Stock-based compensation

Accounting for stock-based compensation requires that the cost resulting from all stock-based payments be recognized in the financial statements based on the grant date fair value of the award. Our stock-based compensation primarily consists of stock options, restricted stock units (RSUs) and performance-based restricted stock units (PRSUs).

Beginning with the 2018 grant, RSU and PRSU awards are credited with dividend equivalent units on each date that a cash dividend is paid to holders of Common Stock. The fair value of the RSU and PRSU awards granted in 2018 was determined as the closing stock price on the date of grant. Prior to 2018, RSU and PRSU awards were not credited with dividend equivalent units and the fair value was determined by reducing the stock price on the date of grant by the present value of the estimated dividends to be paid during the vesting period. The estimated dividends were based on Caterpillar's quarterly dividend per share at the time of grant.

We recognized pretax stock-based compensation expense of $52 million and $164 million for the three and nine months ended September 30, 2018, respectively, and $48 million and $165 million for the three and nine months ended September 30, 2017, respectively.

The following table illustrates the type and fair value of the stock-based compensation awards granted during the nine months ended September 30, 2018 and 2017, respectively:




	Nine Months Ended September 30, 2018					Nine Months Ended September 30, 2017
	Shares Granted	Weighted-Average Fair Value Per Share		Weighted-Average Grant Date Stock Price		Shares Granted	Weighted-Average Fair Value Per Share		Weighted-Average Grant Date Stock Price
Stock options	1,605,220	$	46.11	$	150.90	2,701,644	$	25.01	$	95.66
RSUs	722,521	$	150.64	$	150.64	924,421	$	90.11	$	96.01
PRSUs	344,866	$	150.93	$	150.93	437,385	$	86.78	$	95.66

The following table provides the assumptions used in determining the fair value of the stock-based awards for the nine months ended September 30, 2018 and 2017, respectively:




	Grant Year
	2018		2017
Weighted-average dividend yield	2.70	%	3.42	%
Weighted-average volatility	30.2	%	29.2	%
Range of volatilities	21.5-33.0%		22.1-33.0%
Range of risk-free interest rates	2.02-2.87%		0.81-2.35%
Weighted-average expected lives	8 years		8 years

As of September 30, 2018, the total remaining unrecognized compensation expense related to nonvested stock-based compensation awards was $217 million, which will be amortized over the weighted-average remaining requisite service periods of approximately 1.9 years.