Australian Accounting Review
Volume 26, Issue 4 pp. 429-438
Original Article
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Managing Employee Stock Option Expense: A Fair Value Approach

Ming-Cheng Wu, 

Corresponding Author

Ming-Cheng Wu

National Changhua University of Education, Taiwan

Correspondence: M.-C. Wu, Department of Finance, National Changhua University of Education, No.1, Jin-De Road, Changhua 500, Taiwan, R.O.C. Tel: +886 4 723 2105 ×7321; fax: +886 4 721 1290; email: [email protected]Search for more papers by this author
I-Cheng Lin, 

I-Cheng Lin

National Changhua University of Education, Taiwan

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Yi-Ting Huang, 

Yi-Ting Huang

Taiwan Academy of Banking and Finance

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First published: 14 December 2016
https://doi.org/10.1111/auar.12111
Citations: 1

Abstract

Owing to special characteristics, classic option pricing models are not well suited to the valuation of employee stock options (ESOs). This paper attempts to conduct a more general fair value estimation based on attaching performance targets to option vesting. Considering a setting that includes factors such as options that may be exercised early at employee discretion, employee exit rates and firm default risk, this paper presents a sensitivity analysis and empirical tests of option value. The results highlight the importance of considering the characteristics of ESOs in the design of performance-vested option plans so as to provide the most attractive incentives for employees.

An employee stock option (ESO) is awarded for two purposes: to compensate for past performance and to provide an incentive for future performance. Hall and Murphy (2000) point out that stock options provide a direct link between executive expected utility and shareholder wealth. Top executives understand how their actions affect share prices. Option holdings provide incentives for executives to take actions that increase share prices and avoid actions that decrease share prices. As compensation, companies prefer granting stock options over cash compensation. Cash provides little incentive as it can be immediately expended, and executives cannot be compelled to invest that cash back into the firm. Moreover, options also give executives a stake in the future of the firm. Because executives have some control over the performance of a firm, options provide an incentive for executives to remain employed by the firm and exert their best efforts to improve its performance. Taylor (1994) suggests that gains from exercising executive share options are not simply rewards for the period in which they are realised and reported. Stock options are more likely to form a relatively important part of executive remuneration in firms that face valuable, but relatively risky, investment opportunities.

According to Hall and Liebman (1998), stock options have emerged as the single largest component of compensation for United States (US) executives. In fiscal year 1999, 94% of Standard & Poor's (S&P) 500 companies granted options to their top executives, compared to 82% in 1992. Moreover, the grant date value of stock options accounted for 47% of total pay for S&P 500 chief executive officers (CEOs) in 1999, up from 21% of total pay in 1992 (Hall and Murphy 2002). Based on a sample of 100 large listed Australian corporations, Carlin and Ford (2006a) report that about 80% of large listed corporations in Australia have grant stock options schemes, but that the size and cost of these schemes decline significantly after 2000.

Since the granting of employee stock options to top executives has an incentive effect in improving company performance, stock options have become the most important part of compensation for employees in US firms. However, a number of financial scandals have broken out in the US in recent years, such as when Enron and WorldCom did not disclose the truth in their financial reports and caused their investors to suffer significant losses. Research indicates that the main cause of these events was firms abusing ESOs. Therefore, how to value ESOs has become more and more important. While Brown and Howieson (1994) argue that the value of stock options is difficult to estimate reliably, option awards are valued at 25% of their exercise price (Coulton and Taylor 2002). Furthermore, the Financial Accounting Standards Board (FASB) makes some suggestions as to how employee stock options should be valued from the viewpoint of the company granting them.

In October 1995, the FASB issued a Statement of Financial Accounting Standards (SFAS) No. 123, which encouraged but did not require firms to adopt the fair value based method of accounting for ESOs. Carlin and Ford (2006b) provide evidence that executives are granted with option-based remuneration in about 73% of firms, but by 2002 this had grown to about 94%. They also note that while the use of options in executive compensation packages has grown substantially, the consequences of this trend have not been analysed from a governance perspective. In light of financial scandals, the revised Statement of Financial Accounting Standards 123 (SFAS 123R), issued by the FASB and International Financial Reporting Standard 2 (IFRS 2), requires companies to expense stock-based compensation schemes in their financial statements in 2004 and 2005, respectively. SFAS 123(R) explicitly permits firms to use either a lattice model or the Black and Scholes' (1973) model to value ESOs.

