I own a Tred Barta leather arm guard with a round elastic cord. The cord is pretty worn and needed replacing. I stopped at a local fabric store to look for replacement cord, and ended up buying two relatively inexpensive options.
One option was a round cord, one was flat cord. I tried the flat cord first on my arm guard and see no reason to try the round cord. The flat cord is much more comfortable than the original round cord. It does not bite into my arm nearly as much as the original round cord. You may want to try for yourself.
Good tip. The vintage armguards had flat bands for securing it to your arm. They just need replacing earlier than round.
If the round is Biting into your arm, Maybe it's to tight....
I get my stretch cord from a kite co..