Bruce Karatz, former chief executive of KB Home, spent months denying that stock options had been backdated at the company. Then, in October 2006, pressed with damning e-mails produced by the company’s outside lawyers, Karatz changed his story, admitting that he and the head of human resources, Gary Ray, had been looking back in time to date options.

Those actions, which began about six years earlier, amounted to fraud, Assistant U.S. Attorney Paul G. Stern argued on Thursday as the government closed a month-long trial against Karatz.