Valuation of Employee Stock Options: Applying stochastic volatility in Monte Carlo simulations

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Valuation of Employee Stock Options:

Applying stochastic volatility in Monte Carlo simulations

ROB JELLEMA*

ABSTRACT

An Employee Stock Option (ESO) is an option granted to employees on the shares of their company. ESOs have to be valued at fair value for accounting purposes. The standard ESO valuation methods assume a constant volatility over the maturity of the option. The objective of this paper is to assess the impact of applying stochastic volatility in the ESO valuation, by using Monte Carlo simulations. A stochastic volatility appears to have no influence on the option value, compared to using constant volatility, if there is no correlation between the stochastic processes. If there is correlation between the processes for the volatility and share price, stochastic volatility has a material impact on the option value.

Key words: employee stock options, stochastic volatility, Monte Carlo simulations

Companies make use of so-called compensation and incentive plans. An instrument that is often included in such plans is the Employee Stock Option (ESO). An ESO is a non-transferable long-term option granted to employees on the shares of their company. ESOs are used in an attempt to align the interest of the employee (manager) with the interest of the company (shareholders), and are intended to minimize any potential agency costs. ESOs are also used to retain skilled employees, which is especially important for start-up companies with a high growth potential. Most of these instruments are granted as at-the-money (near-the-at-the-money) options; the strike price is then set equal to the share price at the grant date. In the 1990s, at-the-money-options were considered zero-cost-options. But as explained by Hull (2009), there is no free lunch. Since there is a gain to the employee there should be an expense to the company, which has to be valued for accounting purposes. The typical features of an ESO, such as non-transferability and forfeiture, lead to early exercise behaviour, and will make the valuation of an ESO different from valuing a regular (call) option.

*

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The Financial Accounting Standards Board (FASB) and International Accounting Standards Board (IASB) have proposed accounting rules in which it is specified that ESOs should be recognized at their fair value on the grant date. Determining the fair value at the grant date proved to be a difficult task. The FASB allows for different valuation procedures to be used for valuing ESOs, one of these involves using Black & Scholes by (i) estimating the expected dividends, volatility and life of the option, (ii) use the Black & Scholes model to value the option, (iii) adjust for the expected number of forfeitures to allow for any possibility of the employee leaving the company. Again, it should be noted that the FASB also allows for other valuation methods to be used (FASB No. 123). The Black & Scholes model is typically applied to European options. However, ESOs are different from standard (call) options due to their specific characteristics such as the vesting period, forfeiture rate and non-transferability, as well as their implications for potential exercise during the maturity of the option. Therefore, Hull & White (2004) and Brisley & Anderson (2008) have both proposed a valuation approach based on a lattice model, which has been approved by the IASB. These methods incorporate the vesting period as well as any early exercise behavior due to non-transferability. But, the Black & Scholes, Hull & White and Brisley & Anderson approaches all assume constant volatility. Since it is unlikely that share price volatility will be constant over the life of the ESO, which can be as much as 10 years, I suggest a method that uses stochastic volatility. Previous research by Melino & Turnbull (1995) led to the conclusion that including a stochastic volatility of exchange rates in the valuation model for long-term foreign currency options reduced the overall hedging error. In my paper I will propose a model for valuing employee stock options based on Monte Carlo simulations, including a stochastic volatility. I will compare the results from the Black & Scholes model (for both the full maturity and the expected life of the option), the Hull & White and the Brisley & Anderson approach with the results of my stochastic volatility approach.

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valuation. Heston (1993) applied the technique of stochastic volatility to the valuation of European call options and concludes that the Black & Scholes formula yields virtually identical option values for at-the-money European call options compared to the stochastic volatility model. I wonder whether applying stochastic volatility in the valuation of ESOs will also result in approximately the same option values that would be obtained using constant volatility. Especially considering the special ESO characteristics that cause early exercise. While if there is a difference between calculated option values using stochastic volatility or constant volatility, the important question remains whether this difference is material. I will compare the results of my valuation model with the results from Black & Scholes, Hull & White and Brisley & Anderson. If there is a considerable difference in the prices derived by my model and the ‘standard methods’, this might be a reason to question the regular methods and rethink common practice. It turns out that using stochastic volatility yields similar results compared to using constant volatility if there is no correlation between the stochastic processes. If, however, the correlation is significantly different from zero, this has a material impact on the option value.

This paper is organized as follows. Section I gives a description of the main characteristics of an ESO as well as an overview of the literature on ESO valuation and the valuation methods. Section II discusses the methodology of the various ‘regular’ methods as well as the methodology of my stochastic volatility model. Section III examines the data necessary to perform my analysis. The results are presented in Section IV, and section V concludes.

I. Literature review

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 They have a vesting period, which can be as long as four years, during which the holder is not allowed to exercise.

 When the employee leaves the company (voluntarily or involuntarily) after the vesting period, the option is immediately exercised when it is in-the-money. If the option is out-of-the-money it is forfeited.

 The options are not transferrable. In order to diversify their portfolio employees must exercise the option and sell the shares.

 When ESOs are exercised, new stocks are issued, resulting in possible dilution.

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remaining option term to maturity, vesting schedule and the proportion of the grant remaining unexercised. They emphasize that time alone is a poor predictor of exercise behavior, making the valuation of an ESO difficult. Overall, the ESO characteristics lead to early exercise behavior. Early exercise makes the option worth less than would be the result of valuing a plain vanilla call using the Black & Scholes formula.

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& Scholes model may tend to reduce the value of the option as a consequence of underestimating the expected life of the option.

The use of a lattice model, such as the binomial option pricing model (or binomial ‘tree’), offers a way of dealing with early exercise behavior without having to define the expected life. Binomial models use estimates of expected share price movements over time. The expected share price is calculated for each up and down node. Given the strike price, the option price is calculated at each node by working backwards through the ‘tree’. The binomial model can be adapted in order to incorporate vesting requirements, early exercise behavior of an employee, the possibility of an employee leaving. Hull & White (2004) propose an enhanced binomial model that deals with the specific features of an ESO. It considers both the possibility of an employee leaving the company after the vesting period (based on employee turnover rates) and the employee’s early exercise policy. It assumes that the option is exercised when the share price is a certain fixed multiple of the exercise price. Using a fixed exercise multiple, however, is not entirely rational. Because, as the time to maturity is long the probability of a share price increase, and the option ending up in-the-money, is higher. Therefore, even a risk averse person may want to hold on to his option. As the time to maturity becomes shorter, a risk averse employee will prefer having the certainty of an assured payoff, rather than having the possibility of the option becoming even more in-the-money, or worse still, possibly zero.

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by using the expected life of the option, which is easier to estimate. It turns out that even though the different models derive their exercise policies in different ways, the pricing differences are negligible if the models are calibrated to the same expected life of the option (with the exception of the model proposed by the FASB). Therefore, Ammann and Seiz (2004) argue that the drawback of having to estimate all these different variables can be overcome by using the expected life of the option in order to calibrate the model.

