Using Lattice Models to Value Employee Stock Options Under SFAS 123(R)

By Lookman Buky Folami, Tarun Arora, and Kasim L. Alli

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The 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.

Description of Lattice Models

Lattice 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.

The 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).

In Exhibit 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 2.

By 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 time.

Implementation of Valuation Procedures

The actual implementation of the valuation procedure for ESOs within a lattice model entails three steps:

• Determining the input variables and assumptions for the model;
• Using the lattice model to value the option; and
• Recording the valuation.

The 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.

Suppose company ABC issued stock options to its employees with the following values:

• 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) = 30%
• 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

In 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 respective probabilities.

Case Study

SFAS 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.

Calculation 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 -- p). Equation 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.

Starting 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 3.

The 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.

Note 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

Estimation of Option Value

The 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.

The calculated values are depicted in Exhibit 4.

Calculation 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 10.

The 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.

Using the same steps, the option payoff is calculated for all possible outcomes in segment 10.

Once 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.

Calculation 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.

Equation 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

The value of the option for all the nodes can be computed backwards up to the start of the vesting period.

Calculation 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)] = 1.44

Working backward through the tree leads to the final value of each option as $5.40.

Collection of Input Variables

The Sidebar 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.

Option 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 contractual life.

SFAS 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.

One 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.

Exercise 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.

Vesting 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.

In 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.

Blackout 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.

Exit 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.

In 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 $4.92.

Recording 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 expense $27,333.

Issues to Consider

The 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.

In 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.