I.1a Pay-off diagram

Appendix I.A: Input parameters versus option value in the BSM model (*)

Reproduction of table 3.1: Effect on call option value from an increase in an input parameter.

Input parameter Call option value

American European

Stock price + +

Exercise price - -

Volatility + +

Time to maturity + ?

Risk-free interest rate + +

Expected dividend yield - -

Upon exercise, a call option provides a pay-off equal to the difference between the stock price and the exercise price. When the stock price increases, this pay-off increases. Therefore, the option value increases with an increase in the stock price.

For the second parameter, an analogous argument holds. When the exercise price increases, the pay-off upon exercise decreases. Hence, the option value decreases with an increase in the exercise price.

An increase in volatility increases the value of an option. The volatility of the returns of the underlying asset indicates the uncertainty regarding the expected returns. Assume that the returns on an asset are normally distributed. A return distribution with a higher volatility has fatter tails as compared to a return distribution with a lower volatility, all other things equal. In other words, the probability that an option is far in-the-money or far out-of-the-money increases. Recall from chapter two that an option has an a-symmetric pay-off profile. Hence, an option holder benefits from the increased probability that the option is far in-the-money, while the option holder is protected against the increased probability that the option is far out-of-the-money.

The relationship between the price of a call option and the time to maturity is different for European and American option. For American options, option value increases with time to maturity, since the span of exercise moments increases. This line of reasoning cannot be applied to European call options with different maturities. Consider options A and B to be European call options rather than American call options. Option A grants the option holder the right to exercise at a date at which option B cannot be exercised and vice versa. Therefore, no general relationship between option value and time to maturity can be defined for European call options on dividend- paying stocks. This can be illustrated using the following example. Consider the situation in

which the stock underlying option B distributes a dividend before the maturity of the option, while the underlying stock of option A does not. The dividend distribution causes the stock price to decline on the so-called ex-dividend date. Investors who own shares up to the ex-dividend date receive the associated dividends. Therefore, at the ex-dividend the stock price drops. This stock price drop can cause option A to be worth more than option B, even though option B has a longer maturity.

An increase in the risk-free rate has two opposing effects on the value of a call option. First, as the interest rate rises, the expected growth rate of the stock price increases, thereby making the option more valuable. Second, the increase in interest rates leads to a decrease in the present value of future pay-offs. This effect reduces the value of the option. However, the first effect always dominates the second effect. Therefore, a positive relationship between option value and the risk-free interest rate exists.

The last parameter shown in table 3.1 is the dividend yield. As explained earlier, dividends cause stock prices to drop at the ex-dividend date. Also as established earlier, a reduction in the stock price leads to a reduction in the value of a call option. Therefore, an increase in the expected dividend has a negative effect on option value.

Appendix I.B: Put-call parity (*)

The relationship between the price of a call option and the price of a put option can best be illustrated using an example. Consider setting up a portfolio by buying a stock of $ 30, selling a call option with an exercise price of $30 and buying a put option, also with an exercise price of $ 30. This portfolio and the pay-offs in two scenarios are shown in table I.2 as well as in figures I.1a and I.1b.

Table I.1: Portfolio components

Portfolio components Value Scenario 1: Scenario 2:

=35

St S_t =25

Long stock position S 35 25

Short call option −Max(S−X,0) -5 5

Long put option Max(X −S,0) ^{0 0}

Total X ^{30 30}

This can be seen when considering the two scenarios in table I.1. In scenario one, the stock price rises to $ 35, while scenario two depicts the situation of a stock price drop to $ 25. From table I.1, it can be seen that this portfolio pays off X, or $ 30, under both scenarios. In other words, the portfolio pays off X with certainty. Figures I.1a and I.1b display the pay-off diagrams of the instruments as well as the pay-off diagram of the portfolio. Figure I.1b displays the pay-off diagrams for the stock, the combined option positions and the portfolio. From this figure, it can be seen that the combined pay-off of the two options (the pink and yellow lines in figure I.1a, and the green line in figure I.1b.) exactly offset the pay-off on the stock.

As a result, the present value of the portfolio can be calculated by discounting this future pay-off at the risk-free rate. This leads to the following relationship between the portfolio components:

Xe rT

c p

S₀ + ₀ − ₀ = ⁻ (A1.1)

with:

p₀ = the price of the put option at time t = 0.

