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Assignment 5: SBC – Option-Pricing Models and Volatility
A. Option Pricing Models and Volatility Model
“Valuation techniques meeting SFAS 123(R)s criteria are closed-form models, such as the Black-Scholes-Merton formula, and binomial models, such as the lattice model. The Black-Scholes-Merton formula is generally applied the same way to awards accounted for under SFAS 123 and prior pronouncements, except for the use of expected (as opposed to historical) volatility. Under SFAS 123(R), it must be adjusted to reflect certain characteristics of employee stock options that are not consistent with the models assumptions, such as the ability to exercise the option prior to the end of its contractual term.” [From Basic Principles in the New Accounting for Stock Options (The CPA Journal, September 2005)]
1. Black-Scholes Option Pricing Model
Prepare an EXCEL Spreadsheet Template of the Black-Scholes Option Pricing Model using the explanation contained in the following article:
No Longer an “Option” (The Journal of Accountancy, April 2005)
The EXCEL Template should have separate Input and Output Sections as illustrated in Exhibit 1: The formulas used for taking the Input Numbers and calculating the Output results should be hidden from view so that a user viewing your template will only see the input and output sections. Any numbers inserted in the Input section will automatically result in the output numbers being calculated.
Testing the Black-Scholes Model. Run the following sample data as a test of your model. Correct answers are indicated in RED. Please fill in the remaining blanks.
JofA Data
Situation A
Situation B
Situation C
Current Stock Price (in dollars)
$50.00
$105.65
$18.50
$62.50
Exercise Price (in dollars)
$40.00
$105.65
$18.50
$62.50
Expected Life of Option (in years)
4 years
10 years
4 years
6 years
Risk Free Interest Rate
3.50%
3.25%
7.10%
6.40%
Volatility
Dividend Yield
1.50%
.14%
Number of shares granted
750,000
2,300,000
1,600,000
Value per Option
$18.3416
$46.0806
Total Value of Options
$18.3416
$34,560,441
Continued: Assignment 5 – Option-Pricing Models and Volatility
Exhibit 1
Black-Scholes Option-Pricing Model Template
INPUT Values
Current Stock Price (in dollars)
Exercise Price (in dollars)
Expected Life of Option (in years)
Risk-Free Interest Rate
Volatility
Dividend Yield
Number of shares granted
OUTPUT Values (computed)
Value Per Option
Total Value of Options
2. Binomial Lattice Option Pricing Model
“Lattice-based models use the same basic categories of inputs as Black-Scholes, but they can reflect post-vesting employment termination behavior and other adjustments designed to incorporate certain characteristics of employee share options and similar instruments. Furthermore, lattice models can accommodate changes in dividends and volatility over the options contractual term, estimates of expected option-exercise patterns during the options contractual term, and black-out periods (when options cannot be exercised).
As previously mentioned, SFAS 123(R) requires the same six inputs as used in Black-Scholes; however, the following changes are required in the measurement of these inputs:
Companies must take into consideration assumptions that may vary over the contractual term of the option.
The standard specifies the six inputs as the minimum number of factors to be included in the model. No guidance is offered on any other assumptions that could be incorporated into the model.
Continued: Assignment 5 – Option-Pricing Models and Volatility
A range of reasonable estimates is anticipated for expected volatility, dividends, and option terms. If the likelihood within the range is similar, an average (expected value) of the range should be used. SFAS 123 permits selection of the low end of a reasonable range of assumptions.
Unlike the formula used to calculate option value under Black-Scholes, the lattice model does not use a straightforward mathematical equation. Instead, the lattice model uses an iterative approach that involves generating a large number of possible outcomes and assigning a probability to each outcome.” [From: Accounting for Stock Options (The CPA Journal, August 2005)]
A lattice model produces an estimated fair value based on the assumed changes in prices of a financial instrument during each successive reporting period within the context of the options contractual term. Results will differ if the reporting entity uses the lattice model rather than the Black-Scholes-Merton formula. For example, determining the expected option term using the lattice model requires the consideration of vesting period, employees historical exercise and employment behavior for similar grants, expected share-price volatility, blackout periods, and employee demographic information such as age and length of service. Volatility affects the expected term because although option pricing theory contends that the optimal time to exercise an option is at the end of its term, the option holder may exercise the option early if the price of the stock reaches a certain level. Employee exercise behavior is described as the suboptimal exercise factor. For example, a determination that an appreciable number of individuals will exercise the option if the price of the stock is twice the exercise price