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A Model for Valuing Multiple Employee Stock Options Issued by the Same Company Patrick J. Dennis University of Virginia, McIntire School of Commerce Richard J. Rendleman, Jr. University of North Carolina, Kenan-Flagler Business School January 29, 2003 Abstract In this paper we develop a model in which up to 30 employee stock options issued by the same firm can be valued simultaneously and demonstrate that standard methods of valuation can result in under-valuation, especially for long-dated options and for options where the proportion of outstanding options is large relative to the number of outstanding shares of equity. We also develop a version of the model to assess the valuation impact associated with employee options that are expected to be issued in the future. Our model indicates that the anticipated issuance of employee options can have a very negative impact on the value of a company’s shares, especially if the market views such options as representing excess compensation to employees. Patrick Dennis is Associate Professor, McIntire School of Commerce, The University of Virginia, e-mail: [email protected], phone: 434-924-4050. Richard Rendleman is Professor of Finance, Kenan-Flagler Business School, The University of North Carolina at Chapel Hill, e-mail: [email protected], phone: 919-962-3188, fax: 919-9622068. A Model for Valuing Multiple Employee Stock Options Issued by the Same Company On July 14, 2002, in the wake of Enron, WorldCom, Global Crossings and numerous other financial reporting scandals, Coca Cola announced a change in its accounting policy to begin using the “fair value” method for expensing employee stock options (ESOs). Since the announcement by Coca Cola, many other publicly-held companies have followed suit. As set forth in SFAS no. 123, an option’s fair value “is determined using an option-pricing model that takes into account the stock price at the grant date, the exercise price, the expected life of the option, the volatility of the underlying stock and the expected dividends on it, and the risk-free interest rate over the expected life of the option.” This definition of fair value is interpreted to mean value as determined by a dividend-adjusted Black-Scholes (1973) or binomial (Cox, Ross, Rubinstein [1979] and Rendleman and Bartter [1979]) option pricing model. According to SFAS no. 123, an option’s fair value, computed as of the grant date of the option, is expensed over the option’s life as it becomes vested. An excellent summary of the accounting treatment of employee options, along with a proposed modification of standard option pricing theory to accommodate current accounting practice, is presented in Rubinstein (1995). Rubinstein argues that current accounting standards for expensing employee stock options may create more problems than they solve. Notwithstanding potential problems in accounting treatment, under SFAS no.123, the employee option expense begins with the calculation of an option’s value using the Black-Scholes or binomial model. Both models are designed to value a single option issued on the stock of a single firm. However, almost all companies that issue employee options have granted, or can be expected to grant, many options with potentially different striking prices and maturities. In principle, the values of all such options should be determined simultaneously, and it is not clear that a standard BlackScholes or binomial model that ignores valuation interactions among options issued by the same firm is adequate. Our paper addresses these valuation issues. We develop a model in which all employee stock options are valued simultaneously and demonstrate that using the Black- 2 Scholes or binomial model, and treating the options as if they were independent from each other, results in under-valuation, especially for long-dated options and for options where the proportion of outstanding options is large relative to the number of outstanding shares of equity.1 We also develop a version of the employee option pricing model to assess the valuation impact associated with employee options that are expected to be issued in the future. Our model indicates that the anticipated issuance of employee options can have a very negative impact on the value of a company’s shares, and under certain conditions, can drive the price of the company’s stock to zero. Prior studies have focused on aspects of ESO valuation such as forfeiture, nontransferability and strike-price resetting, but assume that only one option is outstanding. Huddart (1994), Carpenter (1998) and Cuny and Jorian (1995) investigate how forfeiture and non-transferability affect the optimal exercise decision and hence the ESO valuation. The basic premise of these studies is that the holder of an ESO may not follow the optimal Black-Scholes exercise strategy for several reasons, including risk aversion, liquidity needs, diversification concerns, potential departure from the firm or taxes. Since the employee may not follow the optimal Black-Scholes exercise strategy, the Black-Scholes model overstates the value of the employee stock option. These exercise decisions, which are sub-optimal in the Black-Scholes context, can decrease the value of an ESO in relation to that of the Black-Scholes model. Brenner, Sundaram and Yermack (2000) address the valuation impact resulting from the fact that many firms reduce the strike-price conditional on poor firm performance. This treatment, results in a higher option value when compared to that of a fixed-strike-price option. Acharya, John and Sundaram (2000) point out that the higher ESO value does not necessarily lead to a lower firm value, since the reduction in the strike-price increases incentives for the firm’s employees and decreases agency problems between the owners and managers of the firm. Recently, some working papers have also addressed the simultaneity issue associated with valuing multiple ESOs issued by the same company. Darsinos and 1 The standard Black-Scholes and binomial models assume a constant variance of underlying stock returns. However, if the variance of total equity returns is constant for a firm that issues employee options, the variance of the firm’s common stock returns will not be constant. The under-valuation of employee options to which we refer results from the use of a constant variance Black-Scholes or binomial model when the correct specification of common stock return variance is non-constant. 3 Satchell (2002) develop a two-option model where the exercise of the earlier-maturity option can have an impact on the value of the later-maturity option, but they do not account for the fact that the conditional value of the later-maturity option can affect the value of the earlier-maturity option. Kapadia and Willette (2002) and Bodurtha (2002) both develop models with only two options and address the simultaneous valuation of both options. They find that option values determined simultaneously do not differ much from those obtained by treating each option as if it were independent. Valuing multiple options simultaneously is difficult since the problem is pathdependent. We use a depth-first valuation technique that eliminates the problem of having to store an exponentially large number of intermediate path-dependent option values. We also develop pruning methods that significantly reduce the number of computations required in valuing multiple options simultaneously. Our technique can be used to value up to 30 options simultaneously within a reasonable amount of time.2 This allows us to investigate the extent to which dependency among the ESOs causes their values to deviate from those obtained by the Black-Scholes and related models which treat the options as if they were independent. The methodology developed in this paper could potentially be applied to the simultaneous valuation of multiple options or option-like securities in other contexts. For example, most firms that borrow are likely to have numerous debt instruments outstanding with different maturities and different terms. Some instruments may be senior, others may be subordinated, some may be callable or non-callable and others may be convertible to common stock or into other securities. As with multiple employee options, all debt instruments of the same firm should be valued simultaneously, since the potential disposition of each instrument should affect the value of the others.3 The remainder of this paper is organized as follows. Section I develops a twoperiod example of the multiple employee option pricing model as applied to the valuation of existing options which is easily generalized for a larger number of time periods. In Section II, the model is applied to some realistic valuation scenarios to determine the 2 The computational time for a 30-period binomial model with a single option maturing at the end of each binomial period is three to four hours using a 2Ghz Pentium 4 computer. 