Are Employee Stock Option Exercise Decisions Better Explained Through the Prospect Theory?

It is well documented in the empirical literature that employee stock options exercise behavior is driven by economic/rational factors as well as by psychological/behavioral factors. The latter include a set of behavioral biases affecting attitudes towards risk. Perhaps the most comprehensive theory that captures these patterns is the Cumulative Prospect Theory (CPT). The central question examined in this paper is whether or not the CPT leads to better predictions of exercise decisions. Using a distorted lattice approach I show that the CPT broadly outperforms in explaining empirical exercise behavior. Interestingly, my empirical estimates of probability weighting are consistent with those from the experimental literature. I argue that this analysis provides a unifying stream for thinking about issues related to stock options exercise and valuation.


Introduction
Employee stock options exercise and valuation have become a focus of interest in financial economics over the last decades. The related empirical literature has established that employees exercise behaviour obeys both rational and psychological drivers. Rational factors stem directly from the standard framework of the Expected Utility Theory (EUT hereafter). These include liquidity chocks and risk diversification (Hemmer et al., 1996;Core and Guay, 2001;Bettis et al., 2005).
Psychological factors are, however, out of the scope of the EUT. This strand is comprised of a set of behavioural biases that affect beliefs as well as preferences. Anchoring (Huddart and Lang, 1996;Heath et al., 1999;Core and Guay, 2001), overconfidence (Misra and Shi, 2005), miscalibration and mental accounting (Sautner and Weber, 2005) are all part of those factors.
Admittedly, the most comprehensive available theory that elegantly captures some of these psychological biases in experimental settings is Tversky's and Kahneman's (1992) Cumulative Prospect Theory (CPT henceforth). The CPT is viewed as the most prominent descriptive theory of how people formulate their risk preferences. It has broadly accepted applications in mainstream economic analysis that helped explain many puzzling phenomenon (see Barberis (2012) for a review).
In this paper, I build on the CPT to develop a behavioural lattice model for the exercise and the valuation of standard employee stock options. The model handles an exercise policy that maximizes the expected value from the perspective of a representative employee exhibiting preferences as described by the CPT. It also accounts for exogenous non-market factors that may cause early exercise through an exit state that occurs with a given probability 1 . The main goal of this model is to help verify if this behavioural framework leads to better predictions of stock options exercise decisions.
The predictions of the model could be useful to shed light on the economic consequences of the exercise behaviour, specifically those affecting stock options fair value assessment. This paper departs from the observation that the optimal exercise policy issue 2 has been extensively discussed in a literature that heavily loads on the EUT. Most of it develops utilitymaximizing lattice models (Huddart, 1994;Kulatilaka and Markus, 1994;Carpenter, 1998). Basically, 1 Earlier papers that focused on the impact on the stock option value of forfeiture and early exercise due to job termination include Jennergren and Naslund (1993), Cuny and Jorion (1995), Rubinstein (1995), Carr and Linetsky (2000), Hull and White (2004) and Cvitanic et al. (2004). 2 Exercise behavior deviates from the usual value-maximizing exercise policy stated by the standard American options theory because of many reasons. First, opposite to ordinary tradable call options, employee stock options are long-dated nontransferable call options. This implies that the employee has to bear the underlying risk for a long time. Market experience shows however some exceptions to this feature, such like the transferable stock options programs developed by JP Morgan Chase for Microsoft in July 2003 and for Comcast in September 2004. According to this deal, JP Morgan offers the entitled employees to purchase their underwater stock options at a discount premium (i.e. below their potential market value). Second, the employee faces hedging restrictions since he is usually precluded from short selling the company's stock. Finally, the stock option holder may be unable to completely diversify his stock holdings and human capital investment in the firm. Because of these features, the employee will not make the same exercise decision as an unrestricted outside investor. Therefore, the fair value of his stock option contract crucially depends on his endowment and on his risk preferences. these approaches assume that the representative employee chooses an optimal exercise decision as part of a utility maximization problem that may simultaneously cover other issues such as consumption and portfolio choices (Detemple and Sundaresan, 1999), non-option wealth diversification or liquidity needs. Nonetheless, the theoretical decision-making framework underlying these models, namely the EUT, has shown handicaps in capturing some prominent features of employee equity-based compensation. For instance, EUT-based models, taking place in the principal-agent framework, have difficulties accommodating the existence of convex contracts as part of executives compensation packages 3 . Consistently, studies focused on the effect of stock option and restricted stock grants on the managerial efforts, including Jenter (2001), Hall and Murphy (2002) and Henderson (2005), conclude that stock options are inefficient tools for incentivizing risk-averse managers when they are granted by mean of an offset