Timing of Employee Stock Options Exercises and Costs of Stock Options Grants Chris Armstrong, Alan Jagolinzer, and David Larker http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470471921.html
49
Electronic copy available at: http://ssrn.com/abstract=905280 Electronic copy available at: http://ssrn.com/abstract=905280 Timing of Employee Stock Option Exercises and the Cost of Stock Option Grants Christopher S. Armstrong Alan D. Jagolinzer David F. Larcker Stanford University Graduate School of Business 518 Memorial Way Stanford, CA 94305-5015 Revised June 1, 2007 Abstract: This study examines how executives’ and lower-level employees’ option exercise behavior affects firms’ stock option grant cost estimates. Prior research suggests that option grant cost estimates are not materially different when calculated a using utility-based model or a risk- neutral model adjusted for historical exercise rates. This study shows, however, that estimates of exercise times are significantly improved when the model accounts for behavioral and economic determinants of option exercise such as the attainment of performance benchmarks, recent vesting, the intrinsic value of an employee’s option portfolio, and employee rank. Hazard analysis of all executive and employee option grants within a proprietary sample of firms yields lower out-of-sample exercise timing prediction errors relative to utility-based models and estimates using historical exercise patterns. More importantly, option cost estimates are materially different when improved estimates of exercise times are used, which may have implications for financial reporting. JEL Classification: C34, C41, J33, M52 Keywords: Option expense; employee option exercise; SFAS 123; stock option cost; survival analysis We would like to thank Terry Adamson, James Lecher, and Philip Peterson (Aon Consulting) and Sean Scrol (Valtrinsic LLC) for their considerable help with this project. We also thank David Aboody, Mary Barth, William Beaver, John Core, George Foster, Ian Gow, Chip Heath, Leslie Hodder, and workshop participants at Indiana University, the Pennsylvania State University 2006 Accounting Research Conference, and the Stanford University 2006 Summer Accounting Research Camp for valuable comments.
Transcript
1. Timing of Employee Stock Option Exercises and the Cost of
Stock Option Grants Christopher S. Armstrong Alan D. Jagolinzer
David F. Larcker Stanford University Graduate School of Business
518 Memorial Way Stanford, CA 94305-5015 Revised June 1,
2007Abstract: This study examines how executives and lower-level
employees option exercisebehavior affects firms stock option grant
cost estimates. Prior research suggests that option grantcost
estimates are not materially different when calculated a using
utility-based model or a risk-neutral model adjusted for historical
exercise rates. This study shows, however, that estimates
ofexercise times are significantly improved when the model accounts
for behavioral and economicdeterminants of option exercise such as
the attainment of performance benchmarks, recentvesting, the
intrinsic value of an employees option portfolio, and employee
rank. Hazardanalysis of all executive and employee option grants
within a proprietary sample of firms yieldslower out-of-sample
exercise timing prediction errors relative to utility-based models
andestimates using historical exercise patterns. More importantly,
option cost estimates arematerially different when improved
estimates of exercise times are used, which may haveimplications
for financial reporting.JEL Classification: C34, C41, J33,
M52Keywords: Option expense; employee option exercise; SFAS 123;
stock option cost; survivalanalysisWe would like to thank Terry
Adamson, James Lecher, and Philip Peterson (Aon Consulting) andSean
Scrol (Valtrinsic LLC) for their considerable help with this
project. We also thank DavidAboody, Mary Barth, William Beaver,
John Core, George Foster, Ian Gow, Chip Heath, LeslieHodder, and
workshop participants at Indiana University, the Pennsylvania State
University 2006Accounting Research Conference, and the Stanford
University 2006 Summer AccountingResearch Camp for valuable
comments. Electronic copy available at:
http://ssrn.com/abstract=905280
2. Timing of Employee Stock Option Exercises and the Cost of
Stock Option Grants1. Introduction This study examines the degree
to which executive and employee stock optionexercise behavior
affects the estimated cost of employee stock options to the
grantingfirm. Statement of Financial Accounting Standards 123
revised, 2004, Share-basedPayment, (SFAS 123R) requires firms to
recognize an expense for the fair value of theiremployee stock
option grants using an appropriate valuation technique. A key input
toall valuation models is the anticipated timing of option
exercise. 1 Although priorliterature consistently shows that
employees exercise their options before expiry, it isdifficult to
predict exactly when employees exercise their options, to what
degree thisearly exercise affects the option cost to the granting
firm, and how to accurately measurethe cost of the option grant
conditional on the expected exercise timing. Prior research
suggests that the Black-Scholes-Merton model, modified for
earlyexercise, provides reliable estimates of employee stock option
cost (e.g., Carpenter, 1998;Marquardt, 2002; and Bettis et al.,
2005). This evidence has, in part, contributed toregulatory
guidance that outlines acceptable financial reporting methods for
stock optionexpense. 2 These prior studies, however, rely on
estimates of exercise times that do notaccount for many of the
factors known to influence exercise behavior documented inprior
literature (e.g., Heath et al., 1999; Huddart and Lang, 1996,
2003). If an inaccurate1 SFAS 123R, paragraph A14.2 SEC Office of
Economic Analysis Memorandum, Economic Perspective on Employee
Option Expensing:Valuation and Implementation of FAS 123 (R), March
18, 2005. 1 Electronic copy available at:
http://ssrn.com/abstract=905280
3. exercise time is used, then the estimated costs from these
models will likely beinaccurate. This study extends prior research
by developing a firm-specific hazard (or survival)model to estimate
the rate at which executives and employees exercise their
optiongrants, as a function of factors expected to influence
employees exercise decisions.Hazard rates of option exercise are
estimated using proprietary data with detailed grantand exercise
history for all executives and employees at ten publicly-traded
firms. Thiscomprehensive data allows us to examine the factors
associated with early-exercise at agreater organizational depth
than prior studies that focus primarily on senior executive-level
cost estimates. In addition, hazard estimation properly accounts
for the inherentright censoring of the data (since we typically
cannot observe the ultimate outcome forthe more recently granted
options) and provides unbiased estimates of the rate of stockoption
exercise. We find that the rate of option exercise is inherently
price-path dependent and isassociated with a complex mix of
economic and behavioral factors that are unlikely to beaccurately
captured by either a parsimonious utility-based model or a simple
estimate ofhistorical exercise times which is commonly used in
practice. Specifically, we find thatthe rate of option exercise is
associated with prior stock price performance, price levelsrelative
to cognitive benchmarks, and the intrinsic value of an employees
optionportfolio. This is consistent with Heath et al. (1999) and
Huddart and Lang (1996, 2003),who analyze similar grant and
exercise data and find that option exercise patterns areinherently
price-path dependent. We also find that the rate of option exercise
is 2
4. associated with other employee-specific factors that may not
be captured by utility-basedmodels, such as the recent vesting of
options and expected voluntary termination. In addition to
providing evidence regarding factors associated with early
exercise,hazard analysis also provides employee-specific
predictions of option exercise timeswhich can be used as inputs to
an option cost model. If hazard models that account forthe factors
associated option exercise produce more accurate estimates of
exercise timesthan utility-based models or risk-neutral models
adjusted for historical early-exercisetimes, then it might be
possible to improve option cost estimates for financial reporting.
3We therefore assess the hazard models predictive accuracy, by
first computing exercisetime estimates from the hazard model, a
utility-based model, and historical exercise rates.We then compare
these exercise time estimates to realized exercise times in a
holdoutsample. For 67% of the sample firms, the hazard model
results in lower out-of-sampleprediction errors than either
competing model. This suggests that hazard estimation thataccounts
for behavioral and economic factors generally improves the accuracy
ofpredicted exercise times. Regarding potential implications for
financial reporting, Carpenter (1998) and Bettiset al. (2005) do
not report a material difference between cost estimates derived
from autility-based binomial option pricing model and cost
estimates derived from a risk-neutralBlack-Scholes option pricing
model that adjusts for historical exercise rates. Thesefindings
leave open the question of whether exercise timing inputs
materially affectoption cost estimates. We examine this further by
assessing the degree to which option3 The SEC (SAB 107) and the
FASB (SFAS 123R, para. A29) look, in part, to academic research
toprovide guidance for estimating the expected option holding term
as an input to option valuation forfinancial reporting.
Utility-based and risk-neutral models have been studied in prior
literature and risk-neutral models are commonly used by firms for
financial reporting option grant valuation. 3
5. cost estimates differ when adjusted for more accurate
exercise time predictions. MonteCarlo simulation results show that
option grant cost estimates that incorporate expectedexercise times
derived from a hazard model are substantially different from cost
estimatesderived from either a utility-based model or a
risk-neutral model adjusted for historicalexercise rates.
