Two of the nation’s biggest lenders, JPMorgan Chase and Bank of America, are quietly modifying loans for tens of thousands of borrowers who have not asked for help but whom the banks deem to be at special risk.

Rula Giosmas is one of the beneficiaries. Last year she received a letter from Chase saying it was cutting in half the amount she owed on her condominium.

Ms. Giosmas, who lives in Miami, was not in default on her 0,000 loan. She did not understand why she would receive this gift — although she wasted no time in taking it.

Banks are proactively overhauling loans for borrowers like Ms. Giosmas who have so-called pay option adjustable rate mortgages, which were popular in the wild late stages of the housing boom but which banks now view as potentially troublesome.

Before Chase shaved 0,000 off her mortgage, Ms. Giosmas owed much more on her place than it was worth. It was a fate she shared with a quarter of all