of cash compensation. This is obviously inconsistent with actual compensation practices. Moreover, a common prediction of the EUT-based models is that a risk-averse undiversified employee would value his options below their risk-neutral value. This prediction contrasts to several surveys and empirical findings documenting that, frequently, employees are inclined to overestimate the value of their stock options (Lambert and Larcker, 2001;Hodge et al., 2010;Hallock and Olson, 2006;Devers et al., 2007), which is supported by the well documented attractiveness of lottery-like bets, including optional securities, to the "common of the mortals" ( Barberis and Huang, 2008). In essence, these limitations of the EUT provide both rationale and motivation for investigating the ability of the CPT to better predict employee stock option exercise decisions. My first result shows that the models fit the mean data remarkably well. Interestingly, the calibrationsyielded values of the probability weighting parameter lie within the range of estimates from the experimental literature. I argue that this probability weighting evidence may interpret as a manifestation of overconfidence since overconfident employees would be more optimistic about future outcomes, especially when they operate in a context they feel familiar with. This interpretation is supported by Misra and Shi (2005) finding that overconfidence plays an important part in the postponement of exercise for CEOs with high levels of options. It finds also some support in Tate (2005, 2008) who classified CEOs as overconfident when they hold their stockoptions until expiry instead of their lack of diversification. This point needs however further investigations before drawing any final conclusions. The second finding is that the employees at risky firms are more likely to quit and, consequently, are more subject to stock options forfeiture risk. They, therefore, require a higher expected exit amount in order to compensate for the losses of their options.
Another interesting finding in this area is that the reference point estimate from the model calibration increases with the stock price volatility. This means that, on average, stock option holders would be more conservative about the gains and losses breakeven within a risky decision making environment.
The last and main finding from this analysis suggests that the CPT model brings significant improvements over the EU model and the EA model in terms of both in-sample and out-of-sample exercise decisions forecast errors. This result does not suffer a contingency bias raised by the level of risk of the exercise decision environment. It turns also to be robust to the reference point specification in the model.
The last item discussed in this paper relates to the implications in terms of cost to shareholders. I assessed the values of the option contracts in the sample at inception under each of the three competing models. In addition, for the purpose of comparison with some methods recommended by the accounting standards (IFRS and US GAAP) and largely used by practitioners, I computed option fair values using two benchmark models. The first one is the extension of the Black & Scholes model (BS henceforth) that uses the expected lifetime of the option as an input instead of its contractual expiration date. The second benchmark is the Hull  My work aims to provide a straightforward unifying way of thinking about issues related to stock options exercise and valuation. It therefore adds to a growing literature on equity-based compensation that incorporates CPT models. These models have proved successful in explaining some observed compensation practices, specifically the almost universal presence of stock options in the executive compensation packages. Therefore, they have advanced the CPT framework as a promising candidate for the analysis of issues in connection with equity-based compensation. This literature includes Dittmann et al. (2010) who developed a stylized principal-agent model where the agent is loss-averse.
Their model accurately predicts the observed mix of restricted stocks and stock options in executive compensation packages. Bahaji (2011) also used the CPT to elaborate on stock options incentives effect and its implications for the design of the contract. In the same vein, Spalt (2013) showed that stock options may be more attractive to a loss-averse employee subject to probability weighting. He then explained the puzzling phenomenon that riskier firms are prompted to grant more stock options to non-executive employees.
Indeed, this recent line of the literature reveals some new interesting insights, but it leaves aside the effect of exercise behaviour. It actually relies on the simplifying European-style specification of the options contracts so as to avoid representing the optimal exercise policy. The methodology in this paper undertakes this issue using all the CPT components. On the one hand, the effect of probability weighting is parsimoniously represented in a distorted binomial tree. At each level of the tree, the transition decision weights are consistent with the binomial probabilities transformation based on the weighting function of Tversky and Kahneman (1992). On the other hand, the exercise decision is triggered upon maximization of the weighted value from the exercise proceeds relying on the value function featuring loss aversion and reference dependence according to the CPT.
The remainder of the paper is structured as follows. The first section sketches a theoretical framework for stock option exercise from the perspective of a representative CPT employee and describes the piecewise construction of the model. Section 2 presents the empirical comparative analysis. The last section discusses the implications in terms of fair value assessment.

