Valuing Employee Stock Options Part 11 pdf

The analysis takes as inputs the following variables: stock price at grant date, contractual strike price, time to maturity, a single risk-free rate, an optional series of changing risk-

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PARTFourOptions Valuation Results Tables

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Getting Started with the Options

Valuation Results Tables

Part Four provides many solved ESOs using the customized binomial lattice

approach The analysis takes as inputs the following variables: stock price

at grant date, contractual strike price, time to maturity, a single risk-free rate, an optional series of changing risk-free rates over the life of the option, a single dividend rate, an optional series of changing dividend rates over the life of the option, a single volatility, an optional series of changing volatilities, forfeiture rate, suboptimal exercise behavior multiple, vesting period, blackout periods, and number of lattice steps.

These input variables have nonlinear relationships with the option value and intravariable interactions also exist That is, a low suboptimal exercise behavior multiple has little effect on a long-vesting option, but the impact increases at a decreasing rate the lower the vesting period while holding the suboptimal exercise behavior multiple constant Therefore, these tables are required in order to capture these intricate and sometimes complex relationships and interactions among variables.

The idea behind these tables is not to provide the reader with the exact options valuation answers Rather, it provides senior management a ﬁrst- pass look at rough approximations of valuation That is, instead of spend-

ing tens of thousands of dollars on consultants tasked simply to identify the potential impacts of expensing the options or to see if using the customized binomial lattices will in fact reduce or increase the valuation of the options, these tables provide a fast and accurate, but rough guide to the potential impacts a company may face when considering the valuation of its ESOs For instance, suppose your company’s stock price at some future grant date is expected to be $22 and historical volatility indicates that it is hovering around 35 percent, and your ﬁrm’s stocks pay no dividends Further suppose that the employee turnover rate at your ﬁrm is hovering around 10 percent per year and vesting is only 1 month The closest table will be the

Maturity of 3 Years and Vesting of 1 Month on a $20 Stock Price located in

the next few pages Of course without the relevant data you cannot rectly pinpoint the suboptimal exercise behavior multiple, but research has

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shown that the multiple typically tends to hover anywhere between 1.2 and 2.4 So, select 1.6 as the starting point Looking at the table, a forfeiture rate of say 10 percent and a suboptimal exercise behavior multiple of 1.6 indicates a customized binomial lattice valuation of $4.52, as compared to the nạve GBM result of $5.59 The difference is a reduction of $1.07 or

19.2 percent per option Of course you can take haircuts on the nạve

GBM by multiplying it by one less the forfeiture rate or $5.59(1 – 10%) yielding $5.03 Compared to the binomial lattice result, there is still a reduction of $0.51 or 10.2 percent If your ﬁrm issues, say, on average about

15 million options per year (this is pretty standard for a Fortune 100 ﬁrm) the reduction in expenses would be roughly between $7.66 and $16.05 million in valuation per year Another quick way to look at this is to take the discount on last year’s stock options valuation For instance, suppose your ﬁrm granted $100 million in options last year; a 10.2 to 19.2 percent reduction implies a $10.2 to $19.2 million expense reduction going forward.

With these new rough values, it would seem that more detailed analysis

is justified However, if these tables did not exist, clearly one would be able to even grasp the magnitude of potentially over- or underexpensing the options Of course one can create more cursory analysis of these tables Table A.1 illustrates a quick summary of the impact of potential expense reduction across different suboptimal exercise behavior multiples for a 10 percent forfeiture rate Clearly nothing replaces more detailed analysis but this options table method provides quick justification to either pursue or termi- nate potentially more complex and protracted customized binomial analysis Note: These tables were generated using a proprietary customized binomial lattice algorithm with 1,000 steps In addition, the tables are calculated for 35 percent and 70 percent stock volatilities The stock prices used range from $20 to $80, with $20 increments Because in most cases, vesting, suboptimal exercise behavior, and forfeiture rates play a significant role in the option’s final fair-market value, these three variables are allowed to change

un-in these tables If your ﬁrm’s stock is between two sets of tables, calculate the average options valuation price as a rough proxy of the true value.

