Valuation of Employee Stock Options and Other Equity-Based ...

Valuation of Employee Stock Options and Other Equity-Based 

Instruments: Short-Term and Long-Term Strategies for 

Complying with FAS 123R and for Optimizing the Performance of 

Equity-Based Compensation Programs under the New Standard 

Prepared for 

The Silicon Valley Chapter of 

Financial Executives International 

Prepared by 

Ronald D. Rudkin, Ph.D. 

Vice President 

Analysis Group, Inc. 

Two Embarcadero Center, Suite 1750 

San Francisco, California 94111 

(415) 263-2213 

rrudkin@analysisgroup.com 

April, 2006

Abstract 

This report discusses strategic actions firms can take both now and in the future to 

comply with FAS 123R and to optimize the performance of their equity-based 

compensation programs (EBCPs) under the new standard. It discusses challenges that 

firms are expected to face when attempting to comply with the new FAS 123R input 

requirements and recommends strategies for dealing with them. In the report we 

evaluate the key differences between the two most often used valuation models -- the 

modified Black-Scholes (MBS) and the binomial lattice models and show that latticebased 

models produces more accurate and generally lower fair value estimates than 

MBS-based models for both traditional employee stock options (ESOs) and for the 

nontraditional instruments (e.g., premium, capped, indexed and performance-based 

options) analyzed in the report. It discusses changes firms can make to the features of 

traditional ESOs that can enable them to better accomplish their HR objectives and to 

lower compensation expense. Finally, the report discusses how nontraditional 

instruments can enable firms to better accomplish their HR goals and to reduce 

compensation expense compared to traditional ESOs. The nontraditional options 

analyzed are shown to produce lower fair value estimates than traditional ESOs.

Acknowledgements 

I wish to thank the members of the Silicon Valley Chapter of the Financial Executives 

International (FEI) who participated in this project. Special thanks goes to Randy 

Bambrough, Gary Bohe-thackwell, Richard Brounstein, Todd Lowenstein, Michael 

Shahbazian and Dave Wittrock for their support and encouragement during the project and 

for permitting me to present a summary of this paper during one of their dinner meetings. I 

wish to especially thank Todd Lowenstein for engaging me in numerous stimulating 

conversations and email exchanges throughout the course of the project.

Contents 

Executive Summary ................................................................................................................1 

I. Introduction .........................................................................................................................8 

II. Selection of an Appropriate Valuation Model ...................................................................11 

III. Key Issues Concerning the Estimation of Model Inputs and Measures of Employees’ 

Exercise and Post-Vesting Employment Termination Behavior ............................................19 

IV. Strategies for Optimizing the Performance of Equity-Based Compensation Programs 

Under FAS 123R...................................................................................................................22 

V. Conclusions .....................................................................................................................33 

References............................................................................................................................34 

Appendix A............................................................................................................................35 

Appendix B............................................................................................................................39

Executive Summary 

This report discusses and evaluates strategic actions that firms can take both now and in the 

future to comply with FAS 123R and to optimize the performance of their equity-based 

compensation programs (EBCPs) under the new standard. The new standard provides an 

opportunity for firms to take strategic actions that have the potential to reduce compensation 

expense and improve the effectiveness of their EBCPs. These actions include determining 

the most appropriate: 

• Valuation models to use; 

• Data and methods for estimating model inputs and measures of employees’ exercise 

and post-vesting termination behavior (calibration measures); and 

• Instruments firms can use to accomplish their human resources goals (attract and 

retain employees, align employee and shareholder interests and reduce dilution) and 

to reduce compensation expense. 1 

The report is not intended to provide a comprehensive assessment of all aspects of FAS 

123R. Rather, it is intended to focus on the key implementation challenges firm’s are 

expected to face and those aspects of FAS 123R that are expected to have the greatest 

impact on the cost and effectiveness of EBCPs. 

1. Report’s Main Findings 

The report’s main findings are: 

1 

• Initially, most firms will be able to use fairly simple methods to estimate the inputs 

required for the modified Black-Scholes (MBS) model. However, the SEC’s simple 

method for estimating Expected Term (ET) will sunset at the end of 2007. Also, this 

method and methods that produce unbiased estimates based on a firm’s transaction 

data are expected to produce larger estimates of both ET and fair value than those 

produced by the common practice of taking the weighted average of observed 

settlement times. 

• The report discusses challenges that firms are expected to face when attempting to 

comply with the FAS 123R input requirements and recommends strategies for 

dealing with them. 

• Lattice-based models are shown to produce more accurate and generally lower fair 

value estimates than the MBS model for both traditional employee stock options 

(ESOs) and the nontraditional instruments analyzed in this report. 

• The report discusses situations where the MBS model either can’t be used or should 

not be used (e.g., to determine the fair value of a Maximum Value Options or to 

determine the impact on fair value of changing an instrument’s features) to perform 

valuations. 

The nontraditional instruments include premium, capped, indexed and performance-based options with either 

performance or market conditions as well as both standard and performance-based versions of restricted stock, 

restricted stock units and stock appreciation rights. The features include an instrument’s contractual term, type and 

length of the vesting schedule, vesting frequency (e.g., annual or monthly) and strike price. 

1

• Using a lattice model that allows inputs, especially stock price volatility, to vary with 

time is shown to produce more accurate and, under today’s conditions, substantially 

lower fair value estimates than would result if the inputs were held constant. 

• Changing the features of traditional ESOs can enable firms to better accomplish their 

HR objectives and, for both types of models, to produce lower fair value estimates 

than are were produced under the base case. 

• Nontraditional instruments have the potential to enable firms to better accomplish 

their HR goals and to reduce compensation expense compared to traditional ESOs. 

For example, stock settled stock appreciation rights are preferred by employees, 

reduce dilution and have the same fair value as traditional ESOs. Lastly, for both 

types of models the nontraditional options analyzed were shown to produce lower fair 

value estimates than traditional ESOs. 

The following pages provide added detail about the report’s main findings. 

2. Selection of an Appropriate Valuation Model 

The report discusses the key differences between, as well as, the pros and cons of the two 

most often used types of valuation models—the modified MBS and the binomial lattice 

(lattice) models. 2 It also compares the fair value estimates produced by both types of models 

for a traditional employee stock option (ESO). 3 The report shows (see Figure 1) that the fair 

values produced by the MBS model are 20% to 40% (average 27%) greater than those 

produced by the lattice model for a three-year graded vested option. 4 For comparability the 

Expected Term (ET) output by the lattice model is used in the MBS model. 5 

2 The MBS model is the traditional Black-Scholes model with the instrument’s expected term substituted for its 

contractual term. 

3 All of the valuations shown in this report are based on MBS-based and lattice-based models that are specifically 

designed to reflect the features of the traditional employee stock options and nontraditional instruments being valued. 

4 The valuations are based on model inputs that are similar to those used in Illustration 4 of FAS 123R. 

5 The calibration measures, such as expected term, that are output by the lattice model are based on what are known 

as “risk-neutral” probabilities (i.e., the probability of the stock price increasing depends on the risk-free rate). For 

completeness we also provide estimates of expected term and other calibration measures based on risk-adjusted 

probabilities (i.e., the probability of a stock price increase depends on the risk-adjusted return). Using risk-adjusted 

probabilities to estimate expected term, the fair values produced by the MBS model are roughly 15% to 30% (average 

22%) greater than those produced by the lattice model. 

2

$16.00 

$14.00 

$12.00 

$10.00 

$8.00 

$6.00 

$4.00 

$2.00 

$0.00 

Lattice 

Figure 1. 

Fair Values for Three-Year Graded Vested Options 

Lattice Model vs. Modified Black-Scholes Model 

Modified Black-Scholes 

38.84% 

$9.21 

$13.19 25.98% 

$11.07 

$13.94 

18.16% 

$12.46 

Tranche 1 Tranche 2 Tranche 3 

$14.72 

In FAS 123R firms are asked to consider the effect on fair value of expected changes in 

model inputs. By allowing the risk-free rate and volatility to vary with time, we show that the 

fair value estimates are more accurate and 13% lower than those that result from holding 

these inputs constant. The fair value produced by the MBS model (with inputs based on the 

averages of the time-varying values for the risk-free rate and volatility) are 40% greater than 

those produced by the lattice model when these two inputs are allowed to vary with time. 

3. Estimation of Model Inputs and Measures of Employees’ Exercise and 

Post-Vesting Employment Termination Behavior 

Following are the key challenges facing firms with respect to the FAS 123R input 

requirements: 

• 

• 

Inputs are to be “forward-looking” and reflect expected changes during the 

instrument's contractual term (lattice model) or ET (MBS model); 

When estimating volatility, firms are to consider “mean reversion” 6 and the implied 

volatility method; 7 

6 

As discussed in Footnote 58 of FAS 123R, mean reversion is the tendency of volatility to converge to some long-term 

equilibrium value. 

7 

With this method, volatility is inferred from the market price of traded instruments. 

3

• 

• 

• 

Firms are required to estimate the number of options expected to vest. Under the old 

standard, firms could assume that all of the options would vest and were then “trued 

up” based on actual experience; 

When estimating ET, firms are required to aggregate awards into relatively 

homogeneous groups with respect to exercise and post-vesting termination behavior 

for all types of valuation models; and 

When estimating ET, firms are to consider the effect of factors such as length of the 

vesting period, volatility, blackout dates, path of the firm’s stock price and employee 

characteristics in order to reflect differences between historical and expected future 

conditions. 

In Staff Accounting Bulletin 107 (SAB 107), the SEC staff provided additional guidance that 

has the potential to simplify the estimation of some of the key model inputs, at least initially. 8 

SAB 107 provides a simplified method that firms can use to compute ET for “plain vanilla” 

options. Also, SAB 107 provides guidance concerning when firms are allowed to rely 

exclusively on either the implied or historical methods for estimating volatility. The 

implications of the FAS 123R standard and additional guidance given by SAB 107 are 

discussed below. 

a. SAB 107 simplified method for estimating ET 

The estimation of expected term will be greatly simplified for firms that are able to use this 

method. However, the report recommends that firms carefully assess the pros and cons of 

using this method compared to methods that utilize a firm’s transaction data. 

b. Estimating volatility 

The process of estimating volatility will be greatly simplified for firms that are able to place 

exclusive reliance on either the implied method or the historical method. In the event that a 

firm is either unable, chooses not to place exclusive reliance on either of these methods, or 

there is a considerable difference between the estimates of short-term and long-term 

volatility, then we recommend that the firm combine the estimates produced by these two 

methods. The report recommends the use of a statistical method for estimating the 

convergence or mean reversion from short-term volatility (based on implied volatility) to longterm 

volatility (based on the historical method). 

c. Estimating pre- and post-vesting terminations 

Under the new standard, firms are required to estimate the number of options expected to 

vest. A key input to this calculation is the pre-vesting departure rate or turnover rate. The 

report discusses both simple and complex methods that can be used to estimate the 

departure rate. The more complex methods are able to reflect the influence of factors, such 

as the level of the firm’s stock price on a firm’s departure rate. The termination rate is also 

used to reflect the effect of post-vesting termination behavior on fair value. 