Specifically, hazard exercise timing option cost estimates are, on
average19% and 27% different from utility-based cost estimates and
adjusted Black-Scholesestimates, respectively. We also find that
option grant cost estimates derived from autility-based model are
substantially different from cost estimates derived from a
risk-neutral model adjusted for historical exercise rates.
Specifically, utility-based costestimates are, on average, 26%
different from adjusted Black Scholes estimates. Theevidence
indicates that accurate estimates of exercise time materially
affect option grantcost estimates, and this has implications for
financial reporting. The remainder of the paper is composed of six
sections. Section 2 provides a reviewof the existing literature on
employee stock option exercise behavior. Section 3 describesthe
sample of firms included in the analysis. Section 4 develops the
specification of thehazard model of option exercise and presents
the company-specific estimates of the rateof option exercise
derived from this model. Section 5 presents the results of an
out-of-sample validation of the estimated exercise times for
alternative option pricing models.Section 6 presents estimates of
the cost of a hypothetical option grant for each of thecompanies in
our sample using the hazard model of option exercise and
alternativeemployee stock option pricing models. Section 7 provides
a summary of our evidence, adiscussion of its limitations, and a
discussion of potential future research. 4
6. 2. Background Prior research consistently shows that
employees exercise their options before expiry(e.g., Huddart, 1994;
Heath et al., 1999; Huddart and Lang, 2003; Bettis et al., 2005),
andSFAS 123R expressly recognizes that this empirical regularity
has implications fordetermining the cost of a stock option grant.
There has been some ambiguity, however,regarding how option
valuation models should account for early exercise whencomputing
the cost of an option grant. Some valuation models specify expected
exerciseas a function of an employees assumed utility function
(e.g., Huddart, 1994; Kulatilakaand Marcus, 1994; Carpenter, 1998;
Bettis et al., 2005), an exogenous stopping rateevent that triggers
exercise or forfeiture (e.g., Jennergren and Naslund, 1993;
Carpenter,1998), or a predetermined price-to-strike multiple that
automatically triggers exercise(e.g., Hall and Murphy, 2002 and
Hull and White, 2004). Some models also adjust theoption life
downwards in an American option pricing model to reflect historical
earlyexercise rates (e.g., Bettis et al., 2005) or to reflect
exogenous stopping rates (e.g.,Carpenter, 1998). In its guidance,
the SEC cites results from Carpenter (1998),Marquardt (2002), and
Bettis et al. (2005) which conclude that American option
pricingmodels, adjusted for early exercise, provide a reliable
estimate of the cost of an optiongrant. 4 The reliability of option
pricing models that rely on an unconditional estimate ofexercise
times, however, is uncertain given that many of the factors found
to beassociated with employees early exercise decisions are not
included in these models.For example, Huddart and Lang, (1996),
Heath et al. (1999), Huddart and Lang (2003),4 SEC Office of
Economic Analysis Memorandum, Economic Perspective on Employee
Option Expensing:Valuation and Implementation of FAS 123 (R), March
18, 2005. 5
7. and Bettis et al. (2005) provide evidence that an employees
exercise decision is acomplicated function of both behavioral and
economic factors, including several factorsthat depend on the
price-path of the underlying stock. In particular, Heath et al.
(1999)find that option exercise is more likely following positive
stock price performance andwhen the underlying stock price exceeds
certain cognitive thresholds (e.g., the prior 52-week high). There
is also evidence that option exercise rates vary across employee
rank(Huddart and Lang, 1996), following the recent vesting of
options from the option grant(Heath et al., 1999), in the
volatility of returns of the underlying stock (Huddart andLang,
1996; Hemmer et al., 1996; Bettis et al., 2005), and in future
stock priceperformance (Huddart and Lang, 2003). Therefore, it is
important to account for thesefactors when estimating the timing of
early exercise because this presumably impacts theaccuracy of cost
estimates produced by stock option valuation models. This
studyexamines this directly by assessing whether accounting for
these factors improvesexercise time prediction accuracy and
materially affects option grant cost estimates.3. Sample Our
proprietary stock option data were gathered from ten
publicly-traded firms. Foreach company, we obtained a file with
dates and the number of options that were granted,exercised, and
cancelled for every employee who received options during the
periodcovered by the data file. The dataset also contains
information about the strike price,term, and vesting schedule for
each of the option grants. We also know whether canceledoptions
were due to voluntary or involuntary termination from the firm at
any time during 6
8. the sample period. 5 For nine of the companies, we also know
the age and gender of eachemployee. All of the sample companies
voluntarily agreed to participate in this study. Eachcompany
collected and made this data available to a large actuarial and
benefitsconsulting company to develop cost estimates for employee
stock options to satisfy therequirements of SFAS 123R. Our sample
consists of relatively small firms (measured byaverage market
capitalization and number of employees over the sample period) from
adiverse set of industries with some concentration in the
technology sector (Table 1, PanelA). The typical firm has data for
10 to 15 years ending in mid-2005 (Table 1, Panel B).Each firm has
positive revenue growth, most firms have positive net income, and
eachfirm has been publicly traded between six and 33 years.
Although we have no reason tosuspect any confounding selection bias
in our sample, the firms are a non-random groupthat self-selected
into our study. Thus, generalizing our results to a broader
populationshould take this into account. 6 Table 1 (Panel B)
provides descriptive statistics for the individual stock option
plans.The typical employee stock option grant has a ten year life,
is granted with an exerciseprice equal to the fair market value at
the date of grant (i.e., at-the-money), and vests intranches over
three to five years. A large percentage of employees receive and
exerciseoptions within each firm. In some cases, the number of
employees receiving grants5 Specifically, the option cancellation
information provides the number and date on which an
employeesunvested options were cancelled in connection with
termination. We infer the employees termination datefrom this
information. Since only unvested options are cancelled, however, we
are unable to infer thetermination of an employee who holds only
vested options. Since most middle- and senior-level
managersperiodically receive new option grants, this is likely to
be a problem only for lower-level employees.6 Due to the paucity of
employee grant and exercise data available, literature in this area
draws primaryinferences from small sample analyses. Huddart and
Lang (1996), for example, draw inferences from dataprovided by one
private and seven public firms. Similarly, Carpenter (1998) draws
inferences from dataprovided by forty firms. . 7
9. exceeds the average (mean) number of employees. This occurs
because of employeeturnover and subsequent new employee hiring
during the sample period.4. Analysis of Employee Exercise Behavior
In order to determine the affect of anticipated exercise timing on
the cost of an optiongrant, we first present estimates from a
hazard model of the rate of option exercise. Thisestimation
provides estimates of the exercise time which are later used as
inputs tocalculate the cost of option grants. We then validate the
exercise time estimates byassessing out-of-sample prediction
accuracy relative to exercise time estimates impliedby alternative
option pricing models. Finally, we use the hazard model estimates
as aninput for a Monte Carlo simulation of the cost of an option
grant to assess the affect ofconditional exercise timing estimates
relative to alternative models of option cost.4.1. Hazard Analysis
Estimation Hazard analysis allows us to estimate the rate of option
exercise while avoiding a biasin the estimates that would otherwise
result from the right censored nature of the data. 7, 8Models for
survival data specify a probability density function, f(t), for the
length of timeuntil an event occurs. If the model is estimated in
continuous time (as is the case in our7 Although logit estimation
addresses censored observations, it ignores the role of time in the
analysis, andthus precludes the proper inclusion of time-varying
covariates. A key distinction between hazard and
logitspecifications is that the former method is concerned with the
instantaneous (conditional) rate of the eventwhile the latter
method is concerned with estimating the odds ratio (i.e., the ratio
of the probability of anevent to the probability of a nonevent).8
We observe the option grant date for all observations, however,
there are still many options that areneither cancelled, expired nor
exercised as of the last date for which we have available data.