3 Barth, Landsman and Rendleman (1998) discuss the problem of simultaneously valuing multiple debt instruments in greater detail and propose an heuristic procedure for valuing each instrument individually, given the presence of the other instruments. 4 extent to which option values computed via the model compare with those computed using more standard methods. In Section III the model is modified to reflect the pricing impact associated with the expectation that a firm will follow a policy of issuing at-themoney options on a regular basis in the future. Section IV provides concluding comments. I. The Multiple Employee Option Pricing Model The problem of valuing multiple options over multiple maturities is inherently path-dependent, meaning that the prices of options associated with a given value of equity at any future date depends upon the prior pattern of movement in equity value. For example, suppose a company with $100 million in total equity today can potentially reach a value of $120 million five years from now. If the firm’s equity value eventually reaches $120 million by first falling and then rising toward then end, options with maturities less than 5 years are less likely to be exercised than if the firm’s value reaches $120 million by rising first and then falling. Therefore, the number of shares outstanding when the firm’s equity value eventually reaches $120 million will depend upon the pattern of prior equity value movements. Given the path-dependency associated with the simultaneous valuation of multiple employee stock options, the binomial model is used as the basis for valuation. As is well documented, the binomial model can be parameterized to provide approximate values when no closed-form solutions exist for Black-Scholes-type option valuation problems.4 It is in that sense that the binomial model is used here. In the multiple employee option pricing model, the value of total equity, prior to the infusion of new equity capital from the exercise of employee options, is assumed to follow a standard multiplicative binomial process with constant up and down factors of u and d, respectively. The company issues m employee options. Option 1 expires at binomial time t = 1 , option 2 at binomial time t = 2 and so on. The striking prices of the m options are X i = X 1 , X 2 ... X m , and the number of shares associated with each is 4 A Black-Scholes-type valuation problem is any option pricing problem for which the value of the underlying asset is assumed to follow a lognormal distribution. 5 ni = n1 , n2 ...nm , with n0 denoting the number of common shares originally outstanding.5 The company pays no dividends. Each employee option is valued as a European-style option, even though in actual practice, employee options are typically of the American type. As pointed out by Emanuel (1983), Constantinides and Rosenthal (1983) and Cox and Rubinstein (1985), the optimal exercise of American-style warrants, which in principle are equivalent to fully-vested marketable employee options, depends upon the mix of ownership among the outstanding options. An optimal exercise strategy for a single individual who owns an entire warrant issue can involve piecemeal exercise in which a portion of the warrants are exercised early, with the remaining portion held in anticipation of subsequent exercise. However, the optimal exercise strategy for the same warrant, held among two or more individuals, will be the same as that of a single individual who is constrained to exercise all the options at the same time. Rather than overlay another level of complexity to an already complex option pricing problem, all employee options are assumed to be European. The key to valuing options via the multiple employee option pricing model is simultaneously keeping track of binomial-based increases and decreases in total equity value and optimal exercise decisions for the m options issued by the firm. To facilitate this process, let jt denote a binary index value, applicable to binomial time t, that indicates whether total equity increases or decreases in value from time t − 1 to time t. A value of jt = 1 indicates an increase in the value of total equity at time t, and a value of jt = 0 indicates a decrease in value. Also, let kt denote a binary index value that indicates whether or not the option that matures at time t is exercised. If kt = 1 , option t is exercised, and if kt = 0 , option t is not exercised. Although we could present a detailed mathematical formulation of an m-period model, we have chosen, instead, to present a two-period example that is easily generalized for m periods. 5 If a company has issued no employee options for a given maturity, the number of shares for that maturity can be set to $0. 6 A. A Two-Period Example A two-period example is presented in Figure 1. Before the specific details of the example are presented, however, it is useful to consider the general structure of Figure 1. The structure of Figure 1 is that of a non-recombining binomial decision tree. The first branch of the tree indicates potential change in the value of total equity as of binomial time 1. The upper portion of the initial branch is associated with an increase in total equity value ( j1 = 1 ), and the lower portion is associated with a decrease in the value of total equity ( j1 = 0 ). Note that each value of j1 is associated with two possible values of k1 , with k1 = 1 indicating that the option that matures at time 1 is exercised, and k1 = 0 indicating that the same option is not exercised. It is also important to note that passage of time from time t = 0 to time t = 1 is implied by the j1 index values, but no further passage of time is implied by the values of k1 . At time 1 there are four possible states, or sequences of events, { j1 = 0, k1 = 0} , { j1 = 0, k1 = 1} , { j1 = 1, k1 = 0} and { j1 = 1, k1 = 1} . From each of these sequences, the value of total equity can, again, go up ( j2 = 1 ) or down ( j2 = 0 ), as of time 2, and from each resulting up or down state, option 2 can be exercised ( k2 = 1 ) or not exercised ( k2 = 0 ). Note that there are 16 possible sequences of events as of time 2, { j1 = 0, k1 = 0, { j1 = 1, k1 = 1, j2 = 0, k2 = 0} , { j1 = 0, k1 = 0, j2 = 0, k2 = 1} , … j2 = 1, k2 = 1} . This general structure, when extended to m binomial time periods, results in 4m possible sequences through time m. In the example, a company has issued two employee stock options. Option 1 matures at binomial time 1 and gives an employee the right to purchase 1,000,000 shares of stock at $100 per share. Option 2 matures at binomial time 2 and gives an employee the right to purchase 500,000 shares of stock at $100 per share. Presently there are 10,000,000 shares of common stock outstanding, and the total value of equity, including common equity and employee options, is $1 billion. The value of total equity follows a standard multiplicative binomial process with u = 1.2 and d = 0.9 , and the risk-free rate of interest is 0.03 per binomial period. These parameters imply a risk-neutral probability 7 associated with an increase in total equity value of π = 1 + r − d 1.03 − 0.9 = ≈ 0.4333, u−d 1.2 − 0.9 where r denotes the risk-free rate of interest per binomial period. In the model, the value of each option is determined recursively using the depthfirst technique of Dennis (2001). To illustrate the valuation process, consider the uppermost portion of Figure 1 as of time 2, labeled as state 1 (shown as 1 in the figure). This portion of the tree establishes the maturity value of option 2 for sequence { j1 = 1, k1 = 1, j2 = 1, k2 = 1} , the