Optimal payment of mortgages

This article provides a borrower's optimal strategies to terminate a mortgage with a fixed interest rate by paying the outstanding balance all at once. The problem is modelled as a free boundary problem for the appropriate analogue of the Black-Scholes pricing equation under the assumption of the Vasicek model for the short-term rate of investment. Here the free boundary provides the optimal time at which the mortgage contract is to be terminated. A number of integral identities are derived and then used to design efficient numerical codes for computing the free boundary. For numerical simulation, parameters for the Vasicek model are estimated via the method of maximum likelihood estimation using 40 years of data from US government bonds. The asymptotic behaviour of the free boundary for the infinite horizon is fully analysed. Interpolating this infinite horizon behaviour and a known near-expiry behaviour, two simple analytical approximation formulas for the optimal exercise boundary are proposed. Numerical evidence shows that the enhanced version of the approximation formula is amazingly accurate; in general, its relative error is less than 1%, for all time before expiry.