I'm on board with those that say it won't be fixed by simply gelding.
OP this horse has no idea what is going on when he sees other horses run because it's not something he usually sees in his riding career.
When horses are out in a field together, some have roles of protecting the rest of herd. When one runs, no questions are asked.. they usually all run as they are following that horse's leadership. Also horses run and buck around to play with one another.
Now take this into the arena and you have a horse that is confused as to why some are running and some aren't.
It's all about exposure. My horse used to lose his noodle when others would be cantering or trotting or jumping around him. Now, that he's been exposed to it, he's fine. He knows it's nothing to be afraid of and that I'll look after him.
Your stallion is still green and inexperienced. The best thing to do would be to start exposing him to situations with noise, other horses, weird sights, etc and curb his reactions by remaining his leader and re-directing his energy.
Stud or not, he needs to respect you.
"Strength is the ability to use a muscle without tension"