Table:
|  |  | Employee and Director Options |  |
| --- | --- | --- | --- |
|  |  | For the Years Ended December 31, |  |
|  | 2019 | 2018 | 2017 |
| Expected life (in years) | 7.85 | 7.64 | 7.01 |
| Volatility | 30.00% | 32.37% | 35.00% |
| Risk free interest rate | 1.90% | 3.10% | 2.14% |
| Dividend yield | 0.76% | 0.61% | 0.50% |
| Weighted-average grant date fair value | $17.12 | $21.03 | $21.11 |
| Stock price volatility range: |  |  |  |
| Low | 30.00% | 31.72% | 28.19% |
| High | 38.17% | 36.73% | 35.00% |
Paragraphs:
Fair Value Valuation Assumptions
Valuation of Stock Options
The fair value of stock options granted are principally estimated using a binomial-lattice model. The inputs in our binomial-lattice model include expected stock price volatility, risk-free interest rate, dividend yield, contractual term, and vesting schedule, as well as measures of employees’ cancellations, exercise, and post-vesting termination behavior. Statistical methods are used to estimate employee termination rates.
The following table presents the weighted-average assumptions, weighted average grant date fair value, and the range of expected stock price volatilities:
Expected life The expected life of employee stock options is a derived output of the binomial-lattice model and represents the weighted-average period the stock options are expected to remain outstanding. A binomial-lattice model assumes that employees will exercise their options when the stock price equals or exceeds an exercise multiple. The exercise multiple is based on historical employee exercise behaviors.
Volatility To estimate volatility for the binomial-lattice model, we consider the implied volatility of exchange-traded options on our stock to estimate short-term volatility, the historical volatility of our common shares during the option’s contractual term to estimate long-term volatility, and a statistical model to estimate the transition from short-term volatility to long-term volatility.
Risk-free interest rate As is the case for volatility, the risk-free interest rate is assumed to change during the option’s contractual term. The riskfree interest rate, which is based on U.S. Treasury yield curves, reflects the expected movement in the interest rate from one time period to the next (“forward rate”).
Dividend yield The expected dividend yield assumption is based on our historical and expected future amount of dividend payouts. Share-based compensation expense recognized is based on awards ultimately expected to vest and therefore has been reduced for estimated forfeitures. Forfeitures are estimated at the time of grant based on historical experience and revised, if necessary, in subsequent periods if actual forfeitures differ from those estimates.
Question:
How was the fair value of stock options granted estimated by?
Answer: using a binomial-lattice model.