* Mexican peso rises for third straight day * Brazil"s real jumps after encouraging services PMI * Russian rouble gains, Hungarian forint leads among CEE FX * EM stocks touch highest level since May 2018 By Shreyashi Sanyal Nov 5 (Reuters) - Mexico"s peso touched its highest level in nearly eight months against a weakening dollar on Thursday, while emerging market stocks hit multi-year highs as Democratic candidate Joe Biden maintained a narrow lead in a tight U.S. election race. The peso rose for the third straight day, jumping over 1% with a Biden victory seen as positive for the currency, which has suffered from President Donald Trump"s hard line on trade in recent years. The election race is coming down to close contests in five states with Biden narrowly ahead of Trump in some key battleground states, as mail-in and absentee votes were being counted. "The U.S. presidential election race is getting closer to calling it for Biden, but a drawn-out election fight with recounts and legal disputes could mean final results won’t happen until early December," said Edward Moya, senior market analyst at OANDA. Brazil"s real jumped 1.6% after a purchasing managers" survey showed the country"s services sector expanded in October at the fastest rate since January, driving the strongest private sector growth in Latin America"s largest economy in over a decade. The real has fallen 27% so far this year, with record low interest rates adding pressure on the currency. Emerging market currencies also got a lift from a fading dollar which was headed for its biggest three-day drop since March ahead of the outcome of a U.S. Federal Reserve policy meeting that might hint at more stimulus. The Russian rouble was among the smallest of gainers, rising just 0.2% as traders returned from a public holiday on Wednesday. Russia has benefited from Trump"s commitment to oil and close ties with President Vladimir Putin. "The Russian currency is closer to fair value, which we see in the 70-75 range," analysts at ING wrote. "However, obstacles to this include lingering uncertainty over the U.S. election, the risk of increased oil supply, and, most recently, lower than expected FX sales." South Africa"s rand firmed against the dollar, while Hungary"s forint led gains across central European currencies with a 1.1% jump against the euro. Emerging market stocks hit their highest level since May 2018, with bourses in Russia, South Africa and Brazil rising between 2.5% and 3%. The MSCI"s index of emerging market stocks was set for its best day since July. Key Latin American stock indexes and currencies at 1424 GMT: Stock indexes Latest Daily % change MSCI Emerging Markets 1165.27 2.64 MSCI LatAm 1947.05 2.99 Brazil Bovespa 99779.73 1.95 Mexico IPC 0.00 0 Chile IPSA 3522.63 0.13 Argentina MerVal 47602.88 1.222 Colombia COLCAP 1156.41 0 Currencies Latest Daily % change Brazil real 5.5621 1.68 Mexico peso 20.6920 1.01 Chile peso 752.7 0.19 Colombia peso 3766.99 1.24 Peru sol 3.5887 0.31 Argentina peso (interbank) 79.0800 -0.06 Argentina peso (parallel) 157 5.10