the pricing of risk and sentiment: a study of executive...

The Pricing of Risk and Sentiment:A Study of Executive Stock Options

Charles Chang, Li-jiun Chen, and Cheng-der Fuh∗

Option pricing models accounting for illiquidity generally imply the options are valued at adiscount to the Black-Scholes value. Our model considers the role of sentiment, which offsetsilliquidity. Using executive stock options and compensation data from 1992 to 2004 for S&P 1500firms, we find that executives value employee stock options (ESOs) at a 48% premium to theBlack-Scholes value. These premia are explained by a sentiment level of 12% in risk-adjusted,annualized return, suggesting a high level of executive overconfidence. Subjective value relatesnegatively to illiquidity and idiosyncratic risk, and positively to sentiment in all specifications,consistent with the offsetting roles of sentiment and risk aversion.

In a world where diversification has relatively little cost of where diversified assets are tradeable,risk-averse investors require higher returns as compensation for risks associated for illiquidity.For those with investments in illiquid assets, however, illiquidity costs may be offset by positiveprivate information or confidence in future prospects where one believes future returns willoutpace the market’s expectations and provide the necessary risk compensation. One product forwhich this tradeoff can be explicitly modeled is employee stock options (ESOs). Because of theilliquid nature of ESOs, the value perceived by employees (henceforth “subjective value”) maybe quite different from the cost of issuance (the market or “objective” value). Academia has putforth a number of approaches and modeling techniques to account for this difference, virtually allof which conclude that the subjective value of ESOs should be less than the usual Black-Scholesvalue. Empirical evidence of the same, however, has been elusive owing to the lack of a clearclosed-form solution and appropriate data to apply. Indeed, if ESOs are generally worth lessto an employee than its market value, why do employees continue to covet options as part oftotal compensation when doing so implies less cash compensation? One possible explanation isthat employees believe the market to have undervalued the options either because they possesspositive private information and/or suffer from behavioral overconfidence regarding the futurerisk-adjusted returns of the firm (henceforth termed “sentiment”). If employees believe that thefirm will generate positive risk-adjusted returns over and above that which is priced into the

We would like to take this opportunity to thank Bill Christie (Editor) and an anonymous referee for excellent commentsthat helped to shape and refine this work. We are also grateful to seminar participants at the National Taiwan Universityand Peking University finance seminars as well as at the Financial Engineering and Risk Management InternationalSymposium, and the Conference on Statistical Models and Methods in Quantitative Finance and Related Topics for helpfuldiscourse. This work would not have been possible without the early research assistance of Ya-hui Hsu and the supportof National Science Council (Taiwan) grants NSC 99-2811-M-008-031 (Chang), NSC 100-2811-M-008-069 (Chen), andNSC101-3113-P-008-005 and NSC 100-2118-M-008-002-MY3 (Fuh). All remaining errors are ours.

∗Charles Chang is an Associate Professor at the Shanghai Advanced Institute of Finance and an Adjunct AssociateProfessor of Finance at the Chinese University of Hong Kong. Li-jiun Chen is a Post Doctorate at the National CentralUniversity in Chungli, Taiwan. Cheng-der Fuh is a Professor of Statistics at the National Central University in Chungli,Taiwan.

Financial Management • Spring 2013 • pages 79 - 99

80 Financial Management � Spring 2013

options, even as illiquidity tends to generate a discount, sentiment effects may make ESOs asdesirable as, or even more desirable than, the equivalent market value in cash.

Applying a comprehensive set of executive options and compensation data, we are able toexplicitly test these notions and price illiquidity and sentiment. The data set used includes13 years of executive options issuances in the United States and nearly 82,000 observations. Theapplication of executive options data in particular is noteworthy as options issued to executives areparticularly illiquid, are generally a larger portion of total income than those offered to rank andfile employees, and are most closely monitored by regulatory officials.1 In addition, executivesare most likely to believe that they have private information. Each of these characteristics willtend to generate relatively pronounced effects for this subset of assets. The specificity of the dataallows us to compute the proportion of total income that is attributed to options each year foreach executive. Our data also includes information for each executive’s title, rank, and industry,allowing for relatively specific parsing of the data. This information is necessary as as we thensplit the data into groups by title, year, and industry. We further control for the size of eachfirm measured by the firm’s total market value, number of employees in each firm, nonoptioncompensation, and the immediate exercise value of the option using the K-means approach forhierarchical clustering to split executives into comparative groups. Then, by assuming that allexecutives within the same cluster receive the same total compensation, we are able to calculatethe implied subjective value each employee places on their options as compared to the averagecompensation in their employment cluster, a notion described in detail later. It is this subjectivevalue that we relate to key variables, including investor sentiment, in our main tests. In contrast,Bergman and Jenter (2007) analyze options issuance, positing without the aid of a model thatoptimism should coincide with more issuances. Related work by Oyer and Schaefer (2005) alsolimits its investigation to issuance policy, not pricing. As we demonstrate here, issuance behaviorand subjective value are not closely related. While the former has fluctuated a great deal over ourperiod of study, the latter has been relatively stable.

A number of papers address employee valuation of options using extensions of the Black-Scholes model. Mozes (1995), Hull and White (2004), Carpenter (1998), and Bettis, Bizjak,and Lemmon (2005) examine early exercise and its impact on standard American option pricingmodels. Huddart and Lang (1996), Hemmer, Matsunaga, and Shevlin (1996), and Core andGuay (2001) further link early exercise behavior to under-diversification of employees, but donot explicitly price the premium associated with under-diversification. Lambert, Larcker, andVerrecchia (1991) and Hall and Murphy (2002) estimate the subjective value of ESOs througha certainty equivalent approach, finding it to be lower than market value owing to exogenouslyconstrained fixed holdings in the underlying stock. Ingersoll (2006) also considers the effectsof fixed holdings, presenting a constrained portfolio problem where employees allocate wealthbetween the company’s stock, a market portfolio, and a risk-free security. Each paper, however,presents different modeling limitations on the underlying stock diffusion process and none ofthem models the role of employee sentiment.

We test the model and framework presented by Chang, Fuh, and Hsu (2008) that provides aclosed-form solution for ESO value where investors choose between a risk-free asset, the marketportfolio, and the underlying stock. We calculate partial derivatives from that model, apply it toexecutive compensation data, and are able to empirically and explicitly price both the subjectivevalue discount created by illiquidity and the risk-adjusted excess returns necessary for employees

1Much of the literature in the study of subjective value and sentiment, including that of Oyer and Schaefer (2005) andBergman and Jenter (2007), studies rank and file employees.

Chang, Chen, & Fuh � Pricing of Risk and Sentiment 81

to accept options in lieu of equivalent cash compensation (i.e., the sentiment effect).2 We findthat subjective value is significantly higher than the Black-Scholes value in all but one sectorsuggesting a substantial role for sentiment. In fact, we find that executives, on average, valueESOs at a premium of nearly 48% indicating extremely high levels of sentiment. Although Hodge,Rajgopal, and Shevlin (2009) similarly find in a survey of mid- and entry-level managers thatsubjective values exceed Black-Scholes values, virtually all options pricing models concludethat subjective value should be lower than market value.3 The inclusion of a sentiment variable,however, resolves this puzzle as we find that the average executive prices 12% risk-adjustedexcess return over the expected return of the stock into the ESO value. In other words, theybelieve the firm will significantly outperform the market’s expectations and, as such, value theoptions more highly than the market despite the illiquidity discount.