8 SEC Staff, Staff Accounting Bulletin No. 107, dated March 29, 2005. 

4

d. Estimating ET 

The report discusses reasons why developing unbiased estimates of ET, based on a firm’s 

transaction data, is expected to be difficult and proposes methods for overcoming these 

difficulties. Fairly detailed data and sophisticated estimation methods are expected to be 

required to 1) aggregate data into relatively homogeneous groups, 2) deal with the potential 

bias due to censoring (incomplete life cycle data as of the evaluation date), and 3 control for 

differences between the historical values of key explanatory variables (e.g., length of the 

vesting period, volatility, blackout dates, path of the firm’s stock price and employee 

characteristics) affecting ET and the values of these variables that are used for valuation 

purposes. 

e. Estimating measures of employees’ exercise behavior required by 

lattice models 

In the following report, we also discuss methods that can be used to estimate measures of 

employees’ exercise behavior (e.g., expected time-to-exercise, probability of exercise and 

ratio of the stock price at exercise to the strike price) that are required to calibrate lattice 

models. For the most part, these methods are similar to those that are recommended for 

estimating ET. 

4. Strategies for Optimizing Performance under FAS 123R 

a. Evaluation of impact of changing ESO features 

By changing the features of an ESO (e.g., contractual term, strike price, vesting schedule 

and attribution method), the report discusses ways that firms may be able to better 

accomplish their HR objectives and reduce compensation expense. Using a lattice model, 

we show that reducing the contractual term by three years, reducing the length of the graded 

vesting schedule by one year, increasing the strike price by $3.50 and increasing the vesting 

frequency from annual to monthly reduces the fair value produced by the lattice model by 

roughly 3%, 7%, 9% and 13%, respectively. The MBS model shows directionally similar, but 

generally smaller reductions than the lattice model (See Figure 2). Reducing the option’s 

contractual term is the only exception. The MBS model shows 7.7% reduction in fair value 

compared to a 2.5% reduction in fair value for the lattice model. This occurs because the 

reduction in contractual term is predicted by the lattice model to produce a much greater 

reduction in ET (16%) than in fair value (2.5%). The greater reduction in ET results in a 

greater reduction in fair value predicted by the MBS than that predicted by the lattice model. 

5

$14.00 

$12.00 

$10.00 

$8.00 

$6.00 

$4.00 

$2.00 

$0.00 

$10.91 

Base 

-2.50% 

$10.64 

Reduce 

Contractual Term 

Reduce Vesting 

Schedule Length 

Figure 2. 

Effect on Fair Values of Changing the Features 

of a Traditional Employee Stock Option 


Lattice Model Modified Black-Scholes Model 

-7.20% 

$10.14 

-8.80% 

$9.95 

Increase Strike 

Price 

-12.70% 

$9.53 

Increase Vesting 

Frequency 

$13.19 

Base 

-7.70% 

$12.17 

Reduce 


-4.20% 

b. Evaluation of potential benefits of using nontraditional instruments 

$12.63 

Reduce Vesting 

Schedule Length 

-4.40% 

Firms are expected to make greater use of nontraditional instruments because all 

instruments will be treated the same under the new standard and nontraditional instruments 

have the potential to allow firms to better accomplish their EBCP goals and reduce 

compensation expense. In this report, we discuss the objectives, strengths and weaknesses 

and provide fair value estimates for the types of nontraditional instruments expected to be 

most widely used by firms. We also show that for both types of valuation models, fair values 

are lower for the nontraditional options analyzed than for ESOs (see Figure 3). 

As shown in Figure 4, the MBS-based models generally produce higher fair value estimates 

than lattice-based models. For example, for premium, purchased and indexed options, the 

MBS-based models produce fair value estimates that are 27%, 31% and 47% greater than 

those produced by the lattice-based models. The sole exception is the Maximum Value 

Option (MVO), where the MBS-based model produces a lower fair value estimate than the 

lattice-based model. This understatement occurs because the MBS-based model 

understates the cash flows and overstates the discounting associated with an MVO. 

$12.61 

Increase Strike 

Price 

-6.90% 

6 

$12.28 

Increase Vesting 

Frequency

$14.00 

$12.00 

$10.00 

$8.00 

$6.00 

$4.00 

$2.00 

$0.00 

Figure 3. 

Fair Values for Various Non-Traditional Instruments Compared to a Traditional ESO 


$10.91 

Traditional ESO 

-8.80% 

$9.95 

Premium Option 

$9.88 

Purchased Option 

Lattice Model Modified Black-Scholes Model 

-9.44% 

-15.72% 

$9.19 

Maximum Value Option 

-43.63% 

-46.01% 

$6.15 

Indexed Option 

$5.89 

Market-Based Option 

$13.19 

Traditional ESO 

-4.40% 

$12.61 

Premium Option 

-1.59% 

$12.98 

Purchased Option 

-42.42% 

$7.59 

Maximum Value Option 

-31.69% 

$9.01 

Indexed Option 

NA 

Market-Based Option 

7

$14.00 

$12.00 

$10.00 

$8.00 

$6.00 

$4.00 

$2.00 

$0.00 

20.90% 

$10.91 

$13.19 

Figure 4. 

Fair Values for Various Non-Traditional Instruments 


27.00% 

$9.95 

$12.61 

$9.20 

$7.60 

Traditional ESO Premium Option Maximum Value 

Option 

-17.40% 

31.00% 

$9.88 

$12.98 

47.00% 

$6.15 

Lattice 

Modified Black-Scholes 

$9.01 

$5.89 

Purchased Option Indexed Market-Based Option 

This report shows how a lattice model can be used to determine exchange ratios that will 

make employees indifferent between a traditional ESO and a particular nontraditional 

instrument. It provides an example that demonstrates how to compute the exchange ratio 

that will make employees indifferent between restricted stock and a traditional ESO. The 

example demonstrates that even though the fair value of the restricted stock is greater than 

that of a traditional ESO, total compensation expense is less, because fewer shares of stock 

are required to make employees indifferent toward the two instruments. 

8

I. Introduction 

This report discusses and evaluates strategic actions that firms may wish to take both to 

comply with FAS 123R and to optimize the performance of their equity-based compensation 

programs (EBCPs) under the new standard. The new standard provides an opportunity for 

firms to take strategic actions that have the potential to reduce compensation expense and 

improve the effectiveness of their EBCPs. These actions include determining the most 

appropriate: 

• Valuation models to use; 

• Data and methods for estimating model inputs and measures of employees’ exercise 

and post-vesting termination behavior (calibration measures); and 

• Instruments and features firms can use to accomplish their human resources goals 

(attraction, retention and alignment of employee and shareholder interests) and to 

reduce compensation expense. 

The instruments include traditional, capped, indexed and performance-based options with 

either performance or market conditions as well as both standard and performance-based 

versions of restricted stock, restricted stock units and stock appreciation rights. The features 

include the instrument’s contractual term, type and length of the vesting schedule, vesting 

frequency (e.g., annual or monthly) and strike price. 

The report is not intended to provide a comprehensive assessment of all aspects of FAS 

123R. Rather, it is intended to focus on the key implementation challenges firm’s are 

expected to face and those aspects of FAS 123R that are expected to have the greatest 

impact on the cost and effectiveness of EBCPs. 

Section II discusses key issues related to the selection of an appropriate valuation model. 

The discussion focuses on the key differences between as well as and the pros and cons of 

the two most often used types of valuation models – the Modified Black-Scholes (MBS) 

model and binomial lattice (lattice) model. Section II and Section IV also provide fair value 

estimates for both traditional employee stock options and nontraditional instruments based 

on MBS-based and lattice-based models. As required in FAS 123R these models are 

specifically designed to reflect the substantive features of the instruments being valued. 9 

Using assumptions that are similar to those used in FAS 123R, the report shows that latticebased 

models generally produce more accurate and lower fair value estimates than the MBS 

model for both traditional ESO and nontraditional instruments. 

Section III discusses the pros and cons of both simple and complex methods that firms can 

employ to estimate model inputs and measures of employees’ exercise and post-vesting 

termination behavior required for both types of models. Finally, Section IV discusses and 

evaluates changes to the features of ESOs and discusses and evaluates the pro and cons of 

various nontraditional instruments. It shows that the nontraditional instruments are expected 

to enable firms to better accomplish their HR goals and to reduce compensation expense. 

This section also discusses method for estimating the exchange ratios that are designed to 

make employees indifferent between a particular nontraditional ESO and traditional ESOs. 

9 The valuations discussed in this report are based on both lattice- and MBS-based models that were specifically 

designed to value both the traditional and nontraditional instruments discussed above. The lattice model and the 

methods used to estimate its inputs recently passed an audit by a Big Four accounting firm under the new FAS 123R 

standard. As part of the audit process, we prepared comparison of the results produced by our model compared to 

the best known models in the literature. These results, which are available upon request, show that our lattice model 

produces valuations and measures of employees’ exercise and post-vesting termination behavior that are either 

identical to or are close to those produced by the models in the literature. 

9

The paper has two appendices. The first appendix discusses the changes to the traditional 

lattice model that are required to comply with the new standard. It also discusses the pros 

and cons of the various lattice models that have been developed to comply with the new 

standard. 10 The second appendix discusses the lattice and Black-Scholes-based models 

that were used in the report. 

10 Analysis Group, Inc. has developed and evaluated the three types of lattice models discussed in Appendix A of this 

report. We have concluded that the first model (Generalized Version of the Traditional Lattice Model) provides the 

most flexible and accurate framework for valuing both traditional and nontraditional instruments. 

10

II. Selection of an Appropriate Valuation Model 

One of the major challenges confronting firms under FAS 123R is the selection of 

appropriate valuation models. 

A. Requirements for Selecting a Valuation Model 

In the absence of market-based instruments, firms are responsible for selecting valuation 

models that comply with the FAS 123R requirements. Under FAS 123R, firms are allowed to 

use different types of models for different types of instruments. However, the model(s) that 

are selected must comply with the measurement objectives listed in Paragraph 8 of FAS 

123R. This paragraph requires that the valuation model: 

• 

• 

• 

• 

• 

• 

• 

• 

• 

Is applied in a manner that is consistent with FASB’s fair value measurement 

objectives (which require firms to estimate, as of the grant date, the fair value of the 

equity instruments that the entity is required to issue when the employees have 

rendered the requisite service and satisfied any other conditions required to benefit 

from the instrument); 

Is based on generally accepted economic and financial theory; and 

Reflects the substantive characteristics of the instrument being valued. 

In addition to meeting these requirements, firms are required to use valuation models that, at 

a minimum, incorporate the following inputs: 

Exercise price; 

Expected term of the instrument; 

Grant date stock price; 

Expected volatility of the firm’s underlying stock price; 

Expected dividends on the underlying shares; and 

The risk-free rate for either the expected term of the award (closed-form model) or 

contractual term of the award (lattice model). 

It should be noted that under FAS 123R firms are allowed to change the type of valuation 

model they initially select if “…a different technique is likely to result in a better estimate of 

fair value.” 

