* European shares buck bullish mood in Asia * World stocks head for near 14% gain in 2020 * Dollar slips, heads for 7% fall this year * Graphic: 2020 asset performance tmsnrt.rs/2yaDPgn * Graphic: World FX rates in 2020 tmsnrt.rs/2egbfVh (Adds details, latest prices) LONDON, Dec 31 (Reuters) - Stocks fell in European trading on Thursday as lockdowns and rising COVID-19 cases overshadowed the optimism around the rollout of vaccines in the New Year, while the dollar’s descent accelerated to another two-and-a-half-year low. Trading volumes were thin, with many traders away on New Year’s Eve and major European markets closed. Where markets were open, they failed to follow their Asian peers higher. UK’s FTSE 100 fell 1.1% and France’s CAC 40 by 0.1%. U.S. stock futures were flat or slightly higher, ahead of the Wall Street open. The MSCI World Index was flat on the day and slightly below its record level as gains in Asia gave way to the losses in Europe. The index is headed for a near 14% rise in 2020 after surging more than 60% from its March lows. Despite a stimulus-charged rebound since the pandemic-induced market slump in March, most European markets have underperformed the United States and Asia, where a series of record highs have been reached. The pan-European STOXX 600, which was closed on Thursday, recorded a 3.8% drop in 2020 as a rapid surge in coronavirus cases and worries about Brexit curbed the wider recovery in investor sentiment. Still, despite rising COVID-19 case numbers and increasing unemployment, investors are betting the rollout of vaccines in 2021 will unleash an economic rebound spurred by plentiful fiscal and monetary cash. There is a widely-held consensus among investors -- the biggest consensus in years -- that stocks and emerging market assets will rise in 2021 and the dollar and government bonds will perform poorly. “We still see growth slowing around the turn of the year and the recovery will still face headwinds in the coming quarters, but 2021 is shaping up to be better still than our already strong global outlook—led by a stronger U.S.,” JP Morgan economists said in a research note. Among the biggest developments in markets in 2020 has been the drop in the dollar. On Thursday, it hit its lowest since April 2018 and is now down 7.2% against a basket of currencies , its worst annual performance since 2017. The dollar’s weakness, driven by bets that the Federal Reserve will keep interest rates very low, has helped rivals. The euro has been a big beneficiary and is up 10% in 2020. It fell 0.2% on Thursday at $1.2278. The Chinese yuan ends the year at its strongest since mid-2018, with the dollar down 6.5% versus the yuan in offshore markets. Sterling extended its rally versus both the dollar and the euro after Britain’s markets watchdog granted UK market participants the right to use platforms in the European Union to trade swaps for up to three more months, in a bid to avoid disruption. The pound soared in the run-up to Britain and the European Union reaching a Brexit trade deal in Christmas Eve. Asian stock gains were led by Chinese blue chips after the announcement of a trade deal with the European Union. Official data released Thursday also showed activity in China’s service and factory sector expanding in December. Markets barely flickered at the news that China had approved its first COVID-19 vaccine for general public use. Gary Ng, an economist at Natixis in Hong Kong, said the limited impact showed markets had become immune to such news. “For the market to react more strongly in 2021, large-scale (vaccine) rollouts with positive outcomes are needed,” he said. Bitcoin’s rally paused at just under $29,000. The world’s largest cryptocurrency has almost quadrupled in value this year amid heightened interest from bigger investors. Oil prices, which have doubled since April but finish the year more than 20% weaker, retreated as swelling supply discouraged hopes for a swift rebound in economic activity and energy demand. U.S. West Texas Intermediate crude shed 0.93% to $47.95 a barrel and Brent was trading down 1.01% at $51.11. Gold was little changed at $1,893 an ounce, putting it on course for a 25% rise this year, its best showing since 2010 as investors looked to safe havens, protection against inflation and as the dollar wilted. Treasuries were little changed, with benchmark U.S. 10-year yields at 0.9265%.