Also, we find that subjective value is positively related to sentiment level and negatively relatedto the proportion of total wealth held in illiquid firm-specific holdings, even after controlling forkey options pricing variables such as time to maturity, volatility, dividend payout, and whetherthe option is in or out of the money. These results are in accord with the most unique predic-tions of our model, are statistically significant, and suggest that while risk aversion generatesa discount in subjective value, positive sentiment offsets it. As a proxy for sentiment, we com-pute the previous year capital asset pricing model (CAPM) alpha and find that it is positivelyrelated to subjective value, implying higher sentiment levels generated by stronger prior yearperformance. More importantly, we separate our data into “insider” and “true sentiment” groupsbased on whether the sign of sentiment is the same as that of the resulting returns. If they are thesame, we consider these executives “informed” rather than behaviorally biased as per traditionalsentiment-based arguments. We find that sentiment is positively related to subjective value inboth subsets indicating sentiment increases subjective value even if subsequent returns are not inconcert with ex ante sentiment. However, the effect is more pronounced for insiders than for truesentiment. Finally, we also apply Fortune Magazine’s list of Top 100 firms to work as a proxy forsentiment under the assumption that employees of such firms are generally more optimistic abouttheir work environment and prospects.4 Again, these firms enjoy substantially higher subjectivevalues, though generally lower same year returns. These results hold despite numerous variablerespecifications, controls for outliers, and controls for industry effects. The jump specificationused also does not significantly impact our results.

Interestingly, subjective value may be either positively or negatively related to volatility. Theformer result can be explained by the convex payout of the option. DeFusco, Zorn, and Johnson(1991), Nohel and Todd (2005), and Ryan and Wiggins (2001) find that options values increasewith risk. The latter, however, arises because as risk increases, the risk premium related to theunder-diversification caused by illiquidity also increases. The theoretical construct presented inChang et al. (2008) is capable of capturing this result, and Carpenter (2000) and Ross (2004)present examples where convex incentive structures do not imply that the manager is more willingto take risks. Here, we demonstrate that depending upon the parameterization, this correlationmay either be positive or negative, an important departure from the traditional Black-Scholes,

2Chang et al. (2008) do not provide the partials for sentiment or alpha, two of the more critical variables tested empiricallyhere. Moreover, in the interest of completeness, we also generalize the jump specification to allow for bivariate constantand double exponential jump size processes though this does not affect our qualitative results.3While Hodge et al. (2009) uses mid- and entry-level managers, we investigate executives. Interestingly, that paper findsthat risk aversion and stock volatility do not significantly impact subjective values, possibly because of the relativelysmall proportion of total income that options constitute for lower level managers.4Similarly, Cohen (2009) uses this ranking as a measure of loyalty.


moral hazard result. Specifically, we find that there is a strong negative relationship betweensubjective value and idiosyncratic risk. This result contrasts the findings of Chok and Sun (2007),where the authors identify a positive correlation between stock options value and idiosyncraticvolatility when looking at Biotech stocks.

The remainder of this paper is organized as follows. Section I presents our model and SectionII describes our data. Section III proposes an empirical methodology and discusses our results.Section IV addresses robustness checks, while Section V provides our conclusions.

I. ESO Valuation Model

From Chang et al. (2008), we have the following three asset economy where stock price followsa jump-diffusion process:

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

dS

S= (μs + Sen − ds − λsks)dt + σsdW m + νsdW s + (Ys − 1)dqs

dM

M= (μm − dm)dt + σmdWm

dB

B= rdt.

(1)

All variables are defined as per Chang et al. (2008) and appear in the Appendix. Most impor-tantly, sentiment level is denoted by Sen. In other words, the employee overestimates or rationallyadjusts the risk-adjusted return of the company owing to inside information by Sen, which may beinterpreted as a CAPM alpha, a notion we test empirically. The two Brownian motions and the jumpprocess are presumed mutually independent, and the employee’s utility function is assumed to beU (C) = Cγ /γ with a coefficient of relative risk aversion R(C) = −CU ′′(C)/U ′(C) = 1 − γ .Under these assumptions, the subjective value of the ESO is

Fs(S, t) =∞∑j=0

(λsξτ ) j e−λsξτ

j!

{S(t)e−ds

s τ E∗[

j∏i=0

Ys,i�(ds

1

)] − K e−rsτ E∗ [�

(ds

2

)]}. (2)

Taking partial derivatives, we generate the testable predictions of this model. Details can befound in the Appendix.

Subjective value relates positively to stock price, but negatively to dividend payout and strikeprice. These relationships hold in general for the Black-Scholes value of options as well andare not surprising. The critical new variables evaluated in these formulas are sentiment and α,which is defined as the proportion of total wealth that is held in illiquid firm-specific holdings.In this case, α is the illiquid suboptimal holding that the investor holds by accepting ESOs as apart of their compensation package. As shown in the partial derivatives given in the Appendix,subjective value relates negatively to α in all but the knife-edge case where Sen is extremelynegative (i.e., employees are severely pessimistic regarding the outlook for the firm). Simulationresults, available upon request, indicate that the relationship only fails to be negative when negativesentiment reduces subjective value (and, as such, the portfolio weight of the stock) to zero, notan economically meaningful case. Subjective value always relates positively to Sen.

Interestingly, the sensitivity of value to time to maturity τ can be either positive or negative,despite being generally positive in the Black-Scholes setup. This is related to our assumption


that the time to maturity is equal to the vesting period.5 The usual intuition that longer timeto maturity translates into larger time value attributed to the option is offset by the larger riskpremium associated with holding a suboptimal holding for a longer period of time since thevesting period is also longer. Empirically, this correlation will hold so long as the vesting periodis substantively positively related to the time to maturity, an observation addressed later.

Another finding that differs from that of traditional models is that subjective value may notbe positively related to risk, contrary to the convexity and moral hazard arguments. Considerthe partial derivatives with respect to the risk components, where we assume total, idiosyncratic,and jump risks are mutually independent following Ingersoll (2006) and Chang et al. (2008).With respect to total risk, we have the usual finding that the partial is positive (i.e., greater risk,great options value owing to its convex payout). However, with respect to idiosyncratic risk,holding other values constant, we find the opposite. The partial is negative. While this result mayseem counterintuitive, it is consistent with the role of risk aversion that lies at the foundation ofsubjective value’s illiquidity discount as follows.

When employees do not face portfolio restrictions, they allocate wealth between the marketportfolio and risk-free assets and do not make additional investments in individual firm stock.Idiosyncratic risk is diversified away and does not affect the options value so long as total riskis constant. If increases (decreases) in idiosyncratic risk also increase (decrease) total risk, theoptions value will also increase (decrease). However, if some portion of employees’ firm-specificholdings are constrained/illiquid, idiosyncratic risk cannot be fully diversified. As idiosyncraticrisk increases, the risk-premium associated with holding the asset likewise increases and sub-jective value decreases, assuming that total risk remains unchanged.6 This important distinctionsuggests that increasing firm-specific risk may, in fact, reduce the value of the ESO. This may actto reduce the effort and value creation incentives intended by options issuance. Alternatively, theconvex payout of options may also lead to excessive risk-taking and moral hazards, which wouldlikewise be discouraged in our model. Moreover, this correlation is particularly interesting sinceidiosyncratic risk, as opposed to market risk, is most directly affected by management. Meulbroek(2001) identifies nonsystematic, firm-specific risk as more costly to managers. These relationsare tested empirically in detail in our main findings.