Thus, manyobservations (i.e., option grants) in our datasets are
right censored because we do not observe theultimate outcome for
every observation. Survival analysis provides unbiased estimates of
the rate of optionexercise, if the source of right censoring is
independent of future, unobserved values of the hazard (alsocalled
noninformative censoring). Noninformative censoring is a reasonable
assumption for studies thatterminate on a pre-defined date which is
the case for our data. We therefore maintain this assumption
ofnoninformative right-censoring throughout our analysis. 8
10. analysis), the resulting hazard function provides the
instantaneous probability that theevent of interest will occur
within a given interval of time (e.g., the exercise of an optionon
a specific day), given that the event has not yet occurred. Hazard
models can be classified as either parametric or non-parametric,
depending onwhether a functional form is assumed for the baseline
hazard function. Since we use thecompany-specific estimates of the
hazard function to simulate employee exercise eventsin subsequent
analyses, we require an estimate of the baseline hazard. Thus, we
adopt aparametric specification of the hazard in our analysis. 9
Specifically, we adopt theWeibull model which specifies the
following hazard rate: h(t) = pt p-1exp{0 + 1x1 + + nxn} (1)The
baseline hazard is pt p-1exp{0} where p is the shape parameter and
0 is the scaleparameter which dictates the relative magnitude of
the baseline hazard. The shapeparameter p indicates whether the
hazard is monotone decreasing (p < 1), monotoneincreasing (p
> 1), or constant (p = 1). 10 Due to its flexibility, the
Weibull model is thepredominant parametric specification adopted in
applied survival analysis. Equation (1) shows that the Weibull
model is multiplicatively separable into twocomponents consisting
of the baseline hazard (i.e., pt p-1exp{0}) and the relative
hazard(i.e., exp{1x1 + kxk}). The former is solely a function of
time and it provides thehazard rate in the absence of covariates
(or when all the covariates are equal to zero),9 We have verified
that our results are robust to the use of a parametric hazard
function by also estimatingour specifications using the Cox (1972)
proportional hazards model, which is the primary
nonparametricalternative model used for survival analysis. The Cox
model is similar to the parametric model, but thebaseline hazard is
not estimated. The main advantage of a parametric model relative to
a nonparametricmodel is the gain in efficiency if the correct form
of the baseline hazard is known. However,misspecification of the
baseline hazard can produce biased estimates, and thus a
non-parametric model hasthe virtue of robustness.10 The exponential
model is a special case of the Weibull model with a constant hazard
(i.e., p = 1). 9
11. while the latter is a function of covariates other than
time. A key feature of the Weibull(or any other proportional
hazard) model is that time is separated from the
explanatorycovariates so the overall hazard is obtained by shifting
the baseline hazard according tothe relative hazard. Thus, the
Weibull model assumes the baseline hazard is the same forall
subjects and the effect of the covariates in the model is to
multiplicatively shift thebaseline hazard (operating through the
relative hazard). We expect heterogeneity in exercise behavior
across the firms in our sample becauseof differences in the firms
contracting environments. It is likely that firms selectrelatively
homogeneous employees (e.g., similar degrees of risk-aversion)
which shouldmanifest in similarities in their exercise behavior. In
addition, all of the employees of afirm face the same price path,
similar firm-level policies (e.g., stock option trainingprograms
and blackout windows) and similar information environments (e.g.,
knowledgeof the firms investment opportunity set, risks, and level
of competition) which shouldaffect their exercise decisions. Within
a firm, however, there is potential forheterogeneity in exercise
behavior across employee ranks based on evidence fromHuddart and
Lang (1996). Therefore, we allow for heterogeneity in exercise
behavioracross the firms in our sample by estimating equation (2)
at the firm level, but weaccommodate potential differences across
employee rank within a company by allowingthe baseline hazard to
vary across employee rank (using stratified estimation). 11
Another11 Stratified hazard models are used when the baseline rate
of an event (i.e., option exercise in this case)differs across
certain subsets of observations in the sample. Stratified
estimation allows for a separatebaseline hazard for each subset
which reflects differences in the effect of time across the
strata.Specifically, stratified hazard model can be expressed as
h(t ) = pr t p 1 exp{ 0,r + 1 x1 + ... + n xn } , where each
rstratum is indexed by r. Thus, a separate shape and scale
parameter (i.e., p and 0, respectively) isestimated for each group.
Although each stratum has its own baseline hazard, the relative
hazard is thesame for each group, so the effect of a covariate is
to multiplicatively shift the separate baseline hazard for 10
12. desirable feature of hazard analysis is its ability to
accommodate the empirical regularityof repeated failure that arises
if employees exercise options from a given grant onmultiple
occasions. We allow for the possibility of multiple exercises since
this reducesthe potential for biased estimation of the hazard
rate.4.2. Hazard Analysis Specification We analyze employee
exercise behavior using the employee-grant-day as the primaryunit
of analysis. The use of each employees individual option grant
enables us topreserve the specific features of each grant (e.g.,
strike, duration, and vesting schedule) inorder to better
understand its influence on the exercise decision. Estimating the
hazardmodel daily, as opposed to weekly or monthly, allows us to
preserve information relatedto grant timing, option exercise, and
option cancellation, and is also more consistent withthe
assumptions of the continuous-time Weibull survival model which we
use forestimation. 12 Daily estimation also preserves information
related to the underlyingstocks price path that would be lost
through aggregation. We calculate and report robuststandard errors
(Huber, 1967; White, 1980) clustered by employee to correct for
anypotential bias resulting from intra-employee dependence in
exercise behavior. We alsoretain only vested employee-grant-day
observations that are in-the-money (i.e., where theoptions have
positive intrinsic value) since options that are either unvested or
have nointrinsic value are not at risk of being exercised and
therefore should be excluded fromthe risk set.each group. In our
case, we expect there to be differences is in the baseline rate of
option exercise acrossemployee levels, so we estimate a stratified
model using lower-level, middle-level, and senior-levelemployees as
the strata.12 An example of the potential for loss of information
when aggregating data in time occurs when there aremultiple
exercises within a month but none on the same day. If we were to
construct monthly variables andthere are three distinct exercises
on three different days within the same month, these exercises
would betreated as a single exercise when estimating the model.
11
13. The empirical model used to estimate the rate of option
exercise is:Exercisei,k,t = p r t p 1 exp{ t + r0r + 1
PriorRet-Post + 2 PriorRet-Negt +3 90Pctt + 4 RecentVesti,k,t + 5
RecentVest-Othi,t +6 Price-to-Strikei,k,t + 7 VestedIVi,k,t + 8
PortIV-Posi,t +9 PortIV-Negi,t + 10 StdDevt + 11 DaysLefti,k,t +12
DivYldt + 13 FutureRett + 14 PrePosEarnst +15 PreNegEarnst + 16
PostPosEarnst + 17 PostNegEarnst +18 Genderi + 19 Agei,t + 20
InvolTermi,t +21 VolTermi,t + 22 Qtr2t + 23 Qtr3t +24 Qtr4t },
(2)where:Exercise is a dichotomous variable equal to one if the
employee exercises (i) at least 25% ofthe vested and unexercised
options from grant k on day t and (ii) at least 10% of the
totaloptions from the specific grant on the specific day, and is
zero otherwise.PriorRet-Pos is the cumulative raw return (excluding
dividends) of the underlying stock overthe 250 trading days prior
to (and excluding) day t if positive and is zero
otherwise.PriorRet-Neg is the cumulative raw return (excluding
dividends) of the underlying stock overthe 250 trading days prior
to (and excluding) day t if negative and is zero otherwise.90Pct is
a dichotomous covariate equal to one if the price of the underlying
stock on day t-1 isat least 90% of the highest price over the prior
250 trading days and is zero otherwise.RecentVest is a dichotomous
covariate equal to one if shares from the current grant
vestedwithin the prior 30 trading days and is zero
otherwise.RecentVest-Oth is a dichotomous covariate equal to one if
shares from another grant vestedwithin the prior 30 trading days
and is zero otherwise.Price-to-Strike is the ratio of the current
price of the underlying stock to the exercise price ofthe
option.VestedIV is the natural logarithm of (1+ intrinsic value of
vested and unexercised options fromcurrent grant).PortIV-Pos is the
natural logarithm of (1 + intrinsic value of both vested and
unvestedunexercised options (except vested and unexercised options
from current grant)) if positive;and is zero otherwise.PortIV-Neg
is the natural logarithm of (1 + absolute value of the magnitude by
which bothvested and unvested unexercised options are underwater
(except vested and unexercisedoptions from current grant)); and is
zero otherwise.StdDev is the standard deviation of returns of the
underlying stock over the 250 trading daysprior to (and excluding)
day t.DaysLeft is the number of trading days remaining until
options expire.DivYld is the dividend yield of the underlying stock
during the 60 trading days prior to the dateof record for a
dividend payment and is zero otherwise.FutureRet is the cumulative
raw return (excluding dividends) of the underlying stock over
the250 trading days that follow day t. 12
14. PrePosEarns is the realized market response to an earnings
announcement if (i) day t is withinthe three trading day window
that immediately precedes an earnings announcement and (ii)
therealized market response is positive. The variable is zero
otherwise. The market response iscomputed by subtracting the
three-day cumulative S&P 500 index return from the
three-daycumulative firm return, centered on the earnings release
date reported by CRSP.PreNegEarns is the realized market response
to an earnings announcement if (i) day t is withinthe three trading
day window that immediately precedes an earnings announcement and
(ii) therealized market response is negative. The variable is zero
otherwise. The market response iscomputed by subtracting the
three-day cumulative S&P 500 index return from the
three-daycumulative firm return, centered on the earnings release
date reported by CRSP.PostPosEarns is the realized market response
to an earnings announcement if (i) day t iswithin the three trading
day window that immediately follows an earnings announcement
and(ii) the realized market response is positive. The variable is
zero otherwise. The marketresponse is computed by subtracting the
three-day cumulative S&P 500 index return from thethree-day
cumulative firm return, centered on the earnings release date
reported by CRSP.PostNegEarns is the realized market response to an
earnings announcement if (i) day t iswithin the three trading day
window that immediately follows an earnings announcement and(ii)
the realized market response is negative. The variable is zero
otherwise. The marketresponse is computed by subtracting the
three-day cumulative S&P 500 index return from thethree-day
cumulative firm return, centered on the earnings release date
reported by CRSP.Gender is a dichotomous covariate equal to one if
the employee is male and is zero otherwise.Age is the age of the
employee.Qtr2, Qtr3, Qtr4 are dichotomous covariates equal to one
if the day is in the second, third, orfourth quarter of the fiscal
year, respectively, and is zero otherwise.InvolTerm is a
dichotomous covariate equal to one during the 60 days prior to the
cancellationof options due to involuntary termination of the
employee, and is zero otherwise.VolTerm is a dichotomous covariate
equal to one during the 60 days prior to the cancellation of
options dueto voluntary termination of the employee, and is zero
otherwise.i, k, and t index the employee, grant, and day,
respectively.r is an index for employee rank within the firm.
