# An Optimal Mortgage Refinancing Strategy with Stochastic Interest Rate

- 153 Downloads

## Abstract

This article puts forward a framework for assessing the optimal refinancing strategy in continuous time when the interest rate is stochastic and follows a Vasicek model. The optimal refinancing time is obtained by minimizing the conditional expectation of the discounted total payment. A moment generating function is used to derive a closed-form approximation to the refinancing function with infinite maturity under the Vasicek model. The approximation is studied both analytically and numerically. The results indicate three different types of behaviour in the refinancing function, depending on the underlying parameters in the model. Two types indicate optimal refinancing in finite time. We outline a strategy by which a borrower can continually evaluate whether to refinance. By providing a systematic way to evaluate the likelihood of refinancing, these results should be of interest to those trading mortgage-backed securities.

## Keywords

Fixed rate mortgage Optimal refinancing Vasicek model Analytical approximation## References

- Agarwal, S., et al. (2002). When should borrowers refinance their mortgages? Working paper, National Bureau of Economic Research, Summer Institute.Google Scholar
- Agarwal, S., et al. (2013). Optimal mortgage refinancing: A closed-form solution.
*Journal of Money, Credit and Banking*,*45*(4), 591–622.CrossRefGoogle Scholar - Bladt, M., & Nielsen, B. (2010). Multivariate matrix-exponential distributions.
*Stochastic Models*,*26*(1), 1–26.CrossRefGoogle Scholar - Cairns, A. J. (2004). A family of term-structure models for long-term risk management and derivative pricing.
*Mathematical Finance*,*14*(3), 415–444.CrossRefGoogle Scholar - Chen, A., & Ling, D. (1989). Optimal mortgage refinancing with stochastic interest rates.
*AREUEA Journal*,*17*(3), 278–299.CrossRefGoogle Scholar - Dunn, K., & Spatt, C. (1999). The effect of refinancing costs and market imperfections on the optimal call strategy and the pricing of debt contracts. Working paper, Carnegie Mellon University.Google Scholar
- Freddie Mac. (2016).
*15-year fixed-rate mortgages since 1991*. http://www.freddiemac.com/pmms/pmms15.htm. Accessed 18 May 2016. - Gan, S., et al. (2012). When to refinance mortgage loans in a stochastic interest rate environment. In
*International multiconference of engineers and computer scientists*, Hong Kong (Vol. 2). Working paper.Google Scholar - Kalotay, A., et al. (2004). An option-theoretic prepayment model for mortgages and mortgage-backed securities.
*International Journal of Theoretical and Applied Finance*,*7*(08), 949–978.CrossRefGoogle Scholar - Klebaner, F. (2012).
*Introduction to stochastic calculus with applications*. London: Imperial College Press.CrossRefGoogle Scholar - Longstaff, F. A. (2004).
*Optimal recursive refinancing and the valuation of mortgage-backed securities*. Working paper 10422, National Bureau of Economic Research.Google Scholar - Siegel, J. (1984). The mortgage refinancing decision.
*Housing Finance Review*,*3*(1), 91–97.Google Scholar - Stanton, R. (1995). Rational prepayment and the valuation of mortgage-backed securities.
*Review of Financial Studies*,*8*(3), 677–708.CrossRefGoogle Scholar - Vasicek, O. (1977). An equilibrium characterization of the term structure.
*Journal of Financial Economics*,*5*(2), 177–188.CrossRefGoogle Scholar - Wackerly, D., et al. (2007).
*Mathematical statistics with applications*(7th ed.). Boston: Brooks/Cole Cengage Learning.Google Scholar - Xie, D., et al. (2017). Simulation solution to a two-dimensional mortgage refinancing problem.
*Computational Economics*. https://doi.org/10.1007/s10614-017-9689-1.