The CPT exercise model
This section develops a model of stock options exercise and valuation from the viewpoint of the issuer (i.e. the firm). The model is based on a theoretical framework where the option payout depends on the exercise behavior of a representative option holder whose believes and preferences fit with the CPT framework.

The economy of the stock option contract
The representative employee is assumed to be granted a stock option contract at time t=0 expiring within a time period "T". The contract is a Bermudan style call option on the common stock of the company denoted by " t S ", with a strike price "K". It includes two restrictions as commonly do standard employee stock options. The first one is the non-transferability, which means that the employee is precluded from selling the option. The second one is the vesting, which implies the Bermudan feature of the option. The vesting restriction requires the cancellation of the contract in case the holder leaves the company before the end of the vesting period " v t ". In addition, I assumed that the employee is not allowed to short-sell the company stock and that he can earn the risk-free rate "r" from investing in a riskless asset. This assumption stands then for a restriction on the option hedging.
Moreover, the underlying stock price is assumed to follow a standard binomial process with "N" time steps. Thus, in each time step " / t T N δ = " the stock price may move up by a factor " t u e σ δ = " with a probability "  " or down by a factor " / d u = 1 ". The parameters "σ", "µ" and "q" are respectively the stock price volatility, the expected return and the dividend yield. Recall that the upward risk neutral probability writes: Furthermore, following Carpenter (1998), I assumed that at each time period there is an exogenous probability " e p " for the employee to be offered a cash amount "y" per option held in order to leave the company. Leaving the company implies that the option contract is stopped at that time either through exercise, provided the option is vested and is in-the-money, or via forfeiture otherwise.

The representative employee
I assumed that the representative employee exhibits preferences and beliefs as described by the CPT (Tversky and Kahneman, 1992). It follows that to each gamble "x" with countable outcomes x ∈ 1 " and their respective probabilities " { } ,..., i n p ∈ 1 ", the employee assigns the value: Where the function ( ) . v θ , called the value function, is assumed of the form: This formulation has some important features that distinguish it from the normative utility specification. The value function is defined on deviations from a reference point, denoted by "θ". It is concave for gains (i.e. implying risk aversion) and commonly convex for losses (i.e. implying risk seeking) due to parameter "α". It is steeper for losses than for gains (i.e. conveying a loss aversion feature caught up in "λ").
The terms " , a i ω " in (1) are decisions weights associated to each outcome. These result from a transformation of the probabilities using a weighting function " ( ) 1 Following Tversky and Kahneman (1992), the probability weighting function is assumed of the form: This function stands for another piece of the CPT, which is the nonlinear transformation of probabilities. Specifically, it captures experimental evidence on people overweighting small probabilities and being more sensitive to probability spreads at higher probability levels. As we can see from equation (3), in contrast with the CPT, but consistent with the Rank Dependent Expected Utility theory (Quiggin, 1982), the described weighting of probabilities is identically performed over gains and losses so that decision weights sum to one. The degree of weighting is controlled by the parameter "a". The more this parameter approaches the lower boundary 4 at 0.279 the more the tails of the probability distribution are overweighted. For instance, when "a" is set to 1, the probability weighting assumption is relaxed.

Setting the model
Using an alternative behavioral framework, the CPT model builds on prior works by Huddart (1994), Kulatilaka and Marcus (1994) and Carpenter (1998) in that it uses the same principle governing the exercise decision. Similar to these models, the CPT model is a two-state lattice where the optimal exercise decision is driven by the maximization of the expected utility from the exercise proceeds. According to this maximization principle, the option is exercised at a given time period if the utility from the exercise proceeds is higher than the expected utility from continuing to hold the option until the next time period. With that said, the representative employee is supposed to exercise as soon as the intrinsic value of the option exceeds the value of the option from his own perspective (i.e. the subjective value). However, opposite to these models, the utility is assessed using the CPT framework instead of the EUT. Therefore, the key differentiating point is that, on the one hand, only the utility of exercise proceeds is considered rather than that of total wealth and, on the other hand, the expected utility is assessed based on weighted probabilities.