ESOs differ from standard exchange-traded options in several aspects. These include vesting periods, limitations on private sales and the feature of early exercise (Rubinstein 1995). All these restrictions and factors have the effect of reducing the value of the options. The indication is that the Black and Scholes' (1973) model (BS model) and Cox et al.’s (1979) binomial model (CRR model) could fail to incorporate some unique features for ESO valuation. Adopting the BS model or the CRR model could over value ESOs and thus over value the incentives created (Hall and Murphy 2000, 2002; Ingersoll 2006). ESOs have a vesting period, which is usually set as two to three years as compared to 10-year maturity. During the vesting period, employees cannot exercise their in-the-money ESOs. Furthermore, if employees leave the firm voluntarily or involuntarily during the vesting period, they get nothing, even if the ESOs are in-the-money. When employees quit their jobs voluntarily or are fired after the vesting period, they have a few days to exercise their in-the-money ESOs.

SFAS No. 123(R) requires that two additional parameters, the employee exit rate and the expected life of options, be estimated. Maller et al. (2002) conducted a binomial model, which mainly incorporates the possibility of employee departure and forfeiture in ESOs issued by Australian listed companies. The employee exit rate during the vesting period refers to the probability that employees will leave the company each year during the life of the options. We follow the setting in Cvitanic et al. (2008) and assume that the rate during the vesting period is smaller than the rate after the vesting period. The expected life of the options is the average time the vested options remain unexercised.

Johnson and Tian (2000) point out that the establishment of an exercisable clause for ESOs can stimulate stock prices to increase. Kuang and Qin (2009) empirically find that approximately 94% of the firms that adopted stock options as part of their CEO compensation contracts had attached performance targets to CEO stock options in 2004, compared to just 54% in 1999. Kuang and Qin (2009), Fels (2010) and Walker (2010) also find that the use of performance-vested schemes in compensation contracts is associated with higher pay–performance sensitivity, suggesting improved alignment of owner and manager interests through the use of performance targets. Wu and Lin (2013) derive an ESO pricing model embedded with performance hurdles and employee forfeiture. However, the present paper tries to further take into account some other key characteristics of ESOs, the vesting period and the feature of early exercise.

The seminal article on valuing ESOs is from Smith and Zimmerman (1976). Jennergren and Naslund (1993) modify the BS model for employee forfeitures and early exercise due to employee departures, and provide a closed-form solution for the special case of a European ESO and for employee departures occurring at a constant proportional rate. However, Hall and Murphy (2002) show why executives often argue that BS model values are too high. Adopting the valuation of the BS model would lead to an over estimation of costs and thus under value the stock price. Companies would not like to see this. The BS approach is appropriate for European options (i.e., options that cannot be exercised prior to maturity). However, the ESO is not entirely a European option. Because of its unique characteristics, the ESO is more like the American option.

One of the vital characteristics used in ESO valuation is the possibility of the option being exercised early. Hull and White (2004) and Bettis et al. (2005) examine early exercise and its impact on standard American option pricing models. Aboody et al. (2008) further suggest that private information triggers the exercise of sell or hold decisions. Huddart and Lang (1996) discuss this issue empirically and point out that the early exercise rule is not uniform, though it is pervasive. Marquardt (2002), Bettis et al. (2005), Aboody et al. (2008) and Leung and Sircar (2009) also point out that for ESOs with 10 years to maturity the average exercise time is between four and five years. To capture the effects of early exercise, the FASB proposes an expensing approach. In particular, it advocates adapting an expected time to maturity instead of an option expiration date. Ingersoll (2006) also considers the effects of fixed holdings, presenting a constrained portfolio problem where employees allocate wealth between company stock, market portfolio and a risk-free security. Each paper presents different modelling limitations on the underlying stock diffusion process; however, none of them models the several features of ESOs in a pricing model. This expensing method is far from accurate but is very simple and convenient.

Johnson and Stulz (1978) first conducted a valuation model considering the issue of counterparty default risk. They assume that the option itself is the only liability of the option writer and that the default occurs when an option writer's assets cannot afford the promised payment in the option contract. Klein (1996) further modified Johnson and Stulz (1978), allowing the option writer to have other liabilities. Under the BS environment, Klein (1996) developed a closed-form solution for pricing option values subject to the option writer's default risk. Empirically, we find that the firm's default happens when the stock price triggers 1% of the initial price. As a result, we simply assume that firms face default risk when stock prices fall below 1% of the exercise price in this paper. Based on the different financial structures of companies, firms could decide different levels of the default risks.

The purpose of this paper is to develop a binomial tree model to price ESOs. First, we adopt the performance-vested framework presented by Johnson and Tian (2000). A performance-vested ESO binomial tree model, an extension of the CRR model, is constructed. Furthermore, we take into account the vesting period, employee exit rates, the firm's default risk and the early exercise features from the approaches provided by Ingersoll (2006) and Hull and White (2004). Just as the BS formula is a limiting case of the CRR model, the Johnson and Tian (2000) and Hull and White (2004) formulas serve as limiting cases of our proposed binomial model. Also, the proposed computational algorithms are simple and efficient for pricing performance-vested ESOs.

