Download A Model for Valuing Multiple Employee Stock Options Issued by the

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
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Barth, Mary E, Wayne R. Landsman and Richard J. Rendleman, Jr., 1998, “Option
Pricing-Based Bond Value Estimates and a Fundamental Components Approach to
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Black, Fischer and Myron Scholes, 1973, “The Pricing of Options and Corporate
Liabilities,” Journal of Political Economy 3, 637-654.
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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,”
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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,
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Path-Dependent Derivatives,” Journal of Applied Business Research 17, 1-11.
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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
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Kapadia, Nikunj., and Gregory. Willette, 2002, “Dilution and the valuation of options and
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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