(3)   Stock-Based Compensation

Liberty grants, to certain of its directors, employees and employees of its subsidiaries, restricted stock, restricted stock units (“RSUs”) and stock options to purchase shares of its common stock (collectively, "Awards"). The Company measures

the cost of employee services received in exchange for an equity classified Award (such as stock options and restricted stock) based on the grant-date fair value (“GDFV”) of the Award, and recognizes that cost over the period during which the employee is required to provide service (usually the vesting period of the Award). The Company measures the cost of employee services received in exchange for a liability classified Award based on the current fair value of the Award, and remeasures the fair value of the Award at each reporting date.

Included in the accompanying condensed consolidated statements of operations are the following amounts of stock-based compensation, as discussed below:

​

​

Liberty—Grants of Awards

Options granted during the three months ended March 31, 2022 are summarized as follows:

​

​

The Company did not grant any options to purchase Series A or Series B Liberty SiriusXM, Liberty Braves or Liberty Formula One common stock during the three months ended March 31, 2022.

​

Liberty calculates the GDFV for all of its equity classified Awards and the subsequent remeasurement of its liability classified Awards using the Black-Scholes Model. Liberty estimates the expected term of the Awards based on historical exercise and forfeiture data. The volatility used in the calculation for Awards is based on the historical volatility of Liberty common stock and the implied volatility of publicly traded Liberty options. Liberty uses a zero dividend rate and the risk-free rate for Treasury Bonds with a term similar to that of the subject Awards.

Liberty—Outstanding Awards

The following tables present the number and weighted average exercise price ("WAEP") of Awards to purchase Liberty common stock granted to certain officers, employees and directors of the Company, as well as the weighted average remaining life and aggregate intrinsic value of the Awards.  

Liberty SiriusXM

​

​

Liberty Braves

​

​

Liberty Formula One

​

​

As of March 31, 2022, there were no outstanding options to purchase shares of Series A Liberty SiriusXM common stock, and there were no outstanding options to purchase shares of Series B Liberty SiriusXM, Liberty Braves or Liberty Formula One common stock.

As of March 31, 2022, Liberty Braves and Liberty Formula One each had 1 thousand Series A options outstanding and exercisable at a WAEP of $12.35 and $12.63, respectively, and each had a weighted average remaining contractual life of 0.7 years.  

As of March 31, 2022, the total unrecognized compensation cost related to unvested Awards was approximately $54 million. Such amount will be recognized in the Company's condensed consolidated statements of operations over a weighted average period of approximately 1.5 years.

As of March 31, 2022, Liberty reserved 7.2 million, 3.2 million and 8.9 million shares of Series A and Series C common stock, as applicable, of Liberty SiriusXM, Liberty Braves and Liberty Formula One, respectively, for issuance under exercise privileges of outstanding stock Awards.

Sirius XM Holdings — Stock-based Compensation

Sirius XM Holdings granted various types of stock awards to its employees during the three months ended March 31, 2022. As of March 31, 2022, Sirius XM Holdings has approximately 161 million options outstanding of which approximately 100 million are exercisable, each with a WAEP per share of $5.33 and $4.88, respectively. The aggregate intrinsic value of Sirius XM Holdings options outstanding and exercisable as of March 31, 2022 is $210 million and $176 million, respectively. During the three months ended March 31, 2022, Sirius XM Holdings granted approximately 12 million nonvested RSUs with a GDFV per share of $6.93. Stock-based compensation expense related to Sirius XM Holdings was $45 million and $51 million for the three months ended March 31, 2022 and 2021, respectively. As of March 31, 2022, the total unrecognized compensation cost related to unvested Sirius XM Holdings stock options and RSUs was $463 million. The Sirius XM Holdings unrecognized compensation cost will be recognized in the Company's condensed consolidated statements of operations over a weighted average period of approximately 2.4 years.