Model construction
The first step of the CPT model construction consists in determining the transition decision weights that will be used to assess the utility expectation at each node of the binomial tree. The purpose is to build in parallel to the share price binomial process a two-state decision weights process that fits with the employee view of probabilities as specified by the CPT probability weighting 4 The lower boundary at 0.279 is a technical restriction to insure that " The second step is about setting the patterns of exercise decisions. Following Carpenter (1998), I accommodate the possibility of option forfeitures or early exercises caused by non-market events, such as liquidity shocks, employment termination or any other forced exercise through the exit states. Note that the exit decision is an endogenous feature of the model in that it is linked to the size of the cash amount "y" offered to the employee to quit. In addition, I assumed that the exercise decision at a given time period depends on whether or not the exit state is prevailing at that time, on the vesting status of the option and on the prevailing level of stock price, but not on the past stock price path. This assumption allows for backward recursion.
Moreover, as stated in assumption 1 bellow, the employee is assumed to set his reference point based on his initial own share price return expectations over the lifetime of the option: Assumption 1: The reference point " i θ " at a time period " i t " is defined as the intrinsic value resulting from a time-adjusted growth rate of the share price based on the annualized return " ρ " reflecting the expectations of the employee: 5 An interesting alternative to the CPT probability weighting function is using weighting functions implied by listed options prices. This approach uses non-parametric methods to estimate state price densities from options market prices. These estimates are then used as building blocks to construct non-parametric estimators of the weighting function without imposing any constraint on the shape of the former. For more details on this approach see Polkovnichenko and Zhao (2013).
While this setting is inconsistent with empirical evidence on employee exercise activity being linked to share price historical maxima (Huddart and Lang, 1996;Heath et al, 1999), it nevertheless conveys an exercise behavior taking into account non-status quo reference points. Actually, under a path-dependent specification of the reference point, backward recursion becomes impossible in our situation. The specification in (5) exceeds the subjective value of the option. However, in the exit state, the employee decides to continue with the option if its subjective value is greater than the sum of its intrinsic value and the cash amount he has been offered to leave the company. By repeating this process using backward recursion, and starting from the end of the tree, we can find out the nodes where the option is exercised. Note that at the levels where the option is still unvested, it is systematically held to the next period provided the employee is in the continuing state. Instead, an unvested option is forfeited in the exit state if its subjective value is lower than the exit-cash amount. A formal algorithm describing the posited exercise rule is provided in appendix B.

How do beliefs and preferences affect exercise decisions?
The joint effect of all the parameters driving beliefs and preferences is somewhat hard to disentangle. Instead, for sake of simplicity, let's focus on the individual effects of probability weighting and loss aversion, all else being equal.

Proposition-3 (Effect of probability weighting):
Ceteris paribus, the representative employee is less likely to early exercise his options at a given state as his degree of probability weighting increases.

Proof. Sketched in appendix C2. ■
The variables of interest for the analysis of the exercise decision implied by the model are the state subjective values of the option. Analyzing these effects on the exercise likelihood boils down to figuring out how the subjective value responds to changes of preferences and believes at a given state where the option is exercisable. q=0% (dividends effect is ignored); tv=1; a=0.65 (unless otherwise stated); α=0.88; µ=7%; λ=2.25 (unless otherwise stated); pe=0 (exit state is left aside); ρ=10% (unless otherwise stated). Panel-1 and panel-3 illustrate how the exercise node level in the lattice is affected by loss aversion and probability weighting respectively. Plots in panels 2 and 4 show how the subjective value responds to changes in loss aversion and in the degree of probability weighting respectively. Proposition 3 states that stock option holders with more optimistic view about future stock price patterns -as displayed by the outcomes probabilities they subjectively draw -are less inclined to exercise (provided the option is exercisable at that time). Put differently, the value the employee assigns to its stock options increases with his degree of probability weighting (or optimism) and may even overstate the risk neutral value of these options. Note that, in this case, the employee is much more interested in receiving stock options than a risk-neutral agent would be. Assuming the company and the employee bargain efficiently over the terms of the compensation, the company will grant him more options by mean of an offset of other compensation components (Bahaji, 2011). These patterns are further depicted in panel-4 of figure-2. The reported analysis adds to proposition 3 in that it confirms that the subjective value decreases with the probability weighting coefficient, irrespective of the level of the reference point. It follows that a higher emphasis put on the tails of the probability distribution manifests as a delayed exercise at higher share price levels as illustrated in panel-3.