Employee rank is determined by themagnitude of the employees
participation in the grant, since actual rank data is not
availablefor most firms. Employees are considered low, mid, or high
rank if their participation in thegrant is below the 85th, between
the 85th and 95th, or above the 95th percentile of grant
size,respectively. We set Exercise equal to one when an employee
exercises an economicallysignificant number of options from the
grant. We define economically significantexercises as those where
the employee exercises both (i) at least 25% of options that
arevested and unexercised on the exercise date, and (ii) at least
10% of total options from thespecific grant. Restriction (i) is
intended to ensure that the employee exercises ameaningful amount
of what is available for exercise on a given day. Restriction (ii)
is 13
15. intended to preclude immaterial exercises. 13 Both criteria
also allow us to retain somesemblance of magnitude in our exercise
measure, since we lose this information bymaking a continuous
outcome variable discrete. 14,15 We expect the rate of option
exercise to be associated with a mix of behavioral andeconomic
factors based on evidence from prior research and economic theory.
In orderto develop expectations regarding the association between
these factors and optionexercise rates, it is important to note
that the employees in our sample typicallyimmediately sell their
underlying shares upon option exercise (generally using some typeof
cashless exercise program). Therefore, option exercises can be
thought of asdivestitures or net sale events. We include
PriorRet-Pos and PriorRet-Neg to assess the association between the
rateof option exercise and prior stock price performance. Huddart
and Lang (1996, 2003)find that option exercise is associated with
prior returns which they attribute toemployees beliefs that recent
historical performance is indicative of future performance.Since it
is not clear whether the rate of option exercise is symmetrically
associated withprior price performance, we separate prior returns
into positive and negative components.We measure prior returns as
the continuously compounded raw return (excluding13 A common
example of an immaterial exercise in our data set is when an
employee has alreadyexercised vested options from a grant and
exercises a small number of remaining options that are in-the-money
after his employment with the company is terminated.14 Huddart and
Lang (1996, 2003) and Heath et al. (1999) use the fraction of a
grant exercised in a givenmonth or week, measured across all
employees who participated in the grant. By design, their
measurepreserves information about the magnitude, or intensity, of
the exercise.15 Table 1 (Panel B) shows that we lose only a small
proportion of actual exercise observations byestablishing
restrictions (i) and (ii). On average, the ratio of Misc. Exercises
(i.e., those that do not meetone or both restrictions) to Exercises
(i.e., those that meet both restrictions) is only 5.1%. 14
16. dividends) over the prior 250 trading days. 16 It is not
clear, ex ante, whether we willobserve a positive or a negative
association between these covariates and the rate ofoption exercise
since employees may possess either trending or contrarian
beliefsregarding expectations of future performance conditional on
past performance. The covariate 90Pct is included to capture the
association between the rate of optionexercise and the occurrence
of the underlying stock price surpassing 90% of its highestprice
over the prior year. Consistent with prospect theory, Heath et al.
(1999) provideevidence that employees increase the magnitude of
exercise after the underlying stockprice exceeds prior cognitive
benchmarks. Therefore, we expect a positive associationbetween
Exercise and 90Pct. 17 We include RecentVest and RecentVest-Oth to
assess the association between the rateof option exercise and the
occurrence of recent events that might remind employees toassess
the value of their option portfolio (i.e., a recency effect).
RecentVest alsoreflects an employees newly obtained ability to
actually exercise options to perhapsrebalance his equity portfolio.
If employees require external triggers to assess the valueof their
option holdings or if employees have pent-up diversification needs
in anticipationof vesting, then we expect to observe a positive
association between Exercise and thesecovariates.16 Our analysis is
conducted in trading days, as opposed to calendar days, because
employee stock optionexercise is extremely rare on weekends or
holidays. In a hazard analysis context, the observations in
oursample are not at risk of failure on either weekends or
holidays, so these observations are excludedfrom the analysis. We
construct our covariates based on trading days to conform to the
period of analysisin our study. In addition, since option
parameters (e.g., strike price and number of shares) are adjusted
forstock splits, we use the split-adjusted stock price series to
construct all price-based covariates.17 Our results are similar
when we replace 90Pct with an analogous covariate 100Pct that
indicates whetherthe underlying stock price is at or above its
highest price over the prior year. 15
17. Price-to-Strike and VestedIV are included to examine the
association between the rateof option exercise and the intrinsic
value inherent in the specified grant. Price-to-Strikeprovides an
easy measure for the employee to gauge the appreciation in the
price of theunderlying stock since the date of the option grant. 18
As discussed further below, Price-to-Strike also captures the point
at which the employee might be indifferent betweenexercising the
options and continuing to hold the options for another period,
outlined inutility-based models from prior research (e.g., Hall and
Murphy, 2002). VestedIVcaptures the realizable cash value of
exercisable options in the grant. If employeesexercise options to
fulfill consumption needs or to diversify their portfolio to
reduceexposure to firm-specific risk, then we expect to observe a
positive association betweenExercise and these two covariates. We
include PortIV-Pos and PortIV-Neg to assess the association between
the rate ofoption exercise in the specified grant and the intrinsic
value of the other grants in theemployees portfolio. PortIV-Pos and
PortIV-Neg provide measures of the employeesoverall and
firm-specific wealth. If insiders become less risk averse when
their overallwealth increases, they may be less inclined to
exercise their option holdings. If, however,insiders view increases
in firm-specific wealth as inducing greater risk to their
portfolios,they may be more inclined to exercise their option
holdings. Since it is not clear whicheffect dominates, we do not
predict a sign for these covariates. StdDev is included to capture
the riskiness of the underlying stock. For a risk-neutralinvestor,
option value is increasing in the volatility of the returns of the
underlying stock.However, Lambert et al. (1991) show that the value
of employee stock options may not18 Because our sample only
includes observations where the stock price is greater than the
exercise price,Price-to-Strike is bounded by one from below.
16
18. necessarily increase in the volatility of returns since
employees are risk-averse and areunderdiversified. Therefore,
employees may exercise options early if the risk imposed
byvolatility is sufficiently high, so we expect a positive
association between Exercise andStdDev. We include DaysLeft to
control for the remaining time value of the options in thegrant. 19
If employees are aware of the opportunity cost of early exercise
(i.e., theforfeiture of the remaining time value of the options),
then we expect to observe anegative association between Exercise
and DaysLeft. DivYld is included to assess the association between
the rate of option exercise andemployees anticipation of a pending
dividend payment. If employees anticipate theimpending decline in
option value associated with an upcoming dividend payment
(i.e.,resulting from a reduction in the price of the underlying
stock), then we expect to observea positive association between
Exercise and DivYld. 20 We include FutureRet to assess the
association between the rate of option exerciseand employees
private information regarding future firm performance. Huddart
andLang (1996) find little evidence of an association between
option exercise activity andfuture returns, yet Huddart and Lang
(2003) find an association between option exerciseand future
returns for all levels of employees in their sample. If employees
utilize private19 An alternative measure for time value can be
computed by subtracting the intrinsic value from the
overallBlack-Scholes value to back into the time value component.
