South Korea warned of a second wave of the new coronavirus as infections rebounded to a one-month high, while new infections accelerated in Germany.

Still, investors seemed determined to stay optimistic and MSCI’s broadest index of Asia-Pacific shares outside Japan .MIAPJ0000PUS firmed 0.1%.

Japan's Nikkei .N225 added 0.7% and South Korean stocks .KS11 0.3%. E-Mini futures for the S&P 500 ESc1 opened soft but bounced as the morning wore on and was last up 0.3%.

Wall Street had rallied on Friday after the April payrolls report proved dire but not quite as awful as analysts’ worst fears.

“Just getting the worst jobs report in history out, is at the margins helpful for risky assets,” said Alan Ruskin, head of G10 FX at Deutsche Bank.

“Since late March there has been an extraordinary divergence between the real economy and financial risk, with the latter helped by unprecedented policy accommodation,” he added.

“Markets know the real economy data is awful. We are less sure of how long markets aided by policy, can defy the real economy, if the growth improvement is slow.”

The bond market certainly seems to think any recovery will be slow with two-year yields US2YT=RR hitting record lows at 0.105% and Fed fund futures <0#FF:> turning negative for the first time ever.

The rally in prices has come even as the U.S. Treasury plans to borrow trillions of dollars in the next few months to plug a gaping budget deficit.

Federal Reserve Chair Jerome Powell is due to give a key note speech on Wednesday and analysts suspect he will rule out taking rates negative, at least for now.

The decline in U.S. yields might have been a burden for the dollar but with rates everywhere near or less than zero, major currencies have been stuck in tight ranges.

The dollar was a shade firmer on the yen at 106.94 JPY= on Monday but well within the 105.97 to 109.37 band that has lasted since late March. The euro was a fraction softer at $1.0830 EUR= but above last week's low at $1.0765.

Against a basket of currencies, the dollar was idling at 99.837 =USD, sandwiched between support at 98.769 and resistance around 100.40.

In commodity markets, gold edged up 0.3% to $1,706 an ounce XAU=.

Oil prices opened about 1% lower as a persistent glut weighed on prices and the coronavirus pandemic eroded global oil demand, even as some governments began to ease lockdowns.

Brent crude LCOc1 futures lost 27 cents to $30.70 a barrel, while U.S. crude CLc1 fell 39 cents to $24.35.

Editing by Sam Holmes