more the

option risk can be hedged in the outside portfolio, the more attractive the option position

becomes. On the other hand, the effect of greater volatility or longer time to expiration is

ambiguous, because of the conflicting effects of employee risk aversion and the convexity of

the option payoff.

2.2 Empirical Model and Estimation Methodology

Although the structural models in the theoretical literature provide guidance, a more flex-

ible, reduced-form model is necessary to capture the complexities of real employee-exercise

5

patterns. It seems natural to use hazard rates to model the exercise of employee stock op-

tions, since they have often been used in the finance literature to model apparently similar

events, such as mortgage prepayment (see Schwartz and Torous, 1989) and corporate-bond

default (see, for example, Duffie and Singleton, 1999). However, whereas it makes sense

to think of the prepayment of one mortgage as independent of the prepayment of another,

conditional on the level of interest rates, ESOs are typically exercised in blocks. As a result,

the exercise of one option in a given grant held by an individual is extremely highly cor-

related with the exercise of another option in the same grant held by the same individual.

It is also quite highly correlated with the exercise of options in other grants held by the

same individual. This high degree of correlation between options makes it difficult to use

standard econometric techniques, which assume independence between events, to estimate

hazard rates at the individual option level.3 Armstrong et al. (2007) use a hazard rate to

model exercise of an entire grant of options above some threshold fraction, but this throws

away important information about the exact fraction of the grant exercised. In practice,

fractional exercise is both prevalent and highly variable (see Huddart and Lang, 1996), so

this approach is unusable for valuation.

A solution to these problems is to abandon the hazard rate approach altogether and

instead to model the fraction of each grant exercised each period. Heath et al. (1999) follow

this approach, regressing the fraction of each grant exercised against various explanatory

variables. However, their regression approach is also problematic. In particular, it may

generate expected exercise fractions that are negative or greater than one, both of which

cause problems for valuation.4 Heath et al. (1999) also aggregate across individuals, thus

discarding potentially important information about the differences in exercise behavior across

individuals, and introducing correlation across exercises of the aggregated grants.

Like Heath et al. (1999), we also model the fraction of each grant exercised by each

holder each period, but we do so in a manner that generates consistent estimates of expected

exercise rates, which are guaranteed to be between zero and one, while explicitly handling the

correlation between option exercises within and between different grants held by the same

individual. Our approach, based on the fractional-logistic approach of Papke and Wooldridge

3This issue also arises in modeling corporate-bond default. One popular solution, when the number offirms involved is small, has been to use “copula functions,” which explicitly model this correlation (see,for example, Li, 2001). However, in our case the number of options (and hence the number of correlationcoefficients) is too high to be feasible.

4Attempting to remedy this by truncating the variables, will lead to biases. Transforming the proportion

exercised, such as by using a logistic transformation log(

y1−y

), which can take on any value between −∞

and +∞ is also problematic. By Jensen’s inequality, the expected proportion exercising is not just the inversetransformation of the expected transformed proportion. More important, this approach cannot handle thenumerous dates on which no options are exercised at all.

6

(1996), also allows for arbitrary heteroskedasticity in the exercise rates.

Let yijt be the fraction exercised at time t of grant j held by individual i, and write

yijt = G(Xijtβ) + uijt, (1)

where Xijt is some set of covariates in It (the information set at date t), G, the expected

fraction exercised at date t, is a function satisfying 0 < G(z) < 1, and

E(uijt | It) = 0,

E(uijt ui′j′t′) = 0 if i 6= i′ or t 6= t′.

From now on, we shall use the logistic function,

G(Xijtβ) =exp(Xijtβ)

1 + exp(Xijtβ).

Note that, while we are assuming the residuals εijt are uncorrelated between individuals and

across time periods, we are allowing for εijt to be arbitrarily correlated between different

grants held by the same individual at a given point in time, and we are not making any

further assumptions about the exact distribution of εijt. In particular, we are allowing a

strictly positive probability that yijt takes on the extreme values zero and one.

We estimate the parameter vector β using quasi-maximum likelihood (see Gourieroux

et al., 1984) with the Bernoulli log-likelihood function,

lijt(β) = yijt log [G(Xijtβ)] + (1− yijt) log [1−G(Xijtβ)] . (2)

We estimate clustered standard errors that are robust both to arbitrary heteroskedasticity

and to arbitrary correlation between exercises by a given individual on a particular date (see,

for example, Gourieroux et al., 1984; Rogers, 1993; Baum et al., 2003; Wooldridge, 2003;

Petersen, 2009).

3 Data

Our estimation strategy is carried out using a proprietary data set comprising complete

histories of employee stock option grants, vesting structures, and option exercise and can-

cellation events for all employees who received options at 104 publicly traded corporations

7

between 1981 and 2009.5 As shown in Table 1, there is considerable heterogeneity in the

sample of firms in terms of their industry type, reported at one-digit Standard Industrial

Classification (SIC) codes, firm size, as measured by market cap and numbers of employees,

revenue growth over the period, and stock return volatility. Sample firm dividend rates are

low and many firms pay no dividend, so early exercise is driven by other factors. Sample

firm return correlations with S&P Composite Index average from 25% to 37% across sectors,

suggesting employees have some scope for hedging their option compensation by reducing

their market exposure in their outside portfolios.

3.1 Proprietary option data

Our unit of analysis is an employee-grant-day. For each option grant we merge the appro-

priate path of daily split-adjusted stock prices and dividends, starting at the initial grant

date, to the path of option vesting and exercise events for all grants and employees. These

daily paths are constructed using detailed information on the contractual option vesting

structure, the exercise events, and the cancellation events recorded for each grant. We track

the employee-grant-days and a series of time-varying covariates until the options in the grant

are fully exercised, the options are canceled, or we reach the end of the sample period of

December 31, 2009.

Table 2 summarizes the size and structure of the sample of option data by industry

and in aggregate. In total there are 891,139 option exercises across 1,245,201 grants to

521,123 employees at 104 firms. On average, there are 2.4 grants per employee, but there is

considerable variability across firms and employees, with some employees receiving dozens

of option grants. For the firms for which we have the employee ranking of the employee, the

largest grant recipients are typically the CEO or senior managers.

Employee option portfolios represent significant sources of wealth and stock price risk.

While the contracting literature has focused on the performance and risk incentives created

by these portfolios, the wealth and stock price risk also affect option exercise behavior. We

approximate employee option wealth on a given day as the total Black Scholes value of all

of his or her options across all grants on that day, a measure which ignores early exercise,

cancellation risks, and nontradability, and thus overstates the option wealth and subjective

value. Panel C summarizes the distribution of BS Employee Option Wealth across employee-

days. Similarly, we approximate employee option risk as the sum of the Black Scholes deltas

of all his or her options on a given day times the stock price times the annualized stock

return volatility. This is a Black Scholes approximation of the effective dollar volatility of

5The data were obtained as part of a research grant written by the authors and funded by the Society ofActuaries. We thank Terrence Adamson at AON Consulting, who also provided data for this study.

8

employee option wealth. Panel D summarizes the distribution of BS Employee Option Risk

across employee-days. The mean wealth and risk are, respectively, $149,044 and $83,353,

and their variability is high, suggesting that wealth and risk effects may be quite significant.