sequence for which the value of total equity increases at time 1 and time 2, and options 1 and 2 are both exercised. Note that if both options are exercised, there will be n0 + n1 + n2 = 10, 000, 000 + 1, 000, 000 + 500, 000 = 11,500,000 shares outstanding. Also, as shown in state 1, the value of total equity is $1,610,000,000, reflecting two increases in equity value and the exercise of both options. The $1,610,000,000 total equity value is computed as follows. Let At denote the value of total equity as of time t, with A0 denoting the initial value of total equity, or $1 billion in this example. Then, the value of total equity as of time t is given by At = At −1u jt d 1− jt + nt X t kt for t = 1, 2,...m . This value reflects natural changes in equity value as implied by the value sequence j1 , j2 , ... jm plus the infusion of new capital associated with exercise sequence k1 , k2 .... km . Any new capital obtained though the exercise of options is assumed to be invested back into the firm and to follow the same binomial process as previously existing equity capital.6 Noting that j1 = 1, k1 = 1, j2 = 1, and k2 = 1 in state 1, the value of total equity as of time 1 is A1 = A0u j d 1− j + n1 X 1k1 = $1, 000, 000, 000 × 1.21 × 0.91−1 + 1,000, 000 × $100 ×1 1 1 = $1, 200, 000, 000 + $100, 000, 000 = $1.3 billion, reflecting a natural increase in total equity value from $1 billion to $1.2 billion plus the infusion of $100,000,000 in new equity capital from the exercise of option 1. With $1.3 billion in total equity value as of j2 1− j2 time 1, the total value of equity, as of time 2, becomes A2 = Au d + n2 X 2 k2 1 6 An alternative assumption would be that the exercise money is used to repurchase stock and that the repurchased stock is then distributed to employees who exercise their options. Although this assumption eliminates the path-dependency of total equity value, it does not eliminate the path-dependency of shares outstanding. 8 = $1,300, 000, 000 ×1.21 × 0.91−1 + 500, 000 × $100 ×1 = $1,560, 000, 000 + $50, 000, 000 = $1,610,000,000. This value, when distributed among 11,500,000 shares, amounts to $1, 610, 000, 000 = $140 per share. Therefore, when option 2 is exercised, the employee 11,500, 000 nets $140 − $100 = $40 per share in state 1. Since the option is for 500,000 shares, the total value of option 2 is 500, 000 × $40 = $20, 000, 000 . In state 2, the value of option 2 is computed for the sequence { j1 = 1, k1 = 1, j2 = 1, k2 = 0} . Since k1 = 1 and k2 = 0 , the entries for state 2 reflect that option 1 is exercised but option 2 is not exercised. Therefore, the value of option 2 in state 2 is $0. The value of option 2 as shown in state 3 reflects the optimal exercise decision with respect to option 2, given that the value of total equity increases twice and option 1 is exercised at time 1. Inasmuch as the $20,000,000 obtained from exercising option 2 exceeds the value associated with not exercising, or $0, the relevant value of option 2 in state 3 is its exercise value. Similarly, the $3,043,478 value of option 2 in state 6 reflects the optimal exercise decision for option 2, given that the value of total equity increases at time 1 ( j1 = 1 ), option 1 is exercised at time 1 ( k1 = 1 ), and the value of total equity decreases at time 2 ( j2 = 0 ). Next, consider states 8 and 9. Since both states are part of the branch that evolves from state 14, a state for which k1 = 0 , both are associated with option 1 not being exercised. The value of option 2 in state 8 reflects that option 1 is not exercised at time 1 but that option 2 is exercised at time 2. Under these circumstances, the total value of equity is $1,490,000,000, and there are 10,500,000 shares of stock outstanding, resulting in a stock price of $1, 490, 000, 000 = $141.904... per share. Therefore, the net value of 10,500, 000 exercising option 2 is $141.904... − $100 = $41.904 per share or $41.904 × 500, 000 = $20,952,380 total. The value of option 2 in state 9 reflects that option 2 is not exercised. Therefore, the value of option 2 in state 9 is $0. The $20,952,380 value for option 2 in state 10 reflects that if option 1 is not exercised ( k1 = 0 ), it will be optimal to exercise 9 option 2, since $20,952, 380 > $0 . Similarly, the 3,809,524 value of option 2 in state 13 reflects that it is optimal to exercise option 2 in this state. Now consider the entries for state 7. Since j1 = 1 and k1 = 1 , this state is associated with an increase in equity value for the first binomial period and option 1 being exercised. In this state, the relevant up-state value of option 2 is $20,000,000 and the relevant down-state value is $3,043,478. Therefore, in state 7, the value of option 2 is $20, 000, 000 ( 0.43333...) + $3,043, 478 (1 − 0.43333...) = $10, 088, 645 . Similarly, the 1.03 entries for state 14 show that if option 1 is not exercised ( k1 = 0 ), the value of option 2 is $20,952,380 ( 0.43333...) + $3,809,524 (1 − 0.43333...) = $10,910, 711 . 1.03 We now turn to the valuation of option 1. In state 7, a state for which option 1 is exercised, the total value of equity is $1.3 billion, reflecting a natural increase in total equity value from $1 billion to 1.2 billion, plus $100,000,000 received from the exercise of option 1. Of this $1.3 billion, $10,088,645 is the value of option 2, which leaves $1,300, 000, 000 − $10, 088, 645 = $1, 289, 911, 355 of value for common stock and option 1. Upon the exercise of option 1, there will be 10,000,000 + 1,000,000 = 11,000,000 shares outstanding, resulting in a stock price of $1, 289,911,355 = $117.264... per share. 11, 000, 000 Therefore, the net value of exercising option 1 is $117.264... − $100 = $17.264… per share or $17.264... × 1, 000, 000 = $17, 264, 668 for all 1,000,000 shares. In state 14, option 1 has a value of $0, since state 14 is a no-exercise state. Inasmuch as exercising creates greater value than not exercising, the relevant value of option 1 in the initial up state ( j1 = 1 ) is $17,264,668, and the corresponding value of option 2 is $10,088,645. These values are shown in state 15. Option values in the lower portion of Figure 1, branching from state 30, are determined in similar fashion. This same valuation process leads to values of $0 and $1,602,712 in state 30 for options 1 and 2, respectively. The initial $7,263,452 value of option 1, shown in state 31, reflects an up-state value of $17,264,668 and down-state 10 value of $0, while the initial $5,126,165 value of option 2 reflects up- and down-state values of $10,088,645 and $1,602,712, respectively. B. Generalization and Computational Considerations The valuation procedure described above is easily generalized for m options and m periods. With the depth-first technique, the computations are carried out in the order that the states are numbered in Figure 1, and only 4m computer memory locations are needed to store intermediate option values. In contrast, if all the values in the final column of Figure 1 are computed first, followed by the calculation of values in the nextto-last column, and so on, 4m computer memory locations are required. Although the depth-first technique solves the computer memory problem associated with pathdependent valuation, it does not reduce the number of required computations. With either method 2 × 4m − 1 computations are required.7 This implies that a m + 1 -option (period) problem will take approximately four times the computational time of an moption problem. To put this into a more practical perspective, if it takes one minute to compute option values for an m-option problem, it would take over a full day to compute values for an m + 6 -option problem. It is possible to reduce the number of computations required by pruning portions of the binomial tree once it becomes evident that those portions will not enter into the final option solution values. We employ two pruning methods that significantly reduce the computational requirements of the model. Consider states 7 and 14 in Figure 1. Values of both options in state 7 are associated with option 1 being exercised ( k1 = 1 ), while values in state 14 are associated with option 1 not being exercised ( k1 = 0 ). In this particular example, the exercise value of option 1, as shown in state 7, is positive. Therefore, once option values in state 7 have been computed, it is clear that option 1 will be exercised and it is unnecessary to carry out the calculations branching from state 14. Thus, we “prune” the portion of the tree branching from state 14 and similar states, since these branches of the binomial tree are not needed in the calculations. 