Lattice Models to Value Employee Stock Options Under SFAS
Lookman Buky Folami, Tarun Arora, and Kasim L. Alli
SEPTEMBER 2006 - SFAS
123(R) requires that employee stock options (ESO) be measured
at fair value. While either the Black-Scholes or the lattice
option-pricing model is acceptable under the standard, the
lattice model is better suited to the unique characteristics
use of a lattice model has broad implications. Auditors
and corporate executives must understand the variables that
are used to calculate the fair value of ESOs. They must
also understand how changes in the required variables drive
ESOs’ fair value and the resulting effect on total
compensation expense. Finally, the additional record-keeping
responsbilities created by implementing SFAS 123(R) require
significant changes on the part of corporate finance executives
for modeling resources, data requirements, and record-keeping
capabilities. If the existing information systems of a company
cannot provide the information necessary to value ESOs,
some modifications to the information system may be necessary
to meet the requirements for applying the lattice model.
Implementing the lattice model within the SFAS 123(R) framework
presents certain practical implications that are discussed
by way of an example, below.
of Lattice Models
models are option-pricing models that involve constructing
a binomial tree representing different paths that might
be followed by the underlying asset during the life of the
option. In the case of ESOs, the underlying asset is the
company stock. The fair value of the option is then derived
by backward induction through the binomial tree.
lattice model of stock valuation divides time into discrete
bits and models prices at these points in time. The period
of time covered by the lattice is broken down into individual
time periods (month, quarter, or annual) and the model predicts
possible stock prices at the end of each time period. In
the lattice model, stock prices will move either up or down
at the end of each period. A probability of occurrence is
assigned to each possible up or down position. As shown
in Exhibit 1, in a short period of time a stock price can
either rise by the up-move factor (u), or decline by the
down-move factor (d).
1, S0 represents the initial stock price at time zero
(i.e., the grant date). Su is the stock price in the next
time segment, assuming the price rose by the up-move factor
of u. Sd is the stock price in the next time segment, assuming
the stock price fell by the down-move factor d. By generalizing
a one-period binomial tree one can construct a multi-period
stock-price tree. Each node represents a probable stock
price in that time period. Therefore, the more we divide
total time into smaller pieces over the life of the option,
the bigger the lattice gets (as more possible stock prices
are modeled), and the more accurate the model can be. This
is depicted in Exhibit
following the logic depicted in Exhibit 2, one can build
a stock-price tree for the option’s contractual life.
The next step is to estimate the fair value of an ESO. To
do this, one builds another tree, starting with the option
payoff at the terminal nodes in the future, when the option
matures, and discounting the option back to the present
of Valuation Procedures
actual implementation of the valuation procedure for ESOs
within a lattice model entails three steps:
Determining the input variables and assumptions for the
Using the lattice model to value the option; and
Recording the valuation.
assumptions used in the model should reflect external and
internal information that is available on the grant date.
These assumptions must be reasonable and supportable, and
must not represent the biases of a particular party. Applying
the lattice model requires inputs for the minimum set of
substantive characteristics specified in SFAS 123(R). The
interplay of these inputs is illustrated using the case
study described below.
company ABC issued stock options to its employees with the
Stock price on the grant date (S): $20
Exercise price of the option (X): $20
Risk-free rate over the life of the option (r): 6%
Time to maturity of the option (T): 5
Expected life of the option (L): 5
Dividend yield of the option’s underlying stock
over the life of the option (D): 3%
Volatility of the underlying stock of the option (s) =
Exercise multiple over the life of the option due to suboptimal
exercise behavior (e): 2
Vesting period of the option grant (V): 1.5 years
Blackout dates (post-vesting period when the options cannot
be exercised): None
Annual exit rate of employees (W): 6%
Number of time steps in the option life (N): 10
addition to these required inputs and assumptions, the lattice
model used in the following example defines other input
variables, such as the up- and down-move factors and the
123(R) requires that companies use observable market prices
of identical or similar equity or liability instruments
to value share-based compensation cost, when such measures
are available. If observable market prices in active markets
are not available, then an appropriate option pricing model
such as a binominal lattice model can be used to estimate
the fair value of ESOs. The standard Cox-Ross-Rubinstein
(CRR) binomial tree model is used to construct a multi-period
stock price tree for ESOs with an expected life of five
years. Using the initial stock price of $20, the stock price
tree lists expected stock prices for each time period, assuming
that the stock price will go either up or down.
of price movement and probabilities. In each
time period, the stock price will go either up or go down.