Table 3 summarizes the size, vesting structure, and maturity of option grants in the

sample. The average grant has $49,680 worth of underlying shares at the grant date, but

this varies widely across industry, with the greatest mean and variance of grant size in the

finance industry. The combined effects of the potentially large number of grants per employee

and the size of these grants implies that individual employees may hold large inventories of

options with different strikes, expiration dates, and vesting structures. This feature of the

data introduces significant correlation across the exercise decisions of individual employees.

There is likely to be high correlation in the exercise decisions across grants that are held by

the same individual. A particular strength of our fractional logistic estimator is that it does

not require assumptions of independence across exercise events. We also pool by employee

and correct our standard errors to account for our pooled structure.

Vesting structures also vary widely, both across and within firms in our sample, and can

be complex. The average grant has 5.8 vesting dates, but some have as many as 60 vesting

dates. An example of a vesting structure that would lead to a large maximum would be a

grant with a 25% vest at the end of the first year and then 2.08% monthly vests over the

next 36 months. The minima are generated by “cliff vests,” where all the options in a given

grant vest on the same day.

The only homogeneous contractual feature of employee stock option grants across firms

is the maturity in months from the issuance date to the date of expiry. The term of executive

stock options is quite uniformly ten years although there are some fifteen-year and four-year

maturity options granted on the part of some firms. At the employee-level, the employees in

our sample are in some cases managing as many as ten different contractual option vesting

structures in their inventory of options.

Table 4 summarizes exercise patterns in the sample. The summary statistics weight

exercise outcomes by the dollar value of the exercise. Options are exercised very early. The

average option has 4.2 years remaining to expiration and has only been in the money and

vested for 206 days. On average, the option is 419% in the money at the time of exercise,

and more than a third of the time, the stock price is near its annual high. At the time of

exercise, an overall average of 71% of a grant’s vested options are exercised, with a standard

deviation of 28%. It is this prevalence of random, fractional exercise of option grants that

motivates the development of our fractional logistic estimation strategy.

In summary, there are three primary features of the stock option exercise patterns ob-

served in our sample. First, many employees hold more than one option grant and make

9

exercise decisions over more than one vested option at any given time. For this reason,

estimation strategies must account for the correlated decision structure of employee option

exercise. Second, both the contractual vesting structure and the exogenous price paths

appear to have strong effects on option exercise patterns. Thus careful controls for daily

realizations of both features must be included in a successful estimation strategy. Finally,

many option positions are exercised fractionally, that is, the proportion of the outstanding

options that are exercised at exercise events can be substantially less than one. For this

reason, a successful econometric methodology must account for path-dependent fractional

exercise behavior if it is to avoid yielding significant misspecification bias and inaccurate

forecasts of exercise timing.

3.2 The covariates

Employees may voluntarily choose to exercise options or they may be forced to do so be-

cause of impending employment termination or option expiration. To estimate the model of

voluntary exercise, we begin with the sample of employee-grant-days on which the option is

in the money and vested and then eliminate those days that are within six months of the

grant expiration date or six months of a cancellation of any option by that employee, because

most cancellations are associated with employment termination. The remaining employee-

grant-days are treated as days on which the employee has a choice about whether and how

many options to exercise. To explain the fraction of options exercised by a given employee

from a given grant on a given day, as specified by Equation (1), we choose as covariates

variables drawn from optimal option exercise theory, such as Carpenter et al. (2010), as well

as behavioral variables identified in empirical studies such as Heath et al. (1999).

Since employee stock options are non-transferable, the optimal exercise policy for these

options can look quite different from that for standard American call options. However, in

virtually every model of optimal exercise, the degree to which the option is in the money

is an important determinant of the exercise decision. In both standard option theory and

in many models of employee option exercise, option holders exercise once the stock price

rises above a critical boundary. Intuition also suggests that in practice, exercise becomes

more attractive as the option gets deeper in the money and more of its total value shifts

to its exercise value. The variable Price-to-strike ratio, the employee-grant-day ratio of the

split-adjusted price of the stock to the split-adjusted option strike price captures the degree

to which the option is in the money.

Carpenter et al. (2010) prove very generally that the dividend effect for employee option

exercise is qualitatively the same for employee option exercise decisions as for standard,

10

transferable options. That is, a higher dividend makes early exercise more attractive, all

else equal. The variable Dividend in next two weeks is the product of an indicator that a

dividend will be paid within the next 14 calendar days and the ratio of the dividend payment

to the current stock price.

The theoretical effect of higher stock return volatility on the exercise decision is more

complicated for employee options than the simple negative effect from standard theory.

Employee risk aversion and the convexity of the option payoff have offsetting effects on

employees’ attitudes toward volatility, and the net effect is an open empirical question. The

variable Volatility is the annualized daily volatility estimated from the stock return over the

66 trading days prior to the given employee-grant-day.

Unlike in standard theory, the degree to which the employee can hedge the option position

in an outside portfolio is an important theoretical determinant of the exercise decision, and

Carpenter et al. (2010) and others have shown that the higher the correlation between the

stock return and the return on a tradable asset, the lower the propensity to exercise early.

The variable Correlation is the correlation between the stock return and the return on the

S&P 500 Composite Index estimated from daily returns over the three months prior to the

given employee-grant-day.

The theoretical effect of more time to expiration on the exercise decision can also be more

complicated for employee options than the simple negative effect from standard theory, and

is thus also an open empirical question. The variable Time to expiration is the number of

calendar days from the given employee-grant-day to the expiration date of the grant, divided

by 365.

Recent empirical studies of employee stock option exercise report links between behavioral

indicators, or “rules of thumb,” that employees appear to rely upon in making their option

exercise decisions. Armstrong et al. (2007) find a statistically significant association between

the timing of vesting events and option exercise. They argue that recent exercise events both

mechanically affect an employees’ ability to exercise their options and may also serve as a

periodic reminder to employees to evaluate the value of their option positions. Heath et al.

(1999) and Armstrong et al. (2007) also find a statistically significant positive association

between option exercise and the occurrence of the current stock price exceeding the 90th

percentile of the past year’s price distribution. They argue that this association is driven by

cognitive benchmarks that employees use in their decision rules. Given the importance of

these variables in prior studies, we also include them as controls in all of our specifications.

To capture the vesting structure of the grant, the variable Vesting event in past two weeks

indicates whether the given employee-grant-day is within 2 weeks since a vesting date for

that grant. Our cognitive benchmark proxy is the variable Price ≥ 90th percentile of prior

11

year distribution, which indicates whether the current stock price is greater than or equal to

90th percentile of the stock price distribution over the prior year of trading.

Barber and Odean (2001) find significant differences in the trading patterns of men and

women, with men the more frequent traders. For 69 firms in our sample, we have information

on the age and gender of the employee. Table 5 shows that the average employee is 42 years

old and 56% of employees are male. The variable Male indicates whether or not the option

holder is male.

