In the US, the Dow Jones was up 123.96 points, or 0.8%, at 15,7890.70 in early trade.
European markets were also higher, and shares in the US were up shortly after trading began.
However, investors remain worried over the continuing slide in oil prices and slowing growth in China.
London’s FTSE 100 index, which measures the share prices of the 100 most valuable companies traded on the London Stock Exchange, was up nearly 1.5%.
The index rose 83.10 points to 5,756.68, and the share indexes in France and Germany were both up by more than 1.5%.
Much earlier, Japan’s main share index closed down by more than 2%.
On January 20, global stock markets suffered hefty losses and London’s FTSE 100 ended the day down 3.5%.
By doing so it entered a “bear market”, having fallen 20% from its record high in April 2015.
Oil prices remained weak on January 21, having hit their lowest levels since 2003 in the previous session.
A brief rally in crude prices quickly ran out of steam, and after climbing back above the $28-a-barrel mark, Brent crude fell back to $27.75.
US crude was 1% lower on the day, trading at $28.09 a barrel, having fallen below $27 on January 20.
Crude oil prices have been falling since mid 2014, but oil-producing countries have maintained output despite the decline, contributing to the excess supplies on the market.
Earlier in the week, the International Energy Agency warned that oil markets could “drown in oversupply” in 2016.
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