Finally, regarding jump risk, the partial derivative with respect to jump frequency is indetermi-nate in sign, while that related to jump size is negative. In order to thoroughly investigate the roleof jumps, we test a variety of jump size models: Y = 0 (no residual value), double exponential,and bivariate constant jump sizes using jumps of ±7%.7

II. Data and Preliminary Results

While our modeling specification is similar to Chang et al. (2008), ours is the first to testsuch a model to empirically price illiquidity and sentiment. Data for this study are collected fromthe Compustat Executive Compensation (Execucomp) database. From this database, data for allexecutive stock options issued from 1992 to 2004 are collected including strike price K, maturitydate T , volatility Vol, and the number of options issued. In addition to options data, we collect total

5Not detailed in this study, we further find that holding vesting period constant, subjective value is monotonicallyincreasing in time to maturity.6If, however, total risk increases with idiosyncratic risk, the extent to which a resulting increase in options value is offsetby the under diversification discount depend upon the level of illiquidity in the holding. Empirically, we find that thepartial is negative 93.4% of the time and is both statistically and economically significantly negative.7The 7% relates to an approximately two standard deviation daily return during our test period.


compensation data from Compustat including salary, bonus, options, and other income earned byexecutives each year. The compensation data are first used to calculate subjective value. Then,we apply our model and the appropriate options data to back out sentiment levels.

A. Calculating Subjective Value

We calculate the subjective value of ESOs implied by total compensation packages using aK-means hierarchical clustering methodology to split executives into like groups based uponindustry, rank, year, the firm market value, nonoption compensation, and the immediate exercisevalue of the options.8 The number of groups is decided by a cubic clustering criterion andthe average total compensation is calculated. Then, assuming that all executives within thesame cluster receive the same total compensation, for each executive in this cluster, the impliedsubjective value is derived by comparing the difference between nonoption compensation and theaverage compensation. We then set all negative implied ESO values equal to zero and recalculateaverage compensation in each cluster with these subjective values, repeating until the total sumof changes in subjective values in a given cluster is less than 0.01. This eliminates some negativesubjective values such that the final number of negative or zero values is about 5.7% of our dataset. Worth noting is the observation that, even in the first iteration of the process, after grouping,only about 7.9% of our data have options with a negative or zero value, lending credence to thestability of our groupings. Without clustering, using the entire panel as a single group, 9.2% arenegative values. While the improvement to 5.7% may seem modest, clustering also reduces thestandard deviation of subjective values by a factor of 5.9

To illustrate this intuition, presume that all executives within the same cluster receive thesame compensation on average, where any differences in salaries, bonuses, and other incomeshould be accounted for by options. If a chief executive officer’s (CEO’s) average total annualcompensation in a given year is $2,000,000, a particular CEO who receives $1,500,000 in salaryplus bonuses must then value options awarded to then at $500,000 in order to agree to continuedemployment. More importantly, it may be the case that the market value of these options is only$100,000, but the CEO subjectively values them at $500,000 because they believe the market tohave undervalued the options. While this method of calculation is clearly not perfectly precise,we offer numerous robustness checks using different grouping criteria, all of which arrive atqualitatively identical results. Included in these checks, we control for potentially systematicdifferences in compensation level related to α (the percent of total wealth held in illiquid firm-specific holdings). Some intangible sources of value such as training, learning opportunities, andadvantageous work environments are not controlled here, but may be relatively unimportant giventhat this is an executive database of listed firms.

As can be seen in Panel A of Table I, the mean total annual compensation for executivesin this data set is a bit over $2 million with a median of just over $1 million. The mean andmedian ESO compensation numbers are roughly $1.2 and $0.4 million, respectively. Not surpris-ingly, chief executives who were also board members received the highest compensation ($4.2million), but options are a substantial portion of that compensation ($2.4 million). Indeed, optionscompensation generally substantially outweighs all other forms of compensation.

8Gabaix and Landier (2008) find that total market value as a proxy for firm size has the strongest predictive poweron compensation. We, however, rerun all tests using the number of employees as the size proxy and find qualitativelyidentical results.9It is important to note that without clustering, high salary recipients (a small part of the data) generate negative subjectivevalues. However, a large group of low salary employees have extremely high subjective values resulting in a systematicbias in subjective value. We conclude that some method of clustering is essential to eliminating this bias.


Table I. Summary Statistics for Employee Stock Options and Compensation

Panel A presents summary statistics for compensation data for four categories of executives: 1) board andCEO (B&C), 2) board and not CEO (B&NC), 3) not board and CEO (NB&C), and 4) not board and notCEO (NB&NC). Numbers are reported in thousands and LTIP represents the long-term incentive pay. Theaccompanying box plots illustrate the natural log of total compensation for the two largest industries in oursample. We group executives according to position, the firm’s total market value, nonoption compensation,and the immediate exercise value of the options for each industry by hierarchical clustering using a K-means approach. Panel B presents summary statistics for ESOs used in this study. Sok, τ , Vol, Div, andN are the ratio of stock price to exercise price, time to maturity, implied volatility, dividend yield, andnumber of executives/firm/year to receive options, respectively. BSOPM is the options value calculatedusing Black-Scholes and α is the proportion of total wealth held in illiquid firm-specific holdings.

Panel A. Compensation Summary Statistics

Aggregate Mean By Title

Mean Median Std. Dev. B&C B&NC NB&C NB&NC

Salary 365 300 234 556 481 335 286Bonus 336 151 816 650 479 289 222Other annual 24 0 179 44 35 22 16All other total 70 11 540 94 131 50 45LTIP 77 0 442 127 128 72 48Restricted stock 163 0 803 366 220 184 101Options 1,178 378 3,407 2,382 1,683 1,264 748Total 2,214 1,074 4,262 4,219 3,158 2,217 1,465

Panel B. ESO Summary Statistics

Sok τ Vol Div α BSOPM N

Mean 1.012 9.308 0.431 1.37% 0.353 13.09 4.39Median 1.000 9.668 0.370 0.62% 0.307 11.05 5.00Std. dev. 0.434 1.728 0.243 1.77% 0.227 10.25 1.30Max. 37.50 25.51 4.120 20.39% 1 328.68 11Min. 0.230 0.100 0.102 0.00% <0.001 <0.001 1


The accompanying figures present box plots of the natural log of total compensation for thetwo largest industries in our sample: Consumer Discretionary and Information Technology. Withthe exception of some outliers, which are subsequently removed in our main tests, the boxedareas generally do not overlap from cluster to cluster demonstrating the relative homogeneity offirms within each cluster and generally distinctly separated from other clusters. As a result, webelieve that compensation characteristics within each cluster should be quite comparable lendinga measure of credence to our method of calculating subjective value.

B. Options Data and Preliminary Results

Execucomp provides the data necessary to calculate options prices using our model. Whethereach option is in or out of the money (Sok) is measured as the stock price at issuance divided bythe strike price. If the option is in (out of) the money, Sok is greater (less) than one. The timeto maturity is denoted τ , volatility Vol, and dividend yield Div. Following Dittmann and Maug(2007), we further define the net cash inflow (NCash) for each year as follows:

NCash = Fixed salary (after tax)+ Dividend income from shares held in own company (after tax)+ Value of restricted stock granted− Personal taxes on restricted stock that vest during the year+ Net value realized from exercising options (after tax)− Cash paid for purchasing additional stock.