Rearranging equation 3.4 yields:

0 0

0 Xe S p

c + ^−rT = + (A1.2)

Equation 3.5 is referred to as put-call parity. Put-call parity describes the relationship between the value of a call option and the value of an equivalent put option. It shows that the price of a put option can be found using the same input parameters used when valuing a call option. The direct formula for pricing Equation 3.6 describes the price of a put option:

) ( )

( d₂ S₀N d₁ N

Xe

p= ^−rT − − − (A1.3)

I.1a Pay-off diagram

($10)

$0

$10

$20

$30

$40

$10 $20 $30 $40

Stock Price ($)

Pay-off ($)

Long Put Short Call

Stock Portfolio

`

I.1b Pay-off diagram

($10)

$0

$10

$20

$30

$40

$10 $20 $30 $40

Stock Price ($)

Pay-off ($)

Combined Options Stock Portfolio

Appendix I.C: Hull & White model (2004)

Hull & White (2004) set forth a binomial model that uses a stock price multiple to take into account early exercise. In addition, the model takes into account forfeiture and early exercise resulting from employees leaving the company by applying pre-vesting and post-vesting exit rates. Two of the assumptions made are that the option can be exercised at any time after the pre- vesting period and that the option holder cannot hold the options after leaving the company.

These assumptions hold for most option plans observed in practice. In addition, the model could be extended to deal with exceptions to these assumptions, although the model would lose the benefit of its initial simplicity.

The model is set up as follows. There are N time steps with equal length of Δt. The pre-vesting and post-vesting exit rates are labeled e₁and e₂ respectively. The stock price is denoted as S_n,j, with n indicating the number of time steps and j indicating the number of up moves in the model, and the strike price is denoted X. Option value is expressed as f_n,j. At time N, at the end of the option’s life, the value of the option is equal to its intrinsic value:

(

)

max _,

, S X

f_{N j} = _{N j} − (IC.1)

At earlier time steps, for 0≤i≤ N−1, there are three different possibilities:

1 The option has vested (iΔt ≥v) and the stock price equals or exceeds the strike price multiple (S_n,j ≥ XM ). In this case the option value equals its intrinsic value:

X S

f_i,j = _n,j − (IC.2)

It should be noted that M denotes the stock price multiple at which exercise is assumed.

2 The option has vested (iΔt ≥v), but the stock price does not equal or exceed the strike price multiple (S_n,j < XM ). In this case the option value equals:

(

) [

(

)

]

(

)

, e t e pf p f e t S X

f_{n j} = − Δ ^−rΔt _{n+ j+} + − _{n+ j} + Δ _{n j} − (IC.3)

In this situation, the option is only exercised if an option holder is forced to exercise as a result of employment termination. If the option holder remains with the company, the

option value is equal to the discounted expected value of the option values at the nodes one time step later.

3 The option has not yet vested (iΔt<v). In this case, if an option holder leaves the company, the option is forfeited. Hence, the option value is equal to the first term on the right hand side of equation 1C.3:

( ) [

( )

]

t r j

n e t e pf p f

f _, = 1− ₁Δ ^−Δ _{+1, +1} + 1− _+1, (IC.4)

The value of the option, f_0,0 , is found by rolling back the tree to the origin of the tree.

In the above model set-up, two employee exit rates, e₁and e₂, are used. If observations regarding employment termination do not support a difference between the employee exit rate during the pre-vesting period and during the post-vesting period, then one exit rate is used and e₁= e₂. These model parameters are calculated as e_i =ln

(

)

In addition to estimating the value of the ESO, the model is able to calculate the implied expected life of the option also referred to as “fugit”. This is done in the following way. At the final node the expected life is set equal to zero:

,_j =0 LN

Again, at earlier time steps, for 0≤i≤ N−1, there are three different possibilities:

1 The option has vested (iΔt ≥v) and the stock price equals or exceeds the strike price multiple (S_n,j ≥ XM ). In this case the expected life of the option equals zero:

,_j =0

Ln (IC.5)

2 The option has vested (iΔt ≥v), but the stock price does not equal or exceed the strike price multiple (S_n,j < XM ). In this case the expected life equals:

(

) [

(

)

]

L_n,j = 1− ₂Δ _n+1,j+1 + 1− _n+1,j +Δ (IC.6) In this situation, the option is only exercised if an option holder is forced to exercise as a result of employment termination, in which case the expected life of the option is zero. If the option holder remains with the company, the expected life is equal to the expected life of the option at the nodes one time step later plus the time that elapses until the next time step.

