We study 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 a Black-Scholes type pricing equation under the assumption of the Vasicek model for the short 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 thefree boundary. Fornumerical simulation, parameters for the Vasicek model areestimated via the method of maximum likelihood estimate using 40 years of datafrom US government bonds. The asymptotic behavior of the freeboundary for the infinite horizon is fully analyzed. Interpolatingthis infinite horizon behavior and a known near expiry behavior, two simple analytical approximation formulas for the optimalexercise boundary are proposed. Numerical evidence shows that theenhanced version of the approximation formula is amazinglyaccurate; in general, its relative error is less than $1%$, forall time before expiry.
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