Fixed salary is the sum of five Compustat data types: 1) Salary, 2) Bonus, 3) Other Annual, 4)All Other Total, and 5) Long-Term Incentive Pay (LTIP).10 Denote the year when the executiveenters the database by tE. The executive’s cumulative wealth for year t is then Wt = NCasht +∑t−1

=tENCash

∏ts=+1 (1 + r s

f ). In other words, assume that the executive has no wealth prior toentering the firm. All NCasht are realized at the end of the fiscal year and invested at the risk-freerate r t+1

f during the next fiscal year. Then, α is the sum of all illiquid firm-specific holdingsincluding unvested restricted stocks and options divided by total cumulative wealth. Included intotal wealth is the aggregate value of previously issued vested in-the-money options calculated,in accordance with ExecuComp and the literature, as the immediate exercise value (i.e., year-endstock price less the strike price). The aggregate value of unvested options is similarly calculated asthe immediate exercise value of the options. In both cases, out-of-the-money options are ignored.Respecifications of options price and α are discussed in robustness checks, though none give riseto any significant qualitative departures.

Note that, when calculating options value, time-to-maturity τ is reduced by 30%, and nor-mal unsystematic volatility is calculated as two-thirds implied volatility following calibrationsapplied by Compustat and by the majority of papers in this area (Bryan, Hwang, and Lilien,2000; Aggarwal and Samwick, 2003; Bettis et al., 2005; Ingersoll, 2006). We address alternateapproaches to calculating α in robustness tests including an iterated approach that synchronizes α

and subjective value simultaneously. Qualitative findings with respect to sentiment are identical.Summary statistics for each of these variables are shown in Panel B of Table I. While the

median option is issued at the money, the mean is in the money (Sok = 1.012). Note thatvirtually all options are issued at the money (Sok = 1). This is true for about 90% of our data

10For cash paid for purchasing additional stock where direct data are unavailable, we use the change in stock holdingstimes the year-end stock price to calculate this value.


set. As a robustness check, we also try using Sok = 1 for all issues, as well as omitting allfirms for which Sok is not equal to one, removing Sok as a variable, and find no qualitativedifferences. Average time to maturity is about 9.3 years and α is about 35%, implying that theilliquid firm-specific holdings account for more than one-third of executive total wealth.11,12

Median values of other model parameters are Vol = 0.37 and Div = 0.62%. The risk-free rateis r = 5.3%, and following Duffee (1999) and Fruhwirth and Sogner (2006), which use USand German bond data, respectively, we estimate the median default intensity as λs = 0.01.We use the common parameterization for the coefficient of relative risk aversion 1 – γ = 2.Throughout our regression analysis, we remove outliers using a standardized residuals approach,removing those with residuals greater than 3 or less than –3. In all, about 0.05% of our sample isremoved.

III. Main Findings

A. Preliminary Findings

Substituting the subjective value implied by compensation data into our model, along withthe options variables given in our data set, we are able to back out sentiment levels Sen. Theresults are presented in Table II. There are about 105,000 options issued by each firm (AvgIss)over the test period with a total of nearly 2,700 firms and 82,000 total observations accountedfor. Industry breakdowns, while exhibiting some fluctuations in point estimates, demonstrate thatresults across industries are qualitatively similar. While the mean Black-Scholes value of optionsBSOPM is about $13.09 with some variation across industries, the mean subjective value Subis more than $19.38, reflecting a 48% premium. That is, although virtually all of the theoreticalliterature implies a subjective value discount, the empirical data indicate that executives generallyvalue ESOs more highly than their Black-Scholes values. Though not reported in the table, t-testsreport that subjective values are statistically significantly higher than Black-Scholes values at the1% level for almost all industries and in aggregate. The only exception is the others industry,where Sen is still significantly positive, but Sub is about equal to BSOPM owing to a particularlyhigh α in this industry.

Given the large proportion of executive income that is attributed to illiquid, firm-specificoptions holdings, this finding suggests substantial overconfidence or positive inside informationregarding their firm’s future prospects. The data confirm that the average executive prices ESOssuch that the firm should outperform the market’s expectations by an average of 12% per annum(Sen); t-tests indicate that these values are significantly different from zero at the 1% level in allindustries and in aggregate.

Panel B reports the mean and median values of Rt and Sub in each subsample, where Rt isthe CAPM alpha. Top is a dummy variable taking a value of one if the executive works for afirm listed in Fortune Magazine’s top 100 companies for which to work. Our results indicate thatfirms with higher previous year returns tend to have significantly higher subjective values. Thisis true of both the mean and median value. Interestingly, subsequent return momentum is notconsistently present in these data, at least in regard to mean values. Firms listed in the top 100

11For some issues for which there is no time stamp, we assume an issuance date of July 1 since this would be the middleof the fiscal year for the vast majority of firms.12Holland and Elder (2006) find that rank and file employees exhibit an α close to 10% and concur that subjective valueis decreasing in α due to risk aversion and under diversification.


Table II. Preliminary Statistics for Subjective Value and Sentiment

Panel A presents, by industry, Black-Scholes value BSOPM , subjective value Sub, sentiment level Sen,average total compensation AvgCom, number of options issued AvgIss, and number of observations byindividual Obs. AvgCom and AvgIss are reported in thousands. Sen is calculated using our model wherethe distribution of jump size follows y = 0. Con. Dis., Con. Sta., Inf. Tec., and Tel. Ser. refer to ConsumerDiscretionary, Consumer Staples, Information Technology, and Telecommunication Services, respectively.Panel B reports the mean and median values of Rt and Sub in each subsample where Rt is the CAPM alphaat time t. Top is equal to one if the firm is listed as a top 100 firm by Fortune magazine in a given year.The p-values measure the significance of difference tests. AvgIss and Sub/BSOPM are graphed in the lastfigure. The y-axis of the histogram is on the left and that of the line chart is on the right.

Panel A. Summary by Industry

Sector BSOPM Sub Sen AvgCom AvgIss Obs

10 Energy 12.589 16.239 0.089 1,774.08 78.46 4,30715 Materials 10.822 17.613 0.076 1,399.90 61.89 6,41220 Industrials 12.569 20.305 0.115 1,591.53 70.64 12,13425 Con. Dis. 12.177 19.799 0.106 2,030.63 98.64 15,92530 Con. Sta. 12.053 18.121 0.076 2,405.29 111.40 4,34735 Health Care 16.290 21.098 0.115 2,338.68 105.61 8,88340 Financials 13.440 21.770 0.066 2,892.28 99.37 10,44145 Inf. Tec. 15.853 18.499 0.229 2,748.47 164.53 14,61450 Tel. Ser. 12.768 22.941 0.162 5,310.29 272.27 1,32455 Utilities 5.475 14.633 0.063 1,370.11 67.66 3,931

Others 9.487 9.583 0.208 1,360.34 111.24 56Total 13.088 19.385 0.120 2,213.73 105.13 82,374

Panel B. Summary for Subjective and Sentiment

Rt−1 > 0 Rt−1 < 0 p-Value Top = 1 Top = 0 p-Value

mean(Rt) 0.00042 0.00041 0.4493 0.00038 0.00059 <.0001median(Rt) 0.00031 0.00028 0.0176 0.00030 0.00044 <.0001mean(Sub) 21.4131 15.6336 <.0001 24.7420 17.6169 0.0003median(Sub) 14.1350 10.9487 <.0001 17.4147 10.9794 <.0001

make significantly lower risk-adjusted returns in the year in which they are so listed.13 However,they enjoy substantially higher subjective value. This indicates that sentiment may generally beindependent of performance, but does significantly affect subjective value.