Additionally, this study presents numerical analyses and uses Tabcorp Holding Ltd as a case that illustrates the relationship between ESO value and the main parameters embedded in the model. This paper also compares the ESO values calculated using the proposed model with the values obtained using the BS model, the CRR model and the Hull and White (2004) model. This study identifies significant differences among the ESO values estimated using the different models. After considering the primary characteristics of ESOs, the proposed model obtains a more rational estimate than the CRR model.

ESO Fair Value Estimation

The assumption

In this section, we discuss the assumptions of the pricing model and its economic foundations. Our objective in this paper is to adopt a performance-vested ESO pricing model, which will lead to several improvements over the existing one and take into account vesting period, early exercises, employee exit rates and firm default risks that are the features of ESOs. The option has an initial strike price K when it is granted. We employ the approach used in Ammann and Seiz (2004) and Hull and White (2004) and adjust the strike price to be MK. We denote the exercise boundary as M. When M < 1, the price triggers the early exercise (S ≥ MK). However, the option delays the exercise when the exercise boundary M is larger than 1. The adjusted strike price MK is used to determine the exercise time but not the payoff. To conclude, the option is early exercised when the stock price hits the level of MK after the vesting period and with a payoff as follows:
(1)
Based on SFAS 123(R), reported by the FASB, the enhanced SFAS 123(R) approach adjusts the option value to be: , which takes into account the employee exit rate q and the vesting period t. Hull and White (2004) consider the possibility of an employee exit rate after the vesting period and further provide the pricing model of ESOs as follows:
(2)
First, after options are vested and , the payoff of ESOs can be written as: . Second, when options are vested but , the payoff can be presented as follows:
(3)

Finally, would stand for the payoff when options are unvested.

Generally, there is a negative relationship between stock price and the employee exit rate. Bajaj et al. (2006) empirically find that employees face the considerable possibility of being fired when a firm's performance is poor. They also suggest that employees would not be likely to quit a job voluntarily when the stock options are at-the-money. The enhanced SFAS 123(R) approach ignores these features of employee stock options and possibly over estimates the option values. We incorporate the approach used in Bajaj et al. (2006) and adjust the employee exit rate to be: .

In this paper, we follow the setting used in the Johnson and Tian (2000) model and set a performance hurdle (B), which is higher than the exercise price and lower than the early exercising barrier (M). The price has to reach the performance hurdle to trigger action on the option. As far as the firm's default risk is concerned, we assume that firm default risks occur when stock prices fall below 1% of the exercise prices.

The pricing model

In this section, we conduct our pricing model based on the binomial tree. In this generalised model, we first consider that the option is vested and triggers the early exercise level . We further take into account the possibility of firm default risk. When the option is vested but the stock price does not hit the early exercise level, , we further consider the possibility of employee exit and firm default risk. We then rewrite the option value as follows:
(4)
where would stand for the payoff when options are unvested.
In this paper, we also employ the utility maximisation model and assume employees maximise their utilities rather than just the expected return of the option. The vested option is exercised if the expected utility derived from exercising the option is greater than the utility gained from holding it. Marcus and Kulatilaka (1994) and Rubinstein (1995) propose the following utility function:
(5)
where W is the total wealth of the employee, consisting of nonoption wealth and the ESOs held, and γ is the coefficient of employee risk aversion. The employee is said to be risk-neutral if γ = 1, risk-averse if γ < 1 and risk-seeking if γ > 1. The model assumes that both nonoption wealth and cash realised from exercising the options are reinvested in the risk-free asset (Huddart 1994). The potential drawbacks of the utility maximisation model include its dependence on a specific utility function and the need to estimate three parameters: the expected stock return m, the initial wealth W0 and the employee risk aversion γ.
Finally, the model takes performance hurdles, early exercise and the utility maximisation model into account simultaneously and then puts the employee exit rate and default risk into the model. The parameters are the same as in the section above. The precise algorithmic description of the complete model is as follows:
(6)
  • if and then ,
  • if ⩾ v and ⩽ < and > then = − K,
  • if ⩾ v and ⩽ < , and < (W)],
  • then ,
  • if , then .

Numerical Results and Empirical Tests

Sensitivity analyses

In order to obtain option values through our approach and carry out a sensitivity analysis, we require the basic inputs of the classical performance-vested stock option pricing models. These inputs are the exercise price and stock price at the grant date, performance hurdles, employee exit rates, vesting periods, life of option, expected volatility, risk-free rates and dividend yield.

