Summers had been expected to move aggressively to rein in the Fed's huge economic stimulus program.
The Dow Jones industrial average rose 118 points, or 0.8 per cent, to close at 15,494 Monday.
The Standard & Poor's 500 index rose nine points, or 0.6 per cent, to 1,697.
The Nasdaq composite fell four points, or 0.1 per cent, to 3,717, pulled down by Apple.
The president is expected to nominate Ben Bernanke's successor as early as this month. The current front-runner is Janet Yellen, the Fed's vice chair.
Two stocks rose for every one that fell on the New York Stock Exchange. Volume was average at 3 billion shares.