13Edmans (2011) uses the same rankings to investigate long-run returns and finds that those on the list enjoy higherreturns. However, this may be explained by differences in time series and other factors as it spans 1984-2009. In contrast,Statman, Fisher, and Anginer (2008) investigate another Fortune list, “America’s Most Admired Companies,” and findthat stocks on the list have lower returns from 1983 to 2006.


The accompanying time-series figure confirms that relative subjective values are greater thanone, but relatively stable over time. In contrast, the number of issuances generally increases. Theindustry with the second highest subjective values (Financials) has a below average number ofissuances. These observations highlight the importance of looking at pricing, rather than issuancealone, as high subjective values do not imply that ESOs will be a more popular financing tool.

B. Regression Results and Variable Sensitivities Implications

We now shift our attention to the testable implications of our model, namely confirmingthe relationship between key options variables and subjective value. Specifically, we apply thefollowing regression equation:

Sub = Int + βαα + βSen Sen + βSok Sok + βτ τ + βVolVol + βDivDiv + ε, (3)

where Int is the intercept term and all variables are defined as before. Note that for all resultspresented here, we use clustered standard errors by firm to calculate significance, although theordinary least square (OLS) results are nearly identical.

First, we apply subjective value per share Sub as the dependent variable. For the first threetests in Panel A of Table III, we use the CAPM risk-adjusted alpha from the year prior to optionissuance, Rt–1, as a proxy for sentiment under the conjecture that those stocks that performedbetter in the previous year generate more positive sentiment prior to options being issued. Notethat our model implies that only the risk-adjusted excess return should be priced since the marketportion of the firm’s return is eliminated via the risk-neutral measure. In contrast, Bergmanand Jenter (2007) test the gross prior year return. Since a year’s worth of data is required tocalculate these alphas, the data set is reduced to about 57,000 observations. We find that α issignificantly negatively related to subjective value. This coincides with our intuition that thelarger the proportion of one’s portfolio held in options, the less diversified the portfolio and theless valuable the ESO. Alternatively, Sen is positively related and significantly so. In other words,positive sentiment is associated with higher subjective value. Note that these results control forthe usual options pricing factors. While Div is significantly negative related as expected, Sokand τ are not consistently significantly related. The lack of a positive coefficient on time tomaturity is likely the result of the link between the vesting period and maturity. As demonstratedin our theoretical discussion, which assumes the two are equal, the additional real option valueof lengthened maturity is offset by the increased risk premium of a longer vesting period. Sincematurity must be equal to or greater than vesting, a close link in the two values may lead to anambiguous relationship between time to maturity and options value, a nontrivial characteristic ofESOs highlighting the importance of taking the vesting period and illiquid holdings into account.Finally, Vol is negatively related. As explored more fully later, this last negative correlation isquite telling and is consistent with our model as the sensitivity of subjective value to idiosyncraticrisk is negative.

We further split the data into two groups according to the sign of the product of Rt and Sen. Apositive (negative) sign implies that the positive sentiment measure is (not) accompanied by strongperformance. The positive case (what we call “insider”) can be explained by nonsentiment-relatedfactors. The executive may have private inside information affording them the ability to forecastfuture returns. They also have the ability to affect future returns, so that optimism may be self-fulfilling. When the sign is negative (what we call the “true sentiment” case), inside informationis not likely to play a role since this result implies that positive (negative) sentiment is followedby poor (good) performance. As it turns out, we obtain similar results in both cases. Sentimentis positively related to the subjective value, while α is negatively related, both significantly so.


Table III. Regression Results for Subjective Value

This table presents the estimated coefficients from the following regressions:

Sub = Int + βαα + βSenSen + βSokSok + βτ τ + βVolVol + βDivDiv + ε,

where Sub, Int, α, Sen, Sok, τ , Vol, and Div refer to the subjective value, intercept term, proportion oftotal wealth held in illiquid firm-specific holdings, sentiment, ratio of stock price to exercise price, timeto maturity, implied volatility, and dividend payout, respectively. In the first three tests of Panel A, Sen =Rt–1, the CAPM alpha. We split the data into two groups according to the sign of the product of Rt and Sen.When Sen correctly forecasts the sign of the CAPM alpha for a given year, we denote this as an “insider.”When Sen and Rt do not match in sign, we denote this “true sentiment.” In the fourth test, Sen is a dummyvariable that takes value one if the firm is in Fortune’s top 100 and zero otherwise. In the next three tests,Sen is calculated from our model with the distribution of jump size following y = 0, a double exponential,and a bivariate constant jump model, respectively. Panel B presents results where DIn is a dummy variabletaking a value of one if the event is insider and zero otherwise, DT takes a value of one if it is true sentimentand zero otherwise, and Sen is again defined as the CAPM alpha. p-Values are in parentheses.

Panel A. Sentiment vs. Subjective Value

Int α Sen Sok τ Vol Div Obs

Sen = Rt–1

Coefficient 1.8988 –0.2126 0.0356 0.0361 –0.2813 –0.3983 –0.0783 56602(p-Value) (<0.0001) (<0.0001) (<0.0001) (0.2476) (0.0489) (<0.0001) (<0.0001)

InsiderCoefficient 1.8650 –0.3001 0.0628 0.0860 –0.2883 –0.3409 –0.0844 23826(p-Value) (<0.0001) (<0.0001) (<0.0001) (0.0401) (0.1712) (0.0004) (0.0017)

True SentimentCoefficient 1.9071 –0.1252 0.0100 –0.0039 –0.2631 –0.4344 –0.0904 21333(p-Value) (<0.0001) (<0.0001) (<0.0001) (0.8583) (0.2858) (<0.0001) (<0.0001)

Sen = TopCoefficient 1.5014 –0.2244 0.0114 0.0802 –0.0560 –0.2318 –0.0807 49090(p-value) (<0.0001) (<0.0001) (0.0169) (0.6641) (0.7216) (<0.0001) (<0.0001)

Y = 0Coefficient 2.2013 –0.4364 0.0076 0.0424 –0.3265 –0.3872 –0.1012 82374(p-Value) (<0.0001) (<0.0001) (<0.0001) (0.1194) (0.3041) (<0.0001) (<0.0001)

Double ExpCoefficient 2.1671 –0.4270 0.0017 0.0429 –0.3056 –0.3786 –0.1005 82374(p-Value) (<0.0001) (<0.0001) (<0.0001) (0.1155) (0.3310) (<0.0001) (<0.0001)

Bivariate ConCoefficient 2.1799 –0.4267 0.0021 0.0428 –0.3075 –0.3883 –0.1023 82374(p-Value) (<0.0001) (<0.0001) (<0.0001) (0.1162) (0.3284) (<0.0001) (<0.0001)

Panel B. Insider vs. True Sentiment Events

Int α DInSen DTSen Sok τ Vol Div

Coefficient 1.9070 –0.2214 0.0372 0.0044 0.0343 –0.2866 –0.3872 –0.0876(p-Value) (<0.0001) (<0.0001) (<0.0001) (<0.0001) (0.3769) (0.0876) (<0.0001) (<0.0001)

While the impact of sentiment is larger in the insider case, sentiment plays a significant role inthe true sentiment case as well, so insider information is unlikely to explain the entirety of ourresults.

Next, we use the Top dummy as a proxy for sentiment. Once again, we find that sentiment issignificantly positively related to subjective value, while α is significantly negatively related. Allother relations are as mentioned above.