First, most ESOs are issued with an exercise price equal to the firm's stock price at the grant date. That is to say, ESOs are usually granted at-the-money (Rubinstein 1995; Marquardt 2002). For calculation purposes, we consider that the stock price and the exercise price are equal to 50. Second, options are usually granted with a 10-year life, and many studies assume a 10-year maturity (Jennergren and Näslund 1993; Hull and White 2004; Bettis et al. 2005; Cvitanic et al. 2008). In this paper, we mimic the approach used in Marquardt (2002) and Vieito and Khan (2012) and calculate the standard deviation of monthly stock returns over the previous five years immediately before the end of each fiscal year as the annualised volatility.

To find the risk-free rate, the employee exit rate and the expected dividend yield, we follow the settings used in Hull and White (2004) and consider that they are equal to 7.5%, 30% and 2.5%, respectively. It is necessary to estimate the parameters used to capture the specific characteristics of option compensation plans. These characteristics are the vesting period, the employee exit rate, the employee wealth and the coefficient of employee risk aversion used to capture the early exercise. The performance hurdle of the barrier B that corresponds to the stock price target for option vesting is also considered.

In Table 1, we show results for the base case and some sensitivity analyses of fair values of ESOs. The expected volatility (σ) takes the value of 30%, 40% and 50%. We take the risk-free rate (r) from 0.05 to 0.1 and the employee exit rate (q) from 3% to 10%. The early exercise multiple (M) is set for M > 1 and equals 1.5, 2.0 and 2.5, respectively. In terms of results, let the expected volatility (σ = 0.3), risk-free rate (r = 0.05) and the employee exit rate (q = 0.03) be unchanged. We can see that when M increases from 1.5 to 2, the ESO value increases from 5.367828 to 5.8692576. Linking to the previous literature, the higher the early exercise multiple (M), the more valuable the option. Similarly, when we leave the expected volatility (σ = 0.3), risk-free rate (r = 0.05) and the early exercise multiple (M = 1.5) unchanged, the employee exit rate (q) increases from 0.03 to 0.1 and the ESO value decreases (C = 5.367828, 4.9177341, 3.976481, respectively). Because the high ratio of mean employees leaving the firm should forfeit their ESOs, the ESO values decrease. To sum up, we can easily find in Table 1 that the volatility of the underlying assets, measured by its standard deviation, has a positive effect on ESO value. We also testify to the impact of risk-free rates on the ESO value. The results are consistent with the vanilla call option, and show that risk-free rates have a positive effect on ESO value. Furthermore, we examine whether moneyness affects the ESO value or not, and we find that stock prices also have a positive effect on option values. Compared to an ESO that is out-of-the-money, an in-the-money ESO seems more likely to reach the exercising boundary and have a greater value.