Panel-1: Exercise nodes implied by loss aversion
The analysis in panel-2 shows how the subjective value is affected by changes in loss aversion. It conveys the intuition that the employee tends to attribute lower value to his option holding as his aversion to losses increases. He will be then led to preempt the exercise and earn lower proceeds on average since he requires a lower amount to compensate for the loss of the subjective time value of his options. This is supportive of the statements in proposition 4. To leave nothing in doubt, these patterns are again illustrated in panel-1.

Testable predictions of the model
I considered that the employee exercise decisions are adequately characterized by the exercise stock price ratio denoted by " S τ ". This variable is defined as the stock price-to-strike price ratio at the time of exercise "τ ". This assumption relies on the argument that the exercise decision is mainly driven by the utility procured by the exercise proceeds. It follows that the model-yielded statistic of interest predicting the exercise decisions is the expectation of the exercise stock price ratio subject to the terminal stock price: where " ( ) is the conditional expectation operator under the real probability measure.
Specifically, the expectation above is determined as the weighted average of the outcomes of the random variable " S τ " across all the stock price paths that result in an exercise and settle at a final stock price level " * T s ". Note that this prediction takes into account the effect of the stock price effective performance on the exercise behavior since it is conditioned on the terminal stock price level.
This makes it comparable to equivalent mean values from empirical data, which enables to test the model prediction power and to estimate the unobservable parameters.
The expected value of the cancellation rate "η " is another model-yielded variable that has to be considered for the purpose of calibration on empirical data: The estimation of this statistic consists in recursively computing the 1-year cancellation probability based on the implied distribution of the cancellation state variable across all share price paths. Therefore, it may be interpreted as the average ratio of cancelled options during a year either through forfeitures or expirations. Note that, opposite to the previous statistic in (6), the mean cancellation rate is not conditioned on a final share price level. Let's also stress that the cancellation rate has to be distinguished from the exit rate. The latter is the model input that drives the frequency of forfeitures prior to the expiry date, whereas the former is a model output that results from the implied exercise policy and, consequently, depends on the model parameters and especially the exit rate.

The empirical analysis
In this section, I examine the performance of the CPT model in predicting actual exercise patterns  8 . As in Huddart (1994) and Kulatilaka and Marcus (1994), this model is based on a binomial stock price tree where the exercise decision is made according to a policy that maximizes the expected utility subject to the hedging restrictions. The option holder utility function is assumed to be of the isoelastic form: " ( ) ( ) where "γ" is the constant relative risk aversion coefficient and "w" is the employee outside wealth (i.e. non-option holding wealth) normalized by the value of his options underlying stocks. Moreover, similar to the CPT model, at each time step there is an exogenous probability "p e " for the employee to be offered a cash amount "y" that motivates him to leave the firm. This allows accommodating the possibility of option forfeiture or an early exercise due to non-market events. In addition, the outside wealth as well as any early exercise proceeds are assumed to be invested in the constant proportion portfolio of the company's stock and a risk-free bond that would be optimal for the employee to hold in the absence of the stock option and the possibility of receiving "y". This optimal portfolio is a binomial version of the continuous-time portfolio developed by Merton (1969Merton ( , 1971. For further details regarding the EU model algorithm the reader is referred to Carpenter (1998).
The EA model is a binomial variant of the continuous-time model of Jennergren and Naslund (1993). The only difference between this model and the standard American option model is the stopping event that occurs with some exogenous probability "p e ". This exit state captures all the factors that might cause a deviation from the standard theory exercise policy such as employment termination, liquidity needs or a desire for diversification. The algorithm of this model is also elaborately described in Carpenter (1998).
The CPT model is expected to outperform the two competing models because of two reasons.
First, the CPT, thanks specifically to the key feature of probability weighting, has proven fertile in explaining stock option subjective valuation patterns (Bahaji, 2011;Spalt, 2013), which makes it a promising candidate for analyzing the employee exercise behavior. Secondly, there exists empirical and experimental evidence on behavioral factors affecting employees exercise policies. Contrary to the normative framework, the CPT is expected to accurately capture these factors since it is a descriptive theory of the way people formulate their choices under risk and uncertainty.
The remainder of this section is organized as follows.  Bermudian-style options) that do not include performance-related vesting provisions. Their vesting is either immediate or achieved after specified periods varying from 2 to 9.5 years. The options maturities are from 5 to 10.5 years.
Recall that the focus of this research is the exercise patterns of standard employee stock options. Therefore, I dismissed from the dataset all the items related to specific grants whose vesting is contingent upon performance conditions. Actually, the initial dataset included 1,251 exercise transactions related to 7 performance stock option plans. All these specific plans were granted to top executives and senior managers. The 542 plans comprised in the scope of this study are Bermudianstyle stock option contracts with vesting schedules subject to service conditions. Some contracts are immediately vested (i.e. do not include vesting periods). The other contracts include either a cliff vesting or a gradual vesting. The vesting periods range from 2 to 9.5 years. All the contracts were granted almost at-the-money and do not comprise buyback provisions. Moreover, the options are not transferable and the beneficiaries are precluded, as insiders, from short selling the firm's stock according to section 16-c of the Securities Exchange Act. This prohibition stands for a hedging restriction.
As pointed out earlier, one of the strengths of this research is the quality of the dataset used. As far as I know, this research is one of the most comprehensive empirical studies on employee stock option exercise patterns in the American context.  (2006). I computed the expected stock price return using the CAPM: the risk free rate plus the risk premium times the beta of the company at the grant date. The latter is computed against the relevant benchmark 10 using daily returns over the 1-year period prior to the grant date. In addition, dividend yields were estimated using recorded dividend data over the year prior to the exercise date. In the same manner, stock return volatilities were computed over the 360-day period prior to the exercise date.