Our sample is the first to contain both a large number of firms and all employees at those

firms. To examine differences in exercise patterns across the ranks, we proxy for employee

rank with Top-decile option holder, an indicator of whether the employee is in the top decile

of option holders at the firm, sorted by the total Black Scholes value of their vested option

holdings at the beginning of the quarter. To the extent that options are granted where their

performance incentives will have greatest benefit, top-decile option holders are likely to be

key decision makers at the firm. For 38 firms in our sample, we also have employee titles.

The variable Senior indicates whether the employee is designated as a senior executive.

Employee wealth and undiversifiable portfolio risk can also have a theoretically impor-

tant effect on the exercise decision (see, for example, Hall and Murphy, 2002; Ingersoll, 2006;

Carpenter et al., 2010). The challenge is in disentangling these effects. Stock-based com-

pensation delivers both wealth, which would reduce the propensity of an employee (with

decreasing absolute risk aversion) to exercise options early, and risk, which would increase

the early exercise rate. We attempt to disentangle these effects by including in our estima-

tion both a proxy for employee wealth and a proxy for employee stock-based risk. For the

full sample, we have BS Employee Option Wealth, the Black Scholes value of the employee

option portfolio, and BS Employee Option Risk, the Black Scholes dollar volatility of the

option portfolio. For nine firms in the sample, we have information about employee salary,

which may also proxy for employee wealth. Table 5 shows that the mean salary in this

subsample is $283,785.

4 Estimation Results

We estimate ten alternative specifications of Equation (1). One uses a base set of covariates,

one adds Gender and Age to the base set, one adds Top-decile option holder, one adds

BS Employee Option Wealth and BS Employee Option Risk, and one adds Salary and BS

Employee Option Risk. For each of these five, we show results with and without one-digit

SIC industry fixed effects. Table 6 reports coefficients and standard errors. The estimators

cluster at the level of the individual employee.

12

As Table 6 shows, the results for the base covariates are strikingly similar across all

specifications, and robust to the inclusion of fixed effects. The parameter estimates are

strongly consistent with the predictions of optimal exercise theory, and also support several

of the leading conclusions of the behavioral finance literature. The rate of voluntary option

exercise is strongly positively related to the level of the stock price, as expected. Exercise

rates are also generally significantly higher when a dividend payment is imminent. Also in

line with the theory of employee option exercise, exercise rates are consistently lower when

the stock return is more highly correlated with S&P Composite Index, so that a greater

fraction of the stock risk can be hedged with reductions in exposure to the market portfolio.

As discussed above, Carpenter et al. (2010) show that stock return volatility and time

to expiration do not have clearly signed theoretical effects on an employee’s optimal exercise

policy, so the empirical effects are an open question. The results reported in Table 6 indicate

that increased levels of stock return volatility and more time to expiration are associated

with smaller fractions of options exercised, which is consistent with the theory for ordinary

American options.

Table 6 also shows that employees exercise a significantly larger fraction of outstanding

options when the stock price is greater than or equal to the 90th percentile of its distribution

over the prior calendar year6 and in the two weeks following a vesting date of the grant in

question. These results are consistent with Heath et al. (1999), Armstrong et al. (2007) and

Klein and Maug (2011), and support the idea that employees tie their exercise decisions to

cognitive benchmarks as a means of reducing monitoring costs.

Based on the subsample of firms for which information on age and gender are available,

specifications 3 and 4, the results in Table 6 indicate that male employees have a greater

propensity to exercise their options than female employees, and older employees are less

likely to exercise options early. These results appear to be inconsistent with the notion that

women and older people are more risk averse. However, they add support to the notion that

men are more over-confident (see, for example, Barber and Odean, 2001).

Table 6 also shows that top-decile option holders exercise faster than lower-decile option

holders. This might be because of the greater exposure of their wealth and human capital to

undiversifiable firm risk. Though Male and Top-decile option holder are positively correlated,

we confirm in unreported results that the gender and top-decile effects are distinct, using

specifications that include both dummies and their interaction. Finally, in the subsample of

38 firms for which we have employee titles, we replace Top-decile option holder with Senior

and find very similar results, confirming the validity of Top-decile option holder as a proxy

for high rank.

6This result is robust to the inclusion of the square of Price-to-strike ratio as an explanatory variable.

13

Top-decile holders are obviously likely to have both high wealth and high exposure to

firm risk, and these portfolio characteristics have theoretically important effects. In partic-

ular, holding all else equal, employee wealth decreases the propensity to exercise, assuming

decreasing absolute risk aversion, and undiversifiable exposure to the stock price increases it.

The challenge is in finding empirical proxies for these portfolio characteristics. To disentan-

gle these portfolio effects using our full sample of employees, we proxy for employee wealth

and portfolio risk using BS Employee Wealth, the total Black Scholes value of the employee’s

option portfolio, and BS Employee Risk, the total Black Scholes delta of the employee’s op-

tion portfolio times the dollar volatility of the stock price. For the subsample of nine firms

for which we have salary data, we use Salary to proxy for employee wealth. Specifications

5 through 8 of Table 6 test for these portfolio risk and wealth effects in exercise decisions.

The results are broadly consistent with the hypothesis that employee risk exposure increases

the propensity to exercise early, and employee wealth decreases it. The BS Employee Option

Risk has either a significantly positive coefficient or a zero coefficient. The coefficients on

the proxies for wealth are significantly negative in three out of four cases.

In unreported results, we find that even after controlling for portfolio risk, top-ranked

employees exercise faster than lower-ranked employees. This may be because top employees

use preplanned exercise policies under SEC rule 10b5-1, which shields against prosecution

for insider trading. It may also reflect their greater exposure of human capital to firm

risk, interactions of option value with corporate decision making, more volatile corporate

information flow, overconfidence, or other psychological factors.

Taken together, our estimation results provide strong empirical support for the predic-

tions of both optimal exercise theory and behavioral finance. To evaluate the economic

significance of these exercise patterns, we now analyze their effect on option cost.

5 Option Cost to the Firm

For an individual option, the exercise function describes the expected proportion of each

outstanding option grant to be voluntarily exercised at a given time and state, conditional

on having survived to that point. Similarly, the termination rate describes the expected

fraction of options stopped through termination. Given the estimated voluntary exercise

rate per period, G(Xβ), and termination rate λ, the value of the option is given by its

14

expected risk-neutral discounted payoff,

Ot = E∗t

{∫ T

t∨tve−r(τ−t) (Sτ −K)+ (Gτ + λ)e−

∫ τt (Gs+λ) ds dτ

+e−r(T−t)e−∫ Tt (Gs+λ) ds(ST −K)+

},

where tv is the vesting date.7 We estimate this value with Monte Carlo simulation, using

antithetic variates and importance sampling to increase precision.

