By Sara Sjolin, MarketWatch

LONDON (MarketWatch) -- The U.K.'s FTSE 100 index headed for its first weekly gain in three weeks on Friday, with risk-sensitive sectors such as banks and mining firms leading the charge north on signs U.S. lawmakers are making progress in solving the budget and debt-ceiling stalemate.

Outside the main index, shares of Royal Mail PLC soared over 30% on its first day of trading.

The FTSE 100 index climbed 0.7% to 6,476.30, on track for its highest closing level since late September.

On Thursday the benchmark jumped 1.5%, as House Republican leaders in the U.S. proposed a six-week extension to the nation's borrowing limit, in efforts to avoid a default on Oct. 17 -- the date Treasury Secretary Jack Lew said is the deadline for raising the debt ceiling.

House Republican leaders and President Barack Obama ended a late-night meeting Thursday with no deal, but did agree to keep talking. "The president looks forward to making continued progress with members on both sides of the aisle," the White House said in a statement.

The U.S. government has been shut down for 11 days after lawmakers failed to agree on a budget for the new fiscal year, which started on Oct. 1.

Stocks markets rallied globally, and in London banks posted some of the biggest gains. Shares of Barclays PLC (BCS) gained 1.6%, HSBC Holdings PLC (HBC) climbed 1.7%, and Lloyds Banking Group PLC (LYG) added 1.4%.

Mining firms, which also tend to be among biggest advancers on upbeat trading days, further posted gains in the FTSE 100. Shares of Glencore Xstrata PLC (GLCNF) picked up 1.9%, Rio Tinto PLC (RIO) gained 1.4% and BHP Billiton PLC (BHP) put on 0.6%.

Shares of Anglo American PLC jumped 2.4% after its subsidiary Anglo American Platinum Ltd. said it resolved strike action associated with restructuring the business.

Whitbread PLC shares moved 3% higher after Citigroup lifted the restaurant operator to buy from neutral, with the analysts saying the firm is "best exposed to the expected rapid economic recovery that we think is underway in the U.K."

Outside the main index in London, shares of Royal Mail soared 32% to 4.37 pounds ($6.98) on its initial public offering. The issue price was GBP3.30 a share, and the solid interest fueled suggestions that the previously state-owned company was sold off too cheap.

On a more downbeat note, shares of Chemring Group PLC sank 21% after the provider of defense systems warned its full-year operating profit will be lower than previously expected because of the U.S. government shutdown and manufacturing problems.

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