Table-2: Variables summary statistics
This table reports summary statistics on the variables used in this empirical study. These include two types of variables: variables describing the exercise patterns and variables characterising the economy of the stock option contracts. The option exercise variables are constructed from a data sample of option exercises at 12 US companies. This set of variables includes the exercise stock price ratio " S τ ", the lifetime of the option " τ ", the cancellation rate "η" and the final stock price ratio " T S * ". The variables specific to the economy of the stock option contract include the maturity of the option "T", the vesting period "tv", the stock dividend yield "q", the stock price returns volatility "σ", the stock price Beta "β", the stock price expected return "µ" and the risk free rate "r".  Panel-B provides the cross-correlations of the variables. As expected, the correlations of the vesting period " v t " with the time to exercise "τ " and with the exercise price ratio " S τ " are positive, which is consistent with the principle that options with longer vesting periods tend to be exercised later and deeper in the money. At the opposite, the exercises seem to occur latter and at higher stock price levels in the companies with lower dividend yields since both " S τ " and "τ " are negatively correlated to "q". This is consistent with the American option rational exercise theory. Also, the volatility "σ" is negatively correlated to the exercise time (while insignificantly correlated to the exercise price ratio) denoting an exercise behavior imbued with risk aversion (Huddart, 1994; Kulatilaka and Marcus, 1994; Carpenter, 1998).   This intuitively interprets as follows: the exercise behavior of an employee working for a risky company indicates that he is much more likely to quit and, therefore, bears a higher risk of giving up his stock options holding. This interpretation is consistent, on the other hand, with the result that such an employee will require a noticeably higher compensation amount to leave his job.  As we can see from the calibration results reported in panel-3 of table-3, the calibration is significant at the 1% level and yields an exit probability estimate consistent with those implied by the two previous models. Note also that the exit probability estimate at the upper volatility quintile is supportive of the finding from the CPT model that risky firms employees are more likely to exit and, consequently, are more subject to stock options forfeiture risk.

The methodology
Where n is the size of the sample.
Recall that the MSPE metric aims to penalize the forecast errors that are higher than 1 and to favor those blow.   The subsequent panels report the results of the analysis based on the subsamples representing the quintiles with respect to the stock price volatility. The purpose here is to verify that the performance of the CPT model is not significantly concentrated in a specific cluster of the sample. If it turns to be the case, then the previous results may suffer a contingency bias raised by the exercise decision environment risk level. As we can notice, the average forecast errors are the highest at the lower volatility quintiles. This means that the predictive performance of the models is poor in a low risk decision making environment. This is also supported by the OLS analysis in that the linear adjustment results are not satisfactory at the lowest quintiles. At the bottom quintile for instance, the regression slopes are even negative. Moreover, the CPT model yields the lowest average forecast errors. The spreads relative to the average forecast errors implied by the other models are statistically significant, except at the 4 th quintile where all the models perform mostly comparably. Furthermore, relying on the OLS analysis, the CPT model shows the most consistent results compared to the other models in the sense that its yielded expected exercise ratios are positively linked to those from the subsamples, in spite of the R² being too low at the second quintile and at the top quintile.  In essence, the CPT model shows a substantial improvement over the EU model and the EA model in terms of the size of the average forecast errors. It also achieves some slight improvements in linearly adjusting the empirical exercise ratios. Therefore, the CPT framework turns out to be the best candidate in explaining the exercise decisions in the data sample.

