Finite Difference Approach to Steady State Problems Arising from Mortgage and Option Pricing

Motivated by mortgage valuation, the paper proposes a finite difference approach to solve a class of free boundary problems which may be useful for option pricing in general. Given certain financially meaningful conditions, a mortgage borrower wishes to find the level of market interest rate at which it is optimal to make prepayment. The problem is an analog of finding the optimal level of stock price for early exercise in American put. Mathematically they both can be formulated as free boundary problems. In this paper an algorithm based on the finite difference scheme is designed to find the numerical solution to the steady state of such problems. The approach is calibrated with the perpetual American put option whose solution is explicitly known. The efficiency of the algorithm is tested by numerical simulations.