Last week, the CBOE Volatility Index <.VIX>, or the VIX, the favoured gauge of investor anxiety, climbed above its long-term average of 20. On Wednesday, a day before the referendum, the index was up 15 percent to 21.17.

Not everyone has been bracing for increased stock gyrations. Recent activity in options on the index points to sizeable trades that are betting that calm will return to U.S. stocks soon.

On Wednesday, bets on the VIX dipping below 15 by mid-July were the most heavily traded VIX contracts.

While higher trading volume in the puts need not necessarily mean that traders expect the VIX to drop, that appears to be the case today, said optionMonster.com senior analyst David Russell.

"By looking for the VIX to go down they are saying that they think there is too much fear priced into the market," he said.

A look at overall open VIX options contracts tells a similar story.

Even though the largest accumulation of contracts are in options that would benefit from a jump in volatility, there are meaningful bets on lower volatility in the July time frame.

Which is more, while large bets on a spike in volatility by mid-July were bought in mid-April, bets on lower volatility have picked-up in recent weeks, MKM Partners derivatives strategist Jim Strugger said.

"Positioning in VIX options ahead of Brexit (referendum) seems to have followed sentiment," Strugger said.

Opinion polls published in the days before Thursday's referendum have mostly shown a shift toward keeping Britain in the European Union, but there are some signs that the "In" camp's momentum has stalled and the race remained too close to call.

(Reporting by Saqib Iqbal Ahmed; Editing by David Gregorio)

By Saqib Iqbal Ahmed