"I think it's going to be 45 days of extreme volatility, it could spike up, it could also go the other way," Bob Dudley told CNBC Wednesday.
It's hard to be precise over how much of Iran's production will be affected by the sanctions. It largely depends on whether the country's oil-buying customers are afraid of secondary sanctions from the US if they do business with Iran. BP's competitor Total announced in August that it was pulling out of a giant oil and gas project in Iran.
But BP and Serica Energy were granted a new license Tuesday to run a North Sea gas field partly owned by Iran showing the US is willing to make some exemptions to the reach of the sanctions.
The prospects for the returning sanctions have made other consumers look for waivers from the US to continue purchasing Iran’s oil.
This, as BP’s chief testified, could help lower oil prices.
"If waivers were granted to others, to big oil consuming countries, you could see it (the price) go down, there's a lot of uncertainty right now," Dudley said.
Some analysts predict as much as 1.5 million barrels per day could be removed from the market, an event that could cause prices to rise further. On Wednesday, Brent crude futures were trading at $84.96 per barrel while US West Texas Intermediate was trading at $74.92.