NEW YORK — The stock market showed signs of life Tuesday following hopeful signs of progress in budget talks being held in Washington. The Standard & Poor’s 500 had its biggest gain this month.

The Dow gained 78.56 points to 13,248.44, after climbing as much as 137 points earlier. The Standard & Poor’s 500 index finished up 9.29 points at 1,427.84. Both the Dow and the S&P have risen for five straight days.

The Nasdaq composite ended up 35.34 points at 3,022.30.

Delta Air Lines rose 52 cents, or 5.1 percent, to $10.66 after the company said it will buy almost half of Richard Branson’s Virgin Atlantic for $360 million as it seeks a bigger share of the lucrative New York-to-London travel market.

AIG gained $1.90 to $35.26 after the U.S. Treasury Department said it has sold the rest of its stake in the insurer. AIG was bailed out by the government after nearly collapsing during the 2008 financial crisis.

Stocks have edged up since the start of the month as investors watch for developments in the budget talks. Tax increases and federal spending cuts are scheduled to start Jan. 1 unless a deal is reached to reduce the U.S.