(Reuters) - The euro dived to its lowest since early 2003 against the dollar on Wednesday, dragging other European currencies with it on the back of the huge differences developing in market interest rates between Europe and the United States.

In a dollar rally that began last July the single currency has lost around a quarter of its value, and there is little sign of that bottoming out. Deutsche Bank on Tuesday forecast a fall to 85 U.S. cents by the end of 2017.

That comes largely courtesy of the collapse in European bond yields, which are set to stay low for the foreseeable future under the program of money-printing launched by the European Central Bank on Monday. Yields on German 30-year government bonds are now lower than those on U.S. two-year paper.

The picture for European stock markets, given the projected 1 trillion of new euros set to flow into the financial sector, is less grim, and the main indexes were all higher in early trade.

"It's the euro versus everything," said Stephen Gallo, European head of FX strategy at BMO Capital Markets in London. "The way these moves look, it's not just speculators piling into euro shorts, it's actually net flow of capital out of the euro."

"There's a pick-up in Eurosystem excess liquidity just as the Fed is playing with its own operations to try to withdraw liquidity and as the rhetoric is tilted towards rate hikes."

The euro sank as low as $1.0638 in early European deals, its weakest since 2003. The Norwegian and Danish crowns both hit similar lows and the Swedish crown a six-year low.

There is little optimism to be found in Asia, where poor data out of China helped stock markets fall to two-month lows.

MSCI's broadest index of Asia-Pacific shares outside Japan was down 0.4 percent after touching its lowest since January. Australian shares lost 0.5 percent and Malaysian and Indonesian stocks also declined.

The falls there and on Wall Street overnight contrasted with gains of around 1 percent in Europe, pointing to the impact of the ECB's quantitative easing program.

"It is remarkable... that European equities like the DAX are still materially cheaper than US equities and still down versus last summer in dollar terms," said BNP Paribas head of global equities and derivatives strategy Gerry Fowler.

"I suspect that in another six months the market may look back and think that a further 20 percent rally (and outperformance) in European equities should not have been considered all that unlikely under the circumstances."

Japan's Nikkei added 0.3 percent as better-than-expected machinery orders helped offset Wall Street losses.

But the deepening decline in the yen, usually a positive factor for Japanese stocks as it buoys exporters, led some to worry about other consequences like the burden placed on importers.

"The market started to worry about side-effects from a further slide in the yen," said Hiroichi Nishi, general manager at SMBC Nikko Securities in Tokyo, adding that there are also concerns that a stronger dollar hurts U.S. multinational companies' earnings.