The Dow Jones industrial average rose 118 points, or 0.8 percent, to close at 15,494 Monday. The Standard & Poor's 500 index rose nine points, or 0.6 percent, to 1,697. The Nasdaq composite fell four points, or 0.1 percent, 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.
Also, oil prices were mostly higher after a deal between the U.S. and Russia over getting rid of Syria's chemical weapons. Crude oil for October delivery fell $1.62 to $106.59 a barrel in New York.
Among companies making big moves:
• Homebuilders rose as investors were encouraged by a decline in long-term interest rates. The biggest gainer in percentage terms was PulteGroup (PHM), up 62 cents, or 3.8 percent, to $17.14.
• Apple (AAPL) continued to slide after investors were disappointed with the company's latest iPhone models introduced last week. Apple lost $14.78 20, or 3.2 percent, to $450.12.
Trading in stock options was halted for less than an hour Monday afternoon because of a problem with their price-reporting system.
Summers' withdrawal helped stocks overseas, too. The FTSE 100 index of leading British shares was up 0.6 percent. Germany's DAX rose 1.2 percent, and the CAC-40 in France was 0.9 percent higher.