Global stocks rallied Monday as investors continued to ride a wave of optimism about the gradual process of reopening the world economy.

Several countries—as well as some U.S. states—have begun allowing businesses to resume operations . The Bank of Japan also took action to support lending markets under stress from the coronavirus shock, promising to buy more bonds and other forms of debt.

The Dow Jones Industrial Average closed up 358.5 points, or 1.5%, in line with gains for the S&P 500. The Nasdaq Composite rose 1.1%. The small-cap Russell 2000 index jumped 4%, again reflecting investors’ economic optimism.

“There is a feeling in the markets that things are starting to go back to normal, albeit at a very slow pace,” said David Madden, market analyst at CMC Markets U.K.

Reopening of New York state could begin in mid-May, according to Gov. Andrew Cuomo. Prime Minister Giuseppe Conte said Italy’s lockdown will be eased from May 4, while France is due to present an exit strategy on Tuesday.

U.K. Prime Minister Boris Johnson, returning to work after being hospitalized for coronavirus, said announcements on reopening would come, but that now is also a time of “maximum risk.”

Even as countries look ahead to reopening their economies, central banks are taking action to limit the damage that continues to result from the shutdowns. The Bank of Japan lifted a cap on its purchases of Japanese government bonds and increased its buying of commercial paper and bank loans. The U.S. Federal Reserve and European Central Bank also are seen as likely to take more action.

Stock indexes overseas also rose. The Stoxx Europe 600 index closed up 1.8% on Monday, while the German DAX rose 3.1%, the French CAC 40 added 2.5%, and the U.K.’s FTSE 100 index ticked up 1.6%.

In Asia, South Korea’s Kospi Composite gained 1.8% and Hong Kong’s Hang Seng climbed 1.9%. Japan’s Nikkei 225 surged 2.7%.

Haven assets were down. The price of gold slipped 0.7% to $1,728.30 an ounce. The yield on the 10-year U.S. Treasury note jumped 6 basis points, or hundredths of a percentage point, to 0.656%, as the price of the securities fell. The U.S. Dollar Index (DXY)—which measures the greenback against a basket of other currencies—lost 0.4%.

The Cboe Volatility Index, or VIX, fell 8.4% on Monday to below 33 points. It’s the lowest the VIX has gone since early March.

The stock gains came despite continued volatility in the oil market. Stocks fell last week when the price for West Texas Intermediate crude for May delivery turned negative, and rebounded when oil recovered later in the week.

On Monday, futures for June delivery of WTI crude fell 24.6% to $12.78 a barrel on continued worries about a lack of storage space. The output of oil is outpacing consumption and producers are on track to run out of places to store their oil in a matter of weeks. Brent, the main international oil-price benchmark, dropped 6.8% to $19.99 a barrel.

Oil-exposed stocks held up relatively well despite oil’s drop. Shares of Exxon Mobil (ticker: XOM) rose 0.4%, Occidental Petroleum (OXY) was closed up 2.5%, and Schlumberger (SLB) slipped 0.4%.

Diamond Offshore Drilling (DO) shares plunged 61% in premarket trading after filing for Chapter 11 bankruptcy protection, following a steep drop in oil-drilling demand. Trading in the stock was halted at the open.

Loews (L), which owns a 53% stake in Diamond Offshore, said it expects to book a “significant” noncash loss this quarter on its stake in the company. Its shares ticked up 0.5%

Reopening won’t mean economies snap back to normal right away, with travel and hospitality demand likely to remain depressed for much longer. InterContinental Hotels Group (IHG) said that at sites where it is open—most of them in China and in the U.S.—occupancy is in the low-to-mid 20% level. Its stock closed up 6.7%.

Deutsche Bank (DB) shares jumped 12.4% as Germany’s leading bank said its first-quarter profit will be higher than market expectations, at €66 million ($71.5 million). Its full first-quarter results will be released on Wednesday.

Boeing (BA) shares slipped 0.3% after the plane manufacturer backed out of a $4.2 billion deal to acquire an 80% stake in Brazil’s Embraer. Boeing is holding its annual meeting on Monday .

Caterpillar (CAT) shares added 1% after Morgan Stanley downgraded the company to Equal-Weight due to an expected downturn in construction activity.

General Motors (GM) shares recovered earlier losses to rise 2.4% after announcing that it was suspending its dividend and share-repurchase program. The auto maker joins a long list of companies that have suspended or cut their dividends to preserve cash.

Write to Steve Goldstein at steven.goldstein@wsj.com, Carleton English at carleton.english@dowjones.com and Nicholas Jasinski at nicholas.jasinski@barrons.com

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