Our subsequent results and results fromprior research suggest that,
because of early exercise, the Black-Scholes value overstates the
total value tothe option. Therefore, estimates of time value
computed from the Black-Scholes value are likely measuredwith
error. To avoid measurement error of this nature, we utilize,
instead, DaysLeft as our proxy for thetime value inherent in the
option grant. DaysLeft has its own limitations, but we expect it to
be sufficientlycorrelated with an options true time value to
provide useful inferences.20 None of the stock options in our
sample are dividend protected. 17
19. information regarding future firm performance in their
exercise decision then we expectto observe a negative association
between Exercise and FutureReturn. 21 PrePosEarns, PreNegEarns,
PostPosEarns, and PostNegEarns are included tocapture the potential
effect of higher trade profit opportunity, higher litigation risk,
andfirm-imposed non-trade windows on the rate of option exercise
around earningsannouncements. Before an earnings announcement,
employees may modify exerciserates to profit from private
information about forthcoming earnings news. Specifically,employees
may exercise options more frequently before negative earnings news
or lessfrequently before positive earnings news. Increased
litigation risk or firm-imposed traderestrictions may, however,
influence employees to exercise options less frequently
beforenegative earnings news. 22 After an earnings announcement,
employees may also altertheir rate of exercise in response to the
earnings disclosure. Specifically, employees mayexercise options
less frequently after negative earnings news or more frequently
afterpositive earnings news. Therefore, we expect to see a negative
association between therate of option exercise and PrePosEarns and
PostNegEarns. We expect to see a positiveassociation between the
rate of option exercise and PostPosEarns. Finally, if
employeestrade to profit from information about forthcoming
earnings, we expect to see a positiveassociation between the rate
of option exercise and PreNegEarns. If employees,however, respond
to increased litigation risk or firm-imposed trade restrictions
before21 Recall that option exercises equate to share sales in our
sample. This implies that we expect to observefewer sales before
positive future returns and greater sales before negative future
returns.22 Trading in proximity to material news events is a key
element for establishing scienter in cases allegingillegal trade
[Freeman v. Decio, 584 F 2d 186, 197 n.44 (7th Cir. 1978)]. Firms
are also known to restricttrade in close proximity to earnings
announcements (Bettis et al. 2000). 18
20. earnings releases, we expect to see a negative association
between the rate of optionexercise and PreNegEarns. Prior research
suggests that older people tend to be more risk-averse and that
femalesare more risk-averse than males (e.g., Bajtelsmit and
Bernasek, 2001, and Bellante andGreen, 2004). If Age and Gender
capture risk aversion, we expect to observe a positiveassociation
between Exercise and Age and a negative association between
Exercise andGender. We include VolTerm and InvolTerm to assess the
association between the rate ofoption exercise and the employees
anticipation of impending employment termination.The rate of option
exercise is likely to be influenced by termination because most
optiongrants have cancellation features tied to termination.
Specifically, when terminationoccurs, the employee typically has 60
days to exercise any vested and unexercisedoptions. Any options
remaining after 60 days are cancelled. If termination
(andsubsequent option cancellation) is imminent, one would expect
an employee to increasethe rate of exercise of available options.
Therefore we expect to observe a positiveassociation between
Exercise and both VolTerm and InvolTerm. Finally, we include Qtr2,
Qtr3, and Qtr4 to control for potential seasonal variation inthe
rate of option exercise since employees consumption needs may vary
during theyear. For example, many companies grant options during
the first quarter of the year.This might result in a higher rate of
exercise of the employees existing optionsincremental to the other
covariates included in the model. Another example would beoption
exercises motivated for tax reasons in the fourth quarter of the
calendar year.4.3. Hazard Analysis Results 19
21. Table 2 reports results for the firm-specific hazard
estimation from equation (2). Tosimplify tabulation and facilitate
the discussion of our results, we report the mean, 20th,50th, and
80th percentiles for the estimated coefficients across the ten
companies. We alsoreport the number of firms for which the
firm-specific coefficient estimates are eitherstatistically
positive or negative (p-value < 0.05, two-sided). In order to
assess theaggregate significance of our results, we report an
overall z-statistic that aggregates theresults across the sample
companies. 23 Consistent with Heath et al. (1999), we find evidence
of a positive associationbetween the rate of option exercise and
positive prior stock price performance.Specifically, we find that
the estimated coefficient on PriorRet-Pos is positive,
whichsuggests that employees are generally contrarian. Also
consistent with Heath et al.(1999), we observe that 90Pct, which
indicates that the firms recent stock priceperformance meets or
exceeds a specific cognitive benchmark, is positively
associatedwith the rate of option exercise. In particular, the mean
coefficient for 90Pct of 0.758suggests that the rate of option
exercise is 113% (e 0.758 1 = 1.134) higher when theunderlying
stock is trading at or above 90% of its highest price over the
prior year. We also find that employees rate of exercise is
positively associated with recentreminders regarding option
portfolios or with pent-up diversification or consumptionneeds.
Specifically, we observe a positive association between the rate of
option exerciseand RecentVest, which indicates that options from
the given grant vested recently.23 The aggregated z-statistic is
calculated as the sum of the individual z-statistics divided by the
square rootof the number of companies for which there is an
estimated coefficient for a given covariate (which, formost
covariates is ten). This aggregated z-statistic assumes that
employee exercise behavior in the samplefirms is independent. To
the extent these observations are correlated, our reported
significance levels willbe overstated. However, many of the
aggregated z-statistics are large, therefore the statistical
results areunlikely to simply reflect cross-sectional correlation
among the observations. 20
22. Similarly, for half of the firms in our sample, we observe
a positive association betweenthe rate of option exercise and the
recent vesting of options from an employees otheroutstanding grants
(RecentVest-Oth). We find that the rate of option exercise is
positively associated with the intrinsicvalue of an option grant.
Specifically, we observe positive coefficients for both
Price-to-Strike and VestedIV. We also find that the rate of option
exercise is decreasing in theintrinsic value of the employees other
option grants when the intrinsic value is positive.In other words,
we find a negative association between Exercise and PortIV-Pos.
Thissuggests that employees slow exercise rates when alternative
grants provide higherrelative realizable value. We also find that
the rate of option exercise is decreasing whenthe employees other
option grants go deeper underwater. Specifically, we find anegative
association between Exercise and PortIV-Neg. This suggests that
employeeswith low portfolio value slow exercise rates, perhaps
because they have less firm-specificwealth at risk. Regarding the
potential for private-information-based exercise, there is no
evidencethat the rate of option exercise increases in anticipation
of dividends, pending pricedeclines (Huddart and Lang, 2003), or
the news in earnings. However, the rate of optionexercise is lower
immediately after both negative and positive earnings surprises.
There is some evidence that senior managers have a lower rate of
option exerciserelative to lower ranking employees, as shown by the
negative coefficient for the high-rank scale parameter shift. Lower
ranking employees may exercise at a greater ratebecause they rely
more on option proceeds to fulfill consumption needs or they
maysimply be more risk-averse. We do not, however, find evidence of
an association 21
23. between the rate of exercise and other demographic
characteristics, Gender and Age,which are included to capture the
employees risk aversion. We also find evidence of a lower rate of
exercise in the third and fourth calendarquarters relative to the
first calendar quarter. It is possible that employees have
greaterconsumption needs during the first calendar quarter (to pay
off accrued holiday-seasondebt, for example) which accounts for the
relatively higher rate of option exercise.Finally, the rate of
exercise is positively associated with pending voluntary
employmenttermination, which is consistent with employees realizing
the intrinsic value of theiroptions prior to their cancellation.
Collectively, our results are consistent with those of Heath et al.
(1999) and Huddartand Lang (1996, 2003), who suggest that option
exercise is associated with a complexmix of behavioral and economic
factors. Our results also demonstrate that employeestock option
exercises appear to exhibit predictable temporal variation,
particularly inshort windows subsequent to earnings announcements
and across calendar quarters. We report a measure of explained
variation in option exercise based on Royston(2006) to assess the
overall goodness-of-fit of our model. The distribution of
thecompany-specific adjusted-pseudo-R2 statistics suggests that the
model described byequation (2) explains a substantial amount of the
variation in the rate of employee stockoption exercise. 2424
Current consulting practice tends to use a more parsimonious
specification to model option exercisebehavior. Our discussions
with several practitioners confirmed that a model consisting of the
price-to-strike ratio, voluntary, and involuntary termination is an
adequate characterization of a model that would beused in practice.
We estimated this model for the ten companies in our sample and
this yielded a mean andmedian adjusted-pseudo-R2 of 10.5% and
10.7%, respectively (compared to a mean and median
adjustedpseudo-R2 of 37.8% and 39.5%, respectively, for the Hazard
Estimation model). Although this model issimple and intuitive, the
reduced explanatory power of this model relative to the model
specified byequation (2) highlights the role of the additional
covariates in our model. 22
24. We also note in Table 2 that, on average, the estimated
shape parameters are greaterthan one, which suggests that a
monotonic increasing baseline hazard generally describesexercise
activity. This is expected because the rate of exercise activity is
likely toincrease the longer a stock option is held (because the
option has a fixed life that istypically equal to ten years).5.