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Black & Scholes value results in a downward sloping exercise boundary above which the option will be exercised. Whereas the Hull & White method would results in a horizontal boundary, and the Black & Scholes method could be depicted as a model having a vertical boundary. A downward sloping exercise boundary requires the option to be at a high multiple of the strike price early in the ESOs life in order to induce voluntary exercise, but allows for voluntary exercise at a low multiple of the strike price later in the option’s life. Brisley & Anderson (2008) show that the percentage-error of their ESO valuation model is minimal compared to the value obtained using utility maximization. For a risk aversion of one the percentage-error is only 1.6% compared to a percentage-error of -20.7% when using the model suggested by Hull & White. This can be explained by the exercise boundaries that are obtained when using the different models. The exercise boundary of the Brisley & Anderson model is similar to the boundary obtained using a utility maximization model. Only for a highly risk averse person the exercise boundary is similar to the exercise boundary of the Hull & White model. The Brisley & Anderson model is preferred over Hull & White because the resulting exercise boundary is approximately equal to the exercise boundary of a utility maximization model, without having to estimate unobservable parameters such as the measure of risk aversion, and also because the estimation of the proportion parameter (from historical data) is less sensitive to historical share price movements.

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the past is representative for what will happen in the future. But, this is not (necessarily) true. Implied volatility is the volatility that is estimated from traded options using Black & Scholes or a similar model (Hull, 2009). The use of implied volatility may be preferred over using historic price information if it contains market information.

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increases. As equity increases in value, leverage decreases and equity becomes less risky (decreasing volatility). This makes volatility a decreasing function of price (Hull, 2009). Figlewski (1997) notes that an objective consideration of the existence of such patterns must lead one to the conclusion that it calls into question the procedure of estimating volatility. Since the existence of a smile or skew implies that the model used to calculate the implied volatility, which is based on a constant volatility, is therefore not a correct description of how the market is pricing options.

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I will develop a model that accounts for vesting conditions, non-transferability, and the possibility that an employee will leave the company, which will result in early exercise, applying stochastic volatility. This model tackles the questionable assumption of constant volatility over the full maturity of the option and, in theory, should yield a more realistic and theoretically convincing valuation. This section will discuss the methodology of the valuation methods that will be applied, starting with the ‘standard methods’ followed by my own Monte Carlo model. In this paper I will value an ESO using a variety of valuation methods. The ESO will be valued based on the following assumptions and methods:

 Full maturity, using Black & Scholes, Binomial Trees and Monte Carlo.

 Expected life, using Black & Scholes, and early exercise based on the exercise triggers defined by Hull & White and Brisley & Anderson using binomial trees as well as Monte Carlo simulation.

 Early exercise, based on the Constant Elasticity of Variance process (with constant volatility) or my stochastic volatility model, calculated using Monte Carlo simulation.

I perform Monte Carlo simulations for the ESO valuation over the full maturity in order to evaluate the accuracy of my methodology. Since the value at full maturity is the value of a plain vanilla European call option, all methods should result in (more or less) the same option value, when disregarding forfeiture. As an alternative to the ‘standard’ (lognormal) share price process I will also use the Constant Elasticity of Variance process (CEV). I use this process to determine whether, and by how much, an alternative share price process (with constant volatility) will affect the valuation of an ESO.

The value of the ESO at full maturity and expected life will be derived using Black & Scholes. The Black & Scholes formula assumes that asset prices change continuously and are lognormally distributed. Since an ESO is a call option on the stock of a company, the following formula should be applied (Hull, 2009),

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( ⁄ ) ( ⁄ )

√ (2)

√ (3)

Where c is the value of the call option, q is the dividend yield, r the risk-free rate, V is the variance rate, T the time to maturity, S0 is the share price at the grant date and K is the

strike price. In order to determine the option value at full maturity, T is set equal to the contractual term of the option.

Having discussed the Black & Scholes model I will now discuss the estimation of the expected life of the option. I derive the expected life of the option by using the binomial tree from Brisley & Anderson. If the intrinsic value at a specific node is above the value of the option value at the same node in the Brisley & Anderson tree, the option will be exercised. Once the moment of exercise is known, the effective life of the option can be derived since the moment of exercise is related to a specific interval. If this number is multiplied by the probability of ending up in that specific node of the tree (state probability), the values can be summed and then multiplied by the size of the interval in order to obtain the expected life of the option. This will be the time to maturity, T, used in the Black & Scholes model to derive the value of the ESO given the expected life.

Hull & White and Brisley & Anderson both make use of binomial trees. A binomial tree is a type of lattice model that can incorporate assumptions about exercise behavior, as well as changes in share price volatility over the life of the option. A binomial model uses estimates of expected share price movements over time. The probability of an up move can be calculated as follows,

( ) (4)

In which u and d are the size of an up and down step respectively and are calculated as follows,

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√ _⁄ ₍₆₎

A down move has a probability of 1-p. With this information the share price at each node in the tree can be determined. The share price at the grant date, the first node, will be multiplied by to get the share price, Su, at the first up node. For the down node the share

price at the first node is multiplied with d to obtain Sd, which is the share price at the down

node. This should be repeated for every node in the tree until maturity. Next the option value can be computed at each node in the tree by working backwards through the tree, given the share price and the strike price at each node, to start at the final node. The option value of either the up or down nodes at maturity can be calculated using the following equation,

( ) (7)

Where S is the share price, K is the strike price, r is the risk-free rate and T is the contractual term of the option. For a European call option, the value at a specific node between full maturity and the grant date can be derived as follows,

_[ _{( )} _] ₍₈₎

Where f is the value at the node considered, p is the probability of an up move, r is the risk-free rate, fu is the option value at the next up node and fd is the option value at the next

down node. The value obtained using a binomial tree will become equal to the Black & Scholes value as the number of time steps is increased (Hull, 2009).

Hull and White (2004) propose a binomial model that considers forfeiture and incorporates early exercise by using a multiple of the exercise price. They use several equations for the backward induction through the tree (as described below). Where N is the number of time steps of length δt, M is the exercise multiple, r is the risk-free rate, p is the probability of an up move, eδt is the probability that the option will be forfeited after the vesting period, Si,j is the share price at the jth node at time iδt and fi,j is the value of the

option at the specific node. At maturity the option is either in-the-money, and exercised, or will expire worthless,

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During the life of the option, when the vesting period has ended, the option may either be exercised if the option is sufficiently in-the-money and the multiple is reached, or may not be exercised. This can be represented by the following equations. If ⁄ , the option is exercised and the option value at the specific node is calculated using (10),

(10)

If the vesting period has ended and the option is not sufficiently in-the-money, ⁄ , the option is not exercised and the value at that node is determined by using,

( ) [ ( ) ] ( ) (11) During the vesting period the option value is derived as follows,

[ ( ) ] (12) During the vesting period the forfeiture rate is not included, a general forfeiture rate that considers potential forfeiture during the vesting period is generally applied by management to the total value of the granted options. By including a multiple, the Hull & White model creates a horizontal boundary above which the option is immediately exercised.

