A striking holiday bargain emerged last week: government warrants in bailed-out banks.

The Treasury Department received a relatively paltry $936 million in an auction for warrants in JPMorgan Chase & Co., affirming that investors want a steep discount to take these kinds of securities off the government’s hands, industry watchers said.

The sale of 88.4 million warrants at $10.75 apiece mirrored the auction a week earlier of warrants in Capital One Financial Corp., which also fetched less than industry experts had expected.

Taken together, the two auctions are setting a market price for the warrants the Treasury took in the hundreds of banks that received federal aid. The value of these warrants has been a major sticking point between the government and banks looking to exit the Troubled Asset Relief Program, and the results last week offer other banks precious intelligence for their negotiations.

“We thought they would go out north of 12 bucks. So less than 11 was lower than we thought, although not surprising in light of the Capital One warrants,” said Espen Robak, the president of Pluris Valuation Advisors LLC, a research firm that specializes in valuing options.

The Treasury decided to auction its warrants in JPMorgan Chase, Capital One and TCF Financial Corp. after the parties failed to settle on a price. (TCF’s warrants are expected to come to market shortly.) Critics had knocked the Treasury early on for not getting enough from banks that repurchased warrants, but these three companies — the first to see their warrants to go to auction — balked at the Treasury’s asking price.

Fred Cannon, a co-director of research at KBW Inc.’s Keefe, Bruyette & Woods Inc., said these auctions should clear up some of the confusion. “What really causes negotiations to break down is uncertainty on both sides,” he said. “We’re moving away from a theoretical discussion to a market price.”

The auctions should also make it easier for banks to decide whether they want to dole out the extra capital to buy and retire their warrants when they return their federal aid, he said. This has been a tough call for some lenders, he said, because they have not been sure what their warrants are worth.

“It makes the decision much more transparent,” he said. “It gives clarity to the decision. It takes the valuation issue off the table.”

Placing a valuation on options and warrants is always tricky, and traders rely on complex formulas that take into account a company’s stock price and the strike price of the warrant, as well as a highly subjective projection of its future volatility.

Guessing the value of the Treasury warrants had been extremely difficult before the JPMorgan Chase and Capital One auctions because no comparable warrants had come to market in recent memory, according to Linus Wilson, an assistant professor of finance at the University of Louisiana at Lafayette.

The Treasury warrants can be exercised until 2018. There were no options — which are similar to warrants — trading in the United States with a call date beyond 2014 two weeks ago, he said.

In the most recent auction, investors showed that they are not willing to pay as much for the longer-dated bank warrants as they are for shorter-dated options. For instance, the JPMorgan Chase warrants auctioned Thursday had an implied volatility of 23.1%. The companies options with a call date of 2012 had volatility of 31.1% that same day. Warrants that do not expire for nine years should, theoretically, have higher volatility — and thus be more valuable — than options that can be called in two years.

“These are indeed trading cheap,” said Nicholas Waltner, managing principal at Kulshan Capital Management LLC, an investment adviser in Seattle. He said the JPMorgan Chase auction is a validation of sorts for the New York lender and its chief executive, Jamie Dimon, who said publicly when the company repaid its $25 billion of federal aid in June that the Treasury was seeking too much for the warrants.

Waltner said the amount Goldman Sachs paid for its warrants around the same time indicates that the Treasury was asking a lot. Goldman paid $1.1 billion, indicating an implied volatility of 43.9%.

With a better understanding of what these warrants are worth, more companies could take JPMorgan Chase’s stance if the Treasury continues to drive a hard bargain, Waltner said. “You might see more auctions because people are saying Treasury is asking too much,” he said. “You would think that the Treasury would start backing off to more realistic assumptions.”

Bank of America Corp. said it had no intention of repurchasing its 211 million warrants from the government when it announced plans on Dec. 2 to repay its $45 billion of federal aid.