"Through" means including the mentioned date. For instance, if Tom Homan is dismissed or resigns on March 10, 2025, then "through January 20, 2025" will resolve YES and all others will resolve NO.

After inauguration, as soon as Homan stops serving as Donald Trump's border czar, all market options will resolve either YES or NO. At that point, resolutions will be final, regardless of whether he later begins serving as Trump's border czar again. If Tom Homan is not serving as Trump's border czar on inauguration day (January 20, 2025), for any reason, then all options will resolve NO.

I don't know if there is a technical way to determine whether someone is "officially" serving as a so-called "border czar" for a president of the United States, so I will rely on press statements and/or announcements from the White House and/or widespread reporting to determine the time, if any, Homan is no longer serving as Trump's border czar.