Table 7 reports option values, labeled ESO Value, for a variety of parameterizations. In

the base case, with stock price and exercise price both equal to 100, the firm volatility is

30%, the average volatility of US firms reported in Campbell et al. (2001), and the firm

correlation with the market is 40%. The base case dividend rate is zero, which is typical for

many option-granting firms, and eases the exposition of the various early-exercise effects in

option values. The vesting period is two years, and the option expiration date is ten years,

which are also typical. Finally, the option holder terminates employment at an annual rate

of 8%, consistent with the option cancellation rates in our sample. Assuming the option

holder voluntarily exercises according to the estimated exercise function in Specification 1,

the base case option cost is 30.55

Panels A, D, E, and F of Table 7 assume voluntary exercise according to the estimated

exercise function in Specification 1 of Table 6, Panel B assumes exercise according to Specifi-

cation 3, and Panel C is based on Specification 5. After the steep decline in the stock market

during the financial crisis, most firms found that the options they granted to employees before

the crash were deeply out of the money during 2008 and 2009. Many firms offered their em-

ployees equal-present-value exchanges of at-the-money options for the old out-of-the-money

options in an effort to restore performance incentives, and this required that they assess the

value of the old options. Therefore, while the left side of Table 7 presents at-the-money

options at their grant date, the right side analyzes so-called “underwater” options, two years

after grant, vested, but 40% out of the money.

The ESO Value columns of Table 7 show that option cost varies significantly with firm

and option characteristics. For example, Panel A shows that as stock return correlation with

the market varies from 0 to 80% and the rate of exercise decreases, the at-the-money option

cost increases 13% from 28.67 to 32.28, while the underwater option value varies 18% from

8.63 to 10.17.

7If the exercise and termination events are sufficiently independent across option holders, conditional onthe given time and state, then in a large enough pool, the fraction of options exercised or terminated willexactly equal their expected values. We assume that such diversification is possible, or, more generally, thatany residual risk is not priced in the market.

15

Panel B shows that option cost varies materially with employee gender. As Table 6

shows, men exercise options significantly faster than women, perhaps reflecting differences

in confidence, as in Barber and Odean (2001). In the base case, the cost of the at-the-money

option granted to a woman is therefore 2.7% higher than the cost of the same option granted

to a man, while the value of the underwater option is 3.5% higher if held by a woman. Panel

C shows that option cost also varies with employee rank, as proxied by rank of employee

option holdings. In particular, because lower-decile option holders exercise more slowly than

top-decile holders, their options are worth 2.9% more at the grant date, and 4.1% more when

40% underwater. We analyze these gender and top-decile effects effects in greater depth in

Section 5.2.

The volatility and dividend effects in Panels D and E are also quite significant. However,

the changes in ESO Value reflect changes in the stock price process as well as changes in

exercise behavior. To isolate the effect of changing exercise behavior, we compare ESO Value

to traditional Black Scholes value assuming expiration at the contractual expiration date,

and to Black Scholes value assuming expiration at the average of the contractual vesting and

expiration dates, as permitted by SAB 110 for firms with limited option exercise data. These

values are multiplied by the probability that the option vests, given the assumed termination

rate. The approximation errors are quite large and vary widely with the firm characteristics.

For example, the errors from traditional Black Scholes vary from 87% to 32% as volatility

varies from 10% to 50%. The errors from the SAB 110 approximation vary from 26% to 6%

for the at-the-money option and −80% to −4% for the underwater option. Similarly, the

errors from traditional Black Scholes vary from 46% to −12% for the at-the-money option

as the dividend rate varies from 0 to 7%, and the SAB 110 approximation errors vary from

10% to −6%. This underscores the importance of properly accounting for the response of

exercise behavior to changes in the environment.

Panel F shows that option cost need not vary monotonically with the length of the vesting

period. The option value first rises, as the longer vesting period prevents voluntary exercises

early in option life, and then declines, as the risk of forfeiture offsets this effect.

5.1 Modified Black Scholes approximation

Most firms use the Modified Black Scholes (MBS) method permitted by FAS 123R, which

approximates option value as the probability of vesting times the Black Scholes value ad-

justed for dividends, with contractual expiration date replaced by the option’s expected

term, conditional on vesting. Though in principle this method could capture some of the

exercise policy effects through adjustments to the option expected term, it suffers from two

16

serious problems. First, the estimation of option expected term is as difficult as estimat-

ing the option cost itself. This is because the realized option term depends on the stock

price path and any other variables that affect exercise decisions, and no single firm is likely

to have a sufficiently long history of option outcomes to perform this estimation with any

precision. Second, even without estimation error in the option expected term, the Modified

Black Scholes approximation error varies widely and systematically with firm characteristics,

because option expected term does not sufficiently summarize the exercise policy. To make

this second point, we compute the option’s expected term using Monte Carlo simulation,

assuming the option holder follows the estimated exercise function and terminates employ-

ment at a constant rate. This expectation is with respect to the true probability measure,

so it depends on the true expected return on the stock. Formally, the option’s true expected

term is

Lt = Et

{∫ T

t∨tvτ (Gτ + λ)e−

∫ τt∨tv (Gs+λ) ds dτ + T e−

∫ Tt∨tv (Gs+λ) ds

}, (3)

where the stock price follows

dS/S = (µ− δ) dt+ σ dZ. (4)

We assume the mean stock return is µ=11%, i.e., a 6% equity premium.

The MBS approximation can either understate or overstate the true option value, de-

pending on the exercise policy. To see why, consider two special cases, and for simplicity

assume immediate vesting. First, if the option holder follows the value-maximizing exercise

policy in the presence of dividends, as in standard theory, then the true option value will be

greater than the Black Scholes value to any deterministic expiration date, so it will exceed

the MBS approximation. Alternatively, suppose the option is stopped, either through exer-

cise or cancellation, at a purely exogenous rate, independent of the stock price. Then the

option value is the average Black Scholes value over possible stopping dates, while the MBS

approximation is the Black Scholes value to the average stopping date, so since the Black

Scholes value tends to be concave in the option expiration date, the MBS approximation will

overstate the true value, by Jensen’s inequality. The exercise policies followed in practice

contain elements of both of these examples, and the MBS approximation can thus either

overstate or understate the true ESO cost.

In the base case, the MBS approximation overstates option cost by 4%, but as Panels

B–D of of Table 7 show, the approximation error varies systematically with firm and option

characteristics. As Table 6 shows, increasing volatility reduces the estimated exercise rate,

and this also increases the option’s expected term. Both the ESO value and its MBS approx-

imation increase with volatility. The ESO value increases more slowly with volatility than

17

the MBS approximation, so the approximation error increases significantly with volatility.

Panel C shows the effect of increasing the dividend rate. This increases the estimated

exercise rate, conditional on the option being vested and in the money, but uniformly reduces

future possible stock prices, and ESO value declines with the dividend rate. The option’s

expected term, conditional on vesting, actually increases slightly, because more stock price

paths stay out of the money longer, but the MBS option value declines even faster than

the true option value, so the error declines in algebraic value. This may be because the

value-maximizing policy calls for some early exercise prior to a dividend payment, and the

estimated empirical exercise policy comes closer to that than the deterministic-time exercise

policy implicit in the MBS approximation.

Panel D illustrates the effect of increasing the vesting period. A longer vesting period in-

creases the risk of pre-vesting forfeiture, which reduces option value. Conditional on vesting,

the option stopping time has less room to vary, so the difference between the option value

and the MBS approximation shrinks.