The methodology used by Brisley & Anderson (2008) creates a downward sloping boundary by taking a fixed proportion of the Black & Scholes value as the exercise benchmark. Since the value of a call option drops when the time to maturity declines, the Black & Scholes value will become lower when moving closer to the final nodes of the tree. As a result, the share price required to trigger early exercise will decline. This induces early exercise at a relatively low multiple later in the life of the option. In this model the conditions for equations (10) and (11) are changed from ⁄ and ⁄ to and respectively. Where µ is the fixed proportion and BSi,j is the Black & Scholes value captured at the jth node at time iδt, for the remaining life

of the option given share price . Equation (10) and (11) remain unchanged.

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Monte Carlo is an appropriate method for valuing options as long as the exercise moments, by solely considering current parameters of the ESO, can be identified. I will define the moment of exercise using the ‘exercise triggers’ specified by Hull & White and Brisley & Anderson. Compared to Monte Carlo simulation, a binomial tree works from the future (back of the tree) to the present (front of the tree). Therefore all possible exercise moments have to be known in order to determine the fair value of the option at the grant date. Since all underlying values are known, one is able to determine the maximum value of the option. For an ESO this is not necessary, because I am not interested in the maximum value of the option. Instead, I want to calculate the option value at the moment of exercise, which is determined given the exercise conditions. Once the option is exercised the option will no longer exist, and any future values will have no impact on the valuation of the option. A method such as Monte Carlo, which works from present to future, is therefore justified because the only share prices that are needed are the prices until the moment of exercise. This means that, as long as the moment of exercise is specified using the exercise triggers from Hull & White or Brisley & Anderson, Monte Carlo is an appropriate method. Having discussed the applicability of Monte Carlo, I will now continue discussing my model.

Throughout this paper I will make use of the following stochastic volatility model (Hull, 2009),

( ) √ (13)

( ) (14)

Where V , a and_L are constant and dzs and dzV are Wiener processes. The variable V is the

asset’s variance rate, the variance rate has a drift that pulls it back to a level V at a rate a _L (mean-reversion). This process assumes that the asset price changes continuously. The level of mean-reversion to the long-term variance rate a, equals 1- α – β and √ for GARCH(1,1). These parameters can be estimated using the maximum likelihood method (Hull, 2009)1. It turns out that for certain values of ξ, the simulated variance might become

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negative. According to Alfonsi (2005), should be restricted to specific values in order to make sure that the variance will always be positive. He mentions that the variance will always be positive as long as √ and Vl > 0. If √ √ the negativity

problem is solved. If however, the long-term variance rate is changed, as for my sensitivity analysis, the condition specified by Alfonsi will change. Causing the ‘original’ ξ ( √ ), which is constant since it is only affected by alpha, to be above the limit for low values of Vl. Hence, I will adjust my original ξ by multiplying it with the ratio of the

applied long-term variance rate divided by the maximum variance rate that is used to determine √ .

The correlation between the two stochastic processes affects the share price distribution, see Heston (1993). Heston found that if there is no correlation between the stochastic processes, stochastic volatility only changes the kurtosis. Correlation between the share price and variance impacts the skewness of the distribution. When volatility is negatively correlated with the share price, a high realization of the stochastic term for the volatility process will on average make it less probable that there will be (large) positive share price changes. And, vice versa, for positive correlation the reverse is true. So, the implied volatility distribution is negatively/positively skewed depending upon the correlation. If the two processes as specified in equation (13) and (14) are uncorrelated, Hull & White (1987) demonstrate that the Black & Scholes formula overprices options that are-at-the-money and underprices options that are deep in or out-of-the-money. In case the volatility is positively correlated with the share price, out-of-the-money options are underpriced by the Black & Scholes formula, while in-the-money options are overpriced. Therefore, correlation between the stochastic processes is included by using the following equation, in which ρ is the correlation, є1 and є2 are the random variables used in

respectively the volatility and share price calculation and and are the ‘original’ random variables (Hull, 2009),

(15)

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The volatility of the share price is calculated using (14) given (15), and this together with (16) will be the input used in (13) to generate random share price paths.

As an alternative to the lognormal share price process described in equation (13), I assume another continuous process with constant volatility and investigate how this impacts the valuation of the ESO. I will use the Constant Elasticity of Variance (CEV) model,

( ) √ (17)

Where r is the risk-free rate, q is the dividend yield, V is the variance rate, and α is a positive constant. If α = 1, this process is similar to the share value process in equation (13). When α < 1 the volatility decreases as the share price increases. This creates a distribution similar to the distribution which would be obtained if there is negative correlation between the two Wiener processes, a heavy left tail and a less heavy right tail (negative skew). When α > 1, the volatility increases as the share price increases, this creates a positively skewed probability distribution (Hull, 2009). However, according to Schroder (1989), α should (originally) be chosen below 1. This makes sense because at alphas above 1, the volatility component of the share price process has an enormous impact. As a result, the share value development may possibly explode. At some point in time the share value might plummet to approximately zero after which it is unable to recover. A derivation of the CEV function is added to the Appendix.

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other below the true price, which on average will result in approximately the true price of the share with half the number of draws (Hull, 2009). This approach will ensure that the random numbers are spread equally around the mean of the standard normal distribution. In theory, this will result in a faster approach of estimating the true share price, because half the amount of random numbers has to be generated. And, as a result of averaging out the over and underestimations, will result in the ‘true’ option value. This technique is not applied to the calculation of the stochastic volatility. For every share price path the option value can be determined at every interval, by subtracting the strike price from the share price belonging to that specific interval and share price path. Next, the present value has to be calculated to arrive at the option value. During the vesting period the option has no value since the employee is not allowed to exercise the option. Once the vesting period has ended and the ESO is exercised, meaning that either the conditions from Hull & White or Brisley & Anderson are fulfilled, the option value at a specific point can be determined for every individual path,

( )( )( ) (18) Where fx,y is the option value and Sx,y is the share price for path x at interval y. K is the

strike price, eδt is the probability that the option will be forfeited, δt is the time interval, v is the vesting period in number of intervals, and r is the risk-free rate. Any potential forfeiture after the ending of the vesting period until the moment of exercise should also be taken into account. For this I apply the following equation,

( ) ₍₁₉₎

In which Sx,y is the share price for path x at interval y, K is the strike price, eδt is the

probability that the option will be forfeited, δt is the time interval and r is the risk-free rate. This value should be added to the option value derived at the moment of exercise, making the overall option value of one simulated path equal to,

∑ (20)

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necessary for that specific share price path. Because, once the option is exercised, further share price developments will no longer influence the option value. The ESO value will be determined by taking the average of the option value from every calculated path. Where f is the ESO value, n is the number of calculated paths and fx is the option value for path x.