3 The option has not yet vested (iΔt<v). In this case, if an option holder leaves the company, the option is forfeited and the expected life is zero. Hence, one would expect the expected life to equal equation 1C.6:

(

) [

(

)

]

L_n,j = 1− ₂Δ _n+1,j+1 + 1− _n+1,j +Δ (IC.7a)

However, note from chapter three that FAS123 makes two corrections. First, contractual life is replaced by expected life in order to account for early exercise. Second, a forfeiture correction is made to account for employees leaving the company during the pre-vesting period. Hence, when comparing expected life as calculated using the Hull & White model with the expected life used in conjunction with BSM, forfeiture should not be taken into account into the Hull & White expected life calculation. To exclude the consequences of forfeiture, the following equation is used:

(

)

pL

L_n,j = _n+1,j+1 + 1− _n+1,j +Δ (IC.7b)

In other words, equation IC.7b is used to calculate the expected life, given that the option vests, whereas IC.7a calculates an unconditional expected life.

Appendix II: IFRS 2 Application guidance (*)

Appendix B of IFRS 2 offers guidance for the practical application of the guidelines set forth in IFRS 2. The most important sections are discussed below. One of the first issues in valuing ESOs is the selection of the option pricing model. An important remark regarding this matter is made in section B2. In selecting which option model to apply, section B2 requires companies to consider factors that knowledgeable willing market participants would consider. As mentioned before, IFRS 2 explicitly recognizes that this might preclude the use of the BSM formula. However, for options with relatively short contractual lives, or that must be exercised within a short period of time after a vesting date, the BSM model may result in a value that is substantially the same as that of a more flexible option pricing model.

Input parameters to option pricing models are prospective. Regarding the measurement of input parameters, two sections of IFRS 2 are worth mentioning with respect to this prospective nature.

First, section B12 of IFRS 2 states that an expected value should be calculated for all variables for which it is likely that there is a range of reasonable expectations. Examples include but are not limited to future volatility, dividends and exercise behavior. Second, section B13 summarizes input-measurement practices in the following way. A company should not simply base estimates of volatility, exercise behavior and dividends on historical information without considering the extent to which past experience is expected to be reasonably predictive of future experience.

One of the phenomena that characterize ESOs is early exercise behavior. As Huddart and Lang (1996) have shown, employees tend to exercise years before maturity, not uncommonly sacrificing more than 50% of the option’s BSM value. Therefore, early exercise behavior is an important value-determining aspect in the valuation process of ESOs. IFRS 2 explicitly lists the factors to consider in estimating early exercise:

• the length of the pre-vesting period;

• the average length of time similar options have remained outstanding;

• the price of the underlying shares;

• the employees’ level within the organization; and

• expected volatility of the underlying shares.

It should be noted that by listing the last factor, IFRS 2 acknowledges the tendency of employees to exercise options on highly volatile shares earlier than on shares with a low volatility. This phenomenon is noted in several studies, discussed in chapter three, that value ESOs from an

Another remark worth noting regarding early exercise is made in sections B20 and B21 which argue that separating an option grant into groups of employees with relatively homogenous exercise behavior is likely to be important, regardless of the option pricing method applied. This difference in exercise behavior stems from the fact that lower-level employees may have different portfolio compositions, risk attitudes and liquidity preferences. Lower-level employees are expected to derive a higher expected utility from exercising early as compared to top executives.

Another important value-driving parameter is expected volatility. A list of factors to consider, similar to the one for early exercise, is put forward for expected volatility. The factors to consider are:

• implied volatility from traded share options;

• the historical volatility of the share price over the most recent period with a time span equal to the (expected) life of the option;

• the length of time a company’s shares have been publicly traded;

• the tendency of volatility to revert to its mean; and

• appropriate and regular intervals for price observations.

The measurement of dividend yield and volatility raises a related issue. Since ESOs typically have long maturities, the model inputs have to be estimated over long horizons. The sensitivity analysis of chapter seven shows that both expected volatility and expected dividend yield can have substantial effects on option value. As a result, a substantial potential for errors in measurement arises. Therefore, the estimation of these inputs requires careful consideration.

