June 12 (Reuters) - U.S. stocks closed higher on Friday as investors clung to hopes of a peace deal between Iran and the United States and SpaceX shares jumped in their first day of trading.

The United States and Iran have indicated that an agreement to end the war between them is imminent. A senior US official said a draft proposal has been prepared and accepted by both sides. US President Donald Trump has said several times since mid-March that an agreement with Iran to end the war is close.

Investor demand surged for SpaceX shares as they began trading on the Nasdaq today. The stock opened at $150, compared to its initial public offering price of $135 per share. By the close of trading, it had reached $166, pushing the company's valuation to over $2 trillion.

Jake Dollarhead, CEO of Longbow Asset Management in Tulsa, Oklahoma, said that the progress reported on peace talks with Iran helped to lift spirits.

He added, "There is still hope for a peace agreement. Trump has stopped the attacks... and third parties confirm that a peace agreement is being prepared."

According to preliminary data, the Standard & Poor's 500 index rose 36.30 points, or 0.51 percent, to close at 7431.83, while the Nasdaq Composite index climbed 77.77 points, or 0.30 percent, to 25887.43.

The Dow Jones Industrial Average rose 353.42 points, or 0.71 percent, to 51,208.20.