Valuing Employee Stock Options Part 3 pptx

In ad-dition, under real-world conditions, ESOs have a time to vesting before theemployee can execute the option, which may also be contingent upon thefirm and/or the individual employee

CHAPTER 3 Impact on Valuation

A BRIEF DESCRIPTION OF

THE DIFFERENT METHODOLOGIES

In options analysis, there are three mainstream methodologies and proaches used to calculate an option’s value:

ap-1 Closed-form models like the Black-Scholes model (also known as the

Black-Scholes-Merton model, henceforth known as BSM) and its ifications such as the Generalized Black-Scholes model (GBM)

mod-2 Monte Carlo path-dependent simulation methods

3 Lattices (binomial, trinomial, quadranomial, and multinomial lattices)

However, the mainstream methods that are most widely used are the form models (BSM and GSM) and the binomial lattices No matter whichtypes of stock options problems you are trying to solve, if the binomial latticeapproach is used, the solution can be obtained in one of two ways The first

closed-is the use of rclosed-isk-neutral probabilities, and the second closed-is the use of replicating portfolios Throughout the analysis, the risk-neutral binomial lat-

use of a replicating portfolio is more difficult to understand and apply, butthe results obtained from replicating portfolios are identical to those ob-tained through risk-neutral probabilities So it does not matter which method

is used; nevertheless, application and expositional ease should be sized, and thus the risk-neutral probability method is preferred

empha-SELECTION AND JUSTIFICATION

OF THE PREFERRED METHOD

Based on the analysis in Chapter 5 and my prior published study thatwas presented to the FASB’s Board of Directors in 2003, it is concluded

19

that the BSM, albeit theoretically correct and elegant, is insufficient andinappropriately applied when it comes to quantifying the fair-market

options without dividends, where the holder of the option can exercisethe option only on its maturity date and the underlying stock does notpay any dividends.3

where the option holder can execute the option at any time up to and cluding the maturity date while the underlying stock pays dividends In ad-dition, under real-world conditions, ESOs have a time to vesting before theemployee can execute the option, which may also be contingent upon thefirm and/or the individual employee attaining a specific performance level(e.g., profitability, growth rate, or stock price hitting a minimum barrierbefore the options become live), and are subject to forfeitures when theemployee leaves the firm or is terminated prematurely before reaching thevested period In addition, certain options follow a tranching or graduatedscale, where a certain percentage of the stock option grants become exer-cisable every year.5Next, the option value may be sensitive to the expectedeconomic environment, as characterized by the term structure of interestrates (i.e., the U.S Treasuries yield curve) where the risk-free rate canchange during the life of the option Finally, the firm may undergo somecorporate restructuring (e.g., divestitures, multinational operations, ormergers and acquisitions that may require a stock swap that changes thevolatility of the underlying stock) All these real-life scenarios make theBSM insufficient and inappropriate when used to place a fair-market value

in-on the optiin-on grant In summary, firms can implement a variety of sions that affect the fair value of the options; the above list is only a fewexamples The closed-form models such as the BSM or the GBM—the lat-ter accounts for the inclusion of dividend yields—are inflexible and cannot

provi-be modified to accommodate these real-life conditions Hence, the mial lattice approach is chosen

bino-It is shown in Chapter 5 that under very specific conditions (Europeanoptions without dividends), the binomial lattice and Monte Carlo simula-tion approaches yield identical values to the BSM, indicating that the twoformer approaches are robust and exact at the limit However, when spe-cific real-life business conditions are modeled (i.e., probability of forfeiture,probability that the firm or stock underperforms, time-vesting, suboptimalexercise behavior, and so forth), only the binomial lattice with its highlyflexible nature will provide the true fair-market value of the ESO Binomiallattices can account for real-life conditions such as stock price barriers (abarrier option exists when the stock option becomes either in-the-money

or out-of-the-money only when it hits a stock price barrier), vestingtranches (a specific percent of the options granted becomes vested or exer-cisable each year), changing volatilities (business conditions changing orcorporate restructuring), and so forth—the same conditions where a BSMfails miserably.