7 m 2 × 4 −1 is the total number of states in the binomial tree after m periods. 11 The second round of pruning can be illustrated by considering state 22. Before the calculations branching from state 22 (and others like it) are made, the exercise value of option 1 (or the option expiring in that state) is calculated, assuming that the values of all non-expiring options (option 2, in the example) are zero. If the exercise value of option 1, calculated in this fashion, is negative, it will be potentially even more negative if the proper values of the non-expiring options are used when calculating option 1’s exercise value. Thus, without even carrying out the detailed calculations of option values in the portion of the binomial tree branching from state 22, if the value of option 1 calculated in this fashion is negative, it is clear that option 1 will not be exercised, and one can skip directly to the calculation of option values branching from state 29. Without pruning, the calculations for a 20-option (period) model would be projected to take 51 days using a 2-Ghz Pentium 4 computer. This projection is based on a 10-option model taking 4.23 seconds of computation time, with computation time increasing by a factor of 4 for each additional option. With pruning, computation time for a 20-option model takes only 11.9 seconds. In addition, computation times for 21option and 22-option models are 23.9 seconds and 47.1 seconds, respectively. 8 Based on these results, it appears that computation time with pruning increases approximately by a factor of 2 per option. The computation time required for a 30-option model is three to four hours. Therefore, we consider 30 options to be the practical limit of our model under present computing standards. II. Application of the Multiple Employee Option Pricing Model to some Realistic Scenarios for Valuing Existing Options A. Standard Valuation Examples According to a recent Business Week article by David Henry, on average, the 200 largest [U.S.] companies grant options on approximately 3 percent of outstanding shares each year. Henry also reports an average annual grant of 1.9 percent for companies in the S&P index and grants of over 10 percent for some high-tech companies, including a 16.5 percent average annual grant by Siebel Systems. Based on these figures, the multiple 8 Computation time with pruning for a given number of binomial periods depends upon the terms of the options being evaluated. The computation times reported here are for the types of options whose values are summarized later in Tables I and II. 12 employee option pricing model is evaluated for annual option grants of 3 percent, 15 percent and 30 percent of the current number of shares outstanding. Although an annual grant of 30 percent is almost twice that of the largest grant reported by Henry, it is a useful amount for assessing the difference in values produced by the multiple employee option pricing model and the standard binomial method. The multiple employee option pricing model also requires an estimate of the volatility of a firm’s returns to total equity which, in turn, is reflected in the values of u and d. Although there are various ways u and d can be expressed in terms of volatility, for the purposes of this section, u = eσ and d = e−σ , where σ is an estimate of the standard deviation of the annual logarithmic return to total equity. A standard deviation of 0.30 is often used as an estimate of volatility for a typical NYSE stock and 0.60 appears to be a reasonable estimate of volatility for the stock of a high-tech firm.9 For each valuation scenario, the company is assumed to have issued options with maturities of one to 10 years inclusively. A common striking price is assumed for each option. This common striking price is computed iteratively as that which equals the theoretical stock price determined by the model.10 For each scenario, the initial value of total equity is $1 billion and there are 10,000,000 shares of common stock outstanding. The risk-free rate of interest is 0.05 per year and each binomial period represents one full year. Table I summarizes valuation results for the multiple employee option pricing model, assuming a volatility for total equity of 0.30. Table II provides a similar summary assuming an equity volatility of 0.60. Both tables consist of three panels of option values, with the values in the first through third panels computed as if each option is for 9 As of May 2002, 86 of 2406 historical volatilities published by the CBOE exceeded 1.00 (100%) and 504 of 2406 exceeded 0.50. The CBOE numbers reflect volatility computed with daily returns over a single month, and it is unlikely that volatility of this high a magnitude would continue over the long term. In addition, there are well-known statistical problems associated with using daily returns for estimating the standard deviation of stock returns. See, for example Blume and Stambaugh (1983). Nevertheless, as a “back-of-the-envelope” estimate, it would seem reasonable to expect some high-tech companies to have long-term equity volatilities on the order of 60 percent per year. 10 If a given common striking price is assumed for each of the 10 options, the model will produce an initial value for each of the 10 options and an initial value of common equity. Dividing the initial value of common equity by the number of shares outstanding, produces an initial stock price per share. If this price, rounded to the nearest dollar, does not equal the common striking price, a new common stock price is assumed, and each option value and the initial stock price are re-computed. This process is continued until the initial stock price, rounded to the nearest dollar, equals the assumed common striking price. 13 300,000, 1,500,000 and 3,000,000 shares of stock, respectively. For each panel, the striking price, assumed to be the same for options of all maturities, is that which equals the theoretical stock price rounded to the nearest dollar. Consider the first panel of option values in Table I. The first column of values within the first panel shows the value of each option computed via the multiple employee option pricing model. The sum of these values is $93,885,144 (not shown in the table). Since the total value of equity is assumed to be $1 billion, the theoretical value of common stock computed via the model is $1 billion - $93,885,144 = $906,114,856 . With 10,000,000 shares of stock initially outstanding, the stock price per share is $906,114,856 ≈ $90.61 . 10, 000, 000 Although not shown in the table, the initial “up” value of common equity (not total equity) is $1,197,084,418, and the initial down value is $698,118,796. Therefore, the initial value of u for common equity is u = of d is d = $1,197, 084, 418 ≈ 1.32112 , and the value $906,114,856 $698,118, 796 ≈ 0.77045 . Together, these values of u and d correspond to an $906,114,856 initial common equity volatility of approximately 0.270.11 Therefore, even though total equity evolves at a volatility rate of 0.30, the initial volatility rate of common equity is only 0.270. As shown in the second column of the first panel, if the option that expires at time 1 is valued as an option on common equity, using the standard binomial model with u = 1.32112 and d = 0.77045 , the option value is $4,163,986, exactly the same as the value of the same option computed via the multiple employee option pricing model. Therefore, this option can be valued via the standard binomial as an option on common equity, ignoring the presence of the other options, without introducing any error to the valuation process. The remaining entries in the second column represent standard binomial option values, assuming the initial values of u and d for common equity remain constant. In 11 Assuming a 0.5 probability associated with an increase or decrease in common equity value, the standard deviation of the logarithmic (continuous) return to common equity in the first binomial period is ln (1.32112 ...) − ln ( 0.77045...) ≈ 0.270 . 