Employee Stock Options Using a Lattice Model
Les Barenbaum, Walt Schubert, and Bonnie O’Rourke
FASB exposure draft on stock-option expensing would require
the valuation of equity-based compensation awards at their
grant date. Option value and the resulting expense are based
upon models that capture the characteristics determining the
value of a particular grant of employee options. The exposure
draft discusses lattice valuation models that accommodate
the often complex attributes of option plans that can change
over time. The lattice model can explicitly capture expected
changes in dividends and stock volatility over the expected
life of the options, in contrast to the Black-Scholes option-pricing
model, which uses weighted average assumptions about option
characteristics. The authors’ objective is to provide
an overview of how lattice models work and to provide insights
into how lattice models can help ascertain the costs and benefits
of various option-granting strategies.
structure, such as the binomial model, incorporates assumptions
about employee exercise behavior over the life of each option
grant and changes in expected stock-price volatility. This
results in more-accurate option values and compensation
stock options have distinct characteristics. For example,
it is typical for a large percentage of employees to exercise
their options upon vesting. Other employees hold their options
and exercise based upon their assessment of the expected
future movement of the stock price. Lattice models provide
a framework to capture the impact of these varying exercise
patterns into the calculation of the value of the option.
This impact can be material.
advantage of the lattice structure is the ability to incorporate
expected changes in volatility over the life of the option.
This is particularly important to young companies that,
while currently recording highly volatile returns, expect
decreased volatility in the future. The lattice model allows
for more precise assumptions, therefore allowing more precise
estimates of option values. The lattice model uses data
collected about employee exercise behavior and stock-price
volatility to project an appropriate array of future exercise
in turn allows more-accurate estimates of option values.
In comparison to the Black-Scholes model, the lattice structure
allows the incorporation of various early-exercise assumptions,
once substantiated by an analysis of employee behavior patterns,
which results in more-accurate, and often lower, option
values and lower expenses.
Logic of Lattice Models
option-pricing models, such as the binomial model, use estimates
of expected stock-price movements over time. The expected
magnitude and likelihood of stock-price movement is predicated
upon the expected volatility of a security’s returns.
1 illustrates a simple two-year lattice model that depicts
the expected price changes of the security, along with their
probability of occurrence. Each node of the lattice reflects
an expected year-end share price. These expectations are
developed through analysis of the security’s historical
volatility and its likely future volatility.
refers to the fluctuations in share returns over time. Volatility
is measured by calculating the expected standard deviation
of the returns of a security. The expected future volatility
then determines expected share price movements over time.
In turn, these potential share price movements are a major
factor in estimating option value. Exhibit 1 illustrates
how the estimated standard deviation results in share price
movements over time.
most common method of estimating future volatility is to
use historical volatility as a proxy. There are no hard-and-fast
guidelines on how far back one should calculate historical
volatility. Future volatility can also be estimated by solving
for the implied volatility of a company’s traded options.
Interpreting implied option volatility requires caution,
as option volatility is impacted by the interaction of:
The option’s expected time to expiration,
the option is trading at-the-money, and
companies generally have higher volatilities than do mature
companies within an industry. Therefore, when calculating
the volatility for a company that has been public for only
a few years, its historical volatility may not be a good
proxy for future volatility. When insufficient data exists
to form estimates of a particular company’s expected
volatility, the FASB Exposure Draft Implementation Guide
suggests using the volatility of comparable companies.
advantage of a lattice model is that it can use different
volatility estimates for different time periods. For example,
options with a four-year expected life can have different
expected volatilities over that period. The implied volatility
of traded options with varying times to expiration can be
used to estimate the various relevant implied volatilities.
The Black-Scholes model requires the use of the same standard
deviation over the entire expected life of an option, thereby
reducing the flexibility of the model and the precision
of the results.
2 illustrates an example with a 64.8% probability that
the price of the security will increase 15% (from $30.00
to $34.50) and a 35.2% probability that the price will decline
by 13% (from $30.00 to $26.09). The probabilities and percentage
price increases are the same for each of the two years.
For example, if the price does go up to $34.50 in year 1,
there is a 64.8% chance that it will go up again in year
2 (to $39.68) and a 35.2% chance that it will decline in
year 2 (back to $30.00).
2 extends the analysis to illustrate how option values are
determined. Assume that fully vested stock options have
been granted with an exercise price of $30.00 and a term
of two years. Therefore, the owner of the option can purchase
shares of stock for $30.00 until the option expires in two
years. If the share price increases in both years 1 and
2, the option holder will net $9.68 ($39.68 -- $30.00) upon
exercise of the option. If the share price stays at $30.00
a share or falls to $22.58 at the end of year 2, the option
holder will not exercise, as the share price does not exceed
the exercise price.
the share price has a value of $30.00 or less at the end
of the two-year period, there is no gain for the holder,
but there is also no loss. The option simply expires unexercised.
