| Editor’s note: This
article uses a simplified example to
illustrate how a lattice model works. In
the exhibits, the option term is only
four years—much shorter than the
10-year life of a typical employee stock
option. So in practice the calculations
will be more extensive than in these
exhibits and companies may have to divide
the time period into additional
intervals. |
he guidance from FASB is clear: Companies must
determine and report the fair value of stock
options they use to compensate employees. But
because employee stock options can’t be
traded publicly, their fair value is not readily
available and must be estimated using
option-pricing models. FASB Statement no. 123(R),
Share-Based Payment (www.fasb.org/pdf/fas123r.pdf), allows entities to use any valuation
model that is based on established principles of
financial economic theory and reflects all
substantive characteristics of the options. Both
the Black-Scholes-Merton and lattice models meet
these criteria. The former’s relative
simplicity makes it popular with smaller
companies—but it may not be adequate for
public companies whose employees often exercise
their options early. That calls for calculations
a lattice model can better accommodate. (For more
information, see “Compare and Contrast.”)
Neil J. Beaton, CPA/ABV,
partner in charge of valuation services at Grant
Thornton LLP in Seattle, said his firm has
performed numerous engagements related to FASB
Statement no. 123(R) and found a lattice model to
be substantially more flexible than a
Black-Scholes model, especially with respect to
restricted employee stock option nuances such as
vesting, early exercise and blackout periods.
“Once we built our initial lattice
model,” he said, “conforming it to the
widely varying requirements of our diverse client
base was fairly easy and has produced results
more accurate than would have been possible with
a Black-Scholes model alone.”
But even if
employers know which valuation model works better
for them, they still may have doubts about how to
build it. An earlier JofA article (see
“No Longer an ‘Option,’ ” JofA, Apr.05, page 63)
explained the workings of the
Black-Scholes-Merton model. This month’s
article provides detailed instructions for
building a lattice model by making the necessary
calculations in Excel.
One company that chose to
implement such a model is the Marysville,
Ohio-based Scotts Co., a manufacturer of
horticultural products. Its CFO, Chris Nagel,
CPA, told the JofA in the April article on
Black-Scholes that he preferred the lattice model
because of its exceptional ability to capture
assumptions about options’ term and
volatility.
“We had adopted
Black-Scholes but now believe a lattice model is
appropriate for valuing options,” Nagel
said. “To value options, you have to make
assumptions about the likely term and volatility,
and I think a lattice model captures those
variables better.”
Because the lattice model makes
it easy to vary assumptions and inputs over time,
entities that grant a great many stock options to
their employees will prefer its flexibility to
the relatively rigid restrictions of the
Black-Scholes-Merton model, which is more
suitable for companies whose employee
compensation includes few stock options. A
lattice model can be complex for a company to
implement, though. “Luckily, I’m not
the one who has to grind through the
numbers,” Nagel said.
But what if, in your company, you
are the CPA who performs that function? If
that’s the case, follow the examples below
that illustrate the structure and functions of a
lattice model.
THE
BASICS
A lattice model
assumes the price of stock underlying an option
follows a binomial distribution, a type of
probability distribution in which the underlying
event has only one of two possible outcomes. For
example, with respect to a share of stock, the
price can go up or down. Starting at a point
we’ll call time period zero, the
assumption of either upward or downward movements
over a number of successive periods creates a
distribution of possible stock prices. This
distribution of prices is referred to as a lattice,
or tree, because of the pattern of lines
used to graphically illustrate it. The lattice
model uses this distribution of prices to compute
the fair value of the option.
Exhibit 1 (below) illustrates an Excel
stock-price tree based on the following
assumptions:
 Current stock price of
$30.
Risk-free interest
rate of 3%.
Expected dividend
yield of 0%.
Stock-price volatility
of 30%.
Option exercise price
of $30.
Option term of four
years.
At the grant
date, year 0, the stock price is $30 (cell B7).
The model assumes that stock prices will increase
at the risk-free interest rate (B15) minus the
expected dividend yield (B16), then plus or minus
the price volatility (B12) assumed for the stock.
Thus, during year 1, the stock price increases by
the risk-free rate, 3%; is unaffected by the
assumed 0% expected dividend yield; and then
either increases or decreases by 30% due to the
expected volatility.