∑ ⁄ (21)

This is the ESO value given (only) one simulation, derived from n paths. My true ESO value will be based upon an average of multiple simulations. I take the average value of multiple simulations in order to reduce any over or underestimation as a result of random sampling. Because the value of one simulation will be either an overestimation or underestimation of the true value of the option. By taking the average of multiple simulations I intend to reduce this error.

Given the methodology underlying the different methods I am able to compute the ESO values2. I will compare the results of the stochastic volatility model with the values derived from the Black & Scholes, Hull & White and the Brisley & Anderson model for options with the same characteristics. I will also perform multiple sensitivity analyses, which I use to be able to assess the robustness of my valuation model and judge the added value of including stochastic volatility. If there is a material difference in option value, this might suggest having to rethink common practice.

III. Data

Since the primary goal of this paper is to investigate whether the different methods lead to different results, it is sufficient to use an approach similar to the one used by Hull and White (2004) and Melino and Turnbull (1995). Their valuation methods are both based on fictitious data. This is permitted as long as the applied parameters such as the vesting period, time to maturity, risk-free rate and dividend yield are used consistently.

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My ‘basic’ ESO will be granted at-the-money with a maturity of 10 years, a vesting period of 3 years and a share value, S, at the grant date of 69.50. The life of the option is divided in 240 equal intervals making equal to a half month (10/240 years). I set the Hull & White multiple, M, at 3. The proportion, µ, from Brisley & Anderson is set at 50%. Huddart & Lang (2006) report an average multiple of 2.22 and a proportion between 55% and 99% based on the exercise behavior of 50,000 employees for eight different companies. In my sensitivity analysis I will investigate the impact of using different multiples and proportions. The continuously compounded risk-free rate is 3.75% and the variance rate is set to 36%, which is equal to a volatility of 60%. I assume that there are no dividend payments throughout the life of the option and that there is no forfeiture. Table 1 gives an overview of all parameters that I will use for the valuation of my ‘basic’ ESO.

Table 1: Overview of the relevant ‘base case’ ESO parameters used in the different valuation models

Parameter ‘Basic’ value

Share value S = 69.50 Strike price K = 69.50 Risk-free rate r = 3.75% Dividend yield q = 0.00% Variance = 36.0% Multiple M = 3 Proportion µ = 50% Forfeiture rate 0%

Vesting period 3 years

Maturity 10 years

Number of Intervals 240

Interval size δt = 0.042 year

IV. Results

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the values obtained using either Hull & White or Brisley & Anderson, based on a constant volatility.

In my presentation of the option values obtained using the ‘standard’ constant volatility methods I will start by valuing a 10 year at-the-money ESO as if it were a European call option, by using the Black & Scholes model. The results are presented in Table 1. These option values are obviously an overestimation of the true ESO value, because the potential forfeiture and non-transferability, which results in early exercise, are not taken into account.

Table 1: Option values of a 10 year ESO using Black & Scholes

This table presents the Black & Scholes values of an at-the-money ESO as if it were a European call option. The results are based on the following parameters: S = K = 69.5, r = 3.75%, q = 0%.

Black & Scholes

36% 25% 10% 1% 49.94 45.08 34.73 22.78

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Table 2: Option values obtained of a 10 year ESO using the ‘standard’ constant volatility models

This table presents the Hull & White, Brisley & Anderson and the adjusted Black & Scholes values of an at-the-money ESO as if it were a European call option. The results are based on the following parameters: S = K = 69.5, r = 3.75%, q = 0%.

Methodology Hull & White Brisley & Anderson

Multiple/ Proportion 1 2 3 4 0% 25% 50% 75% 36% 29.85 40.53 44.27 46.69 29.85 33.55 36.81 41.69 25% 25.84 36.88 40.81 41.84 25.84 29.15 32.16 34.69 10% 19.43 29.62 32.93 33.80 19.43 20.44 23.47 27.62 1% 9.48 21.95 22.74 22.77 9.48 10.40 12.76 17.26 Adjusted Black & Scholes

36% 41.17 42.72 44.05 46.00

25% 35.39 36.92 38.30 40.52

10% 24.77 25.35 27.09 29.54

1% 10.84 11.64 13.74 17.74

without dividends before maturity is never optimal (Hull, 2009). The adjusted Black & Scholes model, based on the expected life derived from the Brisley & Anderson tree (for the same proportion value), yields higher option values compared to the option value from the Brisley & Anderson tree. This would suggest that the adjusted Black & Scholes model overvalues the ESO compared to Brisley & Anderson. Otherwise the adjusted Black & Scholes value would have been equal to the Brisley & Anderson value, because it uses the expected life from the Brisley & Anderson tree. For smaller variances this difference seems to decrease. Having presented the option values based on the ‘standard’ methods I will continue with the valuations using my Monte Carlo model. But, before being able to do so I will first have to determine its accuracy.

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Table 3: Valuation of an at-the-money EU call option using Black & Scholes or a binomial tree

Option values and the differences in option values between Black & Scholes and a binomial tree with 240 equal intervals for multiple maturities, using constant volatility. Based on the following parameter values: S = K = 69.5, r = 3.75%, q = 0%, = 36%.

Maturity 1 year 2.5 years 5 years 7.5 years 10 years

Black & Scholes 17.40 27.43 37.84 44.83 49.94

Binomial tree 17.38 27.41 37.80 44.79 49.91

Difference 0.02 0.03 0.03 0.04 0.04

%-error -0.10% -0.09% -0.09% -0.08% -0.08%

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Table 4: Percentage-error between Monte Carlo and Black & Scholes for a ESO at maturity

The percentage-error of the option value derived using Monte Carlo simulation, compared to the Black & Scholes value for differing maturities and different numbers of simulated paths. The option values are obtained using Monte Carlo with constant volatility. All Monte Carlo option values are based upon the average of 30 simulations. Using the following parameter values: S = K = 69.5,

r = 3.75%, q = 0%, = 36%. Panel A shows the percentage-error without the Antithetic Variable Technique and panel B shows the percentage-error if the AVT is applied. The actual number of calculated values are presented in parenthesis.

