* Mexican peso rises for third straight day * Brazil"s real jumps after encouraging services PMI * Russian rouble gains, Hungarian forint leads among CEE FX * Emerging market stocks touch highest level since May 2018 (Updates prices throughout) 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 U.S. election vote counting. The peso rose for the third straight day, jumping over 1% as a Biden victory would be seen as positive for the currency, which has suffered from Republican President Donald Trump"s hard line on trade. The election is coming down to close contests in five states with Biden narrowly ahead of Trump in some battleground states, as mail-in and absentee votes were being counted. Brazil"s real jumped 1.6% after a purchasing managers" survey showed the 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. There have also been concerns around how Brazil"s government will fund a new fiscal program without overshooting its spending ceiling. Brazil"s central bank president Roberto Campos Neto said the bank could withdraw its "forward guidance" pledge to keep interest rates low for a long time if the government honors its spending cap but uses creative ways to increase spending. Argentina"s central bank will hold interest rates stable at its weekly directors meeting, a bank source told Reuters on Thursday, following a period of adjustments as part of a move to "harmonize" different rates. The country"s peso was flat. 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. South Africa"s rand firmed against the dollar, while Hungary"s forint led gains across central European currencies with a 1.7% 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. Latin American stock indexes and currencies at 1854 GMT: Stock indexes Latest Daily % change MSCI Emerging Markets 1166.88 2.79 MSCI LatAm 1969.66 4.19 Brazil Bovespa 99967.47 2.15 Mexico IPC 38135.91 1.76 Chile IPSA 3675.32 4.47 Argentina MerVal 48214.99 2.523 Colombia COLCAP 1158.40 0.17 Currencies Latest Daily % change Brazil real 5.5645 1.64 Mexico peso 20.6820 1.06 Chile peso 756 -0.25 Colombia peso 3754.97 1.56 Peru sol 3.5938 0.17 Argentina peso (interbank) 79.0900 -0.08 Argentina peso (parallel) 153 5.23 (Reporting by Shreyashi Sanyal in Bengaluru; Editing by Kirsten