Table 1. Sensitivity analyses of ESO fair values
S/K < 1 S/K = 1 S/K > 1
M = 1.5 q = 3% r = 5% 5.3678 6.9544 8.6008 13.9410 17.6360 19.4331 26.6507 29.7891 32.8610
r = 7.5% 6.3143 7.5779 9.1012 15.1387 18.6837 20.2416 28.2044 31.0705 33.9328
r = 10% 7.2065 8.1557 9.5725 16.2122 19.6381 21.0086 29.6243 32.2706 34.9501
q = 5% r = 5% 4.9177 6.4615 8.0587 13.1701 16.6649 18.4527 25.4312 28.4174 31.3510
r = 7.5% 5.7929 7.0514 8.5388 14.3227 17.6757 19.2382 26.9373 29.6622 32.3948
r = 10% 6.6262 7.6030 8.9940 15.3656 18.6034 19.9862 28.3199 30.8319 33.3878
q = 10% r = 5% 3.9765 5.4048 6.8752 11.4560 14.5049 16.2389 22.6369 25.2772 27.8925
r = 7.5% 4.6997 5.9192 7.3071 12.5008 15.4261 16.9658 24.0243 26.4292 28.8633
r = 10% 5.4046 6.4095 7.7224 13.4651 16.2849 17.6636 25.3099 27.5189 29.7914
M = 2.0 q = 3% r = 5% 5.8693 8.3558 9.8164 15.7328 18.9702 21.0816 28.6781 31.6478 35.2049
r = 7.5% 7.1546 9.3406 10.4447 17.4966 20.2646 21.9814 30.7073 33.0926 36.3863
r = 10% 8.4213 10.2683 11.0272 19.0326 21.4283 22.8085 32.4195 34.3892 37.4660
q = 5% r = 5% 5.3593 7.7001 9.1429 14.7641 17.8561 19.9320 27.2483 30.0958 33.4719
r = 7.5% 6.5361 8.6163 9.7403 16.4341 19.0950 20.8037 29.1953 31.4963 34.6224
r = 10% 7.7050 9.4860 10.2983 17.9049 20.2140 21.6101 30.8549 32.7612 35.6793
q = 10% r = 5% 4.2998 6.3183 7.6927 12.6503 15.4056 17.3712 24.0238 26.5824 29.5504
r = 7.5% 5.2498 7.0868 8.2193 14.1102 16.5144 18.1724 25.7771 27.8718 30.6197
r = 10% 6.2109 7.8289 8.7194 15.4274 17.5447 18.9236 27.3041 29.0517 31.6127
M = 2.5 q = 3% r = 5% 5.9300 8.5828 10.6385 16.2724 19.6205 22.5371 29.2615 32.6188 36.4111
r = 7.5% 7.3214 9.7085 11.4282 18.5930 21.1439 23.6400 31.6865 34.2660 37.7485
r = 10% 8.7183 10.7851 12.1639 20.6730 22.5060 24.6393 33.7168 35.7113 38.9458
q = 5% r = 5% 5.4135 7.8965 9.8702 15.2327 18.4263 21.2303 27.7621 30.9558 34.5409
r = 7.5% 6.6837 8.9378 10.6144 17.4019 19.8725 22.2904 30.0706 32.5428 35.8363
r = 10% 7.9678 9.9407 11.3127 19.3642 21.1767 23.2581 32.0240 33.9466 37.0040
q = 10% r = 5% 4.3410 6.4550 8.2301 12.9804 15.8169 18.3502 24.3988 27.2194 30.3432
r = 7.5% 5.3593 7.3171 8.8730 14.8217 17.0882 19.3079 26.4414 28.6609 31.5329
r = 10% 6.4056 8.1603 9.4859 16.5222 18.2564 20.1958 28.2098 29.9576 32.6208
  • Notes:. The parameter M is the early exercise boundary and a multiplier of the exercise price. q is the employee exit rate which is after the vesting period. We assume that the employee exit rate during the vesting period is 10% of q. r is the risk-free rate. σ is the stock price volatility. For calculation purposes, we assume that the stock price equals 30 and the exercise price equals 50 when the option is out-of-the-money (S/K < 1). Furthermore, when the option is at-the-money (S/K = 1), the stock price and the exercise price are equal to 50. While the option is in-the-money (S/K > 1), the stock price is equal to 80 and the exercise price remains at 50. Other parameters such as total employee wealth, the performance hurdle, the option life and the vesting period are equal to 100, 1.2 times of the exercise price, 10 years and two years, respectively.

Table 2 shows the price of the option for different assumptions about early exercise boundary (M) and employee exit rate (q). As might be expected, the value of the option increases as M increases and q decreases (Hull and White 2004). This result is consistent with Hull and White (2004) in that M shows a reverse relationship with employee exit rate (q).

Table 2. The relationship between ESO fair values, the early exercise boundary and the employee exit rate
q = 3% q = 5% q = 7% q = 10%
M = 1.2 13.7758 13.0940 12.4500 11.5491
M = 1.5 15.1387 14.3227 13.5587 12.5008
M = 2.0 17.4966 16.4341 15.4513 14.1102
M = 2.5 18.5930 17.4019 16.3064 14.8217
M = 3.0 18.8043 17.5859 16.4668 14.9525
  • Notes:. The parameter M is the early exercise boundary and a multiplier of the exercise price. q is the employee exit rate, which is after the vesting period. We assume that the employee exit rate during the vesting period is 10% of q. r is the risk-free rate and equal to 5%. σ is the stock price volatility and equal to 30%. In this section, we assume that the option is at-the-money (S/K = 1), the stock price and the exercise price are equal to 50. Other parameters such as total employee wealth, the performance hurdle, the option life and the vesting period are equal to 100, 1.2 times of the exercise price, 10 years and two years, respectively.

Moreover, we try to vary CEO wealth to find out the relationship between CEO wealth and fair value of employee options. As shown in Table 3, the value of the options increases when the CEO is more wealthly. Since employees maximise their utilities rather than the expected return of their options, higher wealth results in higher option value.