In the event of a plan modification pursuant to a merger, acquisition, or restructuring of

option plan or terms, firms need to revalue and sometimes replace existing options, and these

valuations affect contracting terms and re-allocations. The right-hand side of Table 7 presents

option values for the out-of-the-money option. The MBS approximation errors are quite

large. Moreover, comparing SAB 110 and MBS errors for the at-the-money and underwater

options shows how sensitive the errors are to the moneyness of the option. This shows that

there is no easy adjustment to Black Scholes to capture employee exercise patterns, and

further highlights the need for a more robust valuation methodology.

5.2 Gender and Rank Effects in Exercise Patterns and Cost

Specifications 3 and 4 of Table 6 summarize the difference in male and female employee

exercise rates with a single dummy variable, showing that men exercise faster than women.

The resulting option cost differential of options held by women over those held by men is

consistently at least 2–3% for almost all of the parameterizations shown in Table 7, except

in the case of a very high dividend rate, where men’s faster exercise increases option value,

or a very long vesting period, where the cost differential is driven to zero as the period of

exercisability shrinks. The cost implications of the positive top-decile effect summarized by

the dummy variable in specifications 5 and 6 of Table 6 are similar.

To further check the robustness of these cost differentials, and investigate the differential

exercise patterns in greater detail, we estimate specifications with a full set of interaction

variables. Table 8 shows how male employee exercise patterns differ from those of female

18

employees. Men are significantly more responsive to Price ≥ 90th percentile of prior year

distribution, but less responsive to Price-to-Strike Ratio, Correlation, Volatility, and vesting

events. Table 9 shows the differential option cost for men and women based on specification 1

in Table 8. For example, the more negative correlation response of female employees makes

the cost differential increase sharply with correlation. For most parameterizations, the cost

differential remains at least 2–3%.

Table 10 shows how top-decile option holder exercise patterns differ from those of lower-

decile holders. Top-decile holders are more responsive to Dividend in next two weeks, Volatil-

ity, and Price ≥ 90th percentile of prior year distribution than lower-decile holders, but are

less responsive to Price-to-Strike Ratio, Correlation, and vesting events. The option cost

differential between top- and lower-decile holders varies with the parameters, but remains

over 2% for most parameterizations.

6 Conclusions

This paper is the first to perform a complete empirical estimation of employee stock option

exercise behavior and option cost to firms. We develop a methodology for estimating option

exercise and cancellation rates as a function of the stock price path, time to expiration,

and firm and option holder characteristics. Our estimation is based on a fractional logistic

approach, and accounts for correlation between exercises by the same executive. Valuation

proceeds by using the estimated exercise function to describe the option’s expected payoff

along each stock price path, and then computing the present value of the payoff. The

estimation of empirical exercise rates also allows us to test the predictions of theoretical

models of option exercise behavior.

We apply our estimation technique to the first both large and broad-based option dataset

analyzed in the literature, consisting of a comprehensive sample of option grant and exercise

data for over 500,000 employees at 104 publicly traded firms from 1981 to 2009. We find that

firm and option holder characteristics affect option exercise patterns and cost in a manner

consistent with both portfolio theory and results from the behavioral finance literature. For

example, exercise rates are lower, and option cost greater, at firms with higher correlation

with the market, and men exercise faster than women, resulting in lower cost. In addition,

top employees exercise faster than lower-ranked employees, again resulting in lower option

cost.

Finally, our results also indicate that Black-Scholes-based approximations used in practice

can lead to significant pricing errors with no simple fixes. Our paper responds to FASB’s call

for more detailed information on employee exercise behavior by providing a comprehensive

19

analysis of employee exercise rates and implications for option cost. The proprietary data

used in this study were provided by corporate participants in a sponsored research project

funded by the Society of Actuaries, who hope that the results of our study will become the

basis for a standard set of exercise assumptions to be used in calculating ESO values on

firms’ income statements.

20

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22

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23

Tab

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an

dri

sk,as

mea

sure

dby

thei

rB

lack

Sch

ole

sva

lue,

for

ou

rsa

mp

leof

104

firm

s.C

onst

ruct

ion

&M

anu

fact

uri

ng

Tra

nsp

ort

ati

on

,R

etail

Fin

an

ce,

Ser

vic

esA

llC

om

mu

nic

ati

on

sIn

sura

nce

,&

Uti

liti

es&

Rea

lE

state

SIC

(on

ed

igit

)1

&2

34

56

78

Pan

elA

.A

ggre

gate

Nu

mb

erof

Em

plo

yees

,G

rants

,an

dE

xer

cise

Eve

nts

Nu

mb

erof

Em

plo

yees

183,

249

114,2

41

11,9

50

35,1

76

116,5

64

53,2

55

6,6

88

521,1

23

Nu

mb

erof

Gra

nts

365,

458

302,7

68

18,7

17

119,8

93

273,9

46

149,1

69

15,2

50

1,2

45,2

01

Nu

mb

erof

Exer

cise

s31

8,95

3155,1

58

5,2

76

97,1

92

207,7

99

97,1

58

9,6

03

891,1

39

Pan

elB

.N

um

ber

ofG

rants

per

Em

plo

yee

Mea

n1.

992.6

51.5

73.4

12.3

52.8

02.2

82.3

9M

edia

n2.

002.0

01.0

02.0

01.0

02.0

01.0

02.0

0S

tan

dar

dD

evia

tion

1.39

3.1

71.4

13.2

12.6

62.8

92.2

02.5

1P

anel

C.

BS

Em

plo

yee

Op

tion

Wea

lth

($K

)M

ean

46.5

6151.9

225.0

0313.6

7268.4

2198.1

083.6

0152.0

8M

edia

n12

.24

7.8

41.4

718.7

413.1

018.5

07.2

312.1

6S

tan

dar

dD

evia

tion

1,41

7.70

1,4

44.6

4264.8

93,1

83.4

44,9

36.0

92,3

36.6

4616.1

72,8

42.6

9P

anel

D.

BS

Em

plo

yee

Op

tion

Ris

k($

K)

Mea

n8.

52158.7

812.0

4172.8

6100.1

2133.4

847.4

082.5

7M

edia

n0.

566.0

11.0

39.1

16.3

59.2

54.5

01.4

6S

tan

dar

dD

evia

tion

407.

411,8

01.3

2130.7

81,9

69.0

91,8

05.0

41,8

47.3

1378.4

61,4

31.9

5

24

Tab

le3:

Sum

mar

ySta

tist

ics

for

Pro

pri

etar

yO

pti

onD

ata:

Gra

nt

Siz

ean

dC

ontr

actu

alF

eatu

res

Th

ista

ble

pro

vid

essu

mm

ary

stat

isti

csw

ith

inin

du

stri

al

gro

up

ings

at

the

on

e-d

igit

SIC

cod

efo

rth

ed

oll

ar

valu

eof

the

un

der

lyin

gsh

are

sp

ergra

nt,

the

max

imu

mnu

mb

erof

vest

ing

per

iod

sp

ergr

ant,

the

per

cent

of

op

tion

sth

at

vest

on

the

firs

tve

stin

gd

ate

per

gra

nt,

the

nu

mb

erof

month

sfr

om

the

gran

td

ate

toth

efu

llves

tin

gof

all

opti

ons

inth

egra

nt,

an

dth

enu

mb

erof

month

sfr

om

the

gra

nt

date

toth

eex

pir

ati

on

date

per

gra

nt

for

the

104

firm

sin

our

sam

ple

.