Using the CRR specification, with volatility (s = 30%) and
time step (dt = T/N = 5/10 = 0.5), one can compute the up-move
factor (u), the down-move factor (d), the probability of
an up move (p), and the probability of a down move (1 --
1 depicts the output of the lattice model based on management
estimates of volatility and time segment. The probability
estimates derived are then used to develop a stock price
tree for each discrete time specified.
with the initial stock price of $20, one can build a stock-price
tree that lists the probable stock price for each probable
outcome in ten time segments (each time segment represents
six months), as depicted in Exhibit
up-position for the first time period is found at Node (1,
1) = 20 x 1.24 = 24.73. This represents the expected stock
price at time 1 if the stock moves up. The down-position
for the first time period is found at Node (1, -- 1) = 20
x 0.81 = 16.18, represents the expected stock price in time
1 if the stock moves down. Similarly, going forward on the
tree, the up-position for the second time period is found
at Node (2, 2) 24.73 x 1.24 = 30.57. The down-position for
the second time period is found at Node (2, -- 2) = 16.18
x 0.81 = 13.09.
that middle position in time period 2, Node (2, 0), can
be reached in two different ways. The stock could increase
from $20 to $24.73 during the first time period, followed
by a decrease in the next time period to $20. Or, the stock
price could decrease from $20 to $16.18 in the first time
period and then increase to $20 in the second time period.
The stock price can be computed either way: Node (2, 0)
= S (1, 1) x d = 24.73 x 0.81 = 20, or Node (2, 0) = S (1,
-1) x u = 16.18 x 1.24 = 20
of Option Value
next step is to develop an option value tree that is based
on the possible stock prices derived from Exhibit 3. The
possible option values are estimated by calculating the
option payoff in three steps:
The terminal nodes in the final time segment;
The vested time periods; and
The non-vested time periods.
calculated values are depicted in Exhibit
of option values in the final time segment. The
first step in the estimation of the option value is to estimate
the value in the final time segment. In the final time segment
(10), the employee has only one choice: either exercise
the option if it is “in the money,” or let the
option expire if it is “out of the money.” The
option is in the money if the exercise price is less than
the current stock price. At terminal node (10, 10), the
stock value (S) is $166.84, from Exhibit 3 above. This is
the estimate of stock price in time segment 10, assuming
the stock continually rises from time segment 1 through
formula to estimate the option’s payoff in the final
time segment is Node (10, 10) = Max (S – X, 0) = 166.84
– 20 = 146.84. Similarly, Node (10, –2) = Max
(S -- X, 0) = Max [(13.09 – 20), 0] = 0. At this node,
the option is out of the money and the payoff is zero.
the same steps, the option payoff is calculated for all
possible outcomes in segment 10.
the option payoff in the final time segment has been determined,
the option payoff for all prior time segments (nodes) can
be determined. For this step, time is divided into vested
time segments and nonvested time segments.
of option values for vested time segments. For
the vested time-period nodes (3 through 10), the option
can be exercised at any time up to and including maturity.
2 illustrates that after the stock options are fully
vested, the holder will exercise the options in the current
period if the payoff from waiting until the next period
is less than the payoff received by exercising the options
immediately, and vice versa. Applying the above equation
to Node (9, 9) and Node (8, 6) yields the following: Node
(9, 9) = Max[e-(0.06)0.5 (0.48
x 146.84 + 0.52 x 89.16), (134.95 -- (2 x 20)] = Max [113.53,
94.95] = 113.53; and Node (8, 6) Max[e-(0.06)0.5
(0.48 x 67.57 + 0.52 x 37.49), (71.42 -- (2 x 20)] = Max
[50.47, 31.42] = 50.47
value of the option for all the nodes can be computed backwards
up to the start of the vesting period.
of option values for nonvested time segments.
During the nonvested period nodes (1 through 2), such as
Node (2, 2), the option cannot be exercised. The value of
nonvested options, such as at Node (2, 2), would be the
present value of the expected payoff in the next time period,
as follows: Node (2, 2)e-(r)xdt [p
x Up position value + (1 -- p) x Down position
value] = e-(0.06)0.5 [(0.48
x 18.27 + 0.52 x 7.78)] = 12.46. Similarly, Node (2, 0)
= e--(0.06)0.5 [(0.48 x 7.78 + 0.52 x 2.50] = 4.90, and
Node (2, -2)
= e-(0.06)0.5 [(0.48 x 2.50 + 0.52)]
backward through the tree leads to the final value of each
option as $5.40.
of Input Variables
provides a list of the input variables required for a lattice
option-pricing model within an SFAS 123(R) framework, their
effects on the option value, and the possible sources of
information for the required inputs.
life. SFAS 123(R) requires using the expected
life of the option (L) rather than the contractual life.
ESOs cannot be transferred, sold, or exercised before they
vest. Once the options vest, employees can exercise their
ESOs before the end of their contractual life. Some employees
might terminate their employment with the company before
their ESOs vest, resulting in forfeiture of the options.
Therefore, because of early-exercise behavior and employment
termination, the expected life of ESOs differs from their
123(R)’s use of the expected option life reduces the
value of the option because exercise prior to the expiration
of the contractual term will reduce its time value. In a
closed form option-pricing model, expected life is an assumption.
In a lattice model, expected life is an output of the model
based on the contract life, employees’ early-exercise
behavior, the termination rate, and a number of other factors.
practical implication regarding the use of the expected
option life is the need to segregate employees into homogenous
groups based on early-exercise behavior and employee turnover.
For example, one could observe exercise behavior and turnover
for middle management versus upper management, married versus
single, and other demographic factors, in order to gain
insight into how to segregate employees into homogenous
groups. A lattice model can then be used to determine the
fair value of ESOs for employees in the same group. This
approach will yield a more accurate estimate of fair value.
