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Stocks shot higher Tuesday, after the Federal Reserve surprised most traders by cutting its short-term interest rate target by half a percentage point, to 4.75 percent.

Most analysts had expected a quarter-point cut. Response to the bigger-than-expected bite varied between optimism that the Fed has nipped the risk of inflation in the bud to concern that the Fed must see deeper economic woes ahead.

In either case, the half-point cut was good news for financial-services stocks, especially major banks and investment firms struggling to finance loans they are unable to sell.

The Dow Jones industrial average jumped 335.97 points, or 2.5 percent, to 13,739.39. JPMorgan, Citigroup and American Express were among the bigger Dow gainers. All but one of the 30 Dow industrials, Boeing, posted gains.

Broader market indexes posted even larger percentage gains. The Standard & Poor’s 500 index rose 43.13, or 2.9 percent, to 1519.78, the biggest one-day rally since March 2003 and the fourth-best rally in reaction to a Fed rate cut since 1987, according to Bespoke Investment Group.

The Nasdaq composite index rose 70.00, or 2.7 percent, to 2651.66. The Russell 2000 small-company index closed up 30.82, or 4.0 percent, at 806.63.

Stocks opened trading on some positive notes, after better-than-expected quarterly reports from investment bank Lehman Brothers and electronics retailer Best Buy.

Lehman Brothers, which in remarks to analysts minimized the damage to its mortgage-related business, gained $5.87, or 10 percent, to $64.49. Best Buy, which reported strong back-to-school sales of computer gear and improved international results, added $2.92, or 6.6 percent, to $47.46.

New York Stock Exchange trading volume reached a moderate 1.65 billion shares, up from 1.10 billion shares Monday. Advancing stocks outnumbered losers by a nearly 10-1 ratio among NYSE-listed stocks. Nasdaq trading volume totaled 2.09 billion shares, up from 1.41 billion shares Monday, as winners topped losers by nearly a 4-1 ratio.

Commodity futures prices moved higher, led by oil and gold. Oil for October delivery hit a record high for the fifth-straight day, settling up 94 cents a barrel, at $81.51. December gold futures hit a 28-year high of $735.50 in electronic futures trading.