The BSM takes into account only the following inputs: stock price,strike price, time to maturity, a single risk-free rate, and a single volatility.The GBM accounts for the same inputs as well as a single dividend rate.Hence, in accordance with the proposed FAS 123 requirements, the BSMand GBM fail to account for real-life conditions On the contrary, the bino-mial lattice can be customized to include the stock price, strike price, time

to maturity, a single risk-free rate and/or multiple risk-free rates changingover time, a single volatility and/or multiple volatilities changing over time,

a single dividend rate and/or multiple dividend rates changing over time,plus all the other real-life factors including but not limited to vesting peri-ods, changing suboptimal early exercise behaviors, multiple blackout peri-ods, and changing forfeiture rates over time It is important to note that thecustomized binomial lattice results revert to the GBM if these real-life con-ditions are negligible

Therefore, based on the justifications above, and in accordance withthe requirements and recommendations set forth by the proposed FAS 123,which prefers the binomial lattice, it is hereby concluded that the cus-tomized binomial lattice is the best and preferred methodology to calculatethe fair-market value of ESOs

APPLICATION OF THE PREFERRED METHOD

It must be noted here that a standard binomial lattice takes only the sixGBM inputs plus a step size input, and is insufficient and inadequate tomodel ESOs under FAS 123 A special customized binomial lattice was de-veloped to incorporate these additional exotic and changing inputs overtime This customized binomial lattice is used throughout the book Pleasecontact the author for further information about the software and algo-rithms used

In applying the customized binomial lattice methodology, several puts have to be determined, including:

in-■ Stock price at grant date

■ Strike price of the option grant

■ Risk-free rate over the life of the option

■ Dividend yield of the option’s underlying stock over the life of the tion

op-■ Volatility over the life of the option

■ Suboptimal exercise behavior multiple of employees over the life of theoption

■ Forfeiture and employee turnover rates over the life of the option

postvesting period until maturity

The analysis assumes that the employee cannot exercise the option when it

is still in the vesting period Further, if the employee is terminated or cides to leave voluntarily during this vesting period, the option grant will

de-be forfeited and presumed worthless In contrast, after the options havebeen vested, employees tend to exhibit erratic exercise behavior where anoption will be exercised only if it breaches some multiple of the contractualstrike price, and not before This is termed the suboptimal exercise behav-

within a short period if the employee leaves voluntarily or is terminated,regardless of the suboptimal behavior threshold—that is, if forfeiture oc-curs (measured by the historical option forfeiture rates as well as employeeturnover rates) Finally, if the option expiration date has been reached, theoption will be exercised if it is in-the-money, and expire worthless if it is at-the-money or out-of-the-money The next section details the results ob-tained from such an analysis Further, Chapters 9 and 10 provide moredetails on the selection and justification of the input parameters used, whilethe following section provides a theoretical and empirical justification ofthe results

TECHNICAL JUSTIFICATION

OF METHODOLOGY EMPLOYED

This section illustrates some of the technical justifications that make up theprice differential between the GBM and the customized binomial latticemodels Figure 3.1 shows a tornado chart and how each input variable in a

chart, it is clear that volatility is not the single key variable that drives tion value.8In fact, when vesting, forfeiture, and suboptimal early exercisebehavior elements are added to the model, their effects dominate that ofvolatility Of course the tornado chart will not always look like Figure 3.1,

op-as it will change depending on the inputs The chart only illustrates a cific case and should not be generalized across all cases.

spe-In contrast, volatility is a significant variable in a simple BSM as can beseen in Figure 3.2 This is because there is less interaction among inputvariables, due to the fewer input variables, and for most ESOs that are is-sued at-the-money, volatility plays an important part when there are noother dominant inputs

In addition, the interactions between these new input variables arenonlinear Figure 3.3 shows a spider chart9and it can be seen that vesting,forfeiture rates, and suboptimal behavior multiples have nonlinear effects

on option value That is, the lines in the spider chart are not straight butcurve at certain areas, indicating that there are nonlinear effects in themodel This means that we cannot generalize these three variables’ effects

on option value (for instance, we cannot generalize that if a 1 percent crease in forfeiture rate will decrease option value by 2.35 percent, itmeans that a 2 percent increase in forfeiture rate drives option value down4.70 percent, and so forth) This is because the variables interact differ-ently at different input levels The conclusion is that we really cannot say apriori what the direct effects are of changing one variable on the magni-

in-FIGURE 3.1 Tornado chart listing the critical input factors of a customized binomial model.

Critical Input Factors of the Custom Binomial Model

9.8 46 2%

19 91%

9%

2.9 24.5 45%

9.10

8.2 54 9%

91 53%

FIGURE 3.2 Tornado chart listing the critical input factors of the BSM.

Black-Scholes Critical Input Factors

8.2 2%

19 15%

24.5

9.8 9%

91 91%

tude of the final option value More detailed analysis will have to be formed in each case.

per-Although the tornado and spider charts illustrate the impact of eachinput variable on the final option value, its effects are static That is, onevariable is tweaked at a time to determine its ramifications on the optionvalue However, as shown, the effects are sometimes nonlinear, whichmeans we need to change all variables simultaneously to account for theirinteractions Figure 3.4 shows a Monte Carlo simulated dynamic sensitiv-ity chart where forfeiture, vesting, and suboptimal exercise behavior multi-ples are determined to be important variables, while volatility is againrelegated to a less important role The dynamic sensitivity chart perturbsall input variables simultaneously for thousands of trials, and captures theeffects on the option value This approach is valuable in capturing the netinteraction effects among variables at different input levels

From this preliminary sensitivity analysis, we conclude that rating forfeiture rates, vesting, and suboptimal early exercise behavior isvital to obtaining a fair-market valuation of ESOs due to their significantcontributions to option value In addition, we cannot generalize each in-put’s potential nonlinear effects on the final option value Detailed analysishas to be performed to obtain the option’s value every time

incorpo-FIGURE 3.4 Dynamic sensitivity with simultaneously changing input factors in the binomial model.

