Stock markets in Asia sounded a cautionary note on Tuesday after a big rise on Wall Street, reflecting investor jitters over the state of the world economy.

Major markets in the region were up less than 1 percent at midday, though Hong Kong and South Korean shares bucked the trend and rose about 1.5 percent. The caution extended to futures markets, which were predicting modest rises on Wall Street and in Europe later in the day.

Other signs of cheer could be found in the markets. Oil futures rose on Tuesday after tumbling to their lowest levels since 2002. Oil prices offer a handy gauge of how investors expect the economy to perform. Further emphasizing the point, the price of gold, long considered a safe place to park money, fell in Tuesday trading. Prices for longer-term U.S. Treasury bonds, another safe haven, also fell.

Still, signs of worry were not far away. An index that tracks stock market volatility, known as VIX, fell on Monday in the United States but remains at historically high levels.

Stocks on Wall Street rose on Monday as investors bid up shares of health care companies as they reported progress on products that could help with the coronavirus outbreak.

The S&P 500 climbed more than 3 percent, adding to a strong showing last week. The S&P 500 had risen 10 percent last week after a three-day run that was its best since 1933, amid relief over Washington’s $2 trillion spending plan.

But there were lingering signs of caution in the financial markets. Most notably, oil prices tumbled to their lowest levels since 2002.

And in the stock market, Monday’s rally came on relatively light volume, said Matt Maley, chief market strategist at Miller Tabak, a trading and asset management firm. That suggests a lack of conviction among investors, he said.

“It’s a little bit of lack of confidence,” said Mr. Maley, “And you can’t blame them after what’s happened.”

In the oil market, brent crude, the international benchmark, fell more than 6 percent to roughly $26 a barrel on Monday. West Texas Intermediate, the U.S. benchmark, was down more than 5 percent with prices hovering around $20.25 in early afternoon trading. Earlier in the morning the price had briefly dropped below $20 a barrel, a level not seen in almost 20 years.

Facebook and Twitter took down posts featuring Brazil’s president, Jair Bolsonaro, over the past two days after he claimed that the anti-malaria drug hydroxychloroquine was a “cure everywhere” for the coronavirus and called for an end to social distancing and shelter-in-place measures in Brazil.

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