The formula for cell E12, the
year 1 upward path, is
=D21*(1+B15–B16)*(1+B12). For the downward
path, the formula for E29 is
=D21*(1+B15–B16)*(1–B12). The resulting
two possible outcomes for the stock price at the
end of year 1 are an increase to $40.17 (E12) or
a decrease to $21.63 (E29). In lattice
terminology these two possibilities are referred
to as nodes. Two similar possibilities
for the end of year 2 emanate from each of the
year 1 nodes. With the number of nodes doubling
in each successive time period, the tree grows to
16 nodes after four years.
Exhibit 1 also contains the probabilities for
each node on the tree. For example, at the end of
year 2, the stock price of $53.79 (F8) has a
probability of 0.25 (F9), which is the
probability of two successive upward price
movements. With a probability of 0.50 that the
price will increase in any year, the probability
of two successive upward movements is 0.25 (F9).
In fact, two nodes reflect a stock price of
$28.96 at the end of year 2 (F16 and F25). F16
represents the result of an upward movement in
price in year 1 followed by a downward movement
in year 2; F25 reflects a downward price movement
in year 1 followed by an upward movement in year
2. Similar to the probability of two successive
periods of upward price movement, the
probabilities for F17 and F26 are 0.25.
The probability (that is,
0.0625) for each terminal node (column H)
corresponds to four successive movements in the
stock price.
Know
the Options
Unlike stock options that
are traded on an exchange, employee stock
options
Can be exercised, but not
sold or transferred.
Cannot be exercised during
“blackout” periods, which
companies typically declare just before
releasing their earnings or at other
times to prohibit employee purchases or
sales of company stock or options.
n Typically have terms of 10
years, in contrast to most traded
options’ terms of less than one
year.
n Are subject to vesting periods of up to
four years, during which the options
cannot be exercised, and are forfeited by
those who leave the company before
becoming vested.
n Often are exercised early for reasons
such as divorce, separation from service
or financial need. |
CARRY ON CRUNCHING
After developing a
stock-price tree, the next step is to calculate
the intrinsic value of the option at
each terminal node by subtracting the
option’s exercise price (B8) from the stock
price at that node. If the stock price at the
option’s expiration date exceeds the
exercise price, the option is said to have intrinsic
value and the options are assumed to be
exercised. Otherwise, the option has no intrinsic
value.
Exhibit 2, below, presents an Excel template that
calculates the option’s fair value. Columns
J through M are added to exhibit 1’s stock price tree (hidden here
for simplicity). This example presumes that
option holders will not exercise their
options early. Rows 5 through 20 represent the 16
terminal nodes from column H in exhibit 1.
In column K the
intrinsic values of the option at the
corresponding nodes are computed using Excel IF
statements to determine whether the stock prices
at those nodes exceed the option’s exercise
price. For example, cell K5’s formula is
=IF(H5>B8,H5–B8,0). That formula
calculates and displays the option’s
intrinsic value, $66.44, the amount by which the
terminal stock price exceeds the exercise price
for the path reflecting four successive upward
price movements. Column K shows the option is
“in the money” or has intrinsic value
at K5, K6, K7, K9 and K13 of the 16 terminal
nodes.
In column M the intrinsic
values of the option are multiplied by their
respective probabilities (column L). Then the
present value of each is determined using the
risk-free interest rate (B15). The formula in
cell M5, –PV(B15,J5,,K5*L5), computes the
present value of the probability-weighted
intrinsic value for the topmost terminal node
(H5) in exhibit 1. (Editor’s
note: Normally Excel’s PV function returns a
negative value because Excel considers present
value to be the outflow required to pay for
future inflows. To prevent any confusion, cell
M5’s PV statement begins with a negative
sign and therefore expresses the present value as
a positive.) Thus, the $3.69 present value
represents the $66.44 intrinsic value weighted by
its 0.0625 probability and discounted at a 3%
rate for four years. Corresponding formulas in
cells M6 through M20 calculate the intrinsic
value for each of the other 15 terminal nodes in
column H of exhibit 1. The
summation (M22) of column M, $8.56, is the
option’s fair value and the amount of
expense to be recognized. A fuller application of
the lattice model will allow CPAs to consider
changes in stock price and other factors on at
least a weekly basis.
BEYOND
THE BASICS
The lattice model
has a key advantage over its Black-Scholes-Merton
counterpart; it offers CPAs several ways to
incorporate assumptions about the projected early
exercise of options. One approach, demonstrated
in FASB Statement no. 123(R), assumes the options
will be exercised if the stock price reaches a
selected multiple of the exercise price.
