Japan's benchmark Nikkei index was the first major stock market to open for trading at 8 p.m. Eastern time on Sunday. After Obama's televised statement, the Nikkei was up 1.7 percent.
There's also evidence that investors believe the deal Obama announced is likely to pass in Congress.
After the deal was announced to raise the debt limit and cut at least $1 trillion in spending over the next decade, Dow futures were up 182 points, or 1.5 percent. Future contracts for the broader S&P 500 index rose 1.6 percent. When futures are up during off-hours trading, stocks typically rise when the market opens.
If the deal is approved, John Brady, a senior vice president for futures and options at MF Global believes "stocks will rally, and stocks will rally big."
He said Monday could be an up and down day for markets. Stocks will rise if the news out of Washington is that the deal is on track and will fall if news leaks that the deal might be in trouble. If the deal fails to pass in Congress, he said: "The rally will be torpedoed."
Jack Ablin, chief investment officer at Harris Private Bank in Chicago, said the agreement could lead to a rally even before a vote in the House or Senate.
"I think this spells relief on Wall Street," he said shortly after the accord was announced and congressional leaders endorsed it.
A deal would remove a major source of something investors hate: Uncertainty. But there's another reason a so-called relief rally might be a big one. Companies have reported strong quarterly earnings in the past few weeks. But traders have been reluctant to buy stocks on the good news fearing the debt wrangling in Washington might set off a financial crisis.
Thomas Tzitzouris, head of fixed income research at Strategas Research Partners said Sunday that to avoid a steep decline, the market needs to believe there is progress toward the deal.
If not, he said: "When (Congress says) there is progress and then there isn't, that really spooks the market. That would be a double whammy.
That's what happened last week when a series of proposals gave investors hope there would be a deal. But one party shot each one down. Nearly every measure of market confidence fell last week as Tuesday approached without a deal. Gold, which tends to rise when investors aren't confident about other investments, rose 2 percent last week. A measure of stock market volatility, the VIX, jumped 6 percent.
In turn, the yield on the 10-year Treasury note sank to its lowest level of the year on Friday, 2.80 percent. Treasury yields fall when demand for them goes up. And demand tends to rise when investors are worried and want a safe place to put their money. Treasury bonds have long been considered the world's safest investment and are a top holding of the largest pension funds in the U.S., millions of Americans who own mutual funds and many foreign governments.
If the agreement to raise the nation's borrowing limit and defuse the building financial crisis does not pass in both the House of Representatives and the Senate, analysts said Sunday that they expect stock markets across the globe to fall on Monday.
In the U.S. that would add to six straight days of stock losses. The Dow Jones industrial average fell 581 points, or 4.6 percent, in that time.
Brady predicts the S&P 500 could fall as low as 1,200 in the next two days if there is no deal before the market opens Monday. That would be a 7 percent drop from Friday's close of 1,292 on Friday. The S&P was down 3.9 percent last week. A loss of another 7 percent would send the S&P down to a level it hasn't reached since last November.
The Treasury Department has said that after Tuesday the U.S. government won't have enough money to meet all of its financial obligations if Congress doesn't raise the nation's debt ceiling. If a deal isn't approved, the Treasury Department will have to decide which bills to pay and which to delay. Among them: interest payments on bonds, salaries of federal employees and Social Security payments to retirees. The Treasury Department has not indicated which payments will take priority if the debt ceiling is not raised.
"If this issue can be taken out of the headlines and the focus on Washington can be redirected toward corporate earnings and economic fundamentals, the market will have removed a significant obstacle," said Quincy Krosby, chief market strategist at Prudential Financial.
Corporate earnings have been strong so far for the second quarter. Many major U.S. companies have reported their earnings in the last three weeks and others such as consumer goods companies Procter & Gamble and Kraft Foods will report this week.
"When you look at corporate earnings, which are immune to politics, you see that companies have been knocking the cover off the ball," said Douglas Cote, the chief market strategist at ING.
Despite weak economic growth in the U.S., corporations in the S&P 500 are on pace for record profits for the year. Big companies have cut operating costs dramatically the past three years. Many also get nearly half of their revenue overseas. That means they can generate higher profits even if demand for their goods and services isn't increasing as quickly.
Ablin, of Harris Private Bank, said Sunday that even a tentative agreement should energize the markets on Monday-- "just for the resolution alone."
While plenty of other challenges loom ahead politically and economically, Ablin said the market already has built in enough of a negative outlook to be able to absorb those bumps.
Among those challenges: A report Friday said that the U.S. economy grew at an annual rate of only 1.3 percent from April through June. This year the economy has grown at its slowest pace since the recession ended in June 2009.
A debt deal that cuts short-term government spending significantly could further weaken the economy, experts say.
Analysts say companies won't be ready to hire and invest in new projects until some other Washington issues are resolved, like the cost of health care legislation passed last year and financial reform legislation.
Mark Luschini, chief investment strategist at Janney Montgomery Scott investment firm in Philadelphia, said Sunday night that the market was poised to rally Monday even without a vote, as evidence by a surge in U.S. futures indexes in electronic trading immediately after the agreement was announced.
The market "most definitely" will view the breakthrough with relief, he said, predicting a 2 or 3 percent jump in the Dow over a couple of days. That would make up for more than half the Dow's losses last week. Gold, on the other hand, is likely to retreat on the good news.
"Investors shouldn't allow (the market) reaction to persuade them that this is an all-clear signal," Luschini said.
He said attention would turn very quickly to the jobs report that will be released on Friday. Economists expect that it will show 110,000 jobs were added by employers in July, according to FactSet. That's well below the level that would indicate healthy job growth.
"Corporate America's making plenty of money. But we have high unemployment and you see the GDP numbers.," said Cliff Caplan, a certified financial planner with Neponset Valley Financial Partners in Norwood, Mass. on Sunday.
Caplan said an agreement could temporarily bring order to the market, but stocks could go down again because of the economic difficulties ahead.
"No matter what this bill does, it's not going to be enough," he said.