Out-of-Sample Validation We assess the relative ability of the
hazard model to correctly predict the timing ofoption exercise
using an out-of-sample analysis. This is done by first splitting
each ofour sample firms data into mutually exclusive estimation and
holdout subsamples. Theparameters for each model are estimated
using the data in the estimation subsample andthe ability of the
models to predict actual stock option exercise dates is assessed in
theholdout subsample. In order for the validation analysis to not
be confounded by right censoring, it isnecessary to observe the
entire option life. Therefore, our holdout sample consists of
theearliest grants in our data because we can observe the full life
of the option (i.e., timefrom grant until expiry) for these grants.
25 The estimation subsample consists of the latergrants in each
firms data set, for which hazard estimation econometrically
accounts for25 In particular, we include in the holdout subsample
those option grants for which the entire life of theoption is
observable. For certain grants, the entire life of the grant is
observed (i.e., there is no rightcensoring) because all of the
options are either cancelled or exercised prior to the censoring
date of thefirms dataset. These options are excluded from the
holdout subsample (and included in the estimationsubsample) in
order to avoid selection bias by including those grants that, ex
post, are exercised early. 23
25. right-censoring. 26 We have sufficient data for nine of our
ten sample firms to form aholdout subsample so these firms
constitute our sample for the validation. 27 For this analysis, we
estimate three different hazard models. The first model is
aslightly reduced version of the exercise model in equation (2)
that excludes FutureRetbecause we do not want to bias the results
by using forward looking data. We alsoexclude three variables
related to an employees other option grants
(RecentVest-Oth,PortIV-Pos, and PortIV-Neg) because the unit of
analysis for the valuation is theemployee-grant (rather than the
employee). Including these covariates would requireemployee-level
analysis and would also require assumptions about an
employeesdecision model at the portfolio level. Excluding these
covariates biases against thepredictive ability of the hazard
estimation to the extent the excluded covariates haveexplanatory
power. We also estimate hazard models of voluntary and
involuntarytermination because these models are necessary inputs
into the exercise hazard model.The voluntary and involuntary
termination hazard models are described in Appendix A.For these
models, we also exclude three variables that require forward
lookinginformation (FutureRet, IV Future Vest-Pos, and IV Future
Vest-Neg) because we do notwant to bias the results in favor of
hazard estimation. After obtaining the estimated coefficients for
each of the three hazard models for theestimation period, we
computed the fitted voluntary and involuntary termination
hazardrates in the holdout sample using observed covariate values
for each day. The resulting26 We choose to not estimate the
validation in chronological order (where the holdout subsample
wouldchronologically follow the estimation subsample), to avoid
confounding inferences from right-censoring inthe holdout
subsample. Our approach is conservative because it negates
potential for employee learning,which would naturally induce
correlation across subsamples and potentially enhance hazard
modelpredictive ability.27 For Firm H, we do not have a
sufficiently long enough time series to observe the actual outcome
of theearliest grants. 24
26. fitted rates of termination are transformed into daily
probabilities of termination. Wedetermine whether a termination is
expected to occur on each day by comparing theprobability of
termination to a randomly drawn value from a unit uniform
distribution. 28The voluntary and involuntary termination
predictions on each day are used as inputs inthe hazard model for
employee exercise. Given the expected termination and
othercovariate values in the holdout sample, the fitted rate of
exercise and the correspondingprobability of exercise on each day
are then computed. We simulate the incidence ofexercise for the day
by comparing the daily probability of exercise to a randomly
drawnvalue from a unit uniform distribution. If an exercise event
occurs (i.e., the dailyprobability of exercise exceeds the random
probability draw from the unit uniformdistribution), we assume that
the employee exercises all of the vested and unexercisedoptions
from the grant. Thus, we allow for the possibility of multiple
exercises from eachgrant in the holdout subsample. We iterate this
procedure 1,000 times for each grant andutilize the
equally-weighted holding time estimate as our estimated time to
exercise forthe option grant. 29 We compare our hazard estimates of
exercise time to estimates from a utility-basedhazard model and to
estimates from a procedure commonly used by firms in practicewhen
computing option grant costs for financial reporting. To generate
utility-basedexercise time predictions, we estimate a hazard model
that includes only the price-to-28 Specifically, if the random
number drawn from the uniform distribution is less than the
estimatedprobability of termination then a termination event is
deemed to occur.29 The estimated coefficients of our full hazard
model are relatively stable between the estimation andvalidation
subsamples. Specifically, the Pearson (Spearman) correlation
between the estimates in the twosubsamples is 0.865 (0.692) and the
correlations between the estimated coefficient t-statistics is
0.912(0.875). These results suggest that our hazard model exhibits
considerable inter-temporal stability. 25
27. strike (S-to-K) ratio and an exogenous termination
indicator. 30 This S-to-K hazard modelcan be viewed as an empirical
analog to utility-based models examined in prior literaturebecause
utility-based models view the exercise decision as an optimal
stopping problem.The solution to the problem is characterized as a
time-varying exercise boundary that isa function of the underlying
stock price. Similarly, the S-to-K hazard model estimatesthe hazard
rate (which is analogous to the exercise boundary) as a
time-varyingfunction of the underlying stock price. To generate
Common Practice estimates of the exercise time, we compute the
meanobserved holding time for options with realized exercise. We
then adjust this measure byassuming that any unexercised (i.e.,
right censored) options will be exercised after half oftheir
remaining time to expiry has elapsed. 31 Table 3 presents the
predicted stock option holding times relating to the first
realizedexercise event (Panel A) and the weighted average realized
exercise events (Panel B) inthe holdout subsamples for the three
alternative models. The prediction errors arecalculated as the
absolute value of the difference between actual and estimated
optionholding times (in calendar days). Table 3 (Panel A) shows the
prediction errors for eachmodel relative to the hold time that
precedes the first observed exercise in the holdoutwindow. Panel A
shows that, for six of the nine sample firms, Full Hazard
Estimationyields the lowest estimation error in calendar days. 32
Similarly, Panel B shows theprediction errors for each model
relative to the weighted average hold time for all options30 The
S-to-K exogenous termination probability is computed by estimating
a baseline Weibull hazard rateof termination and then converting
the rate into a daily probability of termination.31 Several
consultants noted in conversation that this estimation is commonly
used in practice.32 For Firm D, Full Hazard Estimation error does
not differ statistically from Common Practice EstimationError. To
the extent these errors are equal then both Full Hazard and Common
Practice Estimation tie formost accurate prediction. 26
28. in the holdout window. Panel B also shows that, for six of
the nine sample firms, FullHazard Estimation yields the lowest
estimation error in calendar days. Collectively, thisevidence
suggests that hazard estimation that incorporates economic and
behavioralfactors associated with exercise timing tends to yield
more accurate out-of-sampleestimates of realized option holding
times than alternative models discussed in priorliterature and
commonly used for financial reporting.6. Option Value Using Monte
Carlo Simulation In this section, we develop a firm-specific Monte
Carlo simulation that incorporatesprice path dependency in the
option exercise decision (using the hazard model estimates)into
estimates of the cost of a typical option grant. Our simulation is
based on thefollowing sequential steps: (1) simulate a daily stock
price path; (2) estimate the dailyprobability of employment
termination as a function of simulated stock price and
otherfactors; (3) simulate daily employment termination events
based on estimated dailytermination probabilities; (4) estimate the
daily probability of option exercise as afunction of the simulated
price path, simulated termination events, and other factors;
(5)simulate daily exercise events based on estimated daily exercise
probabilities; (6)compute the realized intrinsic value of the
options exercised on the simulated exercisedates; and (7) discount
the simulated realized intrinsic values of the grant date at the
risk-free rate of return.6.1 Simulation Mechanics We first simulate
200,000 random price paths for the underlying stock over the
tenyear term of the option plus two years prior to the grant and
one year after the expiration 27
29. of the grant. We simulate price paths before and after the
option term period toincorporate the influence of both prior and
future returns on employee termination andexercise activity. We
assume that the price of the underlying stock, St, evolves
accordingto a standard geometric Brownian motion evolution, which
is described by the followingstochastic differential equation: dSt
= Stdt + StdWt, where Wt is a Brownian motion (ora Weiner process),
is the drift, and is the volatility. 33 We operationalize this
byassuming that the price of the underlying stock on the grant date