We also back Sen out of our pricing model under the aforementioned three different jumpsize assumptions. Since our model itself determines the association between subjective value andSen, the purpose of these tests is simply to observe the other variable relationships, as well asthe stability of the model to the specification of the jump process. We see that results are quiteconsistent across the three processes tested here. All other coefficients remain qualitatively asbefore with the coefficient of α, most importantly, remaining significantly negative in all cases.

Finally, in order to more clearly test the difference in impact of sentiment for insider versustrue sentiment events, we interact the event identification dummy with our sentiment proxy asfollows:

Sub = Int + βαα + βInSen DInSen + βT Sen DT Sen + βSokSok + βτ τ

+βVolVol + βDivDiv + ε.(4)

All variables are defined as formerly indicated and Sen is the previous period CAPM alpha. DIn

is a dummy variable that takes a value of one if the event is considered insider and zero otherwise.By analogy, DT takes a value of one if the event is true sentiment event and zero otherwise. Theresults appear as Panel B in Table III.

Note that while sentiment subjectively increases value significantly in both cases, the impactof sentiment when the event is likely to be an insider event is much larger. In other words,when strong prior performance reveals real information regarding future performance that maybe known to managers, the impact on subjective value is strong. When prior performance provesnot to be informative, the impact on subjective value is small. However, the impact is positiveand significant in both cases.

C. Normalized Results

As a normalization, we rerun all tests using the quotient subjective value divided by the Black-Scholes value. The results presented in the first three tests in Table IV confirm key findings.Note that relative subjective value is increasing in Sen and decreasing in α, significantly so inboth cases. That is, the more positive the sentiment, the higher the ESO value, while the largerthe illiquid holdings, the lower the ESO value. However, while the direction of the correlationsremains consistent for both the insider and true sentiment subsets, statistical significance isweaker now in the case of the insider subset. We find that Vol and Div are no longer reliablynegatively related to subjective value, perhaps because the Black-Scholes value now appears inthe quotient, negating the effects. Interestingly, τ is significantly negatively related to subjectivevalue since the longer one has to wait, the greater the risk caused by under-diversification affectssubjective valuation. Sok remains insignificant as before.

The main results are unchanged when Top is used as a proxy for sentiment, and the resultsare not significantly impacted by a change in the jump model used. All of the findings arequalitatively the same as without the use of normalization.

Finally, when testing the impact of insider versus true sentiment events, sentiment remainspositively related to subjective value. However, in true sentiment events, the coefficient is indis-tinguishable from zero. Again, the implication is that insider events dominate the effect.

D. Subjective Value and Risk

We now turn our attention to the sensitivity of subjective value to risk. While we note that ourmodel implies a positive correlation between total risk and subjective value, it further dictates that


Table IV. Regression Results for Normalized Subjective Value

This table presents the estimated coefficients from the following regressions:

Sub/BSOPM = Int + βαα + βSenSen + βSokSok + βτ τ + βVolVol + βDivDiv + ε,

where Sub, BSOPM, Int, α, Sen, Sok, τ , Vol, and Div refer to the subjective value, Black-Scholes value,intercept, proportion of total wealth held in illiquid firm-specific holdings, sentiment, ratio of stock price toexercise price, time to maturity, implied volatility, and dividend payout, respectively. In the first three testsof Panel A, Sen = Rt–1, the CAPM alpha. We split the data into two groups according to the sign of theproduct of Rt and Sen. When Sen correctly forecasts the sign of the CAPM alpha for a given year, we denotethis as an “insider.” When Sen and Rt do not match in sign, we denote this “true sentiment.” In the fourthtest, Sen is a dummy variable that takes a value of one if the firm is in Fortune’s top 100 and zero otherwise.In the next three tests, Sen is calculated from our model with the distribution of jump size following y =0, a double exponential, and a bivariate constant jump model, respectively. Panel B presents results whereDIn is a dummy variable taking a value of one if the event is insider and zero otherwise, DT takes a value ofone if it is true sentiment and zero otherwise, and Sen is again defined as the CAPM alpha. p-Values are inparentheses.

Panel A. Sentiment vs. Subjective Value

Int α Sen Sok τ Vol Div Obs

Sen = Rt–1

Coefficient 2.4945 –0.6085 0.0312 0.0183 –1.1156 0.0542 0.1258 56602(p-Value) (<0.0001) (<0.0001) (0.0305) (0.8771) (0.0005) (0.6842) (0.0151)

InsiderCoefficient 2.4584 –0.8531 0.0716 0.0469 –1.0446 0.2450 0.0758 23826(p-Value) (0.0018) (<0.0001) (0.1511) (0.8566) (0.1023) (0.3911) (0.4775)

True SentimentCoefficient 2.3315 –0.3640 0.0014 –0.0028 –1.1155 –0.0386 0.1880 21333(p-Value) (<0.0001) (<0.0001) (0.0668) (0.9027) (<.0001) (0.1974) (<.0001)

Sen = TopCoefficient 2.4547 –0.4581 0.0035 –0.2289 –1.0200 0.0983 0.1505 49066(p-Value) (<.0001) (<.0001) (0.0392) (0.1010) (<0.0001) (<0.0001) (<0.0001)

Y = 0Coefficient 2.7797 –0.7190 0.0132 0.0252 –1.2932 –0.0627 0.2569 82364(p-Value) (<0.0001) (<0.0001) (<0.0001) (0.3946) (<0.0001) (0.0142) (<0.0001)

Double ExpCoefficient 2.6516 –0.6902 0.0026 0.0229 –1.2425 –0.0536 0.3091 82362(p-Value) (<0.0001) (<0.0001) (<0.0001) (0.3735) (<0.0001) (0.0159) (<0.0001)

Bivariate ConCoefficient 2.6719 –0.6889 0.0033 0.0228 –1.2461 –0.0693 0.3062 82362(p-Value) (<0.0001) (<0.0001) (<0.0001) (0.3768) (<0.0001) (0.0019) (<0.0001)

Panel B. Insider vs. True Sentiment Events

Int α DInSen DTSen Sok τ Vol Div

Coefficient 2.4612 –0.6541 0.0550 0.0004 0.0204 –1.1028 0.1018 0.1181(p-Value) (<0.0001) (<0.0001) (0.0071) (0.9062) (0.8813) (0.0048) (0.5359) (0.0730)

the sensitivity of the subjective value to idiosyncratic risk is negative, a notion supported by ourempirical findings. This indicates that increased levels of risk may negatively affect subjectivevalue owing to the inability of executives to fully diversify their holdings. In contrast, the Black-Scholes, as well as the majority of options pricing models, prescribe no role to idiosyncratic risk(i.e., the sensitivity should be zero) and are generally not able to capture the empirical findingthat subjective value is negatively related to risk.


Table V. Summary Statistics for Vega

This table presents test results for vega. In Panels A, B, and C, the distribution of jump sizes are zero jump,double exponential jump, and bivariate constant jump, respectively. UV(total) and UV(freq) are total riskand jump frequency risk vegas under our model when all holdings are liquid. V(total), V(idio), V(freq), andV(size) refer to total risk vega, idiosyncratic risk vega, jump frequency risk vega, and jump size risk vega,respectively, under our subjective value model where vegas are as follows:

V(total) = ∂ Fs

∂σtotal, V(idio) = ∂ Fs

∂vs, V(freq) = ∂ Fs

∂√

λs

, V(size) = ∂ Fs

∂√

ks,2

.