Maturity 1 year 2.5 years 5 years 7.5 years 10 years Number of paths

Panel A Without AVT

1,000 paths (1,000) -0.18% -0.79% 0.54% -1.39% -0.32%

2,500 paths (2,500) -0.69% -1.51% -0.49% -0.14% -4.69%

5,000 paths (5,000) -0.06% -0.51% -1.15% -1.18% -3.71%

7,500 paths (7,500) -0.28% -0.50% -1.06% -1.94% -1.85%

10,000 paths (10,000) -0.33% -0.67% -1.28% -1.45% -2.89%

Panel B With AVT

1,000 paths (2,000) 0.38% 0.20% 0.66% 0.08% -2.84%

2,500 paths (5,000) -0.10% -0.22% -0.38% -0.65% -1.03%

5,000 paths (10,000) 0.10% -0.02% -0.05% 0.20% -0.23%

7,500 paths (15,000) 0.02% 0.02% -0.02% -0.47% -0.65%

10,000 paths (20,000) -0.03% -0.09% 0.05% 0.03% -0.56%

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calculation. In theory, the Monte Carlo option value converges to the Black & Scholes value if the number of simulated paths is increased. Yet, it might need around a million simulated paths to become accurate up to a couple of decimals. This is incredibly time consuming (see also Benninga, 2008). Especially if the number of computed variables that influence the exercise of the option is increased. For example, the time needed to compute the Brisley & Anderson option value is longer than the calculation time of Hull & White because it involves having to calculate the remaining Black & Scholes value at every interval. Determining the European call option value (using constant volatility) is relatively simple in terms of extra variables that have to be calculated. Yet, the European call value, based upon 30 simulations using 10,000 paths, already takes 60 minutes to compute. Only imagine the computation time needed for Brisley & Anderson using stochastic volatility.

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the average of 30 simulated option values is much lower when using the AVT. The percentage-error is -2.89% without using the AVT compared to an error of -0.23% if the AVT is applied. Although applying the AVT does not result in a smaller spread for all maturities based on the minimum and maximum option value, it does, however, reduce the percentage error. Overall, the percentage-error is smaller for all option values obtained using 5,000 paths when applying the AVT, compared to the option values based on 10,000 paths without using the AVT. This is exactly why this technique is applied.

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Table 5: Minimum and Maximum of the 30 option values underlying the percentage-error in Table 3, with and without applying the AVT

This table presents the Black & Scholes value of a European call option as well as the ESO value at maturity based upon an average of 30 option values using Monte Carlo simulation, with and without applying the AVT for different maturities. It also contains the minimum and maximum option values that are included in the average to obtain the ESO value, as well as the percentage-error which is based upon the averge of 30 option values compared to the Black & Scholes value. The percentage-error compared to the Black & Scholes value is similar to the percentage-error in Table 3 because the ESO values in both tables are based upon the same average of 30 option values. Based on the following parameter values: S = K = 69.5, r = 3.75%, q = 0%, = 36%

Without Antithetic Variable Technique With Antithetic Variable Technique

1 year 2.5 years 5 years 7.5 years 10 years 1 year 2.5 years 5 years 7.5 years 10 years

Black & Scholes 17.40 27.43 37.84 44.83 49.94 Black & Scholes 17.40 27.43 37.84 44.83 49.94

1,000 paths Average 17.37 27.22 38.04 44.21 49.78 1,000 paths Average 17.46 27.49 38.08 44.87 48.53

Min 14.39 23.16 27.76 27.34 29.56 Min 16.49 25.44 33.07 38.15 38.62

Max 20.38 32.14 51.39 58.31 66.38 Max 18.49 29.68 46.52 57.62 64.96

%-erroraverage -0.18% -0.79% 0.54% -1.39% -0.32% %-erroraverage 0.38% 0.20% 0.66% 0.08% -2.84%

Min 15.46 23.27 30.46 33.21 38.12 Min 16.60 25.81 34.88 39.68 41.96

Max 18.69 29.65 42.68 52.98 55.03 Max 18.07 29.26 39.87 55.96 67.07

%-erroraverage -0.69% -1.51% -0.49% -0.14% -4.69% %-erroraverage -0.10% -0.22% -0.38% -0.65% -1.03%

Min 16.22 25.72 33.63 38.34 41.74 Min 16.88 26.16 35.83 40.87 44.27

Max 18.16 28.77 40.46 51.58 62.65 Max 17.87 28.36 39.85 51.81 55.02

%-erroraverage -0.06% -0.51% -1.15% -1.18% -3.71% %-erroraverage 0.10% -0.02% -0.05% 0.20% -0.23%

Min 16.48 25.24 34.19 39.94 43.14 Min 16.90 26.33 35.35 40.13 43.07

Max 18.07 28.55 39.44 47.88 55.54 Max 17.77 28.22 39.78 47.14 53.85

%-erroraverage -0.28% -0.50% -1.06% -1.94% -1.85% %-erroraverage 0.02% 0.02% -0.02% -0.47% -0.65%

Min 16.54 25.89 35.59 40.64 44.36 Min 16.80 26.58 35.62 42.10 45.23

Max 18.01 28.61 39.50 47.77 53.13 Max 17.75 28.25 39.39 47.98 55.39

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If the option value is derived using 30 simulations of 5,000 paths, the maximum percentage-error based upon six option values, is -1.0%. The minimum percentage-error is only -0.2%, which is a range of 0.8%. The range is higher for valuations based upon a smaller number of simulations. But, because the calculation time of using 30 simulations is approximately 30 minutes, I prefer to use less simulations. Since using 10 simulations is still accurate according to Table 6, while being more time efficient, I will use 10 simulations. Reducing the computation time to (only) 10 minutes. I believe this is a reasonable sacrifice in terms of accuracy since using 25 simulations even results in a potential overestimation of 1.8%, or an undervaluation of -1.8%, as can be observed from Table 6. This is caused by the characteristics of a random variable technique such as Monte Carlo, since such a technique needs a very large number of random calculations to become accurate. Although this error is reduced by applying the Antithetic Variable Technique, there still is variability in calculated option values. Overall, the percentage-error is minimal compared to the Black & Scholes value of 49.94 and makes my Monte Carlo model, as well as Monte Carlo itself, a reliable technique for valuing options.

Table 6: Percentage-error of ESO value using Monte Carlo simulation compared to Black & Scholes

The percentage-error for multiple ESO values obtained using Monte Carlo simulation compared to Black & Scholes. The option value is the average of 30, 25, 20, 15, 10 or 5 simulations of 5,000 paths. For an ESO with the maturity of 10 years, using constant volatility and the AVT. Based on the following parameter values: S = K = 69.5, r = 3.75%, q = 0%, = 36%.

Number of simulations 30 25 20 15 10 5 ESO value 1 -0.6% -0.6% 0.6% -2.1% -1.2% -1.6% 2 -0.3% -0.1% -2.4% 1.1% 0.0% -5.9% 3 -0.3% 1.8% -0.6% -1.3% 1.3% 8.4% 4 -0.8% -1.8% -0.3% 0.8% -0.4% -2.3% 5 -0.2% -0.3% -0.4% 0.6% -1.2% 1.2% 6 -1.0% -0.9% 0.6% -2.3% -1.1% -0.2% Minimum -1.0% -1.8% -2.4% -2.3% -1.2% -5.9% Maximum -0.2% 1.8% 0.6% 1.1% 1.3% 8.4% Range 0.8% 3.6% 3.0% 3.4% 2.5% 14.3%

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price of 69.50. Remember that the share value distribution changes for every simulation that is performed due to the random sampling of Monte Carlo. As a result, no simulation will generate exactly the same share value distribution as the one in Figure 1. However, for every simulation, the present value of the share price at maturity is approximately equal to the starting share price of 69.50.