(which is also equal tothe strike price) equals the average actual
grant price for the appropriate level ofemployee (i.e., low-rank
mid-rank, or high-rank) for which we simulate the cost of anoption
grant. We then simulate the eleven subsequent years of the
underlying priceassuming a drift equal to the risk-free rate of 5%
and a company specific volatilityestimate based on the historical
annualized daily standard deviation of returns. 34 Theprice path
simulation process is outlined in Figure 1. We next utilize these
simulated prices as inputs to the hazard models, outlined
inAppendix A, in order to estimate the rate of employment
termination. A hazard rate iscalculated for both involuntary and
voluntary termination as a function of time, thesimulated price
path, and other covariates outlined in the Appendix. We then
calculateeach days probability of voluntary or involuntary
termination by transforming theestimated rate of termination, h(t),
into a daily expected probability of termination,(1.0 eh(t)). This
daily probability of termination is then compared to a random
draw33 Our assumption that the underlying price path is a geometric
Brownian motion is consistent with theassumptions of the
Black-Scholes (1973) option pricing formula. As discussed later,
this allows us toverify the accuracy of our simulations by
comparing the value of the option if it were held to maturity
withthe analytical value of the option from the Black-Scholes
formula.34 Since we require the price of the underlying stock to be
the same on the grant date across all simulationsfor each company,
we simulate the two prior years of returns using the same geometric
Brownian motionwith a negative drift term (i.e., dSt = -Stdt +
StdWt). 28
30. from a unit uniform distribution to determine whether a
simulated termination event hasoccurred. 35 If the random draw is
less than the calculated daily probability oftermination, then we
assume that a termination will occur 60 days later, and we
setVolTerm (InVolTerm) equal to one in the exercise model for the
next 60 days. Next, we use the simulated price path and the
simulated termination events as inputsto the daily hazard model,
outlined in equation (2), which specifies the rate of
optionexercise. 36 A daily hazard rate is calculated for option
exercise as a function of time, thesimulated price path, simulated
voluntary termination, simulated involuntary termination,and other
covariates outlined in equation (2). 37 We then calculate the
probability ofoption exercise for the day by transforming the
estimated rate of exercise, h(t), into adaily probability of
exercise, (1.0 eh(t)). This daily probability of exercise is
thencompared to a daily random draw from a uniform distribution to
determine whether asimulated exercise event has occurred. If the
daily random draw is less than the imputedprobability of exercise,
we assume that an exercise has occurred, and we set Exerciseequal
to one. 3835 Specifically, the comparison random number is a
pseudo-random number generated from the multi-seedrandom number
generator of MATLAB version 7.2.36 We estimate equation (2)
excluding PrePosEarns, PreNegEarns, PostPosEarns, PostNegEarns,
Qtr2,Qtr3, and Qtr4, since it is difficult to simulate the earnings
surprise and for parsimony.37 We assume an employee has a single
option grant outstanding (i.e., no outside holdings of either
theunderlying stock or options from another grant). We also assume
that the grant is held by a typicalemployee within each employee
rank (i.e, a male, whose age is equal to the average age for all
employeesof the same rank). Since most companies in our sample have
more than one vesting schedule, we select themost common schedule
for the simulation. In one instance, we did not use the most common
vestingschedule in the simulations because of its complicated
structure (e.g., 25% on the grant date and theremaining shares
ratably over the next 36 months). In order to simplify the
simulation for this company,we transform the monthly component into
a function that is more discrete (e.g., 25% on the grant date
and25% for each of the next three years). If the modal vesting
schedule includes multiple tranches (e.g., 25%per year for four
years), multiple exercises are allowed in the simulation.38 The
employee is assumed to exercise all vested and unexercised options
when an exercise event occurs inthe simulation. Therefore, the
simulation allows each employee between 0 and exercise events if
hiscompany grants options that vest in tranches. 29
31. To compute the cost of 10,000 hypothetical options, we
discount the intrinsic value ofthe exercised shares at the
risk-free rate for each simulated exercise. For multipleexercises,
we sum the present value of the intrinsic value realized from each
trancheexercised.6.2 Simulation Results We first calibrate our
simulation by computing the cost of the grant if it were held
toexpiry (i.e., no early exercise) with the analytical
Black-Scholes value of a Europeanoption. Specifically, we discount
the intrinsic value on the date of expiry back to thegrant date and
compare the average simulated value to the European
Black-Scholesvalue. If our simulation is calibrated correctly, then
these two values should converge. 39Our calibration (untabulated)
shows that the simulated hold-to-expiry value is alwayswithin
approximately 4% of the European Black-Scholes value of the option
grant, so weconclude that the simulation noise is low. We are
interested in determining the impact of more accurately estimating
the time toexercise on the cost of an option grant, and
specifically the degree to which option costmodels produce
homogeneous output. To quantify this effect, Table 4 (Panels B and
C)compare the simulated cost of the option grant to (1) the cost
computed using aconventional method of adjusting Black-Scholes
formula for historical option exerciserates and (2) the cost
computed using a utility-based hazard model.39 The accuracy of our
estimated values of the options depends on the speed with which the
sample mean ofthe empirical density function of option values
converges to the mean of the true population densityfunction. Our
simulations require a number of intermediate calculations at each
time interval (i.e., tradingday) which limits the number of
simulations that are feasible for each company in our sample. We
use acommon variance reduction technique known as antithetic
covariates in our simulation (i.e., the mirrorimage of each
simulated price path is used). This induces a negative covariance
between the twosimulations, which results in a lower variance (and,
therefore, faster convergence) of the empirical density. 30
32. Not surprisingly, Table 4, Panel B shows that the Full
Duration Black-Scholes modelproduces substantively larger option
grant cost estimates than the other models since itdoes not account
for employees early exercise patterns. Table 4, Panels B and C
alsoshow that, on average, Full Hazard Estimation produces
materially different option grantcost estimates relative to
Black-Scholes estimates adjusted for early exercise and relativeto
utility-based hazard estimates. For example, our simulated option
grant cost estimatesfor grants to mid-rank employees are, on
average, 25% lower than Black-Scholesestimates adjusted for average
historical option holding periods. Further, hazard modeloption
grant cost estimates are, on average, 19% different than option
cost estimates froma utility-based hazard model. Results also show
that cost estimates derived from a utility-based model are
substantially different from cost estimates derived from a
risk-neutralmodel adjusted for historical exercise rates.
Specifically, utility-based cost estimates are,on average, 26%
different from adjusted Black Scholes estimates. The
evidencetherefore suggests that accurate timing inputs materially
affect option grant costestimates, which has implications for
financial reporting.7. Summary and Conclusions This study examines
how option exercise behavior by executives and employeesaffects the
estimated cost of the options to the granting firm. In contrast to
prior research,we model the option exercise decision within a
hazard framework which accounts forright censoring and allows for a
variety of behavioral and economic factors to influencethe rate of
option exercise. We demonstrate that that accounting for these
behavioral andeconomic factors in a hazard framework generally
improves out-of-sample predictions of 31
33. the option exercise times. To our knowledge, prior research
has not attempted to validatethe predictability of exercise models
in an out-of-sample setting. We also show that amodel of option
cost based on this estimate of exercise time produces
substantiallydifferent cost estimates than either a utility-based
model or a risk-neutral model adjustedfor early exercise. Although
these two parsimonious models are appealing, our resultssuggest it
is possible to improve the accuracy of exercise timing estimates.
This, in turn,can materially affect option cost estimates for the
granting firm. It is important to highlight two limitations of our
analysis. First, our model does notaccount for the options ability
to provide performance incentives to employees. That is,similar to
virtually all prior empirical research that analyzes the cost of
employee stockoptions, we take the price path of the underlying
stock to be exogenous and do not modelthe possible incentive
effects which can potentially affect the evolution of the
pricepath. 40 This issue is an important topic for future research.
Second, like prior research inthis area, our inferences are drawn
from a small and self-selected sample, whichpotentially limits the
ability to generalize our results.40 See the analysis in Feltham
and Wu (2001) and Armstrong et al. (2007) for models that
incorporate theincentive features of employee stock options.
32
34. ReferencesArmstrong, C., Larcker, D., Su, C., 2006. Stock
options and incentives. Working paper, Stanford
University.Bajtelsmit, V., Bernasek, A., 2001. Risk preferences and
the investment decisions of older Americans. Research Report No.
