"We were prepared for volatility, we had spent a good deal of time as a responsible financial institution, putting together teams globally that could deal with any result emerging from the Brexit vote and we were able to see that that was working," Wright added.
"Yes, that Friday (following the vote) was a volatile day but things have picked up a great deal since then and what we have to do now is look for is the stability which our financial position gives us."
Banks like Barclays are preparing for a predicted economic slowdown in the U.K. and potentially beyond following the vote. To mitigate the risk of a slowdown, Wright said the bank needed to continue with its strategy, announced earlier this year to streamline the bank, reducing its non-core business and head-count.
"That is the way we are going to build this bank and we will stand by it," he said.
Bank of England Governor Mark Carney tried to reassure markets following the vote by emphasizing the strength and resilience of the U.K.'s economy and banking sector. He also pledged an additional £250 billion ($330.7 billion) of liquidity and said he will not hesitate to support the economy if necessary.
Barclays' Wright said that the way in which central banks around the world handled the situation in the "coming weeks and months" was crucial.
"There's no doubt at all that the vote had an effect on central banking policy but as Mark Carney said in his statement, he spoke strongly about the resilience of the U.K.` economy and its banking system and the capacity of the nation to deal with all this."
As the U.K.'s Conservative party goes through the process of electing a new leader and prime minister, expected to be in early September, Wright said the bank would be "responding to the situation as our clients require."
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