Table 3. The relationship between ESO fair values, total employee wealth and the employee exit rate
q = 3% q = 5% q = 7% q = 10%
W = 100 10.5506 9.7544 9.0337 8.0762
W = 120 15.1974 14.1483 13.1906 11.9046
W = 150 17.9579 16.7771 15.6941 14.2313
W = 170 18.8043 17.5859 16.4668 14.9525
  • Notes:. The parameter W is the early exercise boundary and equal to 1.2 times of the exercise price. q is the employee exit rate which is after the vesting period. We assume that the employee exit rate during the vesting period is 10% of q. r is the risk-free rate and equal to 5%. σ is the stock price volatility and equal to 30%. In this section, we assume that the option is at-the-money (S/K = 1), the stock price and the exercise price are equal to 50. Other parameters such as the performance hurdle, the option life and the vesting period are equal to 1.2 times of the exercise price, 10 years and two years, respectively.

We also compare the existing model with our model (Wu et al. Model); the results are shown in Table 4. Under the parameters of the base case with a performance hurdle 1.2 times of the exercise price, a 30% expected volatility (σ), a risk-free rate of 7.5% and an early exercise multiple (M) of 1.5, the option values of the CRR Model, the Hull and White (2004) Model and the Wu et al. Model are $20.9481, $15.1387 and $15.1387, respectively. The reason for these results is that the 30% expected volatility (σ) is not so high that the stock price might not trigger the early exercise boundary M. When the expected volatility (σ) increases to 40%, the option values of the Hull and White (2004) and Wu et al. models are different ($17.6363 and $17.6360).

Table 4. Model comparison
Cox et al. (1979) Model Hull and White (2004) Model Our Proposed Model
q = 3% r = 5% 18.0604 22.1867 26.0114 13.9410 17.6363 19.4378 13.9410 17.6360 19.4331
r = 7.5% 20.9481 24.4170 27.7594 15.1387 18.6839 20.2459 15.1387 18.6837 20.2416
r = 10% 23.8360 26.6183 29.4611 16.2122 19.6383 21.0124 16.2122 19.6381 21.0086
q = 5% r = 5% 18.0604 22.1867 26.0114 13.1701 16.6651 18.4567 13.1701 16.6649 18.4527
r = 7.5% 20.9481 24.4170 27.7594 14.3227 17.6759 19.2418 14.3227 17.6757 19.2382
r = 10% 23.8360 26.6183 29.4611 15.3656 18.6036 19.9894 15.3656 18.6034 19.9862
q = 10% r = 5% 18.0604 22.1867 26.0114 11.4560 14.5051 16.2415 11.4560 14.5049 16.2389
r = 7.5% 20.9481 24.4170 27.7594 12.5008 15.4262 16.9682 12.5008 15.4261 16.9658
r = 10% 23.8360 26.6183 29.4611 13.4651 16.2850 17.6656 13.4561 16.2849 19.6636
  • Notes:. The parameter M is the early exercise boundary and is equal to 1.5 times of the exercise price. q is the employee exit rate which is after the vesting period. We assume that the employee exit rate during the vesting period is 10% of q. r is the risk-free rate. σ is the stock price volatility. For calculation purposes, we assume that the stock price equals 30 and the exercise price equals 50 when the option is out-of-the-money (S/K < 1). Furthermore, when the option is at-the-money (S/K = 1), the stock price and the exercise price are equal to 50. While the option is in-the-money (S/K > 1), the stock price is equal to 80 and the exercise price remains at 50. Other parameters such as total employee wealth, the performance hurdle, the option life and the vesting period are equal to 100, 1.2 times of the exercise price, 10 years and two years, respectively.

Empirical analysis

In this section, we take Tabcorp Holdings Ltd as an example. Tabcorp Holdings Ltd is Australia's leading wagering, racing media company and Keno operator and one of the world's largest publicly listed gambling companies. The company has more than 120 000 shareholders and has a market capitalisation that puts it in the top 100 Australian companies listed on the Australian Securities Exchange (ASX). Moreover, the information on the ESO system of Tabcorp Holdings Ltd and its data on ESOs are complete. First, we select the data for the Tabcorp Holdings Ltd issued ESOs from the financial annual reports from 2008 to 2012. Then, we collect some of the parameters needed for the model. We collect the stock return volatility and the risk-free rate from the annual report of Tabcorp Holdings Ltd. Table 5 shows the data from Tabcorp Holdings Ltd. We then compare the ESO values calculated from the different estimation models in Table 6.

