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Chris Armstrong, Alan Jagolinzer, and David Larker

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- 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 employee’s 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 employee’s 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 model’s 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 employee’s assumed utility function (e.g., Huddart, 1994; Kulatilakaand Marcus, 1994; Carpenter, 1998; Bettis et al., 2005), an exogenous “stopping rate”event 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 employee’s 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 employee’sunvested options were cancelled in connection with termination. We infer the employee’s 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{β0 + β1x1 + … + βnxn} (1)The baseline hazard is pt p-1exp{β0} where p is the shape parameter and β0 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{β0}) and the relative hazard(i.e., exp{β1x1 + …βkxk}). 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 firm’s 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{β 0,r + β1 x1 + ... + β n xn } , where each rstratum is indexed by r. Thus, a separate shape and scale parameter (i.e., p and β0, 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 employee’s 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 underlyingstock’s 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{ε t + rβ0r + β1 PriorRet-Post + β2 PriorRet-Negt +β3 90Pctt + β4 RecentVesti,k,t + β5 RecentVest-Othi,t +β6 Price-to-Strikei,k,t + β7 VestedIVi,k,t + β8 PortIV-Posi,t +β9 PortIV-Negi,t + β10 StdDevt + β11 DaysLefti,k,t +β12 DivYldt + β13 FutureRett + β14 PrePosEarnst +β15 PreNegEarnst + β16 PostPosEarnst + β17 PostNegEarnst +β18 Genderi + β19 Agei,t + β20 InvolTermi,t +β21 VolTermi,t + β22 Qtr2t + β23 Qtr3t +β24 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 employee’s 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 significant”exercises 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 employee’s 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 theemployee’s portfolio. PortIV-Pos and PortIV-Neg provide measures of the employee’soverall 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 option’s “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 employee’s 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 employee’s 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 firm’s 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 employee’s 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 employee’s 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 employee’s 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 firm’s 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 thefirm’s 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 employee’s 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 employee’sdecision 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 = μStdt + σStdWt, where Wt is a Brownian motion (ora Weiner process), μ is the drift, and σ 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 day’s probability of voluntary or involuntary termination by transforming theestimated rate of termination, h(t), into a daily expected probability of termination,(1.0 − e−h(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 = -μStdt + σStdWt). 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 − e−h(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 θ exercise events if hiscompany grants options that vest in θ 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 employee’s 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 employee’s age if the employee is older than 62 and that is equal to zero otherwise.We expect a greater rate of termination when the firm’s 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, is“retirement.” For these observations, if the employee’s 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 employee’s unvested options (Unvested IV – Pos and Unvested IV – Neg,respectively). We expect the rate of termination is also related to the employee’s 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 year’sperformance 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 employee’s options due to voluntary (involuntary) terminationfrom the company and is zero otherwise. Old is the employee’s 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 employee’s 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 employee’s 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 employee’s unvested options if thisamount is positive and is zero otherwise; Unvested IV – Neg is the natural logarithm of one plus the absolutevalue of the intrinsic value of the employee’s unvested options if the intrinsic value is negative and is zerootherwise; MidMgt is a dichotomous covariate equal to one if the employee’s largest option grant outstandingis within the 85th and 95th percentiles of all option grants and is zero otherwise; SrMgt is a dichotomouscovariate equal to one if the employee’s largest option grant outstanding is above the 95th percentile of alloption grants and is zero otherwise; DaysLeft is the number of trading days remaining until options expire. Panel A: Aggregated Voluntary Termination Model Estimates Dependent variable Aggregate Stat. Stat. VolTerm Mean 20th Pctle Median 80th Pctle z-statistic Pos. Neg. Old 0.009 0.003 0.011 0.019 6.62 8 0 Returnyr-1 0.060 −0.500 0.038 0.410 1.94 4 3 Returnyr-2 −0.077 −0.352 −0.152 0.292 2.06 4 3 FutureRet −0.088 −0.338 −0.002 0.356 −0.10 3 2 IV Future Vest – Pos −0.046 −0.060 −0.034 −0.006 −1.43 1 3 IV Future Vest – Neg −0.036 −0.037 −0.005 0.006 −0.48 1 1 Vested IV – Pos −0.061 −0.105 −0.075 0.001 −1.64 1 4 Vested IV – Neg −0.018 −0.062 −0.029 0.053 −1.76 2 3 Unvested IV – Pos 0.068 −0.044 0.037 0.144 −0.64 3 2 Unvested IV – Neg 0.070 −0.045 0.079 0.167 2.75 5 1 MidMgt −0.219 −0.441 −0.191 0.105 −0.16 2 1 SrMgt −0.458 −0.678 −0.314 −0.185 −0.99 0 2 DaysLeft 0.000 0.000 0.000 0.000 3.00 6 0 Intercept (Scale) −10.090 −12.091 −9.814 −8.141 −12.28 1 9 Shape (p) 1.109 1.004 1.075 1.282 1.38 4 1 Pseudo R-squared 0.334 0.144 0.356 0.444