TABLE A.1 Scenario Analysis on the Option Results Tables (10% Annual Forfeiture)

Suboptimal

Behavior 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Options

Valuation $2.85 $3.99 $4.52 $4.75 $4.88 $4.94 $4.98 $5.00 $5.01 $5.02 Nạve BSM $5.59 $5.59 $5.59 $5.59 $5.59 $5.59 $5.59 $5.59 $5.59 $5.59 Difference $ ($2.74) ($1.60) ($1.07) ($0.83) ($0.71) ($0.64) ($0.61) ($0.59) ($0.58) ($0.57) Difference % –49.08% –28.57% –19.19% –14.94% –12.63% –11.52% –10.96% –10.57% –10.31% –10.12% Modiﬁed BSM $5.03 $5.03 $5.03 $5.03 $5.03 $5.03 $5.03 $5.03 $5.03 $5.03 Difference $ ($2.18) ($1.04) ($0.51) ($0.28) ($0.15) ($0.09) ($0.05) ($0.03) ($0.02) ($0.01) Difference % –43.43% –20.64% –10.21% –5.49% –2.92% –1.69% –1.07% –0.64% –0.34% –0.13%

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THIRTY-FIVE PERCENT VOLATILITY AND 3-YEAR

MATURITY ESO S WITH VARYING STOCK PRICE,

SUBOPTIMAL BEHAVIOR, VESTING PERIOD, AND

FORFEITURE RATES

This section provides ESO valuation results on an underlying stock with 35 percent volatility and a 3-year maturity The analysis is divided into 1- month and 1-year vesting periods, and accounts for stock prices at grant date of $20, $40, $60, and $80.

MATURITY OF 3 YEARS AND VESTING OF 1 MONTH ON A $20 STOCK PRICE

Customized Binomial

Lattice

Forfeiture (0%) $ 5.39 $5.46 $5.50 $5.53 $5.55 $5.56Forfeiture (5%) $ 5.12 $5.19 $5.23 $5.25 $5.27 $5.28Forfeiture (10%) $ 4.88 $4.94 $4.98 $5.00 $5.01 $5.02Forfeiture (15%) $ 4.66 $4.72 $4.75 $4.77 $4.78 $4.79Forfeiture (20%) $ 4.46 $4.51 $4.53 $4.55 $4.56 $4.57Forfeiture (25%) $ 4.27 $4.32 $4.34 $4.36 $4.37 $4.37Forfeiture (30%) $ 4.10 $4.14 $4.16 $4.17 $4.18 $4.19Forfeiture (35%) $ 3.94 $3.98 $4.00 $4.01 $4.02 $4.02Forfeiture (40%) $ 3.79 $3.83 $3.84 $3.85 $3.86 $3.87

Stock Price and Strike Price of $20, Maturity of 3 Years, Risk-Free Rate of 3.5%, 35% Volatility, 0% Dividends,

Vesting of 1 Month, Employee Suboptimal Exercise Behavior from 1.2 to 3.0, and Forfeiture Rate from 0% to 40%

–0.53%–5.54%–10.12%–14.32%–18.18%–21.74%–25.01%–28.03%

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Stock Price and Strike Price of $40, Maturity of 3 Years, Risk-Free Rate of 3.5%, 35% Volatility, 0% Dividends, Vesting of 1 Month,

Employee Suboptimal Exercise Behavior from 1.2 to 3.0, and Forfeiture Rate from 0% to 40%

–0.53%–5.54%–10.12%–14.32%–18.18%–21.74%–25.01%–28.03%

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Customized Binomial

Lattice

Suboptimal (1.2) Suboptimal (1.4) Suboptimal (1.6) Suboptimal (1.8) Suboptimal (2.0) Suboptimal (2.2) Suboptimal (2.4) Suboptimal (2.6) Suboptimal (2.8) Suboptimal (3.0)

Forfeiture (0%) $15.68 $16.16 $16.38 $16.50 $16.58 $16.64 $16.68Forfeiture (5%) $14.94 $15.37 $15.58 $15.68 $15.75 $15.80 $15.84Forfeiture (10%) $14.26 $14.65 $14.83 $14.93 $14.99 $15.04 $15.07Forfeiture (15%) $13.64 $13.99 $14.15 $14.24 $14.30 $14.34 $14.36Forfeiture (20%) $13.06 $13.38 $13.53 $13.60 $13.66 $13.69 $13.72Forfeiture (25%) $12.53 $12.82 $12.95 $13.02 $13.07 $13.10 $13.12Forfeiture (30%) $12.04 $12.30 $12.42 $12.48 $12.52 $12.55 $12.57Forfeiture (35%) $11.58 $11.82 $11.93 $11.99 $12.02 $12.05 $12.07Forfeiture (40%) $11.16 $11.38 $11.48 $11.53 $11.56 $11.58 $11.60

