The market has gone down and I owe more than the home is worth. Had your house appreciated in value, the lender would not have shared in the benefit, so there is no reason why the lender should take a hit if the value declines. Yet the fact is that in many cases where the borrower's equity is wiped out, the lender does take a hit simply because the borrower is unable to pay. A deed in lieu of foreclosure is a kind of friendly foreclosure in which the lender agrees not to pursue you for the balance of what is owed. It will hurt your credit rating, but not as badly as a foreclosure.
Lenders accept a deed in lieu of foreclosure when they believe the only alternative is foreclosure.
If they know the borrower is well-heeled, they will want to be paid in full. Sometimes partial payments are negotiated. You are right. The burden is on the borrower to document the distress. If the lender accepts the borrower's petition, the house is sold at a price the lender finds acceptable.