2 14 theory, if the market values of the 10 options and the corresponding value of common equity are determined by the multiple employee option pricing model, the volatility of common equity, and its associated u and d values, will not remain constant. Nevertheless, if one estimates the immediate volatility of common equity to be the 0.270 value computed via the multiple option pricing model and assumes it remains constant, standard binomial values can be computed for each of the 10 options. Inasmuch as the accounting profession is likely to employ constant-volatility Black-Scholes and binomial methods as starting points for valuing employee options, it is useful to see how option values computed in this manner compare with those of the multiple employee option pricing model, even though such values are not “theoretically correct.” The second column shows standard binomial values for each of the 10 options, assuming an initial stock price of $90.611 per share, as computed by the multiple employee option pricing model, and constant values of u and d that correspond to a common equity volatility of 0.270. Note that the standard binomial option values are all slightly less than those computed via the multiple employee option pricing model, although there is little practical difference between the two sets of values. Total equity value, as computed by the standard binomial model, is $998,632,879, rather than the theoretically correct value of $1 billion, but this error of 0.14 percent is very small. Therefore in this particular situation, one could use standard option pricing theory to value each option individually as a CBOE-type option on common stock, while ignoring the presence and valuation interaction of the other options, and very little error would be introduced into the valuation process. On the surface, one might expect option values calculated via the standard binomial (or Black-Scholes) model to be too high in relation to their theoretically correct values, since the standard model does not take the dilution associated with option interdependence into account. Clearly, a failure to reflect dilution should bias standard binomial values upward. However, there is an offsetting effect. The standard binomial model assumes a constant variance of underlying stock returns. However, if the variance of total equity returns is constant for a firm that issues employee options, the variance of the firm’s common stock returns will not be constant. The under-valuation of employee 15 options results from the use of a constant variance Black-Scholes or binomial model when the correct specification of common stock return variance is non-constant. The first panel of values in Table II summarizes results for the same set of initial conditions, except that volatility for total equity is assumed to be 0.6 rather than 0.3, and the striking price for each option is $86 rather than $91. Although the higher volatility rate causes each option to be worth more than in Table I, the overall error from using the standard binomial model is, again, small and essentially the same as when volatility is 0.3. The second panels in Tables I and II summarize the same type of computations, assuming that each option is for 1,500,000 shares, while the third panels of computations are based on each option being issued on 3,000,000 shares. There appears to be greater error in using the standard binomial approach as the number of shares associated with each option is increased. When each option is for 1,500,000 shares, total pricing error is on the order of 1.5 percent, and the pricing errors for the 10-year option in Tables I and II are 6.7 and 4.9 percent, respectively. When each option is for 3,000,000 shares, total pricing error is approximately 3.5 percent, with the error for the 10-year option equal to 10.2 percent in Table I and 8.4 percent in Table II. Although one might argue that the pricing error in the last case is significant, based on Henry’s Business Week article, it is highly unlikely that annual option grants of this size, 30% of initial outstanding shares, would be made on a regular basis. We have also experimented with various patterns of exercise prices to determine if there is a greater difference between option values computed via the multiple employee option pricing model and those computed using the standard binomial model under an assumption of constant variance for common equity equal to that generated by the multiple option pricing model for the initial binomial period. For example, we assume that the striking price for the option maturing in year m is equal to the current stock price but that the striking price for an option maturing at time t < m is equal to X t = X m / striking _ factor m −t . For values of striking_factor equal to 1.10 and 1 / 1.10, there is actually less difference between option values computed from the two models than the differences shown in Tables I and II. Therefore, our conclusions regarding the potential pricing error associated with using the standard binomial model for pricing 16 multiple employee options remains the same. There appears to be little error associated with using the standard binomial method to determine the values of option grants of typical size. Only when the number of shares associated with each option is exceptionally large in relation to the number of outstanding shares is any significant error introduced with standard constant-volatility-based binomial pricing. B. Valuation Examples when Options do not Mature at the end of Each Binomial Period The model, as formulated, assumes the firm has issued m options, with options 1 through m expiring at binomial times 1 though m, inclusively. If one or more of the m binomial times are not associated with the maturity of an option, the model can easily accommodate this situation by specifying that the number of shares associated with options maturing at these times is zero. For example, suppose a firm has issued 1-year, 3-year, 4-year and 5-year options, each with 300,000 shares outstanding, but has issued no 2-year option. Then the model, as formulated, can accommodate this situation by assuming n1 = n3 = n4 = n5 = 300, 000 and n2 = 0 . Although this method will produce correct option values, it may be computationally inefficient if the number of binomial periods is significantly greater than the number of actual options. Consider the situation in which a firm has issued m options, each of which expires at the end of the next m years, but each year is partitioned into y binomial time periods. Therefore, at the end of the first y − 1 binomial periods of each year, there is no expiring option. Also, during the last y − 1 periods of each year, the binomial valuation process will not be path-dependent.12 Computationally, it is inefficient to value the firm’s m options in the last y − 1 binomial periods using a path-dependent valuation structure. Therefore, we modify our valuation procedure to employ a recombining binomial tree structure for the last y − 1 binomial periods between option maturity dates and to only use a non-recombining structure for the first of the y periods between maturities. With this modified structure, the number of paths after m × y binomial periods is ( 2 [ y + 1]) and the number of m 12 If an option expires at the end of a particular binomial period, it creates a path-dependency at the end of the next binomial period only. Therefore, the remaining y − 1 periods will not be path-dependent. 