Chapter five provides further details regarding the input parameters of the CVC binomial model and the general estimation thereof. Also, the valuations carried out in chapter six specify the approach taken to measure the company-specific parameter values.

The above discussion is related to the accounting treatment of a standard employee stock option plan. However, as mentioned earlier, some ESO plans exhibit market-based vesting requirements.

The term market-based vesting is used to refer to vesting conditional on variables that can be observed on a (financial) market. Recall from chapter two that the CVC binomial model is extended to deal with these market-based vesting requirements. IFRS 2 states the following regarding these requirements. Market-based vesting conditions should be taken into account when estimating the fair value at the measurement date. All other vesting conditions, in other words:

vesting conditions based on non-market variables, should not be taken into account when measuring the fair value per option at the measurement date. Rather these vesting conditions are

the transaction amount. Consider, for instance, a grant of 100 options for which vesting is dependent upon a non-market variable, say a sales target. The number of options on which the expense is based should then be determined depending on the probability that the sales target will be reached. Subsequently, expense adjustments are recorded for any differences between the amount of equity instruments expected to vest (based on the estimated probability of vesting) and the amount that actually vested (based on whether the actual event of meeting the sales target occurs or not). The ultimate expense is then based on the amount of options that actually vested.

This practice is also referred to as ‘truing up’.

Appendix III: Statistical analysis (*)

Some of the data listed in section 5.4 are not used as direct inputs to the model, but are used to calculate other model inputs. The probabilities of all triggering events are calculated by applying regression analysis, except for the probabilities of death and disability. These probabilities are based on actuarial data depending on the age of the grantees. To conduct the regression analysis historical data on ESO plans are needed in addition to the data on outstanding grants. The triggering events retirement, termination and early exercise are examined for a relationship with the stock prick. Previous option grants that are similar to the one under valuation in terms of the age of the grantees, the vesting schedule and the nominal term are used to establish this

relationship.

As pointed out in chapter three, studies on early exercise behavior show a significant relationship between early exercise and the stock price. However, evidence regarding the existence of a relationship between termination and retirement and the stock price is mixed.

The regression analysis consists of an ordinary least-squares regression. The general form of such a regression is shown in equation A3.1:

e X

Y =

α

β

with:

Y = the dependent variable;

X = the independent variable;

α = the intercept or constant ; β = the regression coefficient; and e = the error or disturbance coefficient.

An ordinary least-squares regression analysis makes a number of assumptions. These assumptions can be checked for validity using statistical tests. A heteroscedasticity test checks whether the error term is normally distributed with a mean of zero and a constant variance of σ². The error terms are said to be homoscedastic if they have a constant variance. If the spread of the error terms is not constant, the errors are said to be heteroskedastic. A autocorrelation test can be carried out to check whether the successive error terms are independent of each other. When the

values of e are independent the data are said to be non-autocorrelated. The data are said to be autocorrelated if the e-values are not independent.

When, for instance, estimating the probability of retirement, the general regression equation represented in equation 5.3 takes on the form:

(

)

p_r,i =

α

β

with:

i

pr_, = the conditional probability of retirement at time i;

αr = the intercept;

βr = the regression coefficient;

Si = the stock price at time i;

X = the exercise price of the grant; and e = the error or disturbance coefficient.

If historical data of more than one grant are available, an important adaptation to the analysis is necessary. Since, the exercise prices of the different grants will probably be different, a weighted- average of these exercise prices with weights based on the amount of exercisable options needs to be constructed. However, it is important not to aggregate across grants in a way that obscures the early exercise relationship.

It is important to note that carrying out this analysis as well as the statistical significance depends heavily on the data availability. The frequency of observations plays an important role.

Preferably, monthly data are used. Daily data are in practice almost never available and daily stock prices may be prone to autocorrelation effects. The problem with yearly data lies in the amount of data points. These data points are very likely to be insufficient to base any statistically significant conclusions on. Disregarding the potential pitfalls listed above may lead to a wrong specification of the relationship between the stock price and the probability of early exercise.

Similar data analysis and regression equations apply to the probabilities of employment termination and early exercise.