Your question is as much a legal question as it is a business question. Without going into too many specifics, it is common for lenders to bid the exact dollar amount of their security interest (mortgage) in the property. With that being said, some lenders choose to bid a lesser amount than their security interest for other reasons I will not get into with this comment.
If you allow your mortgage to fall into foreclosure, beweare of the consequences: Lenders commonly seek their legal fees and costs against you as well as interest on past due amounts. Additionally, there is a possibility the lender/creditor will seek to collect a deficiency judgment against you.
Nothing in this comment constitutes legal adivce nor establishes an attorney-client relationship.
The foreclosing mortgage lender will almost certainly bid the total amount owing, which would include late payment penalties, attorney's fees, and other foreclosure costs. Consequently, you will not be able to "redeem" for the home's current market value. If your current monthly mortgage payments are manageable, and if you are quite satisfied living where you're living, you should probably not default on the mortgage. However, from a purely financial standpoint, you would be likely to save a great deal of money by defaulting. However, it would almost certainly entail your moving from your home approximately a year after you stop making mortgage payments (in the meantime, you could be saving the money you would otherwise be paying to your mortgage lender), renting another home for several years while your credit is rehabilitated, and then purchasing another home.
The bid amount is determined solely in the foreclosing party's discretion. I have seen the banks bid in the full amount owed plus allowable fees and costs and I have seen them bid one-third of the outstanding mortgage. You're playing a pretty highstakes game of chicken if the foreclosure proceeds with the hope they will bid in the FMV of the property or less.