17 computations required is on the order of 2 ( 2 [ y + 1]) . However, as in the previous m specification, pruning can significantly reduce the number of required computations. Consider a situation in which four options maturing in 1, 2, 3 and 4 years, respectively, are valued using five binomial time periods per year. Therefore, m = 4 and y = 5 , and the options are valued using a 20-period binomial tree. If a non-recombining valuation structure is employed for each of the 20 binomial periods, the number of calculations required would be 2 × 4 20 − 1 ≈ 2.30 × 1012 before pruning. However, using a recombining structure during the last four binomial periods between each maturity reduces the number of calculations to the order of 2 ( 2 [ y + 1]) = 2 ( 2 [5 + 1]) = 41, 472 . m 4 Table III shows computation times and illustrates model precision when additional binomial time intervals are included between option expirations As in Table I, the volatility of equity returns is assumed to be 0.30 per year, the risk free interest rate is 5 percent per year, and the firm is assumed to have issued ten employee options, the first of which expires at the end of year 1, the second at the end of year 2, and so on. Three sets of option values are computed. In the first set, each option is for 300,000 shares and the striking price of each option is $91. In the second set, the number of shares and striking price are 1,500,000 and $68, respectively. In the third set, the number of shares and striking price are 3,000,000 and $53. The first row of stock values shown in Table III correspond to those shown in the last row of Table I. As in Table I, option values in this row reflect that each binomial period represents a full year and that an option matures at the end of each year. As such, m = 10 , y = 1 , and there are a total of 10 binomial periods employed in the calculations. In the second row, m = 10 and y = 2 , resulting in a total of 20 binomial periods. Note that the option prices in this row correspond very closely to those in the first row. Moreover, this same pattern is seen in the rows for which y = 3 , y = 4 , and y = 5 . Although the option values are calculated with greater precision in these rows, the resulting stock prices per share are very close to those shown in the first row. Computation times shown in Table III are for the option prices in the first column only and are somewhat lower than those required for the options whose prices are shown in the other two columns. The computation time for the 50-period model shown in the 18 last row is 1451.5 seconds, or approximately 24 minutes, indicating that pricing 10 options within a 50-period binomial tree is quite feasible. Table IV shows computation times assuming the final option being valued expires in 100 binomial periods, with the 100 periods being divided into m = 1, 2, 4 and 5 years. All computation times reflect that the striking price of each outstanding option is $91, that each option is for 300,000 shares, and that all expiration dates are exactly one year apart. Note that the longest computation time is 277.9 seconds for five options spread over 100 binomial periods, with 20 binomial periods included between each expiration. These results indicate that it is feasible to use 100 binomial periods in the calculation of up to five interdependent employee option prices. III. The Valuation Impact Associated with the Anticipated Issuance of Employee Stock Options The price of a company’s stock should not only reflect the value of employee options that have been issued but should also reflect the values of options that are expected to be issued in the future. Clearly, if a company is expected to carry out a policy of compensating its employees with newly-issued employee options, this policy should be reflected in the company’s stock price. When a company issues a new option to an employee, the striking price of the option is typically equal to the price of the stock at the time the option is issued, and the typical maturity of an employee option is 10 years (Rubinstein [1995], p. 10). In this section we determine the valuation impact associated with the expected issuance of 1year options with each having a striking price equal to the price of the underlying stock at the time of issuance. To the best of our knowledge, 1-year options are never issued in practice. However, since they are shorter-term than those that are typically issued, the valuation impact associated with their issuance should be substantially less than that associated with more typical longer-term options. As we shall show, the impact associated with the anticipated issuance of 1-year options on the value of a company’s stock can be very significant. Therefore, we can infer an even greater impact on share price if longer-term options are expected to be issued. 19 Figure 2 illustrates the multiple employee option pricing model applied to the anticipated issuance of two 1-year at-the-money options. As in Figure 1, each binomial period is assumed to represent one full year. As such, the first option matures at binomial time 1 and the second at binomial time 2. Each option is for 3 percent of the company’s outstanding shares at the time of issuance. All other parameters are the same as those in Figure 1: u = 1.2 , d = 0.9 , r = 0.03 and initial total equity value is $1 billion. In Figure 2, the stock price per share is shown in bold italics at time zero in state 31 and at time 1 in states 7, 14, 22 and 19. These are the states for which new 1-year employee options are expected to be issued, with each having a striking price equal to the stock price in the state of issuance. Each stock price and associated striking price is determined iteratively. Initially, the 1-year option issued at time zero is assumed to have a striking price of $100. ($100 is calculated as the $1 billion initial equity value divided by 10,000,000 shares of stock initially outstanding.) For each state 7, 14, 22 and 29, the option’s initial striking price is assumed to equal the total equity value in that state divided by shares outstanding. Using these initial guesses for the striking prices, values for options 1 and 2 are computed in each of states 7, 14, 22 and 29, along with the stock price per share. If the resulting per-share stock price is within $10−11 of the assumed striking price, no further calculations are made. If not, the option’s striking price is set equal to the new stock price, and the calculations are repeated. This procedure is continued until the pershare stock price in state 31 is within $10−11 of the striking price assumed for the option expiring at time 1. This extreme degree of precision was used to ensure that there would be no apparent error in any of the numbers displayed in Figure 2. Other than the fact that the striking prices of the two options are determined iteratively and are state-dependent, the logic behind the option values calculated in Figure 2 is the same as for those of Figure 1. Therefore, no further explanation of option value calculations is provided. Although option 2 is not issued until time 1, from the perspective of time zero, the option is equivalent to a 2-year option with a state-dependent striking price determined at time 1. Note that the initial value of this option in state 31 exceeds that of option 1. This appears to be a general result applicable to more than just two binomial periods. 20 Generally, as of time zero, the option that matures at time m is worth more than an option that matures at time m − 1 which, in turn, is worth more than that which matures at time m − 2 , etc. This relationship is illustrated in Table V. The underlying valuation parameters of Table V are the same as those for Table I: the initial value of total equity is $1 billion; the volatility of total equity is 0.30 per binomial period (year); the interest rate is 5 percent per binomial period. Each option is for 3 percent of the number of shares outstanding at the time of issuance, and the striking price of each option is equal to the per-share stock price at the time of issuance within a precision of $10 −4 . This lower degree of precision is used to reduce computation time. Unfortunately, the pruning methods described in the previous section do not apply with iterative search, and determining the valuation impact associated with the expected issuance of at-the-money options requires intensive computations. The first column of figures in Table V shows option and equity values, assuming that the last 1-year at-the-money employee option that is expected to be issued matures at the end of year 2, the second column shows similar values, assuming that the last option expected to be issued matures at the end of year 3, and so on. The option and equity values in the last column are computed as if the final anticipated option maturity date is year 9. Regardless of the final anticipated maturity date, the immediate value associated with the anticipated issuance of a given at-the-money option increases with the option’s maturity date. For example, in the second column, option 3, expected to be issued at time 2 and to mature at time 3, has a greater present value than option 2 which, in turn, has a greater present value than option 1. As more at-the money options are expected to be issued, the value of each is slightly lower, but the total present value of all anticipated options increases. This increase in total option value has the effect of decreasing the stock price. For example, if one-year options with maturing in years 1through 3 are expected to be issued, the stock price is $98.497, but if options maturing in years 1 through 9 are expected to be issued, the stock price is $95.377. 