2001-11, American Association of Retired Persons (Washington,
D.C.).Bellante, D., Green, C., 2004. Relative risk aversion among
the elderly. Review of Financial Economics 13, 269-281.Bettis, J.,
Coles, J., Lemmon, M., 2000 Corporate policies restricting trading
by insiders. Journal of Financial Economics 57, 191-220.Bettis, J.,
Bizjak, J., Lemmon, M., 2005. Exercise behavior, valuation, and the
incentive effects of employee stock options. Journal of Financial
Economics 76, 445-470.Black, F., Scholes, M., 1973. The Pricing of
Options and Corporate Liabilities. Journal of Political Economy 81,
637-659.Carpenter, J., 1998. The exercise and valuation of
executive stock options. Journal of Financial Economics 48,
127-158.Cox, D., 1972. Regression Models and Life-Tables. Journal
of the Royal Statistical Society B. 34, No. 2, 187-220.Cox, J.,
Ross S., Rubinstein, M., 1979. Option pricing: a simplified
approach. Journal of Financial Economics 7, 229-263.Feltham, G.,
Wu, M., 2001. Incentive efficiency of stock versus options. Review
of Accounting Studies 6, 7-28.Heath, C., Huddart, S., Lang, M.,
1999. Psychological factors and stock option exercise. Quarterly
Journal of Economics, 601-627.Hemmer, T., Matsunaga, S., Shevlin,
T., 1996. The influence of risk diversification on the early
exercise of employee stock options by executive officers. Journal
of Accounting & Economics 21, 45-68.Huddart, S., 1994. Employee
stock options. Journal of Accounting & Economics 18, 207-
231.Huddart, S., Lang, M., 1996. Employee stock option exercises:
an empirical analysis. Journal of Accounting & Economics 21,
5-43.Huddart, S., Lang, M., 2003. Information distribution within
firms: evidence from stock option exercises. Journal of Accounting
& Economics 34, 3-31.Huber, P., 1967. The behavior of maximum
likelihood estimates under non-standard conditions. Proceedings of
the Fifth Berkeley Symposium on Mathematical Statistics and
Probability 1, 221-233.Hull, J., White, A., 2004. How to Value
Employee Stock Options. Financial Analysts Journal 60,
114-119.Jennergren, L., Naslund, B., 1993. A comment on Valuation
of executive stock options and the FASB proposal. The Accounting
Review 68, 179-183.Kaplan, E., Meier, P., 1958. Nonparametric
estimation from incomplete observations. Journal of the American
Statistical Association 53, 457-481.Kulatilaka, N., Marcus., A.,
1994. Valuing Employee Stock Options. Financial Analysts Journal
50, 46-56.Lambert, R., Larcker, D., Verrecchia, R., 1991. Portfolio
considerations in valuing executive compensation. Journal of
Accounting Research 29, 129-149.
35. Marquardt, C., 2002. The Cost of Employee Stock Option
Grants: An Empirical Analysis. Journal of Accounting Research 40,
1191-1217.Royston, P., 1980. Explained variation for survival
models. The Stata Journal 6, 83-96.White, H., 1980. A
heteroskedasticity consistent covariance matrix estimator and a
direct test for heteroscedasticity. Econometrica 48, 817-830.
36. Appendix A. Hazard Estimation for Voluntary and Involuntary
Termination Our results suggest that the rate of option exercise is
strongly associated with impendingvoluntary termination. Therefore,
to effectively simulate option exercise, we require anestimate of
the probability of employee termination on a given day. Since we
have actualdata on employee terminations, we develop
company-specific hazard models for bothvoluntary and (where
possible) involuntary employee termination. 41 In our analysis,
weestimate a hazard model similar to the model of employee exercise
described in Section 4.1except that the unit of analysis is the
employee-day. The dichotomous outcome variabletakes a value of one
60 days prior to the cancellation of the employees options for
eithervoluntary or involuntary termination and as a zero on all
other days. Our outcome measurewill correctly code every
termination for an employee that either holds out-of-the-money
orunvested stock options. However, we will not observe the
termination of employees thathold only vested, in-the-money stock
options because these options will be exercised ratherthan
cancelled. We expect that termination is associated with the
following covariates. We expect agreater rate of termination for
older employees, so we include a covariate, Old, that is equalto
the employees age if the employee is older than 62 and that is
equal to zero otherwise.We expect a greater rate of termination
when the firms performance is low, and therefore weinclude the
stock return during the previous two years (Returnyr-1 and
Returnyr-2). We expecta lower rate of termination if expectations
regarding future returns are positive, so we alsoinclude FutureRet.
We expect a lower rate of voluntary termination if the employee
knows41 For most of the companies in our sample, we have
information that classifies the termination as eithervoluntary or
involuntary. Certain companies have more detailed classifications
(e.g., death) which naturally fallinto one of the two categories
(e.g., involuntary in the case of death). One ambiguous category,
however, isretirement. For these observations, if the employees age
was greater than 62 on the date of retirement, theobservation was
classified as involuntary termination, otherwise it was classified
as voluntary termination.
37. that a currently unvested option will have substantial
intrinsic value when it vests in the nearfuture. Therefore, we
include IV Future Vest Pos and IV Future Vest Neg, which capturethe
future intrinsic value of options that vest within the next 60
days. 42 We expect the rateof termination is also related to the
intrinsic value of already vested options, although thedirection of
this relationship is not, ex ante, clear. Employees who hold vested
options withsubstantial intrinsic value may hold enough wealth
(assuming these options will beexercised) to comfortably transition
to new employment. Employees who hold vestedoptions with no
intrinsic value (especially options that are substantially out of
the money),may recognize there is little forfeit cost associated
with transitioning to new employment.Therefore we include Vested
IV-Pos and Vested IV Neg. 43 We also include similar variablesfor
the employees unvested options (Unvested IV Pos and Unvested IV
Neg,respectively). We expect the rate of termination is also
related to the employees rank, so weinclude SrMgt and MidMgt.
Finally, we expect that the rate of termination is increasing inthe
time value for stock options, since these options are more apt to
be recent grants withlong duration to vesting and low intrinsic
value. Therefore, we also include DaysLeft. We present aggregated
results for the voluntary and involuntary termination models
inTable A1 (Panels A and B, respectively). As might be expected, we
find that Age is the mostsignificant covariate associated with the
rates of both voluntary and involuntary employeetermination. We
find a higher rate of voluntary termination when the prior
yearsperformance is positive. This result may be consistent with
employees having better external42 If the future intrinsic value is
positive, then IV Future Vest Pos equals the natural logarithm of
one plus thefuture intrinsic value. Otherwise, IV Future Vest Pos
equals zero. If the future intrinsic value is negative,then IV
Future Vest Neg equals the natural logarithm of one plus the
absolute value of the future intrinsicvalue. Otherwise, IV Future
Vest Neg equals zero.43 If the intrinsic value is positive,
VestedIV Pos equals the natural logarithm of one plus the intrinsic
valueand equals zero otherwise. If the intrinsic value is negative,
VestedIV-Neg equals the natural logarithm of oneplus the absolute
value of the intrinsic value and equals zero otherwise.
38. market value when they are associated with firms that
perform well. We find weak evidencethat the rate of voluntary
termination appears to decrease as the intrinsic value of
soon-to-vest options increases. The coefficient for IV Future Vest
Pos is negative for eight of ourten firms, although the aggregate
z-statistic does not support statistical significance
atconventional levels. We find consistent evidence that the rate of
voluntary termination isgreater when there is more time value
inherent in the options. Specifically, we observepositive
coefficients for both Unvested IV Neg and DaysLeft. Finally, the
hazard modelshave large pseudo R-squared and this gives us some
confidence that our firm-specifictermination estimates are more
accurate than generic termination rates that are publiclyavailable
in actuarial tables.
39. Table A1Aggregated Results for Firm-Specific Termination
Rate Estimation Company-specific Weibull model estimates of the
Voluntary (Panel A) or Involuntary (Panel B) terminationhazard
model. The mean, 20th, 50th, and 80th percentiles of the individual
company-specific estimates arereported. The total number of
positive and negative estimated coefficients that are significant
at the 5%significance level (two-tailed) are reported. The
aggregate z-statistic is calculated as the sum of the
individualcompany-specific z-statistics divided by the square root
of the number of companies for which there is anestimated
coefficient for a given covariate. VolTerm (InvolTerm) is a
dichotomous covariate equal to one on the60th day that precedes the
cancellation of the employees options due to voluntary
(involuntary) terminationfrom the company and is zero otherwise.
Old is the employees age if greater than 62 and is zero
otherwise;Returnyr-1 is the cumulative raw return (excluding
dividends) during the prior 250 trading days; Returnyr-2 is
thecumulative raw return (excluding dividends) starting 500 trading
days prior and ending 250 trading days prior tothe current date;
FutureRet is the cumulative raw return (excluding dividends) during
the subsequent 250trading days; IV Future Vest Pos is the natural
logarithm of one plus the intrinsic value of any options thatvest
within the next 60 trading days if this amount is positive and is
zero otherwise; IV Future Vest Neg is thenatural logarithm of one
plus the absolute value of the intrinsic value of any options that
vest within the next 60trading days if the intrinsic value is
negative and is zero otherwise; Vested IV Pos is the natural
logarithm ofone plus the intrinsic value of the employees vested
and unexercised options if this amount is positive and iszero
otherwise; Vested IV Neg is the natural logarithm of one plus the
absolute value of the intrinsic value ofthe employees vested and
unexercised options if the intrinsic value is negative and is zero
otherwise; UnvestedIV Pos is the natural logarithm of one plus the
intrinsic value of the employees unvested options if thisamou