At the time of the option grant, the option clearly has
value. It is more likely that the stock will have a value
greater than $30.00 at the end of two years, and the holder
will not suffer any loss if it does not.
mechanics of calculating the option value at the time of
grant begin by determining the option value at the expiration
period and working backward to the date of the grant. At
the end of year 1, the share price will have either increased
to $34.50 or fallen to $26.09. If the share price is $34.50
at the end of year 1, the option holder has an asset that
will either rise $9.68 (share price of $39.68) or fall to
$0 (share price of $30.00). The respective probability of
these outcomes is 64.8% and 35.2%. Using a time value of
money discount rate of 5%, the value of the option in year
one will be $5.97, as calculated below:
x $9.68) ÷ 1.05] + [(35.2% x $0) ÷ 1.05] =
to work backward in time, the value of the option at the
grant date is based upon the option values at the end of
year 1. The calculation is the same as in the previous example,
and yields an option value of $3.68, the present value of
$5.97 and $0 weighted by
the probabilities of each outcome occurring:
x $5.97) ÷ 1.05] + [(35.2% x $0) ÷ 1.05] =
the option value is based upon the expected share price
at each node on the lattice. If the historical volatility
is higher, and the future volatility is projected to be
higher, all else being equal, the option will have more
value; the higher the probability of an increase in stock
price, the higher the value of the option. There is no real
risk of loss to the option holder, who will simply not exercise
the option if the stock price declines. Therefore, as long
as there is a positive probability that the price will rise
above the exercise price, the option has value.
analysis above illustrates the value of transferable options
at the grant date. Employee stock options, however, are
not transferable, and this affects their value.
Transferability of Options
the above example, if the share price has risen to $34.50,
the option would be worth $5.97, factoring in the possibility
of a rising price in year 2. But if the option cannot be
sold, the option holder must choose between exercising the
option at the end of year 1 and holding it until the end
of year 2. If the holder chooses to exercise the option
at the end of year 1, the proceeds would be only $4.50.
they cannot sell the option in the open market, many employees
will exercise their options early to realize a gain rather
than take the chance that the share price will fall. In
other words, the option is worth only $5.97 at the end of
year 1 if it can be sold. There is a positive probability
that the stock will rise in year 2 and be worth $9.68, but
it also might decline and become valueless. Employees may
prefer to take a profit of $4.50 rather than risk losing
all the potential value. The result of the potential early
exercise is that the grant date value of the option falls
from $3.68 to $2.78:
x $4.50) ÷ 1.05] + [(35.2% x $0) ÷ 1.05] =
reduced option value is due to the increased likelihood
of early exercise that nontransferability represents.
structures allow expected employee exercise behavior to
be modeled in order to develop more-accurate estimates of
option values. Survey data indicate that many employees
will exercise their options early to realize built-in gains
when the underlying share price reaches 1.5 to 2.5 times
the exercise price. (See Jennifer N. Carpenter, “The
Exercise and Valuation of Executive Stock Options,”
Journal of Financial Economics, 1998.) Exhibit
3 extends the example to a 10-period binomial with the
expectation of early exercise when the underlying share
price reaches 1.5 times the exercise price. As shown, when
the price exceeds $45.00, employees will exercise their
options, effectively stopping the binomial tree from expanding.
blue cells indicate where early exercise takes place. The
gray areas represent the portion of the binomial tree that
is no longer relevant due to early exercise. This results
in the option value falling from $12.34 to $8.83 as a result
of early exercise. The loss in option value is due to the
lost probability of further increases in share value. Data
on the exercise history of employees can be incorporated
into the exercise behavior of employee groups to generate
a more-accurate and often lower option value than with the
Black-Scholes option pricing model.
Exercise with Vesting Requirements
4 adds an assumption of five-year cliff vesting. Cliff
vesting occurs when exercise of the option is not allowed
until the end of a vesting period. This impacts the employee’s
ability to early exercise when the share-price-to-exercise-price
ratio reaches 1.5. Vesting requirements extend the life
of the option, increasing its value and resulting in added
compensation expense. Exhibit 4 illustrates that the vesting
requirements change the timing of employee early exercise.
Even though the share price may reach $45.00 in year 3,
early exercise cannot occur until vesting is allowed in
year 5, as shown by the purple cells. The vesting requirement
increases the expected time to exercise, thereby increasing
the option value from $8.83 to $9.63.
benefit of a lattice model is that it can analyze the trade-offs
between the benefits of increased employee retention through
longer vesting periods versus the added option expense of
delaying early exercise. Other trade-offs, such as the reduced
option cost of issuing options out-of-money, can be more
fully understood using lattice structures.
Exercise with Vesting Requirements and Forfeiture
companies experience option forfeiture due to layoffs or
voluntary exit. Employees that exit prior to vesting do
not impact the value of stock options, because the employee
is not entitled to the option. Nonetheless, such exits impact
the expected option expense through a reduction in the number
of options expected to vest. Once the employee is vested,
employee termination will lead to early exercise if the
option is in-the-money.
lattice tree shows how employee terminations impact the
value of stock options. Exhibit
5 shows the 10-year employee stock option with five-year
cliff vesting and expected employee turnover of 3% annually
for vested employees. Under the exercise rules in the previous
example, the blue highlights show the nodes where normal
early exercise takes place. The cells in orange indicate
additional early exercise nodes due to employee turnover
exit. The value of the option decreases from $8.83 to $7.36
because of the additional early exercise.
Barenbaum, PhD, is a partner at Kroll, Inc., and
a professor of finance at LaSalle University in Philadelphia,
Walt Schubert is a professor of finance at
LaSalle University. Bonnie O’Rourke
is a partner at Kroll, Inc.