For the sake of simplicity, the case study above assumes
that the contractual life equals the expected life.
multiple and suboptimal behavior. The valuation
process for ESOs must take into account that the option
exercise behavior of each employee is different. Some employees
might exercise an option once the stock price doubles, while
others might exercise once the stock price grows by a minimum
of 10%. The behavior of the second group is termed suboptimal
exercise behavior (e). This feature of ESOs can be appropriately
handled by modifying the standard lattice model. The option’s
value decreases with the incidence of suboptimal exercise
behavior, because the option holders who exercise the option
suboptimally will not realize the full gain associated with
the upside potential of the stock price.
period. Under SFAS 123(R), two additional
factors that must be incorporated into the valuation of
ESOs are the vesting period and the exit rate. The vesting
period is usually the service period from the grant date.
Many ESO plans will have an underlying vesting schedule
where only a certain percentage of the grants become exercisable
each year. The longer the vesting period, the more likely
that an employee will not be with the organization and the
ESOs grant will be forfeited. The higher the probability
of forfeitures, the lower the market value of the option.
the case study above, the ESOs can be exercised only after
the vesting period of 1.5 years has elapsed. In a binomial
lattice model, the delay caused by the vesting requirement
is easily handled by modifying the option tree to allow
exercise only after the vesting period.
period. Employee stock options, unlike exchange-traded
options, are not traded in a secondary market. The only
way an employee can liquidate the position is to exercise
the options and sell the stocks received. A blackout period
is a time period during which employees cannot exercise
the option after they are vested. Blackout periods generally
have the effect of lowering the option value. Because options
cannot be traded during this timeframe, the magnitude of
the effect on the option value is comprised of two components.
First, the longer the blackout period, the lower the value
of the ESOs. Second, the inability to exercise may actually
increase the value of the option by preventing holders from
exercising the options suboptimally for some specified time
period. Notwithstanding this contra-effect on suboptimal
behavior, the net effect of blackout periods on ESOs is
generally a reduction in value. The case study above assumes
there is no blackout period.
rate. ESOs are subject to forfeitures when
an employee resigns or is terminated prematurely before
the end of the vesting period. This anticipated forfeiture
rate must be estimated. When employees leave before the
option vests, unvested options are forfeited. Employees
may be forced to exercise vested options upon leaving the
company. This premature exercise behavior leads to suboptimal
exercise; that is, options are exercised before the up-side
potential is completely realized. The implication of this
requirement is that employers must estimate their exit rate,
which can be calculated using company or industry data.
The higher the exit rate, the lower the estimated fair value.
our example, we assume that the exit rate is 6% per time
period. To calculate the value of the option under SFAS
123(R), multiply the value obtained through the lattice
model with the probability that the options are not forfeited
before vesting (for procedural details, see M. Ammann and
Ralf Seiz, “Valuing Employee Stock Options: Does the
Model Matter?,” Financial Analysts Journal,
September/October 2004). This is expressed as (1 –
w)v (where w = annually compounded employee exit rate and
v = the vesting period). In the case study, the option value
under the lattice model is $5.40, so the SFAS 123(R) option
value is 5.40 multiplied by (1 – 0.06)1.5, yielding
ESO fair value. In conclusion, the estimated
fair value of each option in the case study above is $4.92.
Assuming that 100,000 ESOs are granted by the company, the
total value would be $492,000. This amount would be allocated
over the service period of the employees who received the
options. The service period is usually the vesting period,
which in this case is 1.5 years, making the monthly compensation
implementation of SFAS 123(R) has broad implications for
accounting professionals. An appropriate valuation model
must be selected, and the assumptions used within the model
must be determined. Both a lattice model and a Black-Scholes-Merton
model meet the requirement of SFAS 123(R), though the lattice
model can more easily handle the complexities of ESO valuation.
In the case study above, the fair value of the option under
the lattice model is $5.40 ($4.92 after the SFAS 123 adjustment).
In comparison, $6.17 would have been obtained for the same
set of data under Black-Scholes-Merton. The difference of
$0.77 translates to an overstatement of compensation expense
by 14.30% for ESOs. The unique features of ESOs and the
flexibility of the lattice model lead to a more accurate
measure of fair value.
addition to selecting an appropriate valuation model, an
employer must maintain a database containing all of the
employee information necessary to validate the model’s
assumptions. Accounting and finance professionals must look
beyond historical estimates to incorporate adjustments based
on future expectations. The determination of the assumptions
used in the model—such as employee exit rates, dividend
yields, and employee exercise behavior—is critical
because it drives ultimate value.
Buky Folami, PhD, CPA, CMA, CFM, is an assistant
professor in accounting at Bryant University, Smithfield,
R.I. Tarun Arora, MBA, is with the audit
group at KPMG, LLP, in Birmingham, Ala. The
views expressed above are the author’s own and do not
represent those of KPMG.
Kasim L. Alli, PhD, is a professor of finance at
the business school at Clark Atlanta University in Atlanta,