# A Steady State Solution to a Mortgage Pricing Problem

## TL;DR

This paper derives an analytical steady-state solution for a mortgage pricing model with prepayment options, assuming CIR interest rate dynamics, using variational methods to solve a free boundary problem.

## Contribution

It provides the first analytical solution to a mortgage pricing problem with prepayment under CIR interest rate dynamics using variational methods.

## Key findings

- Analytical solution to the free boundary problem is obtained.
- The free boundary is characterized by a transcendental algebraic equation.
- The model captures the impact of CIR interest rates on mortgage prepayment pricing.

## Abstract

This paper considers a mortgage contract where the borrower pays a fixed mortgage rate and has the choice of making prepayment. Assume the market interest follows the CIR model, a free boundary problem is formulated. Here we focus on the infinite horizon problem. Using variational method, we obtain an analytical solution to the problem, where the free boundary is implicitly given by a transcendental algebraic equation.