BarbaraKollmeyer

Markets reporter

NEW YORK (MarketWatch) — U.S. stocks rallied at the end the day Tuesday, sending the benchmark S&P 500 to its highest close this year, during what was another characteristically volatile day for equities.

Investor optimism was fueled by hopes that embattled Greece and its creditors are nearing a compromise to avert a default, even though both sides continued to reiterate statements that suggests an accord is far from imminent.

Earlier in the trading session, European and U.S. equities got a boost after news reports offered hope that the European Commission was considering a six-month debt extension. However, investors remained optimistic even though rumors of an extension were dismissed by Germany’s finance minister Wolfgang Schäuble.

The S&P 500 SPX, -0.61%  closed 21.84 points, or 1.1%, higher at 2,068.58 and turned positive for the year. Nine of the 10 main sectors finished higher, while energy-sector stocks ended with modest losses, following a drop in oil prices.

The Dow Jones Industrial Average DJIA, -0.82%  jumped 139.55 points, or 0.8%, to 17,868.76, with 26 of its 30 member ending with gains.

The Nasdaq Composite COMP, -0.44%  added 61.63 points, or 1.3%, to 4,787.64, helped by a big gain in Apple, Inc. the heaviest-weighted component in the tech-laden index.

Bill Stone, chief investment strategist at PNC Wealth Management, said optimism about a possible solution to the Greek debt problems was driving markets higher.

“News that Greece and the European Union will find a solution before the deadline has spurred some buying. Markets obviously would like to see a deal, rather than a default. But at the same time, the possibility of no deal is also being priced in,” Stone added.

In Greece, officials said late Monday that they plan to propose a debt-compromise plan to creditors in Brussels on Wednesday. The country still plans to reject some of the strictest austerity conditions. and the brinkmanship between Greece and its international creditors has been rattling markets for days. However, according to the Wall Street Journal, Greece will proceed with the privatization of the country’s main port.

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Separately, job openings in the U.S. rose in December to the highest since 2001 — and the number of people hired climbed to the highest level since 2007, government data showed Tuesday. The report follows last week’s strong jobs report, which showed the economy added 257,000 jobs in January.