Few investments at the moment make Wall Street squirm as much as European bank stocks. A contrarian with an appetite for risk might call that a reason to buy.

Prices indeed seem alluring. The Euro Stoxx Banks index has fallen by about half over the past year and now trades below the levels hit in 2009 during the global financial crisis. Many European banks trade at sharp discounts to their "tangible book value," or estimated liquidation price.

In the U.S., by contrast, even battered J.P. Morgan Chase (JPM), which has lost one-quarter of its stock value since the end of March on news of large trading losses, sells for 1.4 times tangible book value, according to FactSet Research Systems (FDS) .

Price isn't the only consideration, however. The risks facing European banks are profound.

Europe's economy isn't growing, and some banks have stubbornly high levels of bad loans. Both factors led Moody's this year to cut its credit rating on dozens of European banks; many more remain under review.

Even more worrisome, the June 17 election in Greece may decide whether that nation is likely to leave the euro zone. If it leaves, defaulting on its debt in the process, banks that hold that debt will suffer losses. Depositors could rush to withdraw funds from suspect banks, triggering a funding crisis.

On the other hand, the stakes are so high that the European Central Bank may step in to bolster the financial system, which could trigger a stock rally. If Greece leaves the euro zone, the ECB will likely cut interest rates and resume its "long-term refinancing operations," injecting low-rate financing into banks, Citigroup (C) predicted on Wednesday in a research note.

When the ECB launched a similar program on Dec. 21, the Euro Stoxx Banks index shot up by 20% over the ensuing two months.

"At some point, I think we'll see a rally where some European bank stocks will double in a matter of weeks, if not days," says Antoine Badel, an analyst at London's Alken Asset Management, which oversees $3 billion. Decisive ECB action might trigger a rise, but "things could get worse before that happens," he says.

Investors who want to buy European banks despite the risks should consider three approaches. The first is easily the safest: Target large, financially strong companies with plenty of exposure to either stable business lines or healthy economies.

For example, Chicago-based asset manager Harris Associates, which manages $74 billion, is betting on Credit Suisse Group (CS), which is based in Switzerland, outside the European Union, and has a large business providing financial services to the wealthy. Such services don't require much bank capital, and clients tend to stay for a long time, says Harris analyst Jason Long. Another Harris favorite: Banco Santander (STD), which is based in Spain but makes half of its profit in Latin America.

Companies like these fetch a premium to the sector, but anxiety surrounding Europe has made them cheaper than they otherwise would be, Mr. Long says. Credit Suisse trades at 1.3 times tangible book value, and Santander, 1.0 times. The median for Europe's largest 20 banks is 0.8.

Mutual-fund investors can follow such an approach through the Oakmark International fund, which is managed by Harris and has a 69% stake in Europe and 25% in financials. It has lost 15.7% over the past year, but the broader universe of foreign, large-company value funds has lost 17.9% during that time, according to Morningstar (MORN) . Over the past 10 years, the fund has returned 6% a year, ranking among the top 8% of peers. There is no upfront sales charge; yearly expenses are 1.06% of assets.

A second approach -- and a more daring one -- is to look for deep discounts among banks with elevated but manageable risks.

Alken's Mr. Badel says that while his firm isn't ready to bet broadly on European banks, it has bought a cluster of Italian ones, including UniCredito Italiano (UCG.MI), which trades at 0.5 times tangible book value; Banco Popolare (BP.MI), 0.4 times; and Banca Popolare di Milano (PMI.MI), 0.3 times.

The reasoning: Italy didn't have a severe housing bubble, and its households are relatively wealthy. Also, it is one of only a handful of European Union countries with a primary budget surplus, meaning the government collects more than it spends, not including debt interest. "We think it's likely to pay its debt," Mr. Badel says.

Morningstar analyst Erin Davis favors two banks she characterizes as "not for the faint of heart": Royal Bank of Scotland Group (RBS), at 0.3 times tangible book value, and France's BNP Paribas (BNP.PA), at 0.6 times. The former is 85% owned by the British government as the result of a bailout, but seems well-capitalized and likely to reinstate its common stock dividend within 12 to 18 months, she says. Paribas has some bad loan exposure, but Ms. Davis views the price as more than low enough to offset the risk for speculative investors.

A third approach is to identify pairs of banks whose prices seem mismatched, and buy shares of one while "selling short," or betting against, the other. Short sales carry considerable risks and are therefore best left to experienced, deep-pocketed traders.

Jon Peace, an analyst at Nomura Equity Research, says BNP is likely to outperform Spain's Banco Bilbao Vizcaya Argentaria (BBVA.MC), which trades at 1.0 times tangible book value, because France's recent election has reduced uncertainty there, while Spanish-asset quality concerns are likely to persist. He also expects the U.K.'s Barclays (BCS), at 0.4 times tangible book value, to outperform Deutsche Bank (DB), 0.9 times, because the former is increasing its earnings faster.

Too nervous to buy European bank stocks, despite the discounts? You won't miss out on the excitement. Once the sector takes a turn, for better or worse, its effects are likely to be felt by investors world-wide.