Abstract
This paper considers in detail a realistic mortgage valuation model (including the potential for early prepayment and the risk of default), based on stochastic house-price and interest-rate models. As well as the development of a highly accurate numerical scheme to tackle the resulting partial differential equations, this paper also exploits singular perturbation theory (a mathematically rigorous procedure, based on the idea of the smallness of the volatilities), whereby mortgage valuation can be accurately approximated by very simple closed-form solutions. Determination of equilibrium contract rates, previously requiring many computational hours is reduced to just a few seconds, rendering this a highly useful portfolio management tool; these approximations compare favorably with the full numerical solutions. The method is of wide applicability in US or other mortgage markets and is demonstrated for UK fixed-rate mortgages, including insurance and coinsurance.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Azevedo-Pereira, J. A. (1997). Fixed Rate Mortgage Valuation Using a Contingent Claims Approach. PhD thesis, University of Manchester, Manchester, UK.
Azevedo-Pereira, J. A., Newton, D. P., & Paxson, D. A. (2000). Numerical solution of a two state variable contingent claims mortgage valuation model. Portuguese Review of Financial Markets, 3, 35–65.
Azevedo-Pereira, J. A., Newton, D. P., & Paxson, D. A. (2002). UK fixed rate repayment mortgage and mortgage indemnity valuation. Real Estate Economics, 30, 185–211.
Azevedo-Pereira, J. A., Newton, D. P., & Paxson., D. A. (2003). Fixed rate endowment mortgage and mortgage indemnity valuation. Journal of Real Estate Finance and Economics, 26, 197–221.
Brunson, A. L., Kau, J. B., & Keenan, D. C. (2001). A fixed-rate mortgage valuation model in three state variables. Journal of Fixed Income, 11, 17–27.
Buser, S.A, & Hendershott, P. H. (1984). Pricing default-free fixed rate mortgages. Housing Finance Review, 3, 405–429.
Chatterjee, A., Edmister, R. O., & Hatfield, G. B. (1998). An empirical investigation of alternative contingent claims models for pricing residential mortgages. Journal of Real Estate Finance and Economics, 17, 139–162.
Collin-Dufresne, P., & Harding, J. P. (1999). A closed form formula for valuing mortgages. Journal of Real Estate Finance and Economics, 19, 133–146.
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1979). Duration and the measurement of basis risk. Journal of Business, 52, 51–61.
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1981). A re-examination of traditional hypotheses about the term structure of interest rates. Journal of Finance, 36, 769–799.
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985a). An inter-temporal general equilibrium model of asset prices. Econometrica, 53, 363–384.
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985b). A theory of the term structure of interest rates. Econometrica, 53, 385–407.
Crank, J. (1984). Free and Moving Boundary Problems. Clarendon Press, Oxford, UK.
Downing, C., Stanton, R., & Wallace, N. (2005). An empirical test of a two-factor mortgage valuation model: How much do house prices matter? Real Estate Economics, 33, 681–710.
Epperson, J. F., Kau, J. B., Keenan, D. C., & Muller III., W. J. (1985). Pricing default risk in mortgages. AREUEA Journal, 13, 152–167.
Foster, C., & Van Order, R. (1984). An option-based model of mortgage default. Housing Finance Review, 3, 351–372.
Foster, C., & Van Order, R. (1985). FHA terminations: A prelude to rational mortgage pricing. AREUEA Journal, 13, 273–291.
Garabedian, P. R. (1998). Partial Differential Equations. AMS Chelsea Publishing, Rhode Island.
Holmes, M. H. (1995). Introduction to Perturbation Methods. Springer, New York.
Kalotay, A., Yang, D., & Fabozzi, F. J. (2004). An option-theoretic prepayment model for mortgages and mortgage-backed securities. International Journal of Theoretical and Applied Finance, 7, 949–978.
Kau, J. B., Keenan, D. C., Muller III, W. J., & Epperson, J. F. (1992). A generalized valuation model for fixed-rate residential mortgages. Journal of Money Credit and Banking, 24, 279–299.
Kau, J. B., Keenan, D. C., Muller III, W. J., & Epperson, J. F. (1995). The value at origination of fixed-rate mortgages with default and prepayment. Journal of Real Estate Finance and Economics, 11, 5–36.
Kevorkian, J., & Cole, J. D. (1996). Multiple Scale and Singular Perturbation Methods. Heidelberg, Springer-Verlag.
Longstaff, F. A. (2005). Borrower credit and the valuation of mortgage-backed securities. Real Estate Economics, 33, 619–661.
Merton, R. C. (1973). The theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183.
Nayfeh, A. H. (2000). Perturbation Methods. New York, Wiley Classic Series.
Pliska, S. R. (2005). Mortgage Valuation and Optimal Refinancing. Stochastic Finance: Proceedings of a 2004 Conference in Lisbon, Portugal, Springer-Verlag, Heidelberg.
Schwartz, E. S., & Torous, W. N. (1989). Prepayment and the valuation of mortgage backed securities. Journal of Finance, 44, 375–392.
Schwartz, E. S., & Torous, W. N. (1992). Prepayment, default and the valuation of mortgage pass-through securities. Journal of Business, 65, 221–239.
Titman, S. D., & Torous, W. N. (1989). Valuing commercial mortgages: an empirical investigation of the contingent-claims approach to pricing risky debt. The Journal of Finance, 44, 345–373.
van Dyke, M. (1975) Perturbation Methods in Fluid Mechanics. Palo Alto, California: Parabolic Press.
Vandell, K. D. (1995). How ruthless is mortgage default? A review and synthesis of the evidence. Journal of Housing Research, 6, 245–264.
Widdicks, M., Duck, P. W., Andricopoulos, A. D., & Newton, D. P. (2005). The black-scholes equation revisited: Asymptotic expansions and singular perturbations. Mathematical Finance, 15, 373–391.
Wilmott, P., Dewynne, J. N., & Howison, S. (1993). Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford, UK.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sharp, N.J., Newton, D.P. & Duck, P.W. An Improved Fixed-Rate Mortgage Valuation Methodology with Interacting Prepayment and Default Options. J Real Estate Finan Econ 36, 307–342 (2008). https://doi.org/10.1007/s11146-007-9055-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11146-007-9055-5