21 Going beyond nine maturities with a precision of $10 −4 is computationally impractical. Nevertheless, it is possible to determine the limit of total option value as the maturity date of the last anticipated option approaches infinity. Given the pattern of option values evident in Table V, it is not surprising that this limiting value is $1 billion, or, in general, the limiting value of total option value is equal to the initial value of total equity. This, in turn, implies that if a company is expected to issue at-the-money employee options indefinitely on a constant proportion of its shares, the price of its stock should be zero! Before taking this result too seriously, it is important to recognize that employee stock options can be viewed as a form of financing. As Rubinstein (1995, p. 11) points out, “Instead of paying for the options in cash, employees pay with their labor services, which leaves additional cash in the firm that can be used for other purposes.” Implicit in Rubinstein’s statement is the notion that a firm with external financing needs can implicitly finance these needs by investing the cash it would otherwise pay its employees and then provide employees with an equivalently-valued indirect claim on equity through the issuance of options. If employee options are viewed as a form of financing, the valuation effects associated with the anticipated issuance of such options should be neutral.13 However, to the extent that the anticipated issuance of employee options is expected to be excessive, the negative impact on the price of an issuing firm’s stock can be substantial. Therefore, if a company is expected to issue at-the-money employee options indefinitely on a constant proportion of its shares, and the options issued are expected to be in excess of the company’s ordinary financing needs, only then should the price of the company’s stock equal zero. IV. Conclusion Current techniques employed in valuing multiple employee stock options for the same firm treat each option as being independent from the others and do reflect that the options should be valued simultaneously. To study the valuation impact of ignoring this simultaneity, we develop a model and valuation technique to account for the 13 This statement assumes that any incentive effects associated with the issuance of employee options are already reflected in the stock price. 22 interdependence of exercise decisions among employee options of the same firm that mature on different dates. We find that for firms with a small fraction of employee options outstanding, not accounting for the simultaneity results in an insignificant degree of under-valuation. However, for firms with a large fraction of employee options outstanding, not accounting for the simultaneity can result in under-valuation on the order ten percent. In these cases, treating multiple options independently can significantly overstate the true value of stockholder equity. The problem of simultaneously valuing multiple employee stock options issued by the same firm is inherently path-dependent and, therefore, computationally intensive. Using the depth-first technique, we are able to eliminate the problem of having to store an exponentially large number of intermediate option values in our computation of option values. In addition, we employ pruning methods that significantly reduce the number of computations required and make it feasible to simultaneously value up to thirty employee options with different maturity dates. Much of the focus in the paper is on the valuation of outstanding employee stock options. However, it is clear that the value of a firm’s stock should not only reflect the value of previously-issued employee options, it should also reflect the expectation that employee options may be issued in the future. Ignoring any incentive effects and productivity gains that result from using ESOs to compensate employees, we show that the expected issuance of employee options can significantly reduce the value of a firm’s stock. Therefore, if the market views such options as representing excess compensation, the price of a firm’s stock can be adversely affected by a general policy of issuing stock options to employees. 23 References Acharya, Viral V., Kose John, and Rangarajan K. Sundaram, 2000, “On the optimality of resetting executive stock options,” Journal of Financial Economics 57, 65-101. Barth, Mary E, Wayne R. Landsman and Richard J. Rendleman, Jr., 1998, “Option Pricing-Based Bond Value Estimates and a Fundamental Components Approach to Account for Corporate Debt,” The Accounting Review 73, 73-102. Black, Fischer and Myron Scholes, 1973, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy 3, 637-654. Blume, Marshall and Robert. F. Stambaugh, 1983, “Biases in Computed Returns: An Application to the Size Effect,” Journal of Financial Economics 12, 387-404. Brenner, Menachem, Rangarjan Sundaram, and David Yermack, 2000, “Altering the terms of executive stock options,” Journal of Financial Economics 57, 103-128. Bodurtha, James N., Jr., 2002, “Dilution and multiple-issue tranches inherent in employee stock option valuation,” Working paper, Georgetown University. Carpenter, Jennifer N., 1998, “The exercise and valuation of executive stock options,” Journal of Financial Economics 48, 127-158. Constantinides, George M. and Robert W. Rosenthal, 1984, “Strategic Analysis of the Competitive Exercise of Certain Financial Options,” Journal of Economic Theory 32, 128-138. Cox, John C., Stephen Ross and Mark Rubinstein, 1979, “Option Pricing: A Simplified Approach,” Journal of Financial Economics 7, 229-263. Cox, John C. and Mark Rubinstein, 1985, Options Markets (Prentice Hall, Englewood Cliffs, NJ). Cuny, Charles. and Philip. Jorion, 1995, “Valuing executive stock options with endogenous departure,” Journal of Accounting and Economics 20, 193-205. Darsinos, Theofanis., and Stephen E. Satchell, 2002, “On the valuation of warrants and ESOs: Pricing for firms with multiple warrants/executive options,” Working paper, University of Cambridge. Dennis, Patrick J., 2001, “A Depth-First Search Technique for the Valuation of American Path-Dependent Derivatives,” Journal of Applied Business Research 17, 1-11. Emanuel, David C, 1983, “Warrant Valuation and Exercise Strategy,” Journal of Financial Economics 12, 211-236. 24 Henry, David, 2002, “An Overdose of Options,” Business Week, July 15, 2002, 112-114. Huddart, Steven J., 1994, “Employee stock options,” Journal of Accounting and Economics 18, 207-231. Kapadia, Nikunj., and Gregory. Willette, 2002, “Dilution and the valuation of options and non-identical warrants,” Working paper, University of Massachusets. Rendleman, Richard J., Jr. and Brit J. Bartter, 1979, “Two-State Option Pricing,” Journal of Finance 34, 1093-1110. Rubinstein, Mark, 1995, “On the Accounting Valuation of Employee Stock Options, Journal of Derivatives 3, 8-24. Figure 1. Two-period example of the multiple employee stock option pricing model. Option 1 matures at binomial time 1 and option 2 matures at time 2. Both options have a striking price of $100 per share. Option 1 is for 1,000,000 shares of common stock and option 2 is for 500,000 shares. There are 10,000,000 shares of stock originally outstanding. Total equity follows a standard multiplicative binomial process with u = 1.2 and d = 0.9. The risk-free interest rate is 0.03 per binomial period. Figure 2. Two-period example of the multiple employee stock option pricing model applied to the valuation of anticipated at-the-money options. Option 1 is issued at time zero and matures at binomial time 1. Option 2 is issued at binomial