In Europe, John Thain, chief executive of the New York Stock Exchange, and Bob Greifeld, the Nasdaq boss, are so far only shadow-boxing over the future ownership of the London Stock Exchange.
But back home in the Big Apple, the pair are about to engage in something closer to hand-to-hand combat.
The NYSE will launch a new electronic trading system this week that allows investors to by-pass its historic floor traders.
The initiative is its latest attempt to claw back some of the share trading that has leaked over to automated trading platforms in recent years, and made Nasdaq a major rival for listings of all sizes.
And the NYSE is also waging a lobbying campaign to deprive Nasdaq of as much as 5 per cent of its revenues, in a dispute over who gets to publish information about off-exchange trading.
The competitive threat from Nasdaq and others, though, is so acute that the NYSE now accounts for barely 70 per cent of the trading in shares of the companies listed on it. And it has prompted Mr Thain to accelerate the roll-out of a new hybrid system that introduces electronic trading, while still allowing some trades to be routed through the trading floor.
Lucent Technologies - the telecoms equipment group - will on Friday become the first company to trade under the hybrid system, which could be extended to cover other stocks this summer.
Mr Thain, a former Goldman Sachs executive who joined the NYSE in 2004, is trying to modernise the 214-year-old exchange in the teeth of traditionalists wedded to its trading floor. Junius Peake, a former trustee of the National Association of Securities Dealers and now professor of finance at the University of Northern Colorado, said: "In the end, I think, the NYSE's customers are going to say the hybrid system is too damn complicated, and its shareholders are going to demand a cheaper system. They will make their big money when the NYSE closes the floor, gets rid of the costs of having all those people running around on it, and replace it with a bank of computers."
Meanwhile, Mr Thain is lobbying regulators to strip Nasdaq of its role in reporting trades that happen, away from the public markets, in-house at the giant investment banks.
These are increasingly settling clients orders using stock that they have on their own books, cutting out the costs of using an exchange. These trades are published via Nasdaq, earning the NYSE's rival some $25m a year, 5 per cent of its income.
The NYSE says this is unfair, and the war of words is raising the temperature between Mr Greifeld and Mr Thain at a time when Nasdaq appears to have outmanoeuvred the NYSE in the battle for the London Stock Exchange. Both men would like to take over the London market, but Nasdaq has bought a blocking stake of 19 per cent in the past few weeks, while Mr Thain has been busy integrating the NYSE with its merger partner, electronic trading group Archipelago.Reuse content