Con

stru

ctio

n&

Manu

fact

uri

ng

Tra

nsp

ort

ati

on

,R

etail

Fin

an

ce,

Ser

vic

esA

llC

om

mu

nic

ati

on

sIn

sura

nce

,&

Uti

liti

es&

Rea

lE

state

SIC

(on

ed

igit

)1

&2

34

56

78

Pan

elA

.D

olla

rV

alu

eof

Un

der

lyin

gSh

are

sp

erG

rant

($K

)M

ean

24.7

345.9

519.9

852.3

2106.4

161.4

341.3

249.6

8M

edia

n5.

7012.5

03.2

111.5

325.8

526.3

810.9

810.0

5S

tan

dar

dD

evia

tion

191.

20281.2

6193.8

5295.1

8715.9

2466.1

2201.5

2383.9

6P

anel

B.

Max

imu

mN

um

ber

ofV

esti

ng

Per

iod

sp

erG

rant

Mea

n15

.07

8.9

03.7

03.3

02.9

04.8

83.2

65.8

0M

edia

n4.

004.0

04.0

03.0

03.0

04.0

03.0

04.0

0S

tan

dar

dD

evia

tion

17.2

510.6

50.4

61.3

41.1

45.8

61.2

58.1

6P

anel

C.

Per

cent

ofO

pti

ons

that

Ves

ton

the

Fir

stV

esti

ng

Date

(per

Gra

nt)

Mea

n38

.98

28.7

527.4

638.7

448.5

936.9

135.8

737.6

0M

edia

n25

.00

25.0

025.0

033.3

333.3

325.0

033.3

325.0

0S

tan

dar

dD

evia

tion

35.7

822.4

53.7

926.7

729.4

228.4

214.3

727.7

9P

anel

D.

Nu

mb

erof

Mon

ths

from

Gra

nt

toF

ull

Ves

tin

g(p

erG

rant)

Mea

n37

.01

47.7

844.4

440.9

932.9

240.7

139.1

740.5

4M

edia

n36

.00

48.0

048.0

036.0

036.0

048.0

036.0

036.0

0S

tan

dar

dD

evia

tion

13.1

821.9

45.4

814.9

117.9

919.1

015.0

119.5

1P

anel

E.

Nu

mb

erof

Mon

ths

from

Gra

nt

toE

xp

irati

on

(per

Gra

nt)

Mea

n11

5.39

114.6

8120.0

0122.4

3120.1

8120.0

0120.0

0118.5

5M

edia

n12

0.00

120.0

0120.0

0120.0

0120.0

0120.0

0120.0

0120.0

0S

tan

dar

dD

evia

tion

16.4

812.9

50.0

011.8

10.3

80.0

00.0

09.8

8

25

Tab

le4:

Sum

mar

ySta

tist

ics

for

Pro

pri

etar

yO

pti

onD

ata:

Exer

cise

Pat

tern

s

Th

ista

ble

pro

vid

essu

mm

ary

stat

isti

csw

ith

inin

du

stri

al

gro

up

ings

at

the

on

e-d

igit

SIC

cod

efo

rth

eth

enu

mb

erof

day

sre

main

ing

toex

pir

ati

on

per

exer

cise

even

t,th

enu

mb

erof

day

sth

eop

tion

was

in-t

he-

mon

eyfr

om

the

vest

ing

date

toth

eex

erci

sed

ate

per

exer

cise

even

t,th

era

tio

of

the

stock

pri

ceto

the

stri

ke

pri

ceat

the

exer

cise

dat

ep

erex

erci

seev

ent,

an

ind

icato

rva

riab

lefo

rw

het

her

the

stock

pri

ceat

exer

cise

was

gre

ate

rth

an

or

equ

al

the

nin

etie

thp

erce

nti

leof

the

pri

orye

ard

istr

ibu

tion,

an

dth

efr

act

ion

of

the

gra

nt’

sve

sted

op

tion

sth

at

are

exer

cise

dp

erex

erci

seev

ent

for

the

104

firm

sin

our

sam

ple

.

Con

stru

ctio

n&

Manu

fact

uri

ng

Tra

nsp

ort

ati

on

,R

etail

Fin

an

ce,

Ser

vic

esA

llC

om

mu

nic

ati

on

sIn

sura

nce

,&

Uti

liti

es&

Rea

lE

state

SIC

(on

ed

igit

)1

&2

34

56

78

Pan

elA

.N

um

ber

ofD

ays

Rem

ain

ing

toE

xp

irati

on

on

Exer

cise

Date

(per

Exer

cise

Even

t)M

ean

1,23

2.97

1,3

96.4

01,9

84.6

21,4

03.9

41,5

36.0

41,8

11.3

42,2

43.3

11,5

48.3

9M

edia

n1,

261.

531,3

82.0

12,2

34.5

11,3

06.5

51,5

15.3

41,8

75.2

62,3

95.3

11,5

11.5

9S

tan

dar

dD

evia

tion

614.

17676.9

0868.7

5792.3

5613.4

7885.4

8729.9

2777.0

4P

anel

B.

Nu

mb

erof

Day

sth

eO

pti

onw

as

In-t

he-

Mon

eyfr

om

Ves

tin

gD

ate

toE

xer

cise

Date

(per

Exer

cise

Eve

nt)

Mea

n32

3.45

118.4

8261.6

5126.0

5267.5

1217.7

5175.5

8206.2

3M

edia

n17

3.50

79.6

780.3

795.0

3155.6

5139.4

4113.7

5113.0

0S

tan

dar

dD

evia

tion

406.

29138.1

1536.8

3119.9

5301.6

3233.3

1215.1

3281.4

5P

anel

C.

Rat

ioof

Sto

ckP

rice

toS

trik

eP

rice

at

Exer

cise

Date

(per

Exer

cise

Eve

nt)

Mea

n2.

962.5

61.7

03.9

66.3

05.2

84.9

74.1

9M

edia

n2.

842.0

91.6

72.6

45.0

63.4

22.2

62.7

2S

tan

dar

dD

evia

tion

1.56

2.4

30.6

74.4

36.0

96.4

08.1

55.0

5P

anel

D.

Wh

eth

erS

tock

Pri

ce≥

90th

Per

centi

leof

Pri

or

Yea

rD

istr

ibu

tion

(per

Exer

cise

Eve

nt)

Mea

n0.

380.3

20.4

90.3

20.3

30.3

50.4

50.3

5M

edia

n0.

280.2

80.5

00.2

70.3

10.2

90.4

30.3

0S

tan

dar

dD

evia

tion

0.32

0.2

20.3

50.2

30.2

30.2

70.2

80.2

7P

anel

E.

Fra

ctio

nof

Gra

nt’

sV

este

dO

pti

on

sth

at

are

Exer

cise

d(p

erE

xer

cise

Even

t)M

ean

0.78

0.6

50.8

90.6

40.6

70.7

10.8

90.7

1M

edia

n0.