Figure 1: Distribution of the share value at maturity for a 10 year European call option

This figure shows the distribution of the share values at maturity for a European call option, based on 5,000 simulated paths using the AVT and one simulation. Based on the following parameter values:

S = K = 69.5, r = 3.75%, q = 0%, = 36%.

Now that I have determined the accuracy of my Monte Carlo model, I can confidently value ‘real’ ESOs that can be exercised at any time beyond the vesting period. In all calculations I will use 10 simulations based upon 5,000 simulated paths and apply the Antithetic Variable Technique.

-4,00 -3,00 -2,00 -1,00 0,00 1,00 2,00 3,00 4,00 5,00 6,00 7,00 8,00 9,00 10,00

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Table 7: ESO option values obtained using constant volatility for the various methods and techniques

This table presents the option values of my ‘base case’ ESO using the various methodologies and different valuation techniques. The option values are based upon following parameter values: S = K = 69.5, r = 3.75%, q = 0%, = 36%, vesting period = 3 years, Multiple = 3, proportion = 50%, expected life = 7.17 years.

Methodology Hull & White Brisley & Anderson Expected Life Technique

Adjusted Black & Scholes 44.05

Binomial trees 44.27 36.81

Monte Carlo simulation 44.61 37.04 44.18

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Table 8: ESO values obtained using the Hull & White binomial tree

This table presents the results of various ESO valuations using different multiples and variance rates while changing the forfeiture rate and dividend yield. All option values are obtained using the Hull & White tree for a 10 year at-the-money ESO. With a strike price of 69.5 a vesting period of 3 years and a risk-free rate of 3.75%. Forfeiture rate 0% 10% Multiple 1 2 3 4 2 3 4 Dividend Yield 0% 36% 29.85 40.53 44.27 46.69 38.52 41.25 42.94 25% 25.84 36.88 40.81 41.84 34.68 37.48 38.71 10% 19.43 29.62 32.93 33.80 27.03 29.32 29.90 1% 9.48 21.95 22.74 22.77 18.59 19.07 19.08 1% 36% 28.35 38.54 41.84 43.78 36.62 39.03 40.35 25% 24.39 34.82 38.12 39.39 32.75 35.07 35.92 10% _18.03 _27.12 _29.25 _29.57 _24.78 _26.21 _26.42 1% 8.00 16.72 16.94 16.94 14.36 14.48 14.49 5% 36% _22.97 _31.26 _33.05 _33.39 _29.73 _30.97 _31.11 25% 19.23 27.32 28.51 28.22 25.73 26.47 26.17 10% 13.14 18.24 17.13 16.23 16.83 15.93 15.30 1% 3.51 3.32 3.27 3.27 3.35 3.32 3.32

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dividends. This is true for options with a 1% variance rate because the chance of the option ending up more in-the-money is relatively small. A similar valuation can be performed using Monte Carlo simulations instead of a binomial tree, based on the Hull & White exercise conditions. This should result in the same option values based on my previous investigation of the accuracy of Monte Carlo, see Table 9.

Table 9: ESO values using constant volatility and Hull & White, Monte Carlo

This table presents the results of multiple ESO valuations using different multiples and variance rates while changing the forfeiture rate and dividend yield. All option values are obtained using the Hull & White exercise conditions for a 10 year at-the-money ESO based on an average of 10 simulations using 5,000 paths. With a strike price of 69.5 a vesting period of 3 years, a risk-free rate of 3.75%.

Forfeiture rate 0% 10% Multiple 1 2 3 4 2 3 4 Dividend Yield 0% 36% 30.82 41.21 44.61 46.99 38.97 42.10 43.87 25% 26.79 37.40 41.04 42.75 35.27 38.58 40.10 10% 18.97 29.79 32.96 33.81 27.69 30.94 32.87 1% 9.33 22.01 22.75 22.80 21.25 22.60 22.68 1% 36% 29.36 38.83 41.90 43.80 36.78 40.20 41.48 25% 25.36 35.44 38.35 39.56 32.96 35.81 37.30 10% 17.45 27.26 29.27 29.60 25.41 27.75 28.92 1% 7.83 16.74 16.95 16.95 16.59 17.20 17.22 5% 36% 23.89 31.40 32.96 33.52 29.79 31.37 31.88 25% 20.22 27.41 28.66 28.19 26.01 26.84 27.07 10% 12.62 18.25 17.31 16.03 17.33 17.13 16.95 1% 3.36 3.29 3.27 3.29 4.00 3.99 3.99

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forfeiture, the highest option values seem to be obtained for options that are exercised earlier in the life of the ESO, which is especially true for options with a low variance rate. This is caused by the impact of the dividend component, which increases compared to the variance component as the variance decreases.

It is interesting to investigate the impact of a similar sensitivity analysis on the option values derived using Brisley & Anderson instead of Hull & White. I will start with the valuation results obtained from the binomial tree, which are presented in Table 10. Overall, the option values obtained using Brisley & Anderson are lower compared to Hull & White. Except for the option values derived using either a multiple of 1 or a proportion of 0%. This is logical since under this assumption the option is exercised the first time it is in-the-money beyond the vesting date, which is the same for either a multiple of 1 or a proportion of 0%. Decreasing the variance rate will decrease the value of the option since it will lower the chances of the option becoming more in-the-money. While increasing the dividend yield decreases the value of the option because granting dividends will lower the share value. Lowering the proportion, which means that the option is exercised earlier, reduces the value of the option. Delaying the exercise of the option will, in general, result in the highest option value.

Table 10: ESO values obtained using the Brisley & Anderson binomial tree

This table presents the results of various ESO valuations using different multiples as well as other variance rates while changing the forfeiture rate and dividend yield. All option values are obtained using the Brisley & Anderson tree for a 10 year at-the-money ESO. With a strike price of 69.5 a vesting period of 3 years and a risk-free rate of 3.75%.

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Even when the forfeiture rate is set at 10%, contrary to what is observed for Hull & White. This is most likely caused by the fact that the (remaining) Black & Scholes value which is used by Brisley & Anderson is adjusted for dividends. This is a drawback of Brisley & Anderson since it uses the Black & Scholes value of a European call option adjusted for dividends. However, when there are dividends, and early exercise is allowed, optimal exercise before maturity might exist. Therefore it might be more appropriate to use a proportion of the optimal American call option value. This will probably result in an increase of the option value as observed in Table 8 and 9 for Hull & White. The results of a similar analysis for Brisley & Anderson using Monte Carlo simulation are presented in Table 11. The option values should, again, be similar to the option values obtained using the binomial tree.