–0.53%–5.54%–10.12%–14.32%–18.18%–21.74%–25.01%–28.03%

Percentage Difference

between Binomial and

Black–Scholes Model

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Customized Binomial

Lattice

Forfeiture (0%) $12.03 $17.27 $19.76 $20.91 $21.54 $21.85 $22.00 $22.11 $22.18 $22.24Forfeiture (5%) $11.69 $16.59 $18.88 $19.92 $20.49 $20.77 $20.91 $21.00 $21.07 $21.12Forfeiture (10%) $11.38 $15.97 $18.07 $19.02 $19.53 $19.78 $19.91 $19.99 $20.05 $20.09Forfeiture (15%) $11.09 $15.39 $17.32 $18.18 $18.65 $18.87 $18.98 $19.06 $19.12 $19.15Forfeiture (20%) $10.82 $14.85 $16.63 $17.41 $17.84 $18.04 $18.14 $18.21 $18.26 $18.29Forfeiture (25%) $10.56 $14.35 $15.99 $16.71 $17.09 $17.27 $17.36 $17.42 $17.47 $17.50Forfeiture (30%) $10.32 $13.88 $15.39 $16.05 $16.40 $16.56 $16.64 $16.70 $16.74 $16.76Forfeiture (35%) $10.09 $13.45 $14.84 $15.44 $15.76 $15.91 $15.98 $16.03 $16.07 $16.09Forfeiture (40%) $9.87 $13.04 $14.33 $14.88 $15.17 $15.30 $15.37 $15.41 $15.44 $15.47

–0.53%–5.54%–10.12%–14.32%–18.18%–21.74%–25.01%–28.03%

MATURITY OF 3 YEARS AND VESTING OF 1 MONTH ON AN $80 STOCK PRICE

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Forfeiture (0%) $4.74 $ 5.10 $ 5.29 $ 5.41 $5.47 $5.50 $ 5.53 $5.55 $5.56Forfeiture (5%) $4.45 $ 4.77 $ 4.94 $ 5.05 $5.10 $5.14 $ 5.16 $5.17 $5.18Forfeiture (10%) $4.18 $ 4.47 $ 4.62 $ 4.72 $4.77 $4.79 $ 4.81 $4.83 $4.84Forfeiture (15%) $3.93 $ 4.19 $ 4.32 $ 4.41 $4.46 $4.48 $ 4.50 $4.51 $4.52Forfeiture (20%) $3.69 $ 3.93 $ 4.05 $ 4.13 $4.17 $4.19 $ 4.20 $4.22 $4.22Forfeiture (25%) $3.47 $ 3.68 $ 3.79 $ 3.86 $3.90 $3.92 $ 3.93 $3.94 $3.95Forfeiture (30%) $3.27 $ 3.46 $ 3.56 $ 3.62 $3.65 $3.67 $ 3.68 $3.69 $3.70Forfeiture (35%) $3.07 $ 3.25 $ 3.34 $ 3.39 $3.42 $3.44 $ 3.45 $3.46 $3.46Forfeiture (40%) $2.90 $ 3.05 $ 3.13 $ 3.18 $3.21 $3.23 $ 3.24 $3.24 $3.25

Stock Price and Strike Price of $20, Maturity of 3 Years, Risk–Free Rate of 3.5%, 35% Volatility, 0% Dividends, Vesting of 1 Year,

–0.52%–7.24%–13.43%–19.15%–24.42%–29.30%–33.82%–38.01%

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Customized Binomial

Lattice

Forfeiture (0%) $9.48 $10.19 $10.58 $10.81 $10.94 $11.01 $11.06 $11.09 $11.12Forfeiture (5%) $8.90 $ 9.54 $ 9.88 $10.10 $10.21 $10.27 $10.31 $10.35 $10.37Forfeiture (10% $8.36 $ 8.93 $ 9.24 $ 9.43 $9.53 $9.59 $ 9.63 $9.66 $9.68Forfeiture (15%) $7.85 $ 8.37 $ 8.65 $ 8.82 $8.91 $8.96 $ 8.99 $9.02 $9.04Forfeiture (20%) $7.38 $ 7.85 $ 8.10 $ 8.25 $8.33 $8.38 $ 8.41 $8.43 $8.45Forfeiture (25%) $6.94 $ 7.37 $ 7.59 $ 7.73 $7.80 $7.84 $ 7.87 $7.89 $7.90Forfeiture (30%) $6.53 $ 6.92 $ 7.12 $ 7.24 $7.31 $7.34 $ 7.37 $7.38 $7.40Forfeiture (35%) $6.15 $ 6.50 $ 6.68 $ 6.79 $6.85 $6.88 $ 6.90 $6.92 $6.93Forfeiture (40%) $5.79 $ 6.11 $ 6.27 $ 6.37 $6.42 $6.45 $ 6.47 $6.48 $6.50