time 1 and matures at time 2. The striking price of each option is equal to the stock price per share at the time the option is issued. Each option is for 3 percent of the shares outstanding at the time of issuance. There are 10,000,000 shares of stock originally outstanding. Total equity follows a standard multiplicative binomial process with u = 1.2 and d = 0.9. The risk-free interest rate is 0.03 per binomial period. The stock price per share is shown in bold italics at the bottom of each relevant state. Table I A Comparison of Option Values Computed via the Multiple Employee Option Pricing Model and Standard Binomial Model with σ = 0.30 0.3 −0.3 Values of u and d for the multiple option pricing model are computed as u = e and d = e , respectively. The risk-free rate is 5 percent per year, and each binomial period represents one full year. There are 10,000,000 shares of stock initially outstanding. Values of u and d for the standard binomial model are computed using the initial up and down returns per dollar invested for common equity computed via the multiple employee option pricing model. For the purposes of computing standard binomial option prices, the initial stock price is assumed to equal the stock price per share computed via the multiple employee option pricing model and values of u and d are assumed to remain constant over time. Option Maturity 1 2 3 4 5 6 7 8 9 10 Common Stock Total equity Stock price per share Shares per option = 300,000 Strike per option = $91 Multiple option Standard model binomial u = 1.34986 u = 1.32112 d = 0.74082 d = 0.77045 $4,163,986 $4,163,986 4,912,784 4,876,144 7,095,488 7,041,335 7,834,253 7,748,791 9,431,013 9,306,239 10,129,043 9,989,307 11,411,282 11,209,475 12,055,895 11,860,500 13,133,105 12,853,230 13,718,295 13,469,017 Shares per option = 1,500,000 Strike per option = $68 Multiple option Standard model binomial u = 1.34986 u = 1.25695 d = 0.74082 d = 0.83661 $12,558,080 $12,558,080 16,298,416 16,054,629 22,684,304 22,034,577 26,644,951 25,813,059 31,252,863 29,734,816 35,169,323 33,602,521 38,862,954 36,354,962 42,621,001 40,218,998 45,884,361 42,516,630 49,269,855 45,986,371 Shares per option = 3,000,000 Strike per option = $53 Multiple option Standard model binomial u = 1.34986 u = 1.21838 d = 0.74082 d = 0.87638 $16,658,573 $16,658,573 22,988,743 22,708,825 31,611,328 30,126,491 38,482,282 36,868,523 44,913,604 41,272,366 51,666,935 48,319,926 57,525,326 52,960,646 63,503,424 58,133,866 69,041,579 62,999,740 74,340,114 66,748,498 $906,114,856 $906,114,856 $678,753,893 $678,753,893 $529,268,090 $529,268,090 $1,000,000,000 $998,632,879 $1,000,000,000 $983,628,535 $1,000,000,000 $966,065,542 $90.61 $67.88 $52.93 28 Table II A Comparison of Option Values Computed via the Multiple Employee Option Pricing Model and Standard Binomial Model with σ = 0.60 0.6 −0.6 Values of u and d for the multiple option pricing model are computed as u = e and d = e , respectively. The risk-free rate is 5 percent per year, and each binomial period represents one full year. There are 10,000,000 shares of stock initially outstanding. Values of u and d for the standard binomial model are computed using the initial up and down returns per dollar invested for common equity computed via the multiple employee option pricing model. For the purposes of computing standard binomial option prices, the initial stock price is assumed to equal the stock price per share computed via the multiple employee option pricing model and values of u and d are assumed to remain constant over time. Option maturity 1 2 3 4 5 6 7 8 9 10 Common Stock Total equity Stock price per share Shares per option = 300,000 Strike per option = $86 Multiple option Standard model binomial u = 1.82212 u = 1.77518 d = 0.54881 d = 0.57928 $7,527,846 $7,527,846 8,188,424 8,156,888 11,522,384 11,462,947 12,120,502 12,042,080 14,311,941 14,180,081 14,839,025 14,709,013 16,437,068 16,230,211 16,891,844 16,710,993 18,120,451 17,840,568 18,504,504 18,276,127 Shares per option = 1,500,000 Strike per option = $57 Multiple option Standard model binomial u = 1.82212 u = 1.67063 d = 0.54881 d = 0.64714 $21,648,127 $21,648,127 24,902,098 24,733,627 34,110,055 33,447,046 37,430,411 36,723,060 43,296,576 41,795,949 46,472,317 45,092,686 50,657,038 48,239,130 53,600,253 51,475,322 56,796,856 53,413,495 59,451,589 56,544,308 Shares per option = 3,000,000 Strike per option = $42 Multiple option Standard model binomial u = 1.82212 u = 1.60423 d = 0.54881 d = 0.69025 $27,673,030 $27,673,030 32,627,533 32,504,008 45,127,217 43,646,039 50,519,588 49,180,819 58,385,529 55,059,119 63,774,669 60,874,657 69,315,197 63,941,763 74,569,302 69,835,339 79,125,667 71,655,179 83,686,240 76,989,324 $861,536,011 $861,536,011 $571,634,680 $571,634,680 $415,196,028 $415,196,028 $1,000,000,000 $998,672,765 $1,000,000,000 $984,747,430 $1,000,000,000 $966,555,304 $86.15 $57.16 $41.52 y and d = e −0.3 / y , respectively. The risk-free rate is 5 percent per year, or Table III Computation Times and per-Share Stock Prices with Model Modified to Reflect Recombining Binomial Trees Between Option Expirations 0.3 / , and there are 10,000,000 shares of stock initially outstanding. Computation times are for options on 300,000 shares with a striking price of $91. Values of u and d for the multiple option pricing model are computed as u = e 1/ y 1 + r = 1.05 Stock price per share Periods Total Time periods Options per year Strike = 91 Strike = 68 Strike = 53 (seconds) (m)* (y) ( m× y ) Shares = 300,000 Shares = 1,500,000 Shares = 3,000,000 10 1 10 0.0 90.61 67.88 52.93 10 2 20 1.0 90.73 67.93 52.92 10 3 30 19.7 90.64 68.00 53.03 10 4 40 215.4 90.68 67.98 52.99 10 5 50 1451.5 90.64 68.02 53.05 *The m options mature at the end of years 1 through m, inclusively. Therefore, m also represents the number of years until the expiration of the last option. Table IV Computation Times with Model Modified to Reflect Recombining Binomial Trees Between Option Expirations Each option has a striking price of $91 and is issued on 300,000 shares of stock. Values of u and d for the multiple option pricing model are computed as u = e 0.3 / y and d = e −0.3 / y , respectively. The risk-free rate 1/ y is 5 percent per year, or 1 + r = 1.05 , and there are 10,000,000 shares of stock initially outstanding. Computation times are for options on 300,000 shares with a striking price of $91. Options (m)* 1 2 4 5 Periods per year (y) 100 50 25 20 Total periods ( m× y ) 100 100 100 100 Time (seconds) 0.0 0.0 37.7 277.9 *The m options mature at the end of years 1 through m, inclusively. Therefore, m also represents the number of years until the expiration of the last option. 0.3 Table V The Valuation Impact Associated with the Anticipated Issuance of at-the-Money Employee Options −0.3 All entries reflect a volatility of total equity of 0.30 per year. Therefore, values of u and d for the multiple option pricing model are u = e and d = e , respectively. The risk-free rate is 5 percent per year, and each binomial period represents one full year. Each option has one year until its maturity date as of the time of its issuance. The number of shares associated with each option equals 3 percent of the number of shares outstanding at the time of issuance, and the striking price of each option is equal to the per-share stock price at the time of issuance, within a precision of $10-4. Option 1 2 3 4 5 6 7 8 9 Total options 990,008,566 $9,991,434 $1 billion 984,968,433 $15,031,567 $97.988 $1 billion 979,882,211 $20,117,789 $97.475 $1 billion 974,750,244 $25,249,756 $96.957 $1 billion 969,572,892 $30,427,108 $96.435 $1 billion 964,350,552 $35,649,448 $95.908 $1 billion 959,083,633 $40,916,367 8 $4,810,778 4,864,707 4,958,895 5,055,433 5,153,866 5,254,218 5,356,548 5,461,922 $95.377 $1 billion 953,772,579 $46,227,421 9 $4,783,409 4,838,005 4,931,686 5,027,688 5,125,575 5,225,371 5,327,119 5,430,865 5,537,703 2 $4,970,311 5,021,123 Common equity $1 billion $98.497 3 $4,944,282 4,994,821 5,092,464 Total equity $99.001 Maturity Date of Last Anticipated Option 4 5 6 7 $4,918,034 $4,891,558 $4,864,855 $4,837,928 4,969,249 4,943,457 4,917,435 4,891,184 5,065,433 5,039,139 5,012,624 4,985,875 5,165,073 5,137,270 5,110,219 5,082,943 5,238,331 5,209,744 5,181,918 5,312,231 5,282,837 5,386,762 Stock price