Table 11: ESO values using constant volatility and Brisley & Anderson, Monte Carlo

This table presents the results of multiple ESO valuations using different proportion as well as other variance rates while changing forfeiture rate and dividend yield. All option values are obtained using the Brisley & Anderson exercise conditions for a 10 year at-the-money ESO based on an average of 10 simulations using 5,000 paths. With a strike price of 69.5 a vesting period of 3 years, a risk-free rate of 3.75%. Forfeiture rate 0% 10% Proportion 0% 25% 50% 75% 25% 50% 75% Dividend Yield 0% 36% 30.88 33.51 37.04 41.53 32.87 35.98 39.96 25% 26.79 29.42 32.74 37.09 28.70 31.36 35.30 10% 18.84 20.75 23.51 27.86 20.42 22.66 26.00 1% 9.35 10.51 12.79 17.32 10.33 12.28 16.20 1% 36% 29.30 31.57 34.68 38.39 31.13 33.75 36.93 25% 25.36 27.56 30.37 34.14 27.05 29.63 32.15 10% 17.46 19.10 21.35 24.51 18.80 20.71 23.18 1% 7.86 8.76 10.27 13.02 8.60 9.85 12.15 5% 36% 23.84 25.51 27.33 29.16 24.95 26.52 28.06 25% 20.07 21.39 22.80 24.65 21.10 22.27 23.76 10% 12.71 13.51 14.59 15.57 13.35 14.10 15.03 1% 3.36 3.55 3.79 4.05 3.53 3.71 3.89

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will also decrease the value of the ESO. Increasing the proportion will delay the moment of exercise, resulting in a higher option value. This is also true for options with a high dividend yield, because the remaining Black & Scholes value is adjusted for dividends.

Next, I will determine the impact of the CEV process, which is an alternative share value process, while keeping the volatility constant. Before valuing an ESO, I will first investigate the impact of using the CEV process for the valuation of a European call option to get a feel of the impact of this process on the option value, compared to the lognormal process. Table 12 presents the option values of a 10 year European call option for different values of alpha (α) and various variance rates.

Table 12: Valuation of a 10 year European call option using the CEV model and Black & Scholes

This table contains the Monte Carlo and Black & Scholes option values of a 10 year European call option for different (constant) volatilities, while using different values for alpha in the CEV model. Option values are based on an average of 10 simulations using 5,000 paths. Using the following parameter values: S = K = 69.5, r = 3.75%. α 0 0.5 0.75 1 B&S value Dividend Yield 0% 36.0 % 21.73339 22.00756 28.35689 49.97294 49.94455 25.0 % 21.73339 21.83184 26.30074 45.14857 45.08141 10.0 % 21.73340 21.73511 23.12447 34.89569 34.73283 1.0 % 21.73340 21.73356 21.73473 22.77022 22.77682 1% 36.0 % 15.11960 15.80501 22.93079 44.02924 44.19912 25.0 % 15.11960 15.45936 20.97561 39.65010 39.52257 10.0 % 15.11960 15.14418 17.40830 29.39324 29.47186 1.0 % 15.11960 15.11961 15.13128 16.94113 16.94539 5% 36.0 % 0.00000 1.86369 8.97136 26.87406 26.78933 25.0 % 0.00000 1.30664 7.13255 23.23254 22.92186 10.0 % 0.00000 0.41263 3.77417 14.57180 14.53334 1.0 % 0.00000 0.00021 0.31113 3.27016 3.27344

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1 the option value becomes equal to the European call option value under the lognormal process, which is (approximately) equal to Black & Scholes. The biggest impact is observed for the option with the highest variance rate, because the possibility of the option becoming more in-the-money is increased as the variance rate is increased. For alphas smaller than one the volatility component decreases as alpha turns to zero, making higher share prices less likely. Meaning that the chance of the option becoming even more in-the-money decreases. As explained in the methodology section, alphas bigger than one will result in a non-convergent process. If alpha is above one the volatility component has an enormous impact, resulting in either an enormous upward or downward drift. Once the share price decreases it will hardly be able to recover. In general, the average share price at maturity is lower when using the CEV process compared to the lognormal process (α = 1) for alphas smaller than one which is due to the smaller impact of the variance component. This decreases the chances of the option becoming even more in-the-money. It seems that an option with a smaller variance rate is less affected by alpha, probably because it has a more stable share value over time.

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variance rate, forfeiture rate or dividend yield compared to the lognormal share price process.

Table 13: ESO values using the CEV process with α = 0.5 and Hull & White, Monte Carlo

This table presents the results of multiple ESO valuations using different multiples and variance rates while changing the forfeiture rate and dividend yield. All option values are obtained using the Hull & White exercise conditions for a 10 year at-the-money ESO based on an average of 10 simulations using 5,000 paths, with a strike price of 69.5 a vesting period of 3 years and a risk-free rate of 3.75%.

Table 14: ESO values using the CEV process with α = 0.75 and Hull & White, Monte Carlo

This table presents the results of multiple ESO valuations using different multiples and variance rates while changing the forfeiture rate and dividend yield. All option values are obtained using the Hull & White exercise conditions for a 10 year at-the-money ESO based on an average of 10 simulations using 5,000 paths, with a strike price of 69.5 a vesting period of 3 years and a risk-free rate of 3.75%.

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Comparing the ESO values for Hull & White based a lognormal share price process with the ESO values obtained using the CEV process, shows the impact of the variance component of the share price process. For low alphas the option value is much less than the option value obtained using a lognormal share price process. Since the share values that are obtained using this process are lower compared to the lognormal process, the impact of the dividend yield is much bigger. The results from the CEV process are approximately similar to the lognormal share price process in the sense that they are affected by dividends, and variance in a similar manner. Although the impact of the individual parameters is magnified by the value of alpha, as can be observed when using an alpha of 0.75 and a forfeiture rate of 10%. At a multiple of 4, option values tend to be higher for options with a dividend yield of either 0% or 1%, and a high variance rate, compared to the option values obtained without forfeiture. The general impact of dividends is another example of this fact. When using the lognormal share price process, the highest option value would be at the multiple of 4 for an option with a variance rate of 36% when the dividend yield is 5%. For an option with a variance rate of 1% the highest option value would be at a multiple of 1. Under the CEV process, with an alpha of 0.75, the option has its maximum value at a multiple of 2, except for the option with a variance rate of 1% which should be exercised beyond the vesting time as soon as the option is in-the-money.

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Table 15: ESO values using the CEV process with α = 0.5 and Brisley & Anderson, Monte Carlo

This table presents the results of multiple ESO valuations using different proportions as well as other variance rates while changing the forfeiture rate as well as the dividend yield. All option values are obtained using the Brisley & Anderson exercise conditions for a 10 year at-the-money ESO based on an average of 10 simulations using 5,000 paths, with a strike price of 69.5 a vesting period of 3 years and a risk-free rate of 3.75%.