Exhibit 3, below, illustrates this approach using
a 2.0 early exercise factor (cell B9)
that assumes all options will be exercised for
pretermination nodes in years 3 or earlier if the
stock price reaches $60—double the $30
exercise price. With all other assumptions being
held constant in exhibit 3, the
stock-price tree presented in exhibit 1 remains valid. Note that the
stock’s price reaches $60 prior to
expiration only on the path (cell G6 in exhibit 1) that represents three successive years
of upward price movements. In exhibit 3 the options are assumed exercised with
a gain to the employee of $42.02 (K13)—the
difference between the year 3 $72.02 stock price
(G6 in exhibit
1) and the $30 exercise
price (B8).
When early
exercise is considered, each node on the stock
price tree must be examined to determine whether
the options will be exercised early. Thus, exhibit 3 contains 30 rows—one for each node
in the exhibit
1 tree. The formula in
cell L13,
=IF(AND(G6>5(B8*B9),L5=0,L8=0),0.5^J13,0),
examines whether the cell G6 stock price in the exhibit 1 tree equals or exceeds the early
exercise multiple. If the stock price meets this
criterion and early exercise has not occurred in
prior periods, the probability (G7) of this exhibit 1 node is multiplied by the option’s
intrinsic value (K13) and discounted by the
risk-free interest rate (B15) to determine the
path’s present value (M13).
Because the exercise price
multiple is not met for any other pretermination
nodes, a probability of zero is specified in
cells L5 to L6, L8 to L11 and L14 to L20. Of the
16 potential termination nodes in exhibit 1, the uppermost two (H5 and H7) are
exercised early at the end of year 3. Since they
are not outstanding in year 4, their
corresponding cells in exhibit 3 (L22 and L23) have a probability of
zero. In year 4 the intrinsic values for the 14
paths not previously truncated are
probability-weighted and discounted to determine
their present values (that is, the probability is
multiplied by the option’s intrinsic value
and discounted by the risk-free interest rate to
determine the path’s present value). The
total of the present values of all the individual
potential paths (M13 and M24 through M37) is the
option’s fair value, $8.46.
A lattice model also can
accommodate additional expectations regarding
when and the extent to which employees exercise
their options. For example, rather than assuming
that all the options are exercised if the stock
price reaches a selected multiple of the exercise
price, a lattice model also can permit the
assumption that only a certain percentage of
outstanding options are exercised.
MEASURING
UNPREDICTABILITY
Another benefit of
the lattice model is that it can accommodate
assumptions that vary over time. Exhibit 4, below, presents a stock-price tree
that assumes the stock’s volatility
decreases from 30% to 24% over the option’s
four-year life.
Exhibit 4
shows how to specify individual volatility
factors for each year of the option’s term
(cells B12 through B15). The impact of decreasing
the stock’s volatility in later years is
evident on the tree’s top branch. After four
successive periods of stock-price increases, the
stock’s price in cell H5 ($87.78) is less
than it is ($96.44) in the corresponding cell of exhibit 1. The lower volatilities reduced the
magnitude of the stock price increases on the top
branch. A similar tempering effect can be seen in
cell H36 on the bottom branch, where the
stock’s price in exhibit 4
($9.57) is greater than it is in exhibit 1 ($8.11). The lower the volatility, the
lower the option’s fair value.
CHARLES P. BARIL is a professor
and LUIS BETANCOURT, CPA, and JOHN W. BRIGGS are
assistant professors at James Madison
University’s School of Accounting in
Harrisonburg, Va. Their respective e-mail
addresses are barilcp@jmu.edu, betanclx@jmu.ed u and briggsjw@jmu.edu.
| AICPA
RESOURCES |
CPE
 Accounting for
Stock Options and Other Stock-Based
Compensation (textbook, # 732087JA). Infobytes: Stock Options and
Other Share-Based Compensation Accounting
(online courses):
Audit
Considerations.
Disclosures.
Nonpublic Company
Considerations.
Measuring the
Share-Based Payment.
History and
Summary of FASB 123(R).
For information about Infobytes, see
product no. BYTXX12JA at www.cpa2biz.com/infobytes.
Publication
Investment Valuation: Tools and
Techniques for Determining the Value of
Any Asset, 2nd edition (hardcover, #
WI414883P0200DJA).
For more information about these
resources or to place an order, go to www.cpa2biz.com
or call the Institute at 888-777-7077.
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