Table-5: Models out-of-sample comparison
This table presents the results of the out-of-sample comparison of the models based on their middle quintile (Q3) calibrations. Mean variable of interest in this analysis is the exercise stock price ratio. The forecasting performance of each model is assessed based on the Mean Squared Percentage Error (MSPE), the Mean Absolute Percentage Error (MAPE) and the outputs of the OLS regression of the variables observed values over the models predictions. The meaningfulness of the differences between the forecast errors metrics is formally tested across the models using the impaired unilateral Student's t-test. The first panel reports the results of the OLS regression and the values of the forecast errors metrics for the whole sample. In the subsequent panels similar outputs are reported but focusing on 5 subsamples representing each a sample quintile with respect to the stock price volatility. P-values are in parentheses.

Additional robustness analyses: reference point specification
As previously underscored, given that the literature is still largely silent on the way a stock option holder would set and adapt his reference point, the specification of the latter is the main aspect that The first alternative specification (AS#1) is the current BS value of the option, denoted by " ( ) .
π ", based on the stock price corresponding to the employee own expectation regarding the annualized stock price return " ρ " over the option lifetime. The only difference between this specification and the original one is that it uses the BS value of the option rather than its intrinsic value: The second alternative (AS#2) assumes that the reference point is set in a static way at a proportion " ρ " of the BS value at the inception of the option: In the last alternative (AS#3) the reference point is specified as a fraction " ρ " of the current BS value of the option based on the spot price of the stock floored by its price at inception: , , , , , The common point in all these alternative specifications is that the reference point becomes dependent on the volatility of the stock price. However, while the second alternative is a static specification of the reference point that conveys the fact that the starting point is usually privileged in a benchmark role (Spranca et al., 1991), the first and the last alternatives are dynamic. Specifically, the last alternative is a stochastic specification of the reference point. Consistent with Arkes et al.
(2006) findings that people tend to adapt their reference points asymmetrically more completely over gains than over losses, this specification assumes that the reference point is adjusted to the spot price of the stock only if the latter is above its starting level at the option grant date. includes the parameters estimates yielded by the calibration. The middle block comprises the calibrations errors "ε" and the related ‫א‬ ² significance test p-values, where the null hypothesis is "H0: ε ≠0". The bottom block reports the model forecasts of the exercise variables implied by each calibration, including the expected cancellation rate and the conditional expectations of the stock price exercise ratios and the exercise timing, and the resulting value of the average stock option contract. A "*", "**" or "***" indicates significance at the 10%, 5% or 1% level, respectively. In order to compare the alternative specifications in terms of the predictive performance, I performed the same in-sample comparative analysis. Broadly, the results show that the specification originally used in the model do as well as AS#1 but outperforms the other alternatives in terms of the size of the average forecast errors. Specifically, AS#1 yields mean forecast errors that are nearly equivalent to those of the original specification, given that the difference is not statistically significant.

Calibration outputs
Its regression R² is slightly lower than that of the original specification. By contrast, the stochastic specification of the reference point (A#3) turns out to have the highest mean forecast errors, but improves in turn the linear adjustment of the empirical stock price exercise ratio.