Table 5. The assumptions of the performance rights valuations (Tabcorp Holdings Ltd)
No. of Grant Share Price (S) at Expected Volatility (σ) Expected Dividend Risk-free Interest Time to Maturity
Grants Date Date of Grant ($) in Share Price (%) Yield (%) Rate (r) (%) (T) (year)
1 23 Oct 2008 6.95 24.00 5.50 4.37 7
2 19 Oct 2009 7.20 26.00 5.50 5.19 7
3 25 Oct 2010 7.47 24.00 6.50 4.97 7
4 23 Sep 2011 2.61 24.00 7.00 3.46 3
5 26 Oct 2011 2.87 24.00 7.00 3.73 3
6 4 Oct 2012 2.86 22.00 6.00 2.40 3
7 31 Oct 2012 2.84 22.00 6.00 2.57 3
  • Notes:. We present the option grants of Tabcorp Holdings Ltd. The stock price S and strike price K are equal (options are granted at-the-money). T is the maturity of the options; the vesting period is assumed to be 2. The employee exit rate (q) is set equal to 3%. The early exercise boundary (M) is set to be 1.5 times the exercise price. The performance hurdle is assumed to be 1.2 times the exercise price.

We can see in Table 6 that the Johnson and Tian (2000) model and the Hull and White (2004) model over estimated the cost of performance-vested ESOs and traditional ESOs. In 2008 (Case No. 1), the value of the ESO varies with a wide range from 5.12 to 4.02. Obviously, the value reported in the annual report is higher than the value calculated from the Hull and White (2004) model. We can easily find that the Hull and White (2004) model produces the lowest value of ESO because it factors the employee exit rate and the early exercise feature into the pricing model. The Johnson and Tian (2000) model has the largest value because it captures the feature of the performance targets. We do not claim that our model is the more general pricing model because of its lower fair value. In this paper, we address the point that even though performance-vested ESOs have a higher cost for granting, after taking into account the real characteristics of ESOs in the pricing model, we obtain a more realistic and general fair value for performance-vested ESOs, and the cost is not as high as the value calculated from the previous model.

Table 6. Performance rights values, MAPE and SMAPE values of Tabcorp Holdings Ltd calculated from different models
No. of Grant J&T H&W Tabcorp WL&H J&T H&W Tabcorp J&T H&W Tabcorp
Grants Date 2000 2004 Values 2015 MAPE MAPE MAPE SMAPE SMAPE SMAPE
1 23 Oct 2008 5.12 4.02 4.42 4.51 0.1353 0.1086 0.0200 0.1267 0.1149 0.0202
2 19 Oct 2009 4.87 3.51 3.92 4.01 0.2145 0.1247 0.0224 0.1937 0.1330 0.0227
3 25 Oct 2010 5.20 4.11 4.50 4.62 0.1255 0.1104 0.0260 0.1181 0.1168 0.0263
4 23 Sep 2011 2.96 0.98 1.34 1.57 0.8854 0.3758 0.1465 0.6137 0.4627 0.1581
5 26 Oct 2011 3.05 1.10 1.49 1.71 0.7836 0.3567 0.1287 0.5630 0.4342 0.1375
6 4 Oct 2012 3.00 1.03 1.37 1.62 0.8519 0.3642 0.1543 0.5974 0.4453 0.1672
7 31 Oct 2012 2.87 0.93 1.31 1.49 0.9262 0.3758 0.1208 0.6330 0.4628 0.1286
MAPE/SMAPE 56.03% 25.95% 8.84% 40.65% 31.00% 9.44%
  • Notes:. All parameters used in the pricing model are described in Table 5. Johnson and Tian (2000) [J&T 2000] is the performance-vested ESO model, while they ignore several characteristics of ESOs such as early exercise and the employee exit rate. Hull and White (2004) [H&W 2004] is the model that considers the early exercise boundary (M) and the employee exit rate. The Wu et al. [WL&H 2015] model is our proposed model that is based on the binomial tree model and further takes into account early exercise (boundary and utility), the employee exit rate, the performance hurdle and the firm's default risk. Tabcorp values are reported by Tabcorp Holdings Ltd in the financial statement. The mean absolute percentage error [MAPE] and the symmetric mean absolute percentage error [SMAPE] are also calculated and reported in this table to compare the percentage errors generated from each model.