Panel A: Y = 0

UV(total) V(total) UV(freq) V(freq) V(idio) V(size)

Without sentimentMean 11.237 13.715 12.787 –0.739 –24.212 –0.162Median 9.173 11.684 10.159 –0.143 –19.611 –0.077

With sentimentMean 5.968 7.646 16.486 –11.222 –41.638 –0.415Median 3.007 4.906 13.191 –2.857 –31.063 –0.144

Panel B. Double Exponential Jump Model

Without sentimentMean 13.456 16.791 –1.039 –0.564 –25.877 –0.017Median 10.875 14.129 –0.871 –0.449 –20.396 –0.008

With sentimentMean 7.397 10.063 –14.217 –1.213 –44.233 –0.042Median 4.173 7.041 –1.823 –0.806 –32.466 –0.014

Panel C. Bivariate Constant Jump Model

Without sentimentMean 13.366 16.792 –1.194 –0.677 –25.860 –0.017Median 10.879 14.131 –1.015 –0.542 –20.382 –0.008

With sentimentMean 7.396 10.060 –15.984 –1.407 –44.231 –0.042Median 4.173 7.037 –2.103 –0.957 –32.468 –0.014

In applying the empirical data to the formulas for the sensitivities of subjective value to variousforms of risk, we find that our model generates a negative relationship between firm-specificrisk and subjective value, a finding that is also consistent with the empirical observations ofMeulbroek (2001). This finding is particularly important as managers can easily affect the firm’sidiosyncratic risk level through various moral hazard related activities.

In Table V, we calculate risk sensitivities, vegas, for all options issues in our data set assumingthat there are no illiquid holdings (UV) (i.e., α = 0) and using our default value for α (V), withand without consideration of sentiment. In the first two columns, we find that the sensitivitywith respect to total risk is positive for both UV and V, regardless as to whether sentiment isconsidered or not. This is true of all jump specifications. In every case, the sensitivity is higherwhen sentiment is not considered. Looking at the vegas with respect to jump frequency, we findthat UV(freq) can be either positive or negative depending upon the jump specification, whileV(freq) is always negative. Interestingly, UV is positive for the constant jump model, but negative


for the other two models indicating the importance of jump specification when liquidity is notconsidered. The magnitude of UV is always smaller than that of V.

Perhaps the most interesting factor affecting the subjective value in our model is idiosyncraticrisk, for which the estimate is always negative and is significantly larger in magnitude thanthe other vegas. While the jump size vega also plays a role and is likewise always negative,the magnitude of this effect is much smaller. This finding highlights the role of idiosyncraticrisk in our model and explains why the empirical sensitivity of subjective value to volatilityis found to be negative, contrary to generally accepted moral hazard models that dictate thatoption compensation encourages risk taking. If agents are sufficiently under-diversified, the riskpremium from taking on excess idiosyncratic risk offsets gains from convexity and discouragesrisk-taking behavior. The corresponding UVs for idiosyncratic and jump size risk are both zeroas these do not play a role in determining market value when there are no under-diversifiedholdings. Also, the Vs are substantially more negative when sentiment is introduced, pointing outthe sharply offsetting effects of positive sentiment and risk aversion in this model. Which piecedominates is dependent upon the risk aversion parameter and α of the employee.

IV. Robustness Checks

We execute numerous robustness checks. Unless otherwise noted, none yield appreciabledifferences, and our conclusions are unaffected. Numerical results and testing specifics areavailable upon request.

A. Estimation of Subjective Values

To aggregate executives with similar compensation characteristics, we apply a K-means clus-tering method. However, we have rerun all of the tests using a simpler, split sample methodologydetermining groups simply controlling for industry, rank, year, size, immediate exercise value,and α. We then calculate subjective value using these groupings and find no significant qualitativedifferences.

While it is intuitively clear that one should never value an option at less than zero, asmall number of negative implied values are implied in our estimation process. We try re-running all tests allowing for negative implied subjective values assuming that α is simplyα = BSOPM/(Salary + Bonus + Other + BSOPM) where BSOPM is the Black-Scholes valueof the options. Whether negative implied values are equated to zero (the default calibration),allowed to be negative, or entirely removed from the data set, none of our findings are affected.

We also try estimating subjective values using an iterated method, solving for a fixed-point α∗

that uses subjective value as an input to α and vice versa. Specifically, we first calculate:

α∗ = Option

Salary + Bonus + Other + Option,

with Option initialized as the Black-Scholes value. Then, we calculate subjective value by Formula(2) using α∗. Next, we recalculate α∗ using this candidate subjective value and iterate until thedifferences between α’s and the subjective values are both less than 10−5. We rerun the regressionsusing these new subjective values, but with α removed, and find that Sen is always significantlypositively related to subjective value.


B. Subsample Tests and Outlier Controls

As a control for outliers, we use a standardized residuals approach to remove outliers from ourdata set and rerun all tests. Alternatively, to account for differing variable magnitudes, we alsotry normalizing each option pricing variable by its sample mean (centering all variables aboutone). We find no significant qualitative differences in either case. While point estimates vary, Senis always positively related to subjective value and α is negatively related, and significantly so.Again, positive sentiment increases ESO value, while having a large illiquid holding decreases it.

We also rerun all tests with both dependent and independent variables normalized by industryaverage. To be even more thorough with regard to industry effects, we also recreate all testsseparately for each industry. We find that, again, no qualitative differences are noted. In everyindustry, positive sentiment increases value, while an increase in α decreases it. We conclude thatindustry effects are minimal.

As previously mentioned, the results requiring the calculation of the previous year’s CAPMalpha utilize a smaller sample. We rerun all of the results using only this same reduced sample,and all of the results are qualitatively identical to those found when utilizing the full data set.

Finally, we also split our data into positive and negative Sen. When calculated using our model,Sen is predominantly positive (more than 80% of the data points). When applying prior yearreturns, that number is only about 60%. We find that, in all cases, α remains negatively relatedto subjective value. Sen is likewise uniformly positively related to subjective value in the grosscase, but is negatively related in the normalized case when the previous year alpha is used as theproxy for Sen and Sen is found to be negative.

C. Test and Model Respecifications

In our main tests, we calculate Sub/BSOPM as a normalized subjective value. We repeatall of the tests using the arithmetic difference Sub – BSOPM . This method lacks magnitudenormalization, but allows for positive or negative subjective values. The results, however, arequalitatively identical.

Additionally, we are able to amend our model to include jumps in the market portfolio. Theresulting valuation formula and partials do not change the intuition discussed in the paper, thoughsolutions are decidedly more complicated. The empirical results are qualitatively unchanged. Aswell, we recalculate our options pricing formula to accommodate barrier options, rather thanEuropean options. Again, while the form of the result is decidedly more complicated, variablerelations are unchanged and the empirical results are likewise qualitatively identical under allreasonable parameterizations. Derivations, simulations, and the empirical results are availableupon request.

V. Conclusion

This paper investigates the Chang et al. (2008) model for ESOs that incorporates illiquidity ofthe options and the potential roles of employee sentiment and insider information as factors thatimpact ESO pricing. We apply empirical data to calculate the subjective value placed on ESOsimplied by the compensation data. Specifically, using data provided by Compustat, we groupexecutives using a K-means hierarchical method based on a number of firm and individual criteria.By assuming that all executives in the same cluster receive the same total compensation, a notionthat relies on the existence of competitive labor markets, we then back out the valuation placedby each executive on their respective ESO. These groups include consideration of nonoption


compensation, rank, industry, year, firm size, and immediate exercise value. Though the extantliterature predicts that employees should discount the value of their options, we find that executivesvalue their options more highly than implied by Black-Scholes, applying an average premiumof 48%. As such, the cost of issuance for the firm is vastly lower than the benefit perceivedby employees suggesting that ESO compensation should be an even larger part of executivecompensation.