Implications for fair value assessment
The international accounting standards (IFRS 2) and the US GAAP (ASC  15 Note that while the approach used here does not strictly follow the accounting standards guidelines in terms of the measurement of the fair value of the option, instead of the total cost of the grant, it provides however the cost per granted option. Actually, IFRS 2 makes a distinction between market based performance features and nonmarket features. Market based performance features should be included in the grant-date fair value measurement. However, the fair value of the equity instruments should not be reduced to take into consideration non-market factors or other vesting features. By contrast, fair values from our three competing models integrate cancellations due to the non-market factors. 16 Carpenter et al. (2010) showed, however, that the optimal exercise policy needs not to be specified in the form of a single critical stock price boundary. They nevertheless proved under the risk-neutral probability measure the existence of a single stock price exercise boundary for CRRA utility functions with risk aversion coefficient less than or equal to one.
(i.e. 6.30%). The ABS value is based on the average lifetime of the options (i.e. 4.30). In complement to this, I transposed the analysis to the out-of-sample setting using the calibration based on the average data from the middle quintile (Q3). This analysis involves 503 plans instead. . Asymptotic t-statistics are in parentheses. A "*", "**" and "***" indicate significance at the 10%, 5% and 1% levels respectively. .

Conclusion
This research draws on the CPT framework to develop an alternative theoretical model for the The central result of this paper is that the model predictions ascertain the ability of the CPT to better explain the exercise decisions than the aforementioned theories. This finding conveys the main contribution of the paper which is the strong ability of the CPT framework to explain some prominent patterns in the employee exercise behavior. It therefore provides rationale for using the CPT framework in order to get more accurate fair value estimates of employee stock options contracts. By admitting this, I proved that the methods recommended by the accounting standards and largely adopted by practitioners would potentially lead to an underestimate of the stock options cost to shareholders.
The major implication of these results lies within the area of fair value measurement issues. It outlines the relevance of behavioral valuation models for achieving meaningful fair value estimates.
More broadly, the results in this work underscore the importance of gaining a thorough understanding of the behavioral factors underlying employees exercise decisions in order to get comprehensive, and more apparent, perspectives on their economic consequences. These include for instance the cash impacts (resp. dilution impacts) due to the early exercise of cash settled (resp. equity settled) stock options.
Another interesting result in this work is given by the probability weighting coefficients yielded by the model calibrations. These estimates turn out to be consistent with those from the experimental literature, which suggests that the exercise behavior is likely to conceal optimism or pessimism biases.
This finding has a far reaching implication for employee sentiments representation. While some recent papers used stock options exercise timing as a proxy to behavioral biases, such as overconfidence Tate, 2005, 2008), I believe that the information in the probability weighting parameters estimates from stock options exercise time series could be used to build more relevant measures of employee sentiments towards very positive (resp. very negative) events. I leave this investigation to future work.
Due to the model specification, this research has some natural limitations that have to be outlined.
The first one relates to the specification of the reference point. Actually, the literature is still silent on how people set reference points when assessing complex gambles like stock option payoffs, which stands for a handicap in applying the CPT. In the absence of such guidance I tested specifications consistent with both empirical evidence on people setting reference points in a dynamic fashion and firms' widespread use of the BS value in financial and employees compensation disclosers. Moreover, the model does not handle the resetting and the reloading provisions that are used in some stock options contracts. The model could however be enhanced in order to parsimoniously integrate factors representing these provisions as suggested in a recent literature (Brenner et al., 2000;Sircar and Xiong, 2006). The last limitation is due to the representative employee setting that ignores the heterogeneity of beliefs. However, this issue needs to be put into perspective along with the finding in Jouini and Napp (2012) showing that an agent with an inverse-S shaped probability weighting function (as in the CPT) may be represented as a collection of agents with noisy beliefs.

Appendices Appendix A1: Construction of the decision weights tree
In this appendix we address the problem of the construction of a decision weights tree using the CPT probability weighting function. Let " , i j p " be the upward transition decision weight at node (i,j), " , i j p " is the corresponding downward transition decision weight and " , i j λ " is the weighted probability to reach the node. The latter is computed either from compounding transition decision weights related to the previous adjacent nodes or from weighting the binomial probability of reaching the node as described in (3) and (4). The transition decision weights are retrieved over the entire tree starting from level 0 towards level N-1 based on the following algorithm: • For i=0 : Given that the binomial distribution increases stochastically as "i" increases, we have: In the same way, noting that One may easily check that 1,1 1,1 1 p p + = .
• For i>1: Similar, assume the relationship is verified for i and prove it for i+1. The transition decision weights at the top node verify: 1 In addition, let the decision function "   ; 0 . In order to show that the probability of exercise increases in the degree of probability weighting (Resp. decreases in "a"), it suffices to prove that this is true at each node (k,l) leading to node (i,j) where the option is exercisable, which boils down to showing that the state