890.6

71.0

00.6

50.6

80.8

31.0

00.7

6S

tan

dar

dD

evia

tion

0.25

0.2

80.2

10.2

80.2

30.3

10.2

00.2

8

26

Table 5: Employee Characteristics

This table presents summary statistics for the demographic information that is reported by a subset of the

firms in the sample. We summarize the information by employees over the sample period.

Number Mean Median Standardof Employees Deviation

Age 248,448 41.77 41.25 10.11Male 208,598 0.54 1.00 0.50Salary 55,729 283,785.41 38,051.45 3,533,343.90

27

Tab

le6:

Est

imat

ion

Res

ult

s

Th

ista

ble

pre

sents

coeffi

cien

tes

tim

ates

and

stan

dard

erro

rs(i

npare

nth

eses

)fo

ralt

ern

ati

vesp

ecifi

cati

on

sof

the

fract

ion

al-

logis

tic

esti

mato

r.S

pec

ifi-

cati

ons

1,2,

5,6,

7,an

d8

are

esti

mat

edu

sin

gth

efu

llsa

mp

leof

104

firm

s.S

pec

ifica

tion

s3

an

d4

are

esti

mate

du

sin

gth

esu

bsa

mp

leof

69

firm

sth

at

rep

orte

din

form

atio

non

gen

der

and

age.

Sp

ecifi

cati

on

s9

an

d10

are

esti

mate

du

sin

gth

esu

bsa

mp

leof

9firm

sth

at

rep

ort

edsa

lary

data

.In

du

stry

clas

sifi

cati

onis

bas

edon

one-

dig

itS

IC.

Th

ees

tim

ato

rcl

ust

ers

at

the

leve

lof

the

ind

ivid

ual

emp

loyee

.

Alt

ern

ati

veS

pec

ifica

tion

s1

23

45

67

89

10

Cov

aria

tes

Con

stan

t-5

.871

8-4

.9232

-5.9

913

-6.3

018

-5.9

610

-4.9

921

-6.0

209

-5.1

736

-6.2

088

-5.1

596

(0.0

086)

(0.0

096)

(0.0

209)

(0.0

272)

(0.0

089)

(0.0

101)

(0.0

194)

(0.0

081)

(0.0

268)

(0.0

333)

Pri

ce-t

o-st

rike

rati

o0.

0059

0.0

064

0.0

162

0.0

159

0.0

057

0.0

063

0.0

051

0.0

032

0.1

041

0.1

086

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

001)

(0.0

002)

(0.0

001)

(0.0

006)

(0.0

006)

Div

iden

din

nex

ttw

ow

eeks

1.84

743.2

360

7.7

375

6.0

740

1.8

534

3.2

046

0.0

626

-0.0

819

0.2

189

5.3

747

(0.0

700)

(0.0

797)

(0.9

302)

(1.0

643)

(0.0

720)

(0.0

807)

(0.0

909)

(0.1

4951)

(1.9

225)

(2.2

071)

Cor

rela

tion

-0.9

591

-0.6

616

-0.6

305

-0.6

323

-0.9

119

-0.6

353

-0.7

219

-1.2

190

-1.8

081

-1.0

099

(0.0

099)

(0.0

103)

(0.0

152)

(0.0

158)

(0.0

100)

(0.0

104)

(0.0

116)

(0.0

070)

(0.0

319)

(0.0

414)

Vol

atil

ity

-0.6

697

-0.0

829

-0.6

576

-0.5

103

-0.7

109

-0.8

575

-0.0

825

0.0

694

-1.1

009

-1.2

239

(0.0

106)

(0.0

1202)

(0.0

197)

(0.0

205)

(0.0

107)

(0.0

1203)

(0.0

151)

(0.0

068)

(0.0

421)

(0.0

508)

Tim

eto

exp

irat

ion

-0.1

039

-0.0

973

-0.1

364

-0.1

375

-0.1

000

0.0

945

-0.0

631

-0.0

778

0.0

252

0.0

054

(0.0

009)

(0.0

009)

(0.0

015)

(0.0

015)

(0.0

009)

(0.0

009)

(0.0

008)

(0.0

007)

(0.0

028)

(0.0

029)

Pri

ce≥

90th

per

centi

leof

pri

oryea

rd

istr

ibu

tion

0.35

590.3

639

0.3

716

0.3

661

0.3

554

0.3

635

0.4

619

0.4

613

0.4

201

0.3

226

(0.0

042)

(0.0

042)

(0.0

058)

(0.0

058)

(0.0

042)

(0.0

042)

(0.0

031)

(0.0

030)

(0.0

107)

(0.0

116)

Ves

tin

gev

ent

inp

ast

two

wee

ks

2.94

522.8

197

3.3

738

3.4

023

2.9

438

2.8

208

2.6

325

2.5

866

2.2

608

2.3

023

(0.0

053)

(0.0

055)

(0.0

072)

(0.0

072)

(0.0

053)

(0.0

055)

(0.0

043)

(0.0

041)

(0.0

157)

(0.0

168)

Age

-0.0

048

-0.0

048

(0.0

003)

(0.0

003)

Mal

e0.1

858

0.2

057

(0.0

059)

(0.0

061)

Top

-dec

ile

opti

onh

old

er0.1

838

0.1

222

(0.0

046)

(0.0

047)

BS

Em

plo

yee

Op

tion

Ris

k($

M)

-0.0

672

0.0

652

0.0

066

-0.0

045

(0.0

647)

(0.0

041)

(0.0

009)

(0.0

028)

BS

Em

plo

yee

Op

tion

Wea

lth

($M

)-0

.1655

-0.0

817

(0.0

556)

(0.0

065)

Sal

ary

($M

)0.0

093

-0.0

068

(0.0

001)

(0.0

020)

Ind

ust

ryfi

xed

effec

tsN

oY

esN

oY

esN

oY

esN

oY

esN

oY

esN

um

ber

ofob

serv

atio

ns

(em

plo

yee-

gran

t-d

ays)

363M

363M

216M

216M

363M

363M

363M

363M

63M

63M

28

Tab

le7:

Val

uat

ion

Res

ult

s

Th

eta

ble

show

sop

tion

valu

esb

ased

onth

ees

tim

ate

dex

erci

sefu

nct

ion

sin

Table

6.

Pan

els

A,

D,

E,

an

dF

are

base

don

Sp

ecifi

cati

on

1,

Pan

elB

isb

ased

onS

pec

ifica

tion

3,an

dP

anel

Cis

bas

edon

Sp

ecifi

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29

Table 8: Estimation Results with Controls for Employee Gender

This table presents coefficient estimates and standard errors (in parentheses) for alternative specifications of

the fractional logistic estimator that include controls for the gender of the employee. Specification 2 includes

controls for industry fixed effects. The estimator clusters at the level of the individual employee.

