Stocks averted a precipitous fall on Wall Street on Tuesday after the Federal Reserve announced an emergency rate cut in the face of a worldwide sell-off. But the market still declined to its lowest level in 15 months.

After falling by nearly 4 percent at the start of trading in New York, the Standard & Poor’s 500 stock index rebounded to within a few points of its previous close, then continued to fluctuate. At the close, it was down 1.1 percent, or 14.69 points.

The Dow Jones industrial average followed a similar pattern and closed at 11,971.19, down 128.11. It was the lowest close since Oct. 16, 2006, and left the index down almost 10 percent since Jan. 1. The Nasdaq composite index ended down 2 percent.

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It was the most volatile day of trading in five years as investors tried to sort out the significance of the Fed’s rate cut, announced about an hour before markets opened. Before the cut, some overseas markets had declined 10 percent in two days, and stock index futures were pointing to a decline of nearly 5 percent when Wall Street reopened after a holiday.

The move cut the Fed funds rate, the central bank’s target for overnight interbank interest rates, to 3.5 percent, from 4.25 percent. It was the biggest percentage point cut in almost 23 years.

The Fed cut, a week before a regularly scheduled meeting, also helped push European markets to a higher close. The FTSE 100 stock index in London was up 2.9 percent after having fallen as much as 4.3 percent in early trading. It had lost 5.5 percent on Monday, when frantic selling swept markets from Japan to Canada.

Traders and analysts in Europe said rumors about the Fed rate cut started circulating before the actual announcement was released, helping to build momentum for a rally there.

Philippe Gijsels, senior equity strategist at Fortis Global Markets in Brussels, said the atmosphere in European markets was quite anxious in early trading, as stocks dived more than 5 percent at the opening. “Stocks came off their lows soon afterward,” he said, “because there were rumors that there would be coordinated action by the E.C.B. and the Fed.”

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