According to Betfair, the odds on a Leave vote have risen from 19% to 28% since the end of last month.
The cost of insuring against such wild swings has soared.
Today the one-month sterling-dollar volatility index, which mirrors the cost of insuring the pound through the referendum vote on June 23, hit 22.02 for the first time since December 2008, when the global financial crisis was at its height.
The longer-dated three-month volatility index was just above 15, which is a level it has not touched since mid-2010.
“The market has become too comfortable with its ‘things will work out alright’ approach,” said Ulrich Leuchtmann at Commerzbank. “The market would be far less prepared for a Brexit outcome than for a ‘Bremain’. That is the reason why these polls receive far more attention and move the market much more than those polls that still see a lead for ‘Bremain.’”