Although neither FAS 123R nor SAB 107 state a preference for any particular type of model, 

they do discuss advantages that they believe lattice-based models have over the MBS 

model. For example, when comparing the two types of models, Paragraph 15 of FAS 123R 

states: 

“A lattice model can be designed to accommodate dynamic assumptions of 

expected volatility and dividends over the option’s contractual term, and 

estimates of expected option exercise patterns during the option’s contractual 

term, including blackout periods. Therefore, the design of a lattice model more 

fully reflects the substantive characteristics of a particular employee share option 

or similar instrument.” 

Also, SAB 107 states that for certain instruments, the MBS model may not be able to satisfy 

the third FAS 123R measurement requirement, because it is not designed to reflect certain 

11

characteristics of the instrument. SAB 107 gives as an example that if exercise depends on 

a specific increase in the price of the underlying shares, then the MBS model would not 

generally be appropriate, because it is not designed to take market conditions into account. 

B. Pros and Cons to Consider When Evaluating MBS and Lattice Models 

Since most companies currently use the MBS model, their choice is expected to be either to 

continue using it or to switch to another type of model, such as a lattice model. This section 

discusses issues that should be considered when deciding to stay with or to switch to 

another type of model. 

The key difference between lattice and the MBS model is that lattice models are able to 

explicitly reflect the features of the instrument (e.g., vesting schedule, contractual term and 

blackout dates) being valued as well as employee exercise and post-vesting termination 

behavior. They are also able to accurately assess the effects of dynamic or time-varying 

inputs on fair value. With the MBS model, the effects of these factors can only be implicitly 

reflected by changing the instrument’s ET. Also, the MBS model can only use fixed or static 

inputs. As a result, it has limited ability to accurately reflect the effect of time-varying or 

dynamic inputs. 

Pros and cons of the MBS include: 

Pros: 

Cons: 

1. Both the firm and its auditor are familiar with this type of model; 

2. This type of model can usually be implemented by the firm’s own staff; 

3. The cost of implementing this type of model is generally less than that of a 

lattice model; 

4. The MBS model uses a single standard formula, which simplifies both 

valuation and the audit process. This is to be contrasted with the lattice 

model, where there are at least three different types of lattice models. These 

models differ primarily with respect to the methods used to reflect employees’ 

exercise and post-vesting termination behavior (see Appendix A for a 

description of these methods as well as the pros and cons of each); and 

5. With the MBS model, firms are potentially able to use simple methods for 

estimating ET and stock price volatility. 11 

1. As is discussed in more detail below, in the longer term firms using the MBS 

model are expected to be required to use detailed data and sophisticated 

methods to estimate ET; 

2. This model is generally not appropriate for certain types of instruments (e.g., 

options with caps or market-based performance conditions); 

11 Firms using the MBS model will be able to use the simplified method developed by the SEC staff for estimating ET 

until December 31, 2007 and may be able to place exclusive reliance on either the implied volatility method or the 

historical method when estimating stock price volatility. 

12

3. The MBS model is not able to accurately reflect the effect of time-varying or 

dynamic inputs; and 

4. As shown in this report, the MBS-based model tends to produce less 

accurate and generally higher fair value estimates than lattice models. 

Pros and cons of a lattice model include: 

Pros: 

Cons: 

1. Lattice models are viewed by both FASB (in FAS 123R) and the SEC (in SAB 

107) as being better able to reflect the important features of the instrument 

being valued and the impact of dynamic or time-varying inputs on fair value 

than the MBS model; 

2. Lattice models are able to accurately value most, if not all, equity-based 

instruments, including complex instruments with caps or market conditions; 

3. Lattice models are able to accurately value instruments for which model 

inputs are expected to experience significant changes during the instrument’s 

contractual term; 

4. As shown later in this report, using typical inputs, a lattice model is generally 

more accurate and provides lower estimates of fair value for both traditional 

ESOs and nontraditional instruments than the MBS model; and 

5. Lattice models can be calibrated to measures of exercise and post-vesting 

termination behavior that are generally easier to estimate (e.g., expected 

time-to-exercise and probability exercise) than ET (the measure of exercise 

and termination behavior required for the MBS model). 

1. Initial audits are expected to be more involved for a lattice model than for the 

MBS model; 

2. Lattice models are more costly to implement than the MBS model; and 

3. Lattice models generally required a greater level of expertise than the MBS 

model. As a result, firms may be required to use outside experts. 

C. Comparative Evaluation of Fair Value Estimates Produced by Lattice and 

Closed-Form Models 

This section provides numerical evaluations of the lattice and MBS models. Table 1 shows 

the inputs and calibration measures used. 

13

Table 1. 

Model Inputs 

Parameter Value 

Stock price $30 

Strike price $30 

Option duration 10 years 

Volatility 50 percent 

Dividend yield 1.0 percent 

Risk-free rate 4.1 percent 

Annual departure rate 3.0 percent 

Vesting period 3.0 years 

Expected time-to-exercise 4.1 years 

The inputs are similar to those used in Illustration 4 in FAS 123R. 12 However, instead of 

calibrating the lattice model to a sub-optimal exercise factor or Exercise Multiple (EM) of 2.0, 

we calibrated it to an estimate of Expected Time-To-Exercise (ETTE) of 4.1 years. The 4.1 

year value was based on the results of a statistical model that was estimated from company 

data. The model is designed to provide forward looking estimates of ETTE that reflect the 

influence of the factors shown in Table 1 (e.g., volatility, departure rate, contractual term and 

length of the vesting period). 

We used ETTE rather then EM as a measure of employees’ exercise behavior, because this 

measure is generally easier to estimate accurately than EM. Also, we used a slightly 

different risk-free rate than is used in Illustration 4. The risk-free rate in Table 1 is the 

average of the time-varying risk-free rates that are used in Section II.E of the report. 

However, changing the risk-free rate from the average value used in Illustration 4 of 2.9% to 

the value shown in Table 1 has virtually no impact on the values of the calibration measures. 

Table 2 shows the fair value estimates and the calibration measures produced by the MBS 

and lattice models for a three-year graded vested option. For comparability the estimates of 

ET that are output by the lattice model are used as inputs to the MBS model. The method 

we used to calculate ET is similar to the method recommended in Paragraph A27 of FAS 

123R. We use a three-year graded vested option, because, in essence, it also provides 

valuations for one-year, two-year and three-year cliff vested options. 

12 In his comments, one of the reviewers of a prior draft stated that the inputs were not typical of those generally found in 

practice. In an attempt to deal with this concern, we have selected inputs that are similar to those used in FAS 123R. 

It should be noted that changing the inputs did not change our prior conclusions. 

14

Tranche 

Lattice 

Model 

Table 2. 

Valuations Based on Risk-Neutral Probabilities 


Modified 

Black- 

Scholes 

Model 

Percent 

Difference 

Expected 

Time-To- 

Exercise 

Expected 

Term 

Exercise 

Multiple 

1 $9.21 $12.79 38.84% 2.28 4.52 1.48 

2 $11.07 $13.94 25.98% 3.23 5.56 1.67 

3 $12.46 $14.72 18.16% 4.10 6.38 1.90 

Average $10.91 $13.82 27.66% 3.20 5.49 1.68 

Table 2 shows that the values produced by the MBS model are 18% to 39% greater than 

those produced by the lattice model, with an average difference of 28%. Notice that the 

model is correctly calibrated, because the ETTE for the third tranche equals the required 

value of 4.10 years. 

Notice also that increasing the length of the vesting period, increases fair value for both 

types of models. It occurs for a lattice model because increasing the length of the vesting 

period prevents risk-averse employees from exercising their options as early as they would 

like. Delaying exercise will increase the time value of the option (opportunity for the stock 

price to increase) and thus the option’s fair value. For the MBS model, increasing the length 

of the vesting period increases the option’s expected term, which in turn, increases the 

option’s fair value. 

D. Should the Calibration Measures Be Based on Risk-Neutral or Risk- 

Adjusted Probabilities? 

There is an unsettled debate as to the correct probability measure to be used to compute the 

calibration measures, especially ET. Two types of probability measures have been 

advocated in the literature: risk-neutral and risk-adjusted. The risk-neutral probability 

measure is the one typically used to value instruments. This measure assumes that a firm’s 

stock price will increase at the risk-free rate. The risk-adjusted probability measure assumes 

that a firm’s stock price will increase at the risk adjusted rate. The risk-adjusted rate includes 

a premium above the risk-free rate for the additional risk associated with holding a firm’s 

stock. 

Most of the academic literature has used risk-neutral probabilities to compute various 

calibration measures. 13 According to Mark Rubinstein, it is also the method that practitioners 

use to compute ET. 14 FASB has also advocated the use of this probability measure. In FAS 

123R, firms that use lattice models are required to output expected term. Paragraph A27 of 

FAS 123R discusses a method, based on risk-neutral probabilities, for computing ET. Also 

Paragraph 282 of FAS 123, FASB discusses a method, which is also based on risk-neutral 

13 See for example J. Hull and A. White; M. Rubinstein; J. Ingersoll; S. Huddard; and M. Garman. 

14 He states: “…we can use a binomial tree to calculate the (risk-neutral) expected life of the option, known in the trade 

as the option ’fugit.’” 

15

probabilities, that firms can use to compute expected option life indirectly (using a lattice 

model) instead of directly using transaction data. 

Table 3 provides the same information shown in Table 2 using risk-adjusted probabilities to 

compute ET and the other calibration measures. 

Tranche Lattice 

Model 

Table 3. 

Valuations Based on Risk-Adjusted Probabilities 


MBS 

Model 

Percent 

Difference 

Expected 

Time-to 

Exercise 

Expected 

Term 

Exercise 

Multiple 

1 $9.21 $12.07 30.99% 2.30 3.96 1.50 

2 $11.07 $13.30 20.33% 3.23 4.96 1.72 

3 $12.46 $14.19 15.03% 4.10 5.81 1.97 

Average $10.91 $13.19 22.12% 3.21 4.91 1.73 

A comparison of Table 2 and Table 3 shows that the use of risk-adjusted probabilities does 

not affect the fair value produced by the lattice model and has only a minor effect on ETTE 

and EM. However, it does cause ET to decrease. The decrease in ET causes the fair value 

produced by the Black-Scholes model to also decrease. Table 3 shows that if risk-adjusted 

probabilities are used, the fair values produced by the MBS model are 15% to 31% greater 

than those produced by the lattice model, with an average difference of 22% (compared to 

18% to 39% for risk-neutral probabilities). To be conservative, the remaining valuations 

shown in this report are based on risk-adjusted probabilities. 15 

It should be noted that for the inputs in Table 1, calibrating the model to an ETTE of 4.1 

years is equivalent to calibrating the model to an EM of 1.97, which is virtually identical to the 

EM of 2.0 used in Illustration 4. 16 However, the fair value estimates produced by the lattice 

model are not directly comparable with the fair value estimate shown in Illustration 4 of FAS 

123R for two reasons. First, the fair value estimate in Illustration 4 does not reflect postvesting 

employment termination behavior, which, as required in FAS 123R, is reflected in our 

analysis. 17 

Second, the model used in Illustration 4 assumes that, for vested options, exercise occurs 

whenever the stock price equals or exceeds twice the strike price. In essence, the model 

15 One of the reviewers that provided comments on a prior draft contended that using risk-adjusted probabilities would 

cause the difference between the results produced by the two types of models (MBS and lattice models) to essentially 

disappear. As can be seen from Table 2 this is not the case. As will be shown later, the difference between the fair 

value estimates produced by two types of models will increase even further when the model inputs are allowed to 

change during the instrument’s contractual term. 