- 40. Panel B: Aggregated Involuntary Termination Model Estimates Dependent variable InvolTerm Aggregate Stat. Stat. Mean 20th Pctle Median 80th Pctle z-statistic Pos. Neg. Old 0.046 0.014 0.022 0.034 9.83 6 0 Returnyr-1 −0.376 −0.741 −0.140 0.022 −0.60 1 2 Returnyr-2 0.078 −0.239 0.041 0.230 0.15 0 0 FutureRet −0.286 −1.005 −0.213 0.141 −0.94 0 2 IV Future Vest – Pos −2.204 −0.062 −0.043 −0.026 −3.92 0 1 IV Future Vest – Neg 0.022 −0.042 0.014 0.132 0.55 2 0 Vested IV – Pos −0.047 −0.087 −0.053 −0.023 −1.14 0 1 Vested IV – Neg −0.048 −0.138 −0.075 −0.062 −1.22 1 1 Unvested IV – Pos 0.144 −0.027 0.155 0.234 1.36 2 0 Unvested IV – Neg 0.150 −0.016 0.178 0.191 1.49 3 0 MidMgt −3.493 −0.689 −0.400 −0.157 −6.97 0 1 SrMgt −3.014 −1.086 −0.449 0.206 −1.77 1 2 DaysLeft −0.001 −0.001 −0.001 0.000 −1.28 0 2 Intercept (Scale) −12.761 −13.171 −12.479 −10.209 −7.61 0 5 Shape (p) 1.083 0.817 1.130 1.376 0.33 1 0 Pseudo R-squared 0.616 0.322 0.753 0.836Observations for each firm-specific estimation:Firm A B C D E F G H I JVolTerm = 1employee-days 60 411 89 634 101 122 935 358 677 355VolTerm = 0employee-days 362,774 979,808 335,065 2,330,091 386,623 449,756 1,208,679 446,965 848,268 587,968Total number ofobservations 362,834 980,219 335,154 2,330,725 386,724 449,878 1,209,614 447,323 848,945 588,323InvolTerm =1employee-days 0 92 0 233 5 74 78 19 10 0InvolTerm = 0employee-days 0 980,127 0 2,330,492 386,719 449,804 1,209,536 447,304 848,935 0Total number ofobservations 0 980,219 0 2,330,725 386,724 449,878 1,209,614 447,323 848,945 0
- 41. Figure 1Price Path Simulation Process Simulated price paths before and after grant dates (to average-age male middle managers),assuming geometric Brownian motionBegin Price Sim Grant Expiration Date End Price SimDate = t0 − 500 Date = t0 T = t0 + 2,500 Date = T + 250 S0 = average dSt = −0.05Stdt + σStdWt firm price on dSt = 0.05Stdt + σStdWt grant dates to male middle managers
- 42. Table 1Summary Statistics Industry describes the primary industry in which the firm operates. Start Sample Coverage is the first date in which our sample has data (provided bythe firm) that tracks employee option grant activity. End Sample Coverage is the last date in which our sample has data (provided by the firm) thattracks employee option grant activity. Employees is the sample period mean of Compustat DATA29. Market Cap is the sample period mean ofCompustat DATA25 multiplied with Compustat DATA199. Market-to-book is the sample period mean of Market Cap scaled by Compustat DATA60.Revenue is the sample period mean of Compustat DATA12. Revenue Growth is the sample period mean of Revenuet/Revenuet-1 – 1. Net Income issample period mean of Compustat DATA172 . Years Public is the number of years for which the company was publicly traded (as reported on thefirm’s website) as of the End Sample Coverage date. Age is the company’s age in years (as reported on the firm’s website) as of the End SampleCoverage date. Option Term is the distance in years between the grant date and the expiration date. Employee-grantees is the number of differentemployees who receive a grant during the sample period. Grants is the total number of grants during the sample period. Positive Intrinsic Value is thepercentage of Grants for which firm price exceeds strike price at any single point in time before expiration. Exercises is the number of economicallymeaningful exercises (i.e., at least 25% of the options available to exercise and at least 10% of the total options in the grant). Misc. Exercises is thenumber of exercises that do not meet the economically meaningful threshold to be included in the analysis. Modal Annual Vesting Rate is the mostcommonly observed grant vesting rate. Terminations is the number of observed voluntary or involuntary employee terminations during the sampleperiodFirm A B C D E F G H I JPanel A: Firm characteristicsIndustry Financial Services Healthcare Education Industrial Technology Technology Technology Technology TechnologyStart Sample 07/01/1993 11/08/1989 8/31/1988 07/26/1988 11/09/1983 08/15/1991 06/13/1983 04/03/1999 01/25/1994 10/10/1990CoverageEnd Sample 05/27/2005 07/05/2005 05/26/2005 06/09/2004 05/31/2005 05/12/2005 06/07/2005 11/21/2005 03/08/2005 08/31/2005CoverageEmployees 2,700 6,700 900 3,100 4,000 1,200 1,300 1,500 800 800Market Cap 900 963 282 1,025 274 1,315 479 946 3,414 208($mill.)Market-to- 1.03 2.69 3.38 16.45 1.72 5.35 3.78 2.50 13.21 0.67BookRevenue 1,146 1,495 64 259 610 264 248 446 218 121($mill.)Revenue 8.76 16.07 25.10 11.51 11.10 24.02 27.24 25.45 24.83 16.59Growth (%)Net Income 76 37 7 25 18 49 5 34 −47 3($ mill.)Years Public 33 19 14 14 33 14 15 6 14 12Age 89 20 25 40 60 37 28 31 22 31

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