Alternative Specifications1 2

Covariates

Constant -5.7275 -6.0242(0.0315) (0.0362)

Price-to-strike ratio 0.0254 0.0245(0.0002) (0.0002)

Dividend in next two weeks 10.4642 9.3907(1.6464) (1.6873)

Correlation -1.0854 -1.0688(0.0238) (0.0240)

Volatility -0.9329 -0.8917(0.5852) (0.0380)

Time to expiration -0.1326 -0.1326(0.0025) (0.0025)

Price ≥ 90th percentileof prior year distribution 0.3508 0.3454

(0.0092) (0.0092)Vesting event in past two weeks 3.4645 3.4895

(0.0119) (0.0118)Age -0.0055 -0.0056

(0.0005) (-0.2191)Male × Constant -0.2411 -0.2191

(0.0414) (0.0416)Male × Price-to-strike ratio -0.0112 -0.0107

(0.0002) (0.0002)Male × Dividend in next two weeks -2.8935 -3.7979

(1.9700) (2.1531)Male × Correlation 0.7641 0.7247

(0.0308) (0.0307)Male × Volatility 0.3442 0.4886

(0.0436) (0.0435)Male × Time to Expiration -0.00201 -0.00686

(0.0031) (0.0031)Male × Price ≥ 90th percentileof prior year distribution 0.0321 0.0318

(0.0118) (0.0118)Male × Vesting event in past two weeks -0.1389 -0.1346

(0.0149) (0.0148)Male × Age 0.0008 0.0008

(0.0006) (0.0006)Industry fixed effects No YesNumber of observations 216,347,669 216,347,669

30

Table 9: Valuation Results by Employee Gender

The table shows option values based specification 1 in Table 8. The left side of the table shows valuations

for a 10-year at-the-money option with a $100 strike price. The right side shows the case of an 8-year option

that is 40% out of the money. The base case assumes 8% annual termination rate, two-year vesting period,

30% volatility, zero dividend rate, and correlation of 0.40. The riskless rate is 5% and the expected market

return is 11%.

At-the-Money Option Underwater OptionChanging Male Female Pct Diff Male Female Pct DiffParameterPanel A. Correlation Effects0.00 30.48 29.70 -2.6 9.48 9.26 -2.40.20 30.79 30.76 -0.1 9.61 9.69 0.90.40 31.09 31.76 2.2 9.73 10.09 3.60.60 31.38 32.69 4.2 9.85 10.44 6.00.80 31.67 33.53 5.9 9.97 10.75 7.9Panel B. Volatility Effects0.10 18.75 19.17 2.3 1.84 1.91 3.40.20 24.69 25.23 2.2 5.50 5.70 3.60.30 31.09 31.76 2.2 9.73 10.09 3.60.40 37.42 38.24 2.2 14.16 14.68 3.70.50 43.48 44.43 2.2 18.58 19.26 3.7Panel C. Dividend Rate Effects0.00 31.09 31.76 2.2 9.73 10.09 3.60.01 28.02 28.52 1.8 8.52 8.79 3.20.03 22.65 22.87 1.0 6.46 6.61 2.40.05 18.16 18.20 0.2 4.83 4.91 1.60.07 14.47 14.39 -0.5 3.57 3.60 0.7Panel D. Vesting Period Effects0.00 30.85 31.86 3.3 9.73 10.09 3.61.00 30.85 31.69 2.7 9.73 10.09 3.62.00 31.09 31.76 2.2 9.73 10.09 3.63.00 30.79 31.33 1.7 9.58 9.93 3.64.00 30.16 30.58 1.4 9.58 9.88 3.2

31

Table 10: Estimation Results with Controls for Rank of Option Holder

This table presents coefficient estimates and standard errors (in parentheses) for alternative specifications of

the fractional logistic estimator that include controls for whether the employee’s option holdings, measured

by their Black Scholes value, are in the top decile of employee holdings at the firm. Specification 2 includes

controls for industry fixed effects. The estimator clusters at the level of the individual employee.

Alternative Specifications1 2

Covariates

Constant -5.8022 -4.8214(0.0010) (0.0013)

Price-to-strike ratio 0.0009 0.0009(0.000006) (0.000006)

Dividend in next two weeks 1.8068 3.0895(0.0006) (0.0075)

Correlation -1.2053 -0.0913(0.0011) (0.0012)

Volatility -0.0631 -0.0785(0.0013) (0.0015)

Time to expiration -0.0110 -0.0108(0.0001) (0.0001)

Price ≥ 90th percentileof prior year distribution 0.0342 0.035

(0.0005) (0.0005)Vesting event in past two weeks 2.9977 2.8681

(0.0006) (0.0007)Top-decile option holder × Constant -0.0338 -0.0444

(0.0019) (0.0019)Top-decile option holder × Price-to-strike ratio -0.0005 -0.0006

(0.00001) (0.00001)Top-decile option holder × Dividend in next two weeks 0.088 1.3401

(0.0282) (0.0246)Top-decile option holder × Correlation 0.0954 0.091

(0.0022) (0.0023)Top-decile option holder × Volatility -0.2108 -0.1735

(0.0022) (0.0023)Top-decile option holder × Time to Expiration 0.0035 0.0042

(0.0002) (0.0002)Top-decile option holder × Price ≥ 90th percentileof prior year distribution 0.0037 0.0039

(0.0009) (0.0009)Top-decile option holder × Vesting event in past two weeks -0.0174 -0.0155

(0.0011) (0.0011)Industry fixed effects No YesNumber of observations 216,347,669 216,347,669

32

Table 11: Valuation Results by Rank of Option Holder

The table shows option values based specification 1 in Table 10. The left side of the table shows valuations

for a 10-year at-the-money option with a $100 strike price. The right side shows the case of an 8-year option

that is 40% out of the money. The base case assumes 8% annual termination rate, two-year vesting period,

30% volatility, zero dividend rate, and correlation of 0.40. The riskless rate is 5% and the expected market

return is 11%. Top-decile option holders are those whose option holdings, measured by their Black Scholes

value, are in the top decile of employee option holdings at the firm.

At-the-Money Option Underwater OptionChanging Top Decile Lower Decile Pct Diff Top Decile Lower Decile Pct DiffParameterPanel A. Correlation Effects0.00 29.47 28.42 -3.6 9.04 8.52 -5.80.20 29.71 29.61 -0.3 9.15 9.05 -1.00.40 29.96 30.77 2.7 9.26 9.56 3.20.60 30.21 31.87 5.5 9.36 10.01 6.90.80 30.45 32.87 8.0 9.46 10.41 10.0Panel B. Volatility Effects0.10 17.30 18.38 6.2 1.75 1.81 3.80.20 23.45 24.36 3.9 5.20 5.39 3.80.30 29.96 30.77 2.7 9.26 9.56 3.20.40 36.41 37.12 2.0 13.57 13.93 2.60.50 42.60 43.21 1.4 17.92 18.29 2.1Panel C. Dividend Rate Effects0.00 29.96 30.77 2.7 9.26 9.56 3.20.01 27.19 27.81 2.3 8.14 8.38 3.00.03 22.25 22.56 1.4 6.23 6.38 2.40.05 18.07 18.16 0.5 4.71 4.79 1.80.07 14.56 14.50 -0.4 3.52 3.55 1.1Panel D. Vesting Period Effects0.00 28.31 29.96 5.8 9.26 9.56 3.21.00 29.15 30.36 4.1 9.26 9.56 3.22.00 29.96 30.77 2.7 9.26 9.56 3.23.00 30.09 30.62 1.8 9.15 9.44 3.24.00 29.77 30.09 1.1 9.30 9.52 2.3

33