16 It should be noted that the lattice model used to perform the valuations is able compute the joint probability of either 

exercise or termination occurring at each node. As a result, it is able to output virtually any measure of employee 

exercise and post-vesting termination behavior, including ET, expected time-to-exercise, pre- and post-vesting 

cancellation rates and the probabilities of exercise. 

17 We were able to match the $14.69 figure shown in Illustration 4 by using a trinomial model that is able to place the 

layers of the lattice so that one of them equals the barrier where exercise is assumed to occur ($60) and by assuming 

that the risk free rate and the volatility are the averages of the values shown for each of these inputs in the illustration 

and the post-vesting termination rate is zero. Using a post-vesting termination rate equal to the forfeiture rate of 3.0%, 

shown in the illustration, produces a lower fair value estimate than the $14.69 shown in the illustration. 

16

assumes that the exercise boundary is horizontal. It is well known that the exercise 

boundary (see Hall and Murphy, 2002) monotonically declines as one approaches the 

instrument’s expiration date. Consistent with the literature, our lattice model assumes that 

exercise occurs whenever the stock price equals or exceeds a monotonically declining 

exercise boundary. The exercise boundary is placed so that the calibration measures output 

by the model match those estimated from a firm’s historical data. 

E. The Effect of Time-Varying Inputs 

In FAS 123R firms are asked to consider the effect of expected changes in the key model 

inputs on fair value. This section evaluates the potential impact of allowing inputs to change 

during an instrument’s contractual term. More specifically, we consider the effect of allowing 

the risk-free rate and the stock price volatility to vary with time. We forecast the term 

structure of both of these inputs by using company data and well-known estimation 

techniques. The risk-free rate is forecast by estimating forward rates using the “bootstrap” 

method (see Hull, 2003). When estimating the future path of volatility we estimate short-term 

volatility using the implied volatility method and long-term volatility using the historical 

volatility method. These methods are approved by both FASB and the SEC. Short term 

volatility is estimated to be roughly 35% and long-term volatility is estimated to be slightly in 

excess of 50%. 

We then estimate the rate of convergence or mean reversion from short-term to long-term 

volatility by using the variance targeting technique discussed in Chapter 17 of Hull, 2003. 18 

The method shows that volatility will converge or mean revert from short- to long-term 

volatility in about four to five years from the grant date. 19 For comparability, the MBS model 

is based on the averages of the time-varying values for the risk-free rate and stock price 

volatility. This is also the approach that is recommended in the literature when using the 

Black-Scholes model to value instruments with inputs that vary with time (see Wilmott, 1998, 

p. 121). As shown in Table 1, the resulting average values for the risk-free rate and stock 

price volatility are 4.1% and 50% respectively. 

In most cases, allowing the inputs to a lattice model to vary with time is straight forward. The 

sole exception is volatility. Allowing volatility to vary with time prevents a typical lattice model 

from “recombining.” That is, an “up” move followed by a “down” will not end up at the same 

position as a “down” move followed by an “up” move. When a lattice fails to recombine, the 

number of nodes that must be evaluated at each time step increases exponentially, instead 

of linearly, as in the case of a recombining tree. This prevents typical instruments from being 

accurately valued in a reasonable period of time. 20 The model used in this report is specially 

designed to accurately value instruments with time-varying volatility in a reasonable period of 

time. 

18 It should be noted that the variance targeting (VT) approach is a variant of the GARH method. However, unlike the 

GARCH method that estimates long-term volatility, the VT approach merely estimates the rate of convergence or 

mean reversion from short- to long-term volatility; where short- and long-term volatility can be estimated based on 

methods that have been approved by both the SEC and FASB. The VT approach we recommend in this report has 

successfully passed an audit by a Big Four accounting firm. 

19 This conclusion contradicts the statement by one of the reviewers of a prior draft of this report that “the volatility curve 

tends to be flat over most of the term of an option, only getting steep towards the end of the term when (a) most 

employees have probably already exercised and (b) the effect of discounting over the long term of the option both 

serve to mitigate the effect.” 

20 A valuation problem will typically have 300 or more time steps (e.g., ten years and 30 steps per year). If this is the 

case, then the maximum number of nodes that must be valued for a recombining lattice is 301. For a nonrecombining 

lattice, the maximum number of nodes that must be valued is two multiplied by ten to the 90th power. 

This number is so large that even the fastest computer can’t solve the problem in a reasonable period of time. 

17

Table 4 shows the average fair value estimates (across the three tranches) for the base 

case where the inputs are held constant and the cases where the risk-free rate and stock 

price volatility are allowed to change during the instrument’s contractual term. 

Inputs 

Table 4. 

Average Fair Value Estimates Based on Time-Varying Inputs 

Lattice 

Model 

Percent 

Difference 

from 

Constant 

Inputs 

Modified 

Black- 

Scholes 

Model 

Percent 

Difference 

from 

Constant 

Inputs 

Constant Inputs $10.91 NA $13.19 NA 

Risk-Free Rate Varies $10.80 1.02% $13.19 0% 

Risk-Free Rate and Volatility Vary $9.49 13.0% $13.19 0% 

Allowing the risk-free rate to vary with time causes the fair value produced by the lattice 

model to decline by about 1.0% (from $10.91 to $10.80). Allowing both the risk-free rate and 

volatility to vary with time causes the fair values produced by the lattice model to decline by 

13%. However, the fair value produced by the MBS model does not change, because it is 

based on the constant inputs, which are derived from the averages of the time-varying 

values for the risk-free rate and volatility. Lastly, the fair value produced by the MBS model 

with constant inputs is 40% ((13.19/9.49-1)*100) greater than the fair value produced by the 

lattice model with time-varying inputs (for both the risk-free rate and volatility). 

18

III. Key Issues Concerning the Estimation of Model Inputs and Measures 

of Employees’ Exercise and Post-Vesting Employment Termination 

Behavior 

The following section discusses the key challenges facing firms with respect to the FAS 

123R input requirements as well as the additional guidance given in SAB 107. 

A. FAS 123R Input Requirements 

Below are key FAS 123R input requirements: 

• 

• 

• 

• 

• 

Inputs are to be “forward-looking” and reflect expected changes during the 

instrument's contractual life (lattice model) or ET (closed-form model); 

When estimating volatility, firms are to consider “mean reversion” 21 and the implied 

volatility method; 22 

Firms are required to estimate the number of options expected to vest. Under the old 

standard, firms could assume that all of the options would vest and then be “trued 

up” based on actual experience; 

When estimating ET, firms are required to aggregate awards into relatively 

homogeneous groups with respect to exercise and post-vesting termination behavior 

for all types of models; and 

Firms are to consider the effect of factors such as length of the vesting period, 

volatility, blackout dates, path of the firm’s stock price and employee characteristics 

when estimating ET. 

B. Additional Guidance in SAB 107 23 

The additional guidance in SAB 107 has the potential to simplify the estimation of the key 

model inputs, at least initially. SAB 107 provides a simplified method that firms can use to 

compute ET for “plain vanilla” options. SAB 107 also provides guidance concerning when 

firms are allowed to place exclusive reliance on either the implied volatility method or the 

historical method. Firms are allowed to rely exclusively on implied volatility if: 

• The company’s valuation model is based on a constant volatility assumption (e.g., 

Black-Scholes model); 

• The market prices of both the traded options and the underlying stock are measured 

at similar points in time and the dates are reasonably close to the grant date; 

• The traded options are both “near-the-money” and close to the exercise price of the 

ESO; 

• The maturities (new or remaining) of the traded options are at least one year; and 

• The options are actively traded. 

Firms can place exclusive reliance on the historical method if: 1) there is no reason to 

assume that volatility in the future will differ from what it has been in the past, and 2) 

21 As discussed in Footnote 58 of FAS 123R, mean reversion is the tendency of volatility to converge to some long-term 

equilibrium value. The footnote states that statistical models have been developed that can reflect mean reversion. 

22 With this method, volatility is inferred from the market price of traded instruments. 

23 SEC Staff, Staff Accounting Bulletin No. 107, dated March 29, 2005. 

19

historical data covers a reasonable period of time (at least equal to the expected term of 

MBS-based models and the contractual term of lattice-based models). 

C. Potential Implications for Firms 

1. SAB 107 simplified method for estimating expected term 

The estimation of expected term will be greatly simplified for firms that are able to use this 

method. 24 However, the simplified method can only be used until December 31, 2007. 

Before adopting this method, it is recommended that firms carefully assess its pros and cons 

compared to approaches that utilize a firm’s own transaction data. The SEC method will 

overstate ET (and thus fair value) for firms using the MBS model if the ET, based on 

transaction data, is less than the midpoint between the average length of the vesting period 

and the option’s expiration date. 

2. Estimating volatility 

The process of estimating volatility will be greatly simplified for firms that are able to place 

exclusive reliance on either the implied method or the historical method. 25 In the event that a 

firm is unable or there is a significant difference between short- and long-term volatility, then 

we recommend that firms consider combining the estimates produced by the two methods. 

One way to do this would be to use statistical methods, based on a company’s stock price 

and dividend data, to estimate the rate of convergence or mean reversion from short-term 

volatility (based on implied volatility) to long-term volatility (based on the historical method). 26 

Under today’s conditions, where short-term volatility tends to be less than long-term volatility, 

using a lattice model with time-varying volatility can reduce fair value compared to holding it 

constant at the average of the values that volatility is expected to take during the 

instrument’s contractual term. 

3. Firms will need to estimate the number of options expected to 

vest 

Under the new standard, firms are required to estimate the number of options expected to 

vest. A key input to this calculation is the pre-vesting departure or turnover rate. The 

turnover rate can be estimated as the ratio of the number of employees departing each 

period to the number of employees at the beginning of the period. To produce more 

accurate departure rates rate estimates, firms can use statistical methods that are able to 

reflect the influence of factors, such as the level of the firm’s stock price, health of the 

industry and employee characteristics. 27 The termination rate is also a key measure of the 

effect of post-vesting termination behavior on fair value. 

24 

With the SEC method, ET is computed as the midpoint between the average length of the vesting period and the 

instrument’s expiration date. 

25 

If a firm uses a lattice model and there is significant difference between the implied and historical volatility estimates, 

then the firm should consider the use of year-by-year or time-varying volatility estimates, possibly along the lines 

discussed later on in the paragraph. This view is consistent with that of PWC (see Page 4-33 of their document: FAS 

123(R), Share-Based Payment-a multidisciplinary approach, May 2005. 

