One of Chicago’s largest high-speed traders has taken a central role in the bitcoin market, stepping into the vacuum created by Wall Street’s hesitant response to the booming investor interest in digital currencies.

DRW Holdings LLC uses quantitative models to buy and sell bitcoin, for its own account and for use as a market maker—firms that grease the wheels of finance by buying, selling and quoting prices. Cumberland, DRW’s digital-currency unit, says it has traded more than $20 billion worth of bitcoin, ethereum and other...