Furthermore, we apply Mean Absolute Percentage Error (MAPE) and Symmetric Mean Absolute Percentage Error (SMAPE) to examine errors and differences between our proposed model and other pricing methods. According to the MAPE value from different models, columns 7–9 in Table 6 show clearly that the Johnson and Tian (2000) model has the highest MAPE with a mean of 56.03%, while the reported value has the lowest MAPE with a mean of 8.84%. Furthermore, results of the SMAPE values presented in columns 10–12 in Table 6 are consistent with that of the MAPE values. We then collect 238 option samples of 35 firms (Table 7) in Australia for the statistics tests, and the related variables are collected from the annual reports. The results are shown in Table 8. The MAPE(SMAPE) values of Johnson and Tian (2000), Hull and White (2004), and the reported value are 54.72% (42.88%), 28.72% (30.31%), and 10.44% (11.75%), respectively. As a result, there exists a significantly greatest difference between our model and the Johnson and Tian (2000) model because it ignores several characteristics of ESOs such as employee forfeiture, early exercise, firm's default risks and the vesting period. Although the reported value has the smallest difference among these three models (the Johnson and Tian model, the Hull and White model and the reported value), the difference is statistically significant. Taking for granted the first grant of Tapcorp Holdings Ltd in 2008, the difference of option values calculated from our proposed model and the reported model is 0.09 (4.51 minus 4.42). If the total amount of grant options is 10 000 shares, the expense would have a A$900 misreport in the financial statement.

Table 7. Company names and the GICS industry group
No. Company Name GICS Industry Group
1 Acacia Coal Ltd Energy
2 Alliance Aviation Services Ltd Transportation
3 Bionomics Ltd Pharmaceuticals & Biotechnology
4 Bigair Group Ltd Telecommunication Services
5 Carbon Energy Ltd Energy
6 Central Petroleum Ltd Energy
7 Corporate Travel Management Ltd Consumer Services
8 Domino's Pizza Enterprises Ltd Consumer Services
9 Donaco International Ltd Consumer Services
10 Ecosave Holdings Ltd Commercial Services & Supplies
11 Exoma Energy Ltd Energy
12 Echo Entertainment Group Ltd Consumer Services
13 Fairfax Media Ltd Media
14 Ferrowest Ltd Materials
15 Firestone Energy Ltd Energy
16 Fitzroy Resources Ltd Materials
17 Genera Biosystems Ltd Pharmaceuticals & Biotechnology
18 Geopacific Resources Ltd Materials
19 Grange Resources Ltd Materials
20 Hansen Technologies Ltd Software & Services
21 Harvey Norman Holdings Ltd Retailing
22 Horizon Oil Ltd Energy
23 Incremental Oil and Gas Ltd Energy
24 Iselect Ltd Consumer Services
25 Jupiter Energy Ltd Energy
26 Kathmandu Holdings Ltd Retailing
27 Legend Corporation Ltd Technology Hardware & Equipment
28 Living Cell Technologies Ltd Pharmaceuticals & Biotechnology
29 Macquarie Radio Network Ltd Media
30 New Standard Energy Energy
31 Onthehouse Holdings Ltd Retailing
32 Orca Energy Ltd Energy
33 Pacific Environment Ltd Commercial Services & Supplies
34 Quickflix Ltd Retailing
35 Tabcorp Holdings Ltd Consumer Services
Table 8. MAPE and SMAPE values and the statistics tests results
J&T MAPE H&W MAPE Tabcorp MAPE J&T SMAPE H&W SMAPE Tabcorp SMAPE
MAPE/SMAPE 54.72% 28.72% 10.44% 42.88% 30.31% 11.75%
t-value 25.785 40.946 30.064 31.535 31.337 28.034
  • Notes:. The 95% confidence intervals of the different parameters are calculated with the value given by the t-test. *** Significance at the 1% level. Johnson and Tian (2000) [J&T 2000] is the performance-vested ESO model, while they ignore several characteristics of ESOs such as the early exercise and the employee exit rate. Hull and White (2004) [H&W 2004] is the model that considers the early exercise boundary (M) and the employee exit rate. The Wu et al. [WL&H 2015] model is our proposed model that is based on the binomial tree model and further takes into account the early exercise (boundary and utility), the employee exit rate, the performance hurdle and the firm's default risk. Tabcorp values are reported by Tabcorp Holdings Ltd in the financial statement. The mean absolute percentage error [MAPE] and the symmetric mean absolute percentage error [SMAPE] are also calculated and reported in this table to compare the percentage errors generated from each model.

Conclusions

Based on the settings of performance-vested ESOs, this paper addresses several key characteristics of ESOs and provides a new approach for determining the fair value of performance-ESOs. The key characteristics include early exercise features, employee forfeitures, CEO utility maximisations and firm credit risks.

Because our approach takes the form of the CRR model, it is particularly simple and flexible enough to fit a variety of observed option biases. The comparative analyses and empirical results strongly show that the more general model derived in this paper can more precisely estimate fair values of performance-vested ESOs.

Acknowledgements

The authors are grateful for comments on an earlier draft from the 2011 Annual Meeting of Taiwan Finance Association and the 2013 Joint Annual Meeting of CTFA (Central Taiwan Finance Association). Ming-Cheng Wu acknowledges financial support from NSC 101-2410-H-018-008.

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