26 

See Hull 2003, Chapter 17. 

27 See Green (2003). 

20

4. Estimating ET 

Firms that either can’t, or decide not to use the SEC staff’s simplified method to estimate ET 

will need to acquire fairly detailed data and to use fairly sophisticated statistical techniques in 

order to obtain unbiased estimates of ET. ET is expected to be the most difficult measure of 

exercise and termination behavior to estimate for three reasons. First, this measure requires 

knowledge of all post-vesting events (exercises, cancellations and expirations) occurring 

from the end of the vesting period to the instrument’s contractual term. Second, expirations 

both have a significant effect on ET and are difficulty to estimate precisely, because they 

require data on an event that occurs more than ten years in the past. 

Third, the estimation of expected term is complicated because the transaction data on grants 

are typically incomplete. The use of simple weighted averages, based on both complete and 

incomplete grants, will cause the estimate of ET to be inefficient and to be biased downward. 

This occurs because data pertaining to options that are outstanding at the valuation date are 

not reflected in the analysis and more recent settlements receive too much weight. This 

problem is usually referred to in the statistical literature as “right censoring.” One way to 

obtain unbiased estimates is to use statistical techniques that are designed to deal with 

censoring. Alternatively, one could output ET from a lattice model that has been calibrated 

to historical data. 28 

Estimates based on either the SEC’s simplified method or methods that reflect censoring are 

expected to produce larger estimates of ET than are produced by computing the weighted 

average of observed terminations. This is illustrated on Pages 4-10 through 4-12 of the 

PWC document (PricewaterhouseCoopers, 2005), where estimates of ET based on 

observed settlements are between 3.32 years and 3.53 years; whereas estimates of ET that 

reflect censoring or, as PWC puts it, “partial lifecycle effects,” are between 5.49 years and 

5.59 years. 

5. Estimating measures of employees’ exercise behavior 

required by lattice models 

The measures that are typically used to reflect employees’ exercise behavior include 

expected time-to-exercise, probability of exercise and ratio of the stock price at exercise to 

the strike price. For the most part, all of the recommendations for estimating ET apply to the 

other measures of exercise behavior. In order to obtain unbiased estimates, firms will need 

to control for censoring and for the differences between the historical values and expected 

future values of the key explanatory variables (e.g. volatility, path of the stock price and 

employee charactrisitics) affecting the various exercise measures. 

28 

We have used a variant of the actuarial model described in Appendix A to use a lattice model to output ET. This 

method is about the only one possible unless a firm can rely on proxy data or has substantial historical transaction 

data. 

21

IV. Strategies for Optimizing the Performance of Equity-Based 

Compensation Programs under FAS 123R 

A. Potential Benefit from Changing the Features (e.g., Contractual Term, 

Strike Price, Vesting Schedule and Attribution Methods) of Traditional 

Employee Stock Options 

Given that traditional “at-the-money” ESOs will be expensed under FAS 123R, we 

recommend that firms take a fresh look at the design of these instruments. Ideallly, this 

would include an evaluation of the impact that changing the features of traditional ESOs will 

have on the firms HR goals and compensation expense. Table 5 summarizes the expected 

effect on fair value of making various changes to the features of ESOs. 

Table 5. 

Effect of Changing Option Features 

Option Features Proposed Change Impact on Fair Value 

Contractual Term Shorten Reduce 

Strike Price Increase Reduce 

Length of Vesting Period Decrease Reduce 

Increase Vesting Frequency Adopt Reduce 

Graded Vested Schedule Adopt Reduce 

Reducing the option’s contractual term will reduce fair value by reducing the length of time 

that the stock price is allowed to increase, but it also reduces the value of the instrument to 

employees. Increasing the exercise price increases the level of the stock price at which the 

option is “in-the-money,” which reduces both fair value and the value of the ESO to 

employees. However, it also provides a strong incentive for employees to increase the 

stock price. Decreasing the length of the vesting period will reduce fair value and the 

option’s retention value, but it increases the value of the option to employees. 

Table 6 shows the effect of modifying the base case shown in Table 1 by: 

• 

• 

• 

• 

Shortening the contractual term from ten years to seven years; 

Shortening the maximum length of the vesting period from three years to two years; 

Increasing the strike price from $30 to $33.50; and 

Increasing the vesting frequency from annual to monthly. 

22

Option Parameters 

Changed 

Table 6. 

Effect of Changing Option Features 

Lattice 

Fair 

Value 

Percent 

Difference 

from Base 

Case 

MBS 

Fair 

Value 

Percent 

Difference 

from Base 

Case 

Base Case $10.91 NA $13.19 NA 

Reduce Contractual Term $10.64 -2.5% $12.17 -7.7% 

Increase Strike Price $9.95 -8.8% $12.61 -4.4% 

Reduce Vesting Period $10.14 -7.1% $12.63 -4.2% 

Increase Vesting Frequency $9.53 -12.8% $12.28 -6.9% 

These changes reduce the fair value produced by the lattice model by roughly 3%, 9%, 7% 

and 13%, respectively. For the MBS model, making the same changes to the features of the 

ESO reduces fair value by roughly 8%, 4%, 4% and 7% respectively. 

In most instances, the reductions are greater for the lattice model than for the MBS model. 

The sole exception occurs when the option’s contractual term is reduced by three years. 

This reduces fair value by roughly 8% for the MBS model and only 2.5% for the lattice model. 

The MBS model shows a greater reduction in fair value because the 30% reduction in the 

option’s contractual term results in a predicted reduction of 16% in the ET. The reduction in 

ET, in turn, leads to a greater reduction in fair value predicted by the MBS-based model than 

that predicted by the lattice model. This analysis suggests that the MBS should not be relied 

upon to accurately assess the effect on fair value of incremental changes to the features of 

an ESO. 

The reduction in fair value associated with the increase in the vesting frequency may be 

surprising. The reason for the reduction is that instead of having one third of the options 

vesting at the end of each of three years (which occurs with an annual vesting frequency), 

1/36 of the options vest at the end of each month for 36 months (with a monthly frequency). 

As a consequence, each year, the majority (11/12) of the grant vest earlier under a monthly 

frequency than under an annual frequency. 

The percentages shown above are the separate influences of each option feature. The 

results would have been greater if more dramatic changes were made or the changes were 

made in combination. For example, if all of the changes were made in combination the 

reduction in fair value would have been roughly 27% for the lattice model and 21% for the 

MBS model. 

23

B. Impact of Changing Vesting Schedules and Attribution Methods 

Table 7 shows the total compensation expense associated with the two basic types of 

vesting schedules (cliff and graded) and attribution methods (FIN 28 and straight line), 

assuming 1000 options are granted. 

Table 7. 

Lattice Model-Based Compensation Expense 

Under Alternative Vesting Schedules and Attribution Methods 

Vesting Schedule and 

Attribution Method 

Timing of Compensation 

Expense 

2005 2006 2007 

Total 

Cost 

Percent 

Difference 

Graded Vesting - FIN 28 $6,299.49 $3,228.77 $1,384.53 $10,912.79 0.00% 

Graded Vesting - 

Straight Line $3,637.60 $3,637.60 $3,637.60 $10.912.79 0.00% 

Cliff Vesting $4,153.59 $4,153.59 $4,153.59 $12,460.78 14.19% 

The rows labeled “Graded Vesting - FIN 28” and shows total compensation expense based 

on the tranche-by-tranche calculation of compensation expense. That is, each tranche is 

treated as if it were a separate award. Tranche-by-tranche compensation expenses are 

computed by multiplying the values in Table 3 by the number of options allocated to each 

tranche (333.33 [1000/3]). In addition to the tranche-by-tranche method, FAS 123R 

(Footnote 86) also allows firms to compute total compensation expense by using a single 

weighted-average expected life to value the entire award. 29 

Under the FIN 28 method, total compensation expense is allocated on a straight line basis 

over each tranche’s vesting period (e.g., total compensation expense for tranche one is 

allocated over one year, total compensation expense for tranche two is allocated over a two 

year period etc.). With the “Straight Line” method, total compensation for all three tranches 

is allocated on a pro-rata basis over the maximum vesting period. . 

Total compensation expense is roughly 14% greater for a three-year cliff vested option than 

for a three-year graded vested option. This occurs because as shown in Table 3, fair value 

increases with increases in the length of the vesting period. Hence, total compensation 

expense for a three-year cliff vested option, which allocates 100% of the grant to the third 

tranche, will generally be greater than the total compensation expense for a three-year 

graded vested schedule, which allocates options to all three tranches. 30 For the MBS model, 

total compensation is 7.58% greater for the three-year cliff-vested option than for the threeyear 

graded vested option. 

29 

As shown in Paragraphs 303 and 304 or FAS 123, this method tends to result in a greater compensation expense 

than the tranche-by-tranche method. 

30 

The fair value for a three-year cliff vested option is the same as the third tranche of a three-year ratable graded vested 

option. The only difference between the third tranche of a three-year ratable graded vested schedule and a three-year 

cliff vested schedule is that a fraction of the grant is allocated to the third tranche with a three-year graded vested 

schedule, whereas 100% of the award is, in essence, allocated to the third tranche for a three-year cliff vested 

schedule. 

24

It should be noted that graded vesting does not always produce the lowest compensation 

expense. The exception can occur in situations where the departure rate is very high or the 

length of the vesting period is very long. In these situations, the number of options expected 

to vest (product of the number of options granted and the probability that the options vest) 

can become sufficiently small so that total compensation expense (the product of the number 

of options expected to vest and fair value) is lower for cliff vesting than for graded vesting. 

With respect to the allocation of total compensation expense, firms are allowed to make a 

one-time election as to the attribution method they will use for options that are subject to 

graded vesting. As stated in Footnote 85, “The choice of attribution method for awards with 

graded vesting schedules is a policy decision that is not dependent on an enterprise’s choice 

of valuation technique.” However, there are important nuances that must be considered 

when making this election. For example, the amount of compensation expense recognized 

as of a particular point in time must be at least as great as the vested portion of the award up 

to that point. Also, the straight line attribution method is only applicable to options with 

service conditions. As a consequence, the straight line method can not be used for options 

that are subject to either performance or market conditions. 

As shown in Table 7, compensation expense is the same for both the FIN 28 or “tranche-bytranche” 

method and the straight line attribution methods; however, the timing of 

compensation expense is front loaded for the FIN 28 method. Hence, total compensation 

expense will be greater, on a present value basis, for the FIN 28 attribution method. 

C. Impact of Using Nontraditional Instruments to Accomplish Attraction, 

Retention and Shareholder Alignment Goals 

1. The new standard is expected to create a more level playing field 

for non-option equity-based instruments 

Under FAS 123, nontraditional instruments were expensed and were generally subject to 

variable accounting. With the new standard, all equity-based instruments will be expensed 

and only liability instruments (e.g., cash settled stock appreciation rights) will be subject to 

variable accounting. As a result, it is expected that under the new standard, firms will make 

greater use of nontraditional instruments because all instruments will be treated the same 

and nontraditional instruments have the potential to allow firms to better accomplish their 

EBCP goals and reduce compensation expense. For example, instruments with market 

conditions (measures derived from the firm’s stock price) have the potential to better align 

employee and shareholder interests than traditional ESOs. Also, stock appreciation rights 

settled in stock appear to dominate traditional ESOs because they have the same fair value 

as traditional ESOs, produce less dilution and have a greater perceived value to employees. 