Forfeiture rate 0% 10% Proportion 0% 25% 50% 75% 25% 50% 75% Dividend Yield 0% 36% 8.32 11.61 16.95 20.18 11.14 15.91 19.29 25% 7.94 10.77 16.07 19.62 10.38 15.15 18.69 10% 7.48 9.01 14.02 18.30 8.87 13.42 17.44 1% 7.39 7.42 11.09 16.38 7.42 10.91 15.76 1% 36% 6.70 9.67 13.22 15.00 9.16 12.41 14.53 25% 6.25 8.85 12.47 14.45 8.45 11.70 13.91 10% 5.59 7.24 10.83 13.43 7.04 10.27 12.82 1% 5.34 5.41 7.99 11.54 5.41 7.85 11.12 5% 36% 2.16 3.02 2.52 2.01 2.83 2.54 2.35 25% 1.67 2.30 1.81 1.39 2.14 1.85 1.69 10% 0.77 1.01 0.66 0.45 0.94 0.72 0.61 1% 0.02 0.02 0.00 0.00 0.02 0.01 0.00

Table 16: ESO values using the CEV process with α = 0.75 and Brisley & Anderson, Monte Carlo

This Table presents the results of multiple ESO valuations using different proportions as well as other variance rates while changing the forfeiture rate as well as the dividend yield. All option values are obtained using the Brisley & Anderson exercise conditions for a 10 year at-the-money ESO based on an average of 10 simulations using 5,000 paths, with a strike price of 69.5 a vesting period of 3 years and a risk-free rate of 3.75%.

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When comparing the results of Brisley & Anderson using the CEV process with the lognormal process it is remarkable to observe that early exercise increases the option value for an alpha of 0.5 and a dividend of 5%. Additionally it seems that if the exercise moment is delayed, the option value is also increased as a result of forfeiture for a dividend yield of 5% when using an alpha of 0.5. This increase is not observed under the lognormal process, probably because the dividend yield has such a big impact under CEV process for small values of alpha. All other effects are similar compared to the lognormal share price process, decreasing the variance rate lowers the option values and increasing the dividend yield also decreases the option values.

Having derived ESO values using constant volatility, I will now continue with the derivation of ESO values using my stochastic volatility model, which is the main objective of my paper. In order to be able to compute the stochastic volatility I will need some value for a based on the GARCH(1,1) model. Pederzoli (2006) estimated three models based on UK stock data. One of the studied models is a GARCH(1,1) model. He used the British equity index FTSE100 to estimate the GARCH parameters and found a to be 0.00838 and α = 0.05077. This means that the variance rate is slowly reverting towards the long-term variance rate. In my stochastic volatility model I will use a somewhat higher mean-reversion factor of 5%, slightly increasing the speed of mean-reversion to the long-term variance rate. Based on a = 5%, α = 5%, and my ‘base case’ variance rate of 36% it seems that √ √ . Thus, according to Alfonsi (2005), the variance rate will never become negative.

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Table 17: Valuation of a 10 year European call option using stochastic volatility and Black & Scholes

This table contains the Monte Carlo and Black & Scholes values of a 10 year European call option for different volatilities, while changing the dividend yield and the correlation between the volatility and share price process. Based on an average of 10 simulations using 5,000 paths. Using the following parameter values: S = K = 69.5, r = 3.75%, ξ = √ , α = 5% and a = 5%, .

Correlation -0.75 0 0.75 B&S value

Dividend Yield 0% 36% 38.66 51.67 60.97 49.94 25% 38.52 45.53 53.80 45.08 10% 32.94 35.24 34.71 34.73 1% 22.65 22.84 22.97 22.78 1% 36% 34.15 44.07 57.84 44.20 25% 33.46 41.40 46.36 39.52 10% 27.91 29.74 31.05 29.47 1% 16.80 17.03 17.12 16.94 5% 36% 20.18 27.78 33.01 26.79 25% 19.26 23.47 26.67 22.92 10% 13.51 14.90 15.61 14.53 1% 3.22 3.28 3.32 3.27

correlation yields a lower option value. In addition, the size of the difference in value depends on the variance rate, being highest for options with a high variance rate. These results are reasonable since a negative correlation will make it less probable that there will be (large) positive share price changes, resulting in a positively skewed distribution (Heston, 1993). A positive skew means that the distribution has a heavier left tail and a less heavy right tail compared to the normal distribution. Vice versa, a positive correlation between the processes produces a negative skew. The results of a negative correlation are similar to what is observed for equity options as a result of leverage, which is explained by Hull (2009). A positive correlation, resulting in a negative skew, would mean that a decrease in leverage (due to an increase in the share value) would result in an increase in volatility, which is not observed in practice.

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the maximum variance rate used to determine √ . In this case the maximum variance rate is 36%, thus for a variance rate of 1%, ξ is equal to √ .

Table 18: ESO value using stochastic volatility and Hull & White, Monte Carlo (no correlation)

This table presents the results of multiple ESO valuations using different multiples as well as variance rates while changing the percentage forfeiture rate as well as the dividend yield. All option values are obtained using the Hull & White exercise conditions for a 10 year at-the-money ESO based on an average of 10 simulations using 5,000 paths. Using the following parameter values: S = K = 69.5, vesting period = 3 years, r = 3.75%, a = 5% and α = 5%, ρ = 0.

According to Table 18, the option value increases as the exercise of the option is delayed. Except for the situation in which the dividend yield is 5%, if the dividend yield is high it might be worthwhile to exercise the option early since the share value decreases as a result of the dividend payments.

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The ESO valuations using Brisley & Anderson are presented in Table 19 and are (also) approximately equal to the results obtained using Monte Carlo simulations for Brisley & Anderson based on a constant volatility. Early exercise does not seem to be valuable due to the dividend adjustment in Black &, similar to using constant volatility.

Table 19: ESO value using stochastic volatility and Brisley & Anderson, Monte Carlo (no correlation)

This table presents the results of multiple ESO valuations using different proportions as well as other variance rates while changing the percentage forfeiture rate as well as the dividend yield. All option values are obtained using the Brisley & Anderson exercise conditions for a 10 year at-the-money ESO based on an average of 10 simulations using 5,000 paths. Using the following parameter values: S = K = 69.5, vesting period = 3 years, r = 3.75%, a = 5% and α = 5%, ρ = 0.

Forfeiture rate 0% 10% Proportion 0% 25% 50% 75% 25% 50% 75% Dividend Yield 0% 36% 30.82 33.33 36.76 41.88 32.94 36.22 39.81 25% 27.01 29.52 32.52 36.99 29.12 31.47 35.14 10% 18.69 20.65 23.65 27.83 20.54 22.67 25.85 1% 9.35 10.49 12.78 17.28 10.27 12.23 16.16 1% 36% 29.90 31.89 34.63 38.44 31.16 33.94 36.96 25% 25.14 27.53 30.55 34.16 27.37 29.48 32.25 10% 17.28 18.96 21.21 24.45 18.88 20.71 23.13 1% 7.79 8.69 10.23 12.99 8.55 9.81 12.12 5% 36% 23.86 25.23 26.95 29.26 24.96 26.36 27.88 25% 20.38 21.53 22.91 24.49 21.02 22.57 23.90 10% 12.74 13.55 14.58 15.63 13.28 14.24 15.10 1% 3.31 3.52 3.77 4.06 3.51 3.67 3.86