We then relate subjective value to sentiment levels and generate the novel finding that executivesmust expect their firm’s risk-adjusted returns to outpace that predicted by the market by 12% inorder to justify the subjective value placed on ESOs. This expectation may be the result of privateinformation regarding the growth prospects of the firm. Moreover, in controlling for the sign ofsentiment and resulting returns, we find that even when the former does not match the latter,subjective value is positively related to ex ante sentiment. Additionally, given the magnitude ofthe return and the observation that options account for an enormous part of total compensation, itis unlikely that executives project such a large sentiment premium for signaling purposes alone.

Testing subjective value and its relation to pertinent variables, we find that subjective value isnegatively related to the proportion of wealth held in illiquid firm-specific holdings and positivelyrelated to sentiment. In other words, the larger the illiquid ESO position is, the larger the discountrisk aversion prescribes and the lower the subjective value implied in the compensation package.Alternatively, the more positive the employee’s view of future risk-adjusted returns, the morevaluable the ESO. This is robust regardless as to whether this view is likely to be generated byinside information or pure sentiment. Though both factors are significantly and positively relatedto subjective value, the impact of the former appears to be quite a bit larger. In addition to theprevious year’s CAPM alpha as a proxy for positive sentiment, we also consider inclusion onFortune’s list of 100 best firms and find the same results. We confirm that specification of thejump model does not affect the results.

Interestingly, we find that subjective value may be negatively related to risk as the inabilityof executives to fully diversify their holdings may lead to risk premia that outweigh the valueplaced on risk by the convexity of options payouts. We note that this correlation is particularlynegative with regard to idiosyncratic risk and is also empirically negative for risk associatedwith both jump frequency and size. Since these aspects of return are precisely those that may bemost directly controlled by executives, traditional moral hazard arguments relating solely to theconvexity of the options payout may not hold.

We conclude that employee sentiment is a necessary consideration when issuing options andthose executives may be substantially over valuing ESOs because of it. Firms that have performedwell should issue more options, and firms should place effort and attention into maintainingpositive sentiment within a firm, especially when offering ESO compensation. Moreover, ESOsmay not generate the sort of risk-taking behavior implied by more traditional options pricingformulas owing to illiquidity. The more illiquid the ESO, the larger the proportion of total wealthESOs represent, the less likely employees are to engage in risk-taking behavior. Options are meantto promote incentive effort and value-creating risk taking. However, managers holding illiquidESOs are discouraged from taking firm-specific risk as it may erode the ESO value. Alternatively,options may also lead to moral hazard since equity holders are insulated from downside risk in thecase of bankruptcy. Once the illiquid nature of ESOs discourages value destroying idiosyncraticrisk-taking, it acts as a protection against moral hazards.

All in all, we provide evidence that the subjective value of an ESO departs significantly fromthe Black-Scholes value, and offer a framework with which to investigate these concerns andopportunities.


Appendix: Equations and Proofs

As per Chang et al. (2008), μs , μm , and r are the instantaneous expected rates of return for thestock S, the market M , and the risk-free asset B, respectively. Let ds and dm be the dividend payoutfor S and M , respectively. The Brownian motion process dWm represents the Normal systematicrisk of the market portfolio, and the Brownian motion process dWs and jump process dqs are theidiosyncratic risk of the company stock where dqs follows a Poisson distribution with averagefrequency λs such that dqs ∼ Poi(λsdt). Let σs and σm be the Normal systematic volatility forS and M , respectively, while νs is the Normal unsystematic volatility of S. Additionally, Ys is thejump size such that E(Ys − 1) = ks and E(Ys − 1)2 = ks,2, a generalizing assumption, but notone that qualitatively impacts results.

The subjective value of the ESO is

Fs(S, t) =∞∑j=0


j!

{S(t)e−ds

s τ E∗[

j∏i=0

Ys,i�(ds

1

)] − K e−rsτ E∗[�[(ds

2

)]},

where ds1 = ln[(S(t)

∏ ji=0 Ys,i )/K ]+(rs−ds

s + 12 σ 2

N )τ

σN√

τ, ds

2 = ds1 − σN

√τ , τ = T − t, σN = √

σ 2s + ν2

s ,

ξ = E[α(Ys − 1) + 1]γ−1, r s = r + αSen − (1 − γ )(αλsks + 12γ λsks,2α

2 + α2ν2s ) − λs(ξ −

1), dss = ds − (1 − α)Sen − (1 − γ )[αλsks + 1

2γ λsks,2 α2 −(1 − α)αν2s ] − λs(ξ − 1) + λsks,

and

α ={ Sen

(ν2s +λs ks,2)(1−γ )

T ′ ≤ t < T ′′ or t ≤ T′ and Sen ≥ (ν2s + λsks,2)(1 − γ )α

α t ≤ T ′ and Sen < (ν2s + λsks,2)(1 − γ )α.

T ′ and T ′′ are vesting and maturity dates, respectively. Note that the general modeling method-ology allows for separate consideration of vesting risk, though we equate them here. Illiquidholdings in this instance generally includes unvested ESOs, restricted stock holdings, vestedESOs for firms that do not have actively traded options in the secondary market, or holdings ofexecutives who are reluctant to sell options/stock for signaling reasons.

The resulting partial derivatives are as follows, where σ 2total = σ 2

s + ν2s + λsks,2:

∂ Fs

∂α= −(1 − γ )τν2

s

∞∑j=0

(λsξτ ) j e− λsξτ

j!

{S(t)e−ds

s τ E∗[

j∏i=0

Ys,i�(ds

1

)]}

+ [(1 − γ )

(λsks + γ λsks,2α + 2αν2

s

) − Sen]τ Fs,

∂ Fs

∂Sen=

∞∑j=0


j!τ

{S(t)(1 − α)e−ds

s τ E∗[

j∏i=0

Ys,i�(ds

1

)]

+ αK e−rsτ E∗ [�

(ds

2

)]}> 0,

�σtotal = ∂ Fs

∂σtotal= σtotal

σN·

∞∑j=0


j!S(t)e−ds

s τ E∗[

j∏i=0

Ys,i�′(ds

1

)√τ

]> 0,


�νs = ∂ Fs

∂νs

= − 2τ (1 − γ )ανs ·∞∑j=0


j!

×{

(1 − α)S(t)e−dss τ E∗

[j∏

i=0Ys,i�

(ds

1

)] + αK e−rsτ E∗ [�

(ds

2

)]}< 0,

�√λs

= ∂ Fs

∂√

λs

= τ [(1 − γ )(2α√

λsks) + γ√

λsks,2α2) − 2

√λs − 2

√λsks]Fs(S, t)

− 2√

λsksτ

∞∑j=0


j!K e−rsτ E∗ [

�(ds

2

)]

+ 2√λs

∞∑j=1


j!

{S(t)e−ds

s τ E∗[

j∏i=0

Ys,i�(ds

1

)] − K e−rsτ E∗ [�

(ds

2

)]},

�√ks,2

= ∂ Fs

∂√

ks,2= (1 − γ )γ λsα

2τ√

ks,2 Fs(S, t) < 0.

References

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