I'm not sure we are on the same page when you say reverse mortgage so let me clarify. A reverse mortgage is a type of mortgage that someone takes out when they have equity in their home and they say to the bank - My house is worth xyz - I want you to start making payments to me based on the equity I have in my home.
I don't think that's what you meant in your question but just tell me if I'm wrong. I think you're asking how to reduce your mortgage payment when you owe more than your house is worth? Unfortunately, the answer is that you can't. Many people are in this situation right now. They purchased at the peak of the market. The value of their home dropped. The interest rates dropped. Then they find that they can't get the lower rate because of the value of their home. The reason the bank won't refinance is because they have underwriting guidelines that tell them to avoid high risk. When you're asking to borrow more money than your house could be sold for, they see big red flags and say no. They allowed you to borrow that amount before because at that time, your house was worth the amount they were lending.
I would like to talk more to you about your specific situation and how far upside down you are on your mortgage. If you have some equity, just not enough for their guidelines, I may be able to help. If you have zero equity and really owe more than your home could sell for, then I could give you some advice depending your personal situation. Give me a call at (518) 591-4629 to go over the specifics of your situation.