OPTIONS WITH VESTING AND SUBOPTIMAL BEHAVIOR

Employee stock option holders tend to execute their options suboptimallybecause of liquidity needs (pay off debt, down payment on a home, vaca-tions), personal preferences (risk-averse perception that the stock price will

go down in the future), or lack of knowledge (firms do not provide ance to their employees on optimal timing or optimal thresholds to exer-cise their options) Therefore, further investigation into the elements ofsuboptimal exercise behavior and vesting is needed, and the analysis yields

be-havior multiples (within the range of 1 to 6), the stock option value can besignificantly lower than that predicted by the BSM With a 10-year vestingstock option, the results are identical regardless of the suboptimal behaviormultiple—its flat line bears the same value as the BSM result This is be-cause for a 10-year vesting of a 10-year maturity option, the option reverts

to a perfect European option, where it can be exercised only at expiration.The BSM provides the correct result in this case

However, when suboptimal exercise behavior multiple is low, the tion value decreases This is because employees holding the option willtend to exercise the option suboptimally—that is, the option will be exer-cised earlier and at a lower stock price than optimal Hence, the option’s

op-FIGURE 3.5 Impact of suboptimal exercise behavior and vesting on option value

in the binomial model.

Impact of Suboptimal Behavior and Vesting on Option Value

Suboptimal Behavior Multiple

Black-Scholes Value

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

upside value is not maximized As an example, suppose an option’s strikeprice is $10 while the underlying stock is highly volatile If an employeeexercises the option at $11 (this means a 1.10 suboptimal exercise multi-ple), he or she may not be capturing the entire upside potential of the op-tion as the stock price can go up significantly higher than $11 depending

on the underlying volatility Compare this to another employee who cises the option when the stock price is $20 (suboptimal exercise multiple

exer-of 2.0) versus one who does so at a much higher stock price Thus, lowersuboptimal exercise behavior means a lower fair-market value of thestock option

This suboptimal exercise behavior has a higher impact when stockprices at grant date are forecast to be high Figure 3.6 shows that (at thelower end of the suboptimal exercise behavior multiples) a steeper slopeoccurs the higher the initial stock price at grant date.11

Figure 3.7 shows that for higher volatility stocks, the suboptimal gion is larger and the impact to option value is greater, but the effect isgradual.12For instance, for the 100 percent volatility stock (Figure 3.7), thesuboptimal region extends from a suboptimal exercise behavior multiple of1.0 to approximately 9.0 versus from 1.0 to 2.0 for the 10 percent volatil-ity stock In addition, the vertical distance of the 100 percent volatilitystock extends from $12 to $22 with a $10 range, as compared to $2 to $10with an $8 range Therefore, the higher the stock price at grant date and

re-FIGURE 3.6 Impact of suboptimal exercise behavior and stock price on option value in the binomial model.

Impact of Suboptimal Behavior on Option Value with different Stock Prices

Suboptimal Behavior Multiple

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

the higher the volatility, the greater the impact of suboptimal behavior will

be on the option value In all cases, the BSM results are the horizontal lines

in the charts (Figures 3.6 and 3.7) That is, the BSM will always generate the maximum option value assuming optimal exercise behavior, and over- expense the option significantly.

OPTIONS WITH FORFEITURE RATES

Figure 3.8 illustrates the reduction in option value when the forfeiture rate

pe-riod The longer the vesting period, the more significant the impact of feitures will be This illustrates once again the nonlinear interactingrelationship between vesting and forfeitures (that is, the lines in Figure 3.8are curved and nonlinear) This is intuitive because the longer the vestingperiod, the lower the compounded probability that an employee will still

for-be employed in the firm and the higher the chances of forfeiture, reducingthe expected value of the option Again, we see that the BSM result is thehighest possible value assuming a 10-year vesting in a 10-year maturity

option with zero forfeiture The BSM will always generate the maximum

option value assuming all options will fully vest, and overexpense the tion significantly.

op-FIGURE 3.7 Impact of suboptimal exercise behavior and volatility on option value in the binomial model.

Impact of Suboptimal Behavior on Option Value with Different Volatilities

Suboptimal Behavior Multiple

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20