Finally, nontraditional options generally have lower fair values than traditional ESOs. 31 

The nontraditional instruments that will be analyzed in this report include: 

• Nontraditional options 

- Premium options 32 

31 

As demonstrated later in this section, the instrument with the lowest fair value will not necessarily produce the lowest 

total compensation expense. The instrument that produces the lowest compensation expense will be the one for 

which the product of fair value and the number of instruments expected to vest is the lowest. 

32 

We could have also discussed the potential benefits or using discount options. A discount option is similar to a 

premium option, but the strike price is set below the grant date stock price. However, because of an expected change 

in the Internal Revenue Service Code (IRC), it is expected that this type of option will cease to be used by most firms. 

Under the recently proposed IRS ruling, discount options will be both subject to, and in violation of, the provisions of 

25

- Maximum value options 

- Purchased options 

- Indexed options 

- Performance-based options 

- Market-based options 

• Non-performance-based and performance- or market-based versions of non-option 

instruments 

- Restricted stock 

- Restricted stock units 

- Stock appreciation rights settled in stock or cash 

Because of their complexity, more flexible models, such as lattice models, will generally 

required to value such nontraditional instruments as capped, indexed and performancebased 

options with market conditions as well as restricted stock, restricted stock units and 

stock appreciation rights where vesting or the number of instruments granted is contingent 

on market conditions. More flexible models, such as lattice models, are also required to 

determine the appropriate “exchange ratio” that will make employees indifferent between a 

particular nontraditional instrument and traditional ESOs. 

2. Descriptions and evaluations of nontraditional instruments 

This section provides descriptions, objectives, pros and cons and fair value estimates for the 

nontraditional instruments discussed above. All of the valuations are based on inputs shown 

in Table 1 and use lattice-based and MBS-based models have been specifically designed to 

reflect the features of each of these nontraditional instruments. While it is possible to value 

all of the instruments with lattice-based models, because of its complexity, it is not generally 

possible to use a MBS-based model to value options with market conditions. 

a. Premium options 

With Premium Options (POs) the strike price is set above the grant date stock price. POs 

are designed to reduce fair value and to provide a stronger incentive than traditional ESOs 

for employees to increase the firm’s stock price (because these options begin under water). 

As a consequence, more options will need to be awarded to make employees indifferent 

between POs and traditional ESOs. Assuming that the strike price is set $3.50 above the 

grant date stock price, the fair value of the PO will be about 9.0% lower than that of the 

traditional ESO analyzed in Table 3 (from $10.91 to $9.95) for the lattice model and about 

4% lower (from $13.19 to $12.61) for the MBS model. Also, the value produced by the MBS 

model is 27.0% greater for the PO than that produced by the lattice model. 

b. Maximum value options 

Maximum Value Options (MVOs) are designed to reduce fair value by capping the “spread” 

between the stock price and the strike price, where the maximum spread allowed is usually 

expressed as a multiple of the strike price. The objective of MVOs is to reduce the cost of 

the option, compared to a traditional ESO, without significantly reducing the value perceived 

by employees. A key advantage of MVOs, over other nontraditional options, such as an 

indexed option, is that they are easy to understand, have a lower fair value than a traditional 

Section 409A of the Internal Revenue Code (IRC). Under the proposed regulations, both employee stock options and 

stock appreciation rights are generally exempt from Section 409A of the IRC, as long as the strike price is never less 

than the price of the underlying stock at the grant date. 

26

ESO and can be designed so that the value to employees is similar to that of a traditional 

ESO. Table 8 shows fair values for MVOs, where the maximum intrinsic value is based on 

multiples of one, two or three times the strike price. 

Maximum Spread 

Table 8. 

Analysis of Maximum Value Options 

Binomial 

Lattice 

Model 

Percent 

Reduction 

from 

Traditional 

ESO 

Modified 

Black- 

Scholes 

Model 

Percent 

Reduction 

from 

Traditional 

ESO 

Traditional ESO $10.91 N/A $13.19 NA 

$90 (3X) $10.31 5.52% $9.66 26.80% 

$60 (2X) $9.69 11.18% $8.17 38.10% 

$30 (1X) $8.08 25.84% $5.53 58.20% 

The fair value estimates above are based on both lattice-based and MBS-based models that 

have been specifically designed to reflect the features of an MVO. The fair value estimates 

produced by the lattice model are roughly 6% to 26% lower than that of a traditional ESO, 

while the MBS-based model shows reductions in fair value of 27% to 58%. As shown in the 

table, the MBS model tends to understate the fair value of MVOs. This occurs because the 

MBS model assumes that an MVO can be exercised only at its expiration date. 

Consequently, the fair value of the MVO will be based on cash flows that range from zero to 

the maximum intrinsic value allowed and these cash flows will be heavily discounted (from 

the MVO’s expiration date to the grant date). 

This situation is to be contrasted with a lattice-based model, which assumes that an MVO 

can be exercised any time after the option vests. In fact, it can be shown that MVOs tend to 

be exercised earlier than traditional ESOs, typically when the maximum allowed stock price 

is reached. As a result, the cash flows predicted by a lattice model for a MVO will generally 

be greater than those predicted by a MBS-based model (typically equal to the maximum 

intrinsic value allowed) and, because they are exercised earlier, these cash flows will 

generally receive less discounting than occurs with the MBS-based model. 

Because MBS-based models tend to understate the fair value of MVOs, it is generally not 

advisable to use them to value this type of instrument. The same conclusion has been 

reached by PricewaterhouseCoopers (PwC). 33 PwC states: “However, only lattice models 

should be used for certain alternative awards, including certain performance awards (those 

with market conditions), as well as options with payoff functions limited in certain ways (such 

as maximum value options) …” 34 

c. Purchased options 

With Purchased Options (PUROs), the employee pays a fraction of the strike price at the 

grant date and the remainder when the option is exercised. PUROs provide a strong 

incentive for employees to increase the firm’s stock price and to remain with the company, 

33 PricewaterhouseCoopers, 2005, Page 3-2. 

27

ecause PUROs require employees to invest their own money. It should be noted that 

because the employee puts money at risk, the perceived value of the PURO will be less than 

that of a traditional ESO. Consequently, more options will be required to make employees 

indifferent between PUROs and traditional ESOs. 

By requiring employees to pay a fraction of the strike price at the grant date, PUROs reduce 

the fair value of the option. For example, if the employee must pay 5% of the strike price at 

the grant date and the remaining $28.50 at vesting, then the fair value of the PURO would be 

$9.88, or 9.4% less than the fair value of the traditional ESO ($10.91). The estimate 

produced by a MBS model, which has been modified to reflect the features of a PURO, is 

$12.98. This is 1.6% less than the fair value of a traditional ESO ($13.19). Lastly, the fair 

value produced by the MBS-based model is 31% greater than that produced by the latticebased 

model. 

d. Indexed options 

With Indexed Options (IOs), the strike price is not fixed, but instead varies according to an 

index that reflects either market, industry or peer group performance (e.g., S&P 500 index). 

IOs are designed to reward employees when the company performance exceeds that of the 

index. As such, employees can be rewarded even when company performance declines, as 

long as it declines less than that of the index. The benefits of IOs are: 

• 

• 

• 

They provide a strong incentive for employees to improve performance; 

They can provide incentives even when the company’s current stock price is less 

than the grant date price; and 

The fair value of an IO can be significantly lower than that of a traditional ESO. 

To make employees indifferent between IOs and traditional ESOs, additional options will 

need to be awarded. One potential problem with traditional IOs is that they may not be 

exempt from Section 409A of the IRC, because the strike price can become less than the 

stock price at the grant date. The problem could be alleviated by preventing the strike price 

from dropping below the grant date stock price. However, this change would reduce the IO’s 

value to employees. 

Using lattice and MBS models that have been modified to reflect the features of IOs, the fair 

value produced by the lattice model is $6.15, which is 44% lower than the fair value of a 

traditional ESO ($10.91). 35 Similarly, the fair value produced by a MBS model is $9.01, 

which is 32% less than the fair value produced by the MBS model for the traditional ESO 

($13.19). Lastly, the fair value produced by the MBS for an IO is 47% greater than that 

produced by a lattice model. 

e. Options with performance or market conditions 

The new standard is expected to lead to an increase in equity awards whose payoff depends 

on attaining performance targets. As previously noted, under the new standard we expect 

that firms will make greater use of instruments with performance targets because they have 

the potential to provide incentives that will better align employee and shareholder goals and 

to reduce compensation expense. In fact, according to a recent article in the Wall Street 

Journal (2/21/2006), the use of instruments with performance targets has greatly increased. 

In 2003, 17% of the major U.S. companies granted instruments with either vesting or the 

number of instruments awarded tied to performance targets. This percentage increased to 

35 In addition to the inputs shown in Table 1, it is assumed that the volatility and dividend yield of the index are 40% and 

zero percent respectively and the correlation between the index and the stock price is 70%. 

28

24% in 2004, 30% in 2005 and is expected to reach 50% by 2006. Both performance and 

market-based measures are being used. For example, the number of shares of restricted 

stock the CEO of Tyson Foods, Inc. receives is tied to how well Tyson’s shares fare against 

12 other food companies, with the CEO receiving no shares unless the company’s stock 

outperforms at least six of the companies. 

The new standard discusses two types of instruments with performance targets: 

performance conditions and market conditions. A performance condition depends on a 

measure that is based on a firm’s own operations, such as earnings per share or growth in 

revenues. A market condition depends on the firm’s stock price or some measure derived 

from it, such as total shareholder return (TSR). 

Performance conditions that affect vesting are not reflected in the instrument’s grant date fair 

value estimate. Instead, they are viewed as affecting whether the instrument vests. As a 

consequence, the option is not expensed if the performance condition is not met, even if the 

service condition is achieved. Thus, options with performance conditions that affect vesting 

generally have fair values that are the same as traditional service-vested options. 36 

However, performance conditions that affect fair value (e.g., affect the instrument’s strike 

price, length of the vesting period or contractual term) are reflected in the instrument’s grant 

date fair value. At the grant date, the firm is required to estimate the fair value associated 

with each possible outcome of the market condition. The final compensation expense is 

based on the fair value of the outcome that is actually realized. 

A major disadvantage of using market conditions is that the instrument will be expensed as 

long as the service condition is met, irrespective of whether the market condition is achieved. 

As shown in the example below, this disadvantage is partially offset because instruments 

with market conditions usually have significantly lower fair value estimates than those with 

either service or performance conditions. 