between Binomial and

Black-Scholes Model (1.2) (1.4) (1.6) (1.8) (2.0) (2.2) (2.4) (2.6) (2.8) (3.0)

Forfeiture (0%) –15.19% –8.81% –2.12% –1.50% –1.06% –0.75% –0.52%Forfeiture (5%) –20.40% –14.66% –8.67% –8.12% –7.72% –7.44% –7.24%Forfeiture (10%) –25.25% –20.08% –14.71% –14.21% –13.86% –13.61% –13.43%Forfeiture (15%) –29.76% –25.10% –20.29% –19.84% –19.53% –19.31% –19.15%Forfeiture (20%) –33.97% –29.77% –25.44% –25.05% –24.77% –24.57% –24.42%Forfeiture (25%) –37.90% –34.10% –30.22% –29.86% –29.61% –29.43% –29.30%Forfeiture (30%) –41.56% –38.13% –34.64% –34.32% –34.10% –33.94% –33.82%Forfeiture (35%) –44.98% –41.88% –38.74% –38.46% –38.25% –38.11% –38.01%

Stock Price and Strike Price of $40, Maturity of 3 Years, Risk-Free Rate of 3.5%, 35% Volatility, 0% Dividends, Vesting of 1 Year,

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Black-Scholes Model (1.2) (1.4) (1.6) (1.8) (2.0) (2.2) (2.4) (2.6) (2.8) (3.0)

Forfeiture (0%) –15.19% –2.12% –1.50% –1.06% –0.75%Forfeiture (5%) –20.40% –8.67% –8.12% –7.72% –7.44%Forfeiture (10%) –25.25% –14.71% –14.21% –13.86% –13.61%Forfeiture (15%) –29.76% –20.29% –19.84% –19.53% –19.31%Forfeiture (20%) –33.97% –25.44% –25.05% –24.77% –24.57%Forfeiture (25%) –37.90% –30.22% –29.86% –29.61% –29.43%Forfeiture (30%) –41.56% –34.64% –34.32% –34.10% –33.94%Forfeiture (35%) –44.98% –38.74% –38.46% –38.25% –38.11%

MATURITY OF 3 YEARS AND VESTING OF 1 YEAR ON A $60 STOCK PRICE

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Generalized Black-Scholes: $22.36

MATURITY OF 3 YEARS AND VESTING OF 1 YEAR ON AN $80 STOCK PRICE

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SEVENTY PERCENT VOLATILITY AND 3-YEAR

FORFEITURE RATES

This section provides ESO valuation results on an underlying stock with 70 percent volatility and a 3-year maturity The analysis is divided into 1- month and 1-year vesting periods, and accounts for stock prices at grant date of $20, $40, $60, and $80.

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THIRTY-FIVE PERCENT VOLATILITY AND 5-YEAR

FORFEITURE RATES

This section provides ESO valuation results on an underlying stock with 35 percent volatility and a 5-year maturity The analysis is divided into 1- month, 1-year, and 3-year vesting periods, and accounts for stock prices at grant date of $20, $40, $60, and $80.

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Customized Binomial

Lattice

Black-Scholes Model

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Stock Price and Strike Price of $20, Maturity of 5 Years, Risk-Free Rate of 3.5%, 35% Volatility, 0% Dividends, Vesting of 3 Years,

MATURITY OF 5 YEARS AND VESTING OF 3 YEARS ON A $20 STOCK PRICE

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MATURITY OF 5 YEARS AND VESTING OF 3 YEARS ON AN $80 STOCK PRICE

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SEVENTY PERCENT VOLATILITY AND 5-YEAR

FORFEITURE RATES

This section provides ESO valuation results on an underlying stock with 70 percent volatility and a 5-year maturity The analysis is divided into 1- month, 1-year, and 3-year vesting periods, and accounts for stock prices at grant date of $20, $40, $60, and $80.

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