This could be the trade of the decade

U.S. and Japanese flags are displayed at a foreign exchange firm in Tokyo.
Itsuo Inouye | AP
U.S. and Japanese flags are displayed at a foreign exchange firm in Tokyo.

After months of going nowhere, the dollar has started to take off against the yen, and pros say this is just the beginning.

The last time the dollar was this strong against the yen was in the thick of the financial crisis in September 2008.

Diverging policy paths between the Federal Reserve and the Bank of Japan is the trigger for the move, with the U.S. economy strengthening, and the Japanese economy withering. That's raised hopes for the BOJ to ease further, which weakens its currency, and for the Fed to start thinking about tightening or raising interest rates, bullish for the dollar.

So how much farther does this trade have to run?

Read MoreIt's not about the stronger dollar…at least not yet

Dollar-yen could double, according to Jeffrey Gundlach, noted bond guru and founder of DoubleLine Capital, who predicts we'll see the dollar strengthen to 200 yen in three to five years. That's an 87 percent move.

"I think this breakdown in the yen has pretty high momentum. I wouldn't be surprised to see the yen get substantially weaker," he told "Squawk on the Street" this week.

Alan Ruskin, global head of Group of 10 foreign exchange strategy at Deutsche Bank, predicts the dollar finishes the year at 112 against the yen, another 4.5 percent move. He said it will then rise further to finish 2015 at 120.

But he said it has scope to go even higher than that.

"If U.S. rates fully normalize over a three-year period (fed funds near 4 percent by early 2018) and Bank of Japan rates still are at zero, you could perhaps see dollar-yen at 140."

Kathy Lien, longtime forex strategist who started BK Asset Management, predicts the pair will hit 110 by the end of the year and 113 to 115 next year.

"We'll be moving into a period [next year] where central banks are tightening—including Bank of England and the Fed—and the global economy will be more stable. That's the perfect backdrop for carry trades to return," Lien said.

In a carry trade, investors buy higher-yielding currencies with higher interest rates and fund them by selling lower-yielding currencies, typically the yen. Usually they thrive during environments of economic and financial calm and stability, better known as risk-on.

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David Woo's long-held forecast is 108 by the end of 2014, and 112 by the end of 2015, though "the risk is to the upside of our forecasts," said Woo, head of global rates and currencies research at Bank of America Merrill Lynch.

Bears say the decline in the yen will be fueled by the very medicine trying to help the ailing Japanese economy, with the BOJ forced to fight the weakness with more monetary stimulus and quantitative easing, a prescription for a weaker currency.

"They have the worst demographics in the world. They are importing energy, unless they go back to nuclear which there have been some fears of understandably. They have a negative birth rate. They have huge government debt. Do you have any ideas? I think all you pretty much have is debasing, " according to Gundlach.

He says weakening the yen via QE is the only option, "the mother of all debasement."

Read More Gartman: Bull market in dollar just started

Japan's economy shrank 7.1 percent in the most recent quarter at an annualized rate, thanks in large part to frozen consumer spending after the government increased the sales tax by 3 percent.

There are indications Bank of Japan Gov. Haruhiko Kuroda is open to doing more to stimulate the economy.

On Thursday, Kuroda met with Prime Minister Shinzo Abe and assured him that he would do more if needed, especially if the BOJ is still failing to meet its 2 percent inflation target.

"Should conditions emerge where the target becomes difficult to meet, we are ready to make without hesitation adjustments to policy, additional easing or whatever," Kuroda told reporters after the meeting.

Trading the stronger dollar
Trading the stronger dollar   

Societe Generale's Tokyo-based economist Takuji Aida predicts additional QE announcements by April 2015.

"The BOJ is mostly likely to enhance the quality rather than the quantity of its asset purchases. We predict an additional 10 trillion yen monetary base increase from 60-70 trillion per year ... made up of purchases of ETF and other risk assets, fund provisioning measures to stimulate bank lending and JGBs," he wrote this week.

Some say the Bank of Japan could ramp up its QE program as soon as this fall.

Read MoreBOJ steps up easing, drives rates below zero

The other factor pushing dollar-yen higher is a strengthening U.S. economy. Data releases from manufacturing to services to jobs continue to improve in the U.S., and that has investors betting on a more hawkish Fed and eventually higher Treasury yields.

Dollar-yen, in particular, tends to track U.S. Treasury yields.

"For dollar-yen to move higher, we need U.S. rates to back up to get the Japanese to buy more U.S. bonds. I believe U.S. rates have been depressed artificially this year by foreign official buying and U.S. bank buying of Treasuries."

Woo cites a variety of factors.

He says the Fed's recent ruling on the liquidity coverage ratio will lead bank buying of Treasurys to slow down, while Chinese reserve accumulation and foreign buying continue to ease.

Read MoreECB's Liikanen: More stimulus possible if necessary

This week, Treasury prices have been falling, pushing the 10-year yield to its highest level since the end of July.

Investors speculate the Federal Reserve might shift its tone and language about the timing of interest rate increases ahead of next week's two-day policy meeting.

In other words, the Fed is moving in the opposite direction of its Japanese counterpart, and that is what experts say will continue to create a buying opportunity for the dollar against the yen and continue to push the currency pairs to new highs.

—By CNBC's Sara Eisen