Estimating the fair value of an instrument with market conditions usually requires the use of 

lattice- or simulation-based models because these instruments are generally path 

dependent. That is, the payoff of the instrument depends upon the path or the particular 

sequence of changes in the stock price up to the current time, instead of simply the level of 

the stock at the current time. The estimation of the service period is also much more 

complex. According to FAS 123R, Paragraph A60, the derived service period is to be the 

median of the distribution of price paths for which the market condition is satisfied. 

On balance, awards with market conditions are expected to be harder to value, but their fair 

value is fixed as of the grant date. Instruments with performance conditions are expected to 

be easier to value, but are subject to potential fluctuations in compensation expense 

whenever there is a change in the likelihood that the performance condition will be achieved. 

The example below illustrates the valuation of an option with market conditions. The 

example assumes that the option will vest only if the stock price exceeds 150% of the grant 

date stock price for ten consecutive days during a three-year vesting. Both the market and 

36 This method is expected to produce biased estimates of fair value. The potential bias occurs because of a failure to 

reflect the correlation between the level of the performance measure and the fair value of the instrument. If a firm 

believes that the performance condition will be met, then given this knowledge, it would be expected that the stock 

price distribution would shift upward. As a consequence, the appropriate estimate of fair value, conditioned on 

knowledge that the performance measure will be met, would be greater than the unconditional fair value that would be 

estimated for a typical service-vested option. Lattice and simulation models have been developed to reflect the 

correlation between the stock price and the performance index and have the potential to provide more accurate 

estimates of compensation expense associated with instruments with market conditions. 

29

the service conditions must be met for the option to vest. Fair value was computed by using 

a lattice model that is specifically designed to reflect the features of this instrument, including 

its path dependency. This was done by including an extra state variable to keep track of the 

number of consecutive times the market condition was met. The other assumptions are the 

same as those used for the other nontraditional instruments. The lattice model produces a 

fair value of $5.89, which is 46% lower than the fair value of the traditional ESO ($10.19). 

As previously noted, because of its complexity and path dependency, it was not possible to 

use a MBS-based model to value this type of instrument. 

f. Restricted stock and restricted stock units 

With both Restricted Stock (RS) and Restricted Stock Units (RSUs), employees receive 

shares of stock once vesting conditions are met. The vesting conditions can be based on 

performance, market or length of service conditions. Also, both RS and RSUs can vest 

according to either cliff or graded vesting schedules. An RSU is an unfunded promise to 

deliver shares of the company’s stock in the future. As such it does not represent a property 

interest, with some RSU plans allowing the deferral of taxes past the vesting date. Because 

of this feature, RSUs are usually preferred to RS. However, under Section 409A of the IRC, 

RSUs are considered deferred compensation and any deferral past the vesting date must 

comply with this section of the code, which has severe penalties for non-compliance. 37 The 

benefits of RS and RSUs are: 

• 

• 

• 

Unlike options, they continue to have value even if the stock price declines 

significantly; 

They have greater value to employees than ESOs. This reduces the number of 

shares that must be granted to make employees to be indifferent between RS or 

RSUs and ESOs; and 

The reduction in the number of instruments that must be offered will reduce 

compensation expense. 

Both RS and RSUs have been criticized for failing to provide strong incentives for employees 

to accomplish shareholder objectives. It is sometimes alleged that both RS and RSUs are 

merely “payment for pulse.” One way to avoid this criticism would be to make vesting or the 

number of shares awarded contingent on the achievement of either market or performance 

conditions. RS and RSUs with performance conditions are sometimes referred to as 

“performance shares.” Section 162(m) of the IRC provides another reason for firms to 

include performance conditions in either their RS or RSUs. This section precludes public 

companies from deducting the compensation expense it pays to its top officers in excess of 

$1 million per year. However, this limitation can be avoided if the RS or RSUs contain 

performance conditions that meet certain requirements. 

37 Non-compliance with the requirements of Section 409A can lead to an additional 20% tax to the recipient, 

underpayment penalties and an acceleration of taxation. 

30

g. Stock appreciation rights 

With Stock Appreciation Rights (SARs), employees receive the spread (difference between 

the stock price and the strike price) upon exercise. SARs can be settled in either cash or 

stock. Under FAS 123R, SARs that are settled in cash are treated as a liability instrument. 

As such, they reduce dilution because no shares are awarded, but, as liability instruments, 

they must periodically be marked-to-market. If settled in stock, they are treated as an equity 

instrument, which means that they will contribute to dilution, but their fair value is fixed at the 

grant date. 

SARs have the following advantages over traditional ESOs: 

• 

• 

• 

They reduce dilution (compared to ESOs) because only the shares (based on the 

stock price at exercise) required to pay the spread are awarded to the employee; 38 

They have the same fair value as ESOs, but allow employees to acquire shares 

without paying the strike price or the commission on a broker-facilitated cashless 

exercise; and 

Both cash- and stock-setted SARs are valued by the same models used to value 

traditional ESOs. 

Based on the above, it appears that SARs are superior to traditional ESOs in that they have 

the same fair value as ESOs, greater value to employees and result in less dilution. 

3. Determining the number of options required to make employees 

indifferent between nontraditional instruments and traditional 

ESOs 

The fair value of an instrument is only one piece of the compensation expense equation. 

Total compensation expense is the product of fair value, the number of instruments that will 

make employees indifferent between alternative offerings and the number of these options 

that are expected to vest. Lattice models can be designed to estimate both the fair value of 

a particular nontraditional instrument to firms and the perceived value of the instrument to 

employees. The appropriate exchange ratio is determined such that employees are 

indifferent between the number ESOs awarded under the current plan and the number of 

nontraditional instruments awarded under the alternative plan. The value of an instrument to 

an employee is the minimum amount of cash that will make the employee willing to give up 

the right to the instrument. 39 

Table 9 below shows the number of shares that will make employees indifferent between 

restricted stock with a six-year vesting period and 1000 shares of the traditional three-year 

graded vested option analyzed in Table 3. 

38 A simple example may help to illustrate the point. Assume that an employee is to be given either 1000 traditional 

ESOs or 1000 stock settled SARs. The strike price is set at $30 and the instrument is exercised when the stock price 

hits $60. With a traditional ESO, an employee pays $30,000 and receives 1000 shares of stock. With stock settled 

SARs, the employee pays nothing, but receives only 500 shares of stock (1000*(60-30)/60 = 500). 

39 The lattice model assumes that exercise decisions reflect such factors as risk aversion and lack of diversification. As 

such, the perceived value to employees can be viewed as what is usually termed in the economic literature as a 

certainty equivalent. The certainty equivalent is the minimum amount of cash that will give the employee the same 

benefit as 1000 traditional ESOs. 

31

Table 9. 

Determining the Appropriate Exchange Ratio 

That Will Make Employees Indifferent 

Between Alternative Offerings 

Traditional 

ESO 

Restricted 

Stock 

Percent 

Difference 

Compensation Expense $10,910 $8,701 -20.17% 

Perceived Value $3,078 $3,078 0.0% 

Number of Instruments 1,000 348 -65.0% 

The table shows that employees would be indifferent between 348 shares of restricted stock 

and 1000 shares of traditional ESOs (i.e., an exchange ratio of roughly .35 shares of 

restricted stock for each traditional ESO). Employees will be indifferent between the options 

and restricted stock because both instruments have the same perceived value of $3,078. 

However, the total compensation expense would be roughly 20% less for the restricted 

stock. Hence, even though fair value per share is greater for restricted stock ($30.00) than 

for ESOs ($10.91), total compensation expense is less because sufficiently fewer shares of 

restricted stock are required to make employees indifferent between the two types of 

instruments. 40 

40 The reader may have noticed that the product of the number of shares of restricted stock and the price per share does 

not equal the total compensation expense shown in the table. This apparent anomaly occurs because although 348 

shares are granted only 290 shares (348*(1-.03)^6) are expected to vest six years hence. 

32

V. Conclusions 

This report’s main conclusions are: 

1. Initially firms will be able to use simple methods to estimate the required 

inputs for the modified Black-Scholes model. However, the ability to use the 

SEC’s simple methods to estimate ET will sunset at the end of 2007. Also, 

the SEC’s simplified method and the methods required to provide unbiased 

estimates (based on a firm’s own transaction data) are expected to produce 

larger estimates of ET, and hence larger fair value estimates, than would be 

obtained by using the common practice of taking the weighted average of 

observed settlement times. 

2. This report recommends strategies for dealing with the challenges that firms 

are expected to face when attempting to comply with FAS 123R’s input 

requirements. For example, it recommends methods that firms can use to 

develop unbiased estimates of ET as well as the measures of exercise 

behavior that users of lattice models will be required to develop. The 

methods are designed to control for censoring (or the potential bias due to 

outstanding options at the evaluation date) and the effect of differences 

between historical and expected future values for key variables, such as 

volatility, path of the stock price, characteristics of the instrument and 

employee characteristics. 

3. The lattice-based models are shown to produce more accurate and generally 

lower fair value estimates than the MBS-based models for both traditional 

ESOs and the nontraditional instruments analyzed in this report. As shown 

in the report, the MBS model produces inaccurate fair value estimates for 

maximum value options and for incremental changes to the features of 

ESOs. Also, as noted in SAB 107, the MBS model is unable to value certain 

types of instruments (e.g., those with path dependent market conditions). 

4. Using a lattice model and allowing the risk-free rate and volatility to vary with 

time produced more accurate and, under today’s conditions, lower fair value 

estimates than would result if these inputs were held constant. 

5. Changing the features of traditional ESOs can enable firms to better 

accomplish their HR objectives and, for both types of models, to produce 

lower fair value estimate than are obtained with the base features. 

6. The use of nontraditional instruments will enable firms to better accomplish 

many of their HR goals and stock settled SARs are superior to traditional 

ESOs in that they result in the same fair value, have greater value to 

employees and reduce dilution. For both types of models and for all 

instruments analyzed, nontraditional instruments were shown to produce 

lower fair value estimates than traditional ESOs. 

33

References 

1. Bettis, J., Bizjak, J., Lemon, M., Exercise Behavior Valuation and the Incentive 

Effects of Employee Stock Options, Journal of Financial Economics, 2005. 

2. Carpenter, J., The Exercise and Valuation of Executive Stock Options, Journal of 

Financial Economics, 1998. 

3. Carr, P., Linetsky, V., The Evaluation of Executive Stock Options in an Intensity 

Based Framework, European Financial Review, 2000. 

4. Garman, M, “Semper Tempus Fugit,” RISK, May 1989 

5. Green, W., Econometric Analysis, Fifty Ed., Prentice Hall, 2003. 

6. Hall, B., Murphy, K., Stock Options for Undiversified Executives, Journal of 

Accounting and Economics, 2002. 

7. Hull, J., Options, Futures and Other Derivatives, Fifth Ed., Prentice Hall, New Jersey, 

2003. 

8. Hull, J., White, A., How to Value Employee Stock Options, University of Toronto, 

2002. 

9. Ingersoll, J., The Subjective and Objective Valuation of Incentive Stock Options, Yale 

University, White Paper, February, 2005. 

10. Kulatilaka, N., Marcus, A., Valuing Employee Stock Options, The Financial Analysts 

Journal, 1994. 

11. PricewaterhouseCoopers, FAS 123 (R), Share-Based Payment: A 

Multidisciplinary Approach, May 2005. 

12. Rubinstein, M., On the Accounting Valuation of Employee Stock Options, Journal 

of Derivatives, 1995. 

13. Wilmott, P., Derivatives, John Wiley and Sons, 1998. 

34

Appendix A 

Descriptions and Evaluations of Lattice Models that Are Designed To Comply with 

the FAS 123R Valuation Requirements 

A. Changes to the Traditional Lattice Models That Are Required to Reflect the 

Characteristics of Employee Stock Options and Other Equity-Based 

Instruments 

As discussed earlier, FAS 123R requires firms to select valuation models that reflect the 

substantive characteristics of the instrument being valued. Because of its flexibility, a 

binomial lattice model that is designed to value exchange-traded options can be modified to 

explicitly reflect the features of ESOs or other equity-based instruments directly. The 

features include the interplay between early exercise, vesting, blackout dates, departure and 

forfeiture. To reflect the interplay between these factors, the traditional binomial model 

would be modified as follows. For nodes where post-vesting termination does not occur and 

exercise is possible (i.e., the option is vested and the period is not a blackout date), the 

binomial model would be modified to reflect that exercise will occur when it is economic to do 

so (i.e., the benefit from exercise is greater than the benefit from continuing to hold the 

option). 

For nodes where exercise is not possible, the binomial model would be modified by 

precluding exercise (i.e., only the continuation value would be calculated at the node). 

However, in addition to reflecting the possible stock price movements, the continuation value 

would also be modified to reflect the probability of termination occurring during the next 

period. The probability of termination occurring at each node would be based on the 

company’s annual termination rate (i.e., the fraction of employees holding options that leave 

the company each year). For post-vesting nodes where termination occurs, the binomial 

model would be modified to reflect that the option will be exercised if it is “in-the-money” and 

exercise is possible; otherwise it will be forfeited. 

B. Advantages and Disadvantages of Lattice Models that are in Use Today 

This section discusses the advantages and disadvantages of the three types of lattice 

models that are currently in use. 

1. Generalized lattice model 

This is the type of lattice model that was used to perform the valuations shown in this report. 

The model can be viewed as a generalized version of the traditional Cox, Ross and 

Rubinstein model that is used to value exchange-traded American options. The model is 

generalized in the sense that it explicitly reflects the features of ESOs along the lines 

discussed above. In addition, when making exercise decisions employees are assumed to 

reflect risk aversion, wealth effects, lack of diversification and the impact of employment 

terminations. The model is designed to explicitly address both the traditional features of 

employee stock options as well as the features of nontraditional instruments. It includes a 

sophisticated algorithm that allows it to accurately value instruments with time-varying inputs, 

including stock price volatility. The model explicitly reflects the interplay between early 

exercise, vesting, blackout dates, departure and forfeiture based on the method described 

above. 

35

In addition to determining the cost and value of ESOs and other equity instruments, the 

model computes the joint distribution of exercise and termination behavior at each node in 

the binomial tree. As a consequence, it is possible to calibrate the model to virtually any 

measure of observed employee exercise and post-vesting termination behavior, including: 

Expected option term; 

Expected time-to-exercise; 

Expected ratio of the stock price at exercise to the strike price; 

Probability of forfeiture before and after vesting; 

Probability the option expires worthless; and 

Probability of normal and forced exercise each post-vesting period. 41 

Advanced statistical techniques are used to estimate these measures that control for both 

censoring (i.e., bias due to outstanding options at the evaluation date) and the influence of 

variables—such as volatility, path of the stock price length of the vesting period, time 

remaining until expiration—that can cause historical data to differ from expected future 

conditions. 

The model is calibrated to accurately reflect the observed data using the procedure shown in 

the diagram below: 

Inputs 


Volatility 

Dividend Yield 

Stock Price 

Strike Price 

Risk-Free Rate 

Vesting Period(s) 

Calibration Metrics: 

Expected Term 

Departure Rates 

ESOVAL 

Model 

Adjust Calibration 

Parameters 

Does model’s predictions of 

exercise and termination 

behavior equal observed data? 

No 

Yes 

Outputs 

ESO Cost 

ESO Value 

Calibration 

Measures 

The left block shows the model inputs. The first six inputs are the traditional inputs required 

to value exchange-traded options (ETOs). The next three inputs are additional inputs that 

are required to value ESOs. The middle block shows how the model is calibrated to 

observed measures of exercise and forfeiture behavior (e.g., expected term or expected time 

to exercise) by adjusting parameters controlling exercise and post-vesting termination 

behavior. Finally, the right block indicates how the calibrated model can be used to value 

ESOs. 

41 

Forced exercise occurs when employees must either exercise or forfeit vested ESOs, shortly after leaving the firm. 

36

a. Advantages 

The model is able to address the features of both traditional options and nontraditional 

instruments, correctly reflects the effect of time-varying inputs (including stock price volatility) 

and can value instruments from the perspective of both cost to the firm and value to 

employees. The model can be viewed as a generalized version of the best known models in 

the literature in that it has the flexibility to match the values and calibration measures 

produced by these models. 

b. Disadvantages 

Both this model and the Hull and White model, described in 2 below, require that the model 

be calibrated to estimates of employees’ exercise and termination behavior. While this is 

straightforward to do, it does require an extra step. For example, the Actuarial model, 

described below, does not require this step, because exercise is assumed to depend upon 

an exercise function that is estimated from historical data. 

2. Hull and White-based model 

This type of model was originally developed by Hull and White and is described in FAS 

123R. It assumes that employees will exercise their stock options whenever the stock price 

exceeds a given multiple of the strike price (“exercise multiple”). The model reflects the 

interplay between exercise, termination and cancellation based on the methods discussed 

above. The principle inputs required to measure employees’ departure and post-vesting 

termination behavior are estimates of the exercise multiple and post-vesting termination rate. 

a. Advantages 

The advantages of this model are: 

• 

• 

• 

• 

• 

The model is an improvement on the Modified Black-Scholes model discussed in 

FAS 123R since it is able to address more of the features of traditional ESOs than 

the MBS model; 

Fairly simple measures of exercise and termination behavior can be used to calibrate 

the model; and 

The model is described in FAS 123R. 


The disadvantages of this model are: 

The model’s fundamental assumption is that the exercise boundary is horizontal 

(level of the stock price at which exercise occurs) is incorrect. It is well known that 

the exercise boundary continually declines and is equal to the strike price at the 

expiration date. Hence, the exercise multiple should not be constant, but rather 

should decline as one approaches the expiration date. 

Second, this type of model is difficult to implement correctly. To produce accurate 

results, the level of the stock price at which exercise occurs should lie on one of the 

37

• 

• 

• 

• 

• 

• 

• 

• 

layers of the lattice. This will require the use of sophisticated methods that are used 

to value American “up and in” barrier options. 

As a ratio of two variables, it can be difficult to obtain accurate estimates of the 

exercise multiple. 

This model is not designed to value most of the nontraditional instruments discussed 

in Section IV. Nor can this model accurately value, in a reasonably period of time, 

instruments with time-varying stock price volatility. 

3. Actuarial model 

Often referred to as an “actuarial” model, this type of model typically uses regression-based 

methods to estimate the probability of exercise as a function of such variables as the intrinsic 

value of the option, time remaining until expiration, stock price volatility, contractual term, 

dividend yield, length of the vesting period and employee characteristics. This type of model 

is able to model the interplay between exercise, termination and cancellation based on the 

methods discussed above. 

a. Advantages 

The advantage of this method is that the calibration process is simplified because exercise 

behavior is based on exercise functions that are estimated directly from employees’ 

observed exercise behavior. 


The disadvantages of this model are: 

Since the exercise equations are usually estimated based on regression methods, it 

is questionable that they will provide stable and reliable predictions of exercise 

behavior over the instrument’s contractual term, which can be as long as ten years. 

The exercise equations will be biased if, as is typical, the exercise equations are 

based only on options for which exercise has occurred. 

It is well known that regression-based methods will produce unreliable results if 

explanatory variables are omitted, are measured with error or the values of the 

explanatory variables used for prediction depart from the means of the explanatory 

variables used to estimate the model. 

If an aggregate measure is used for the dependent variable (either aggregating 

across time or across employees making exercise decisions), which is often done, 

then the estimated coefficients will be subject to an aggregation bias. 

The model is not able to estimate the value of an instrument to employees. 

This model is not designed to evaluate the more complex nontraditional instruments 

discussed in Section IV. Nor is it designed to accurately value, in a reasonable 

period of time, instruments with time-varying stock price volatility. 

38

Appendix B 

Descriptions of Lattice and Black-Scholes Models Used in the Report 

This section describes the lattice and MBS models that were used to perform the valuations 

shown in this report. For the models used in this report, the original exchange-traded 

versions of these models were modified to reflect the features of the instrument being 

valued. The traditional Black-Scholes model was modified, as required in FAS 123R, by 

replacing the contractual term by the option’s ET. The traditional lattice model was modified 

to explicitly address the features of ESOs (see the discussion in Appendix A) of the 

nontraditional instruments shown below. 

• 

• 

• 

• 

• 

• 

• 

Premium and discount options 

Maximum value options 

Purchased options 

Indexed options 

Restricted stock and restricted stock units 

Stock appreciation rights 

Performance-based versions of stock options, restricted stock, restricted stock units 

and stock appreciation rights 

Reflecting the features of premium or discount options or stock appreciation rights is 

straightforward. Premium and discount options can be valued by changing the strike price in 

the traditional lattice or Black-Scholes models. Stock appreciation rights can be valued by 

using the lattice and Black-Scholes models that are used to value traditional ESOs. 

Traditional restricted stock or restricted stock units can be valued by treating them as 

European options with a zero strike price and a contractual term equal to the length of the 

vesting period. 

Additional modifications were required to the lattice and Black-Scholes models to reflect the 

features of the other nontraditional instruments. For purchased options, maximum value 

options and indexed options, we modified the lattice and Black-Scholes models to reflect the 

features of these options. For example, for index options, both the Black-Scholes and lattice 

models were modified to reflect that the strike price is not constant, but varies according to 

an index that is correlated with the underlying stock price. 

For performance-based instruments with market conditions, we modified the lattice model to 

reflect that vesting or the number of instruments granted will depend upon the attainment of 

a performance condition that is based on the firm’s stock price. For example, vesting can 

depend upon whether the stock price exceeds a target level a stated number of times over a 

prescribed time period. Options with market-based conditions are one of the most complex 

instruments to value because the payoff is path dependent. That is, the payoff depends 

upon the actual path taken by the performance measure as opposed to depending on the 

level of the performance measure at a given point in time. While lattice models have been 

developed to value options with path dependent market-conditions, to our knowledge no one 

has been able to develop MBS-based model that is able to reflect the substantive features of 

this instrument based on generally accepted economic and financial theory. 

39

Valuation of Employee Stock Options and Other Equity-Based ...

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