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A New Prepayment Model (with Default): An Occupation-time Derivative Approach

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Abstract

A new prepayment model is developed, which improves the modeling of the borrowers decision process by incorporating an occupation-time derivative in the valuation framework of a fixed-rate mortgage. This option-theoretic mortgage valuation model is based on stochastic house-price and interest-rate models, and requires a particularly subtle technique to incorporate a new type of occupation-time derivative, where the barrier (which activates the derivative) is in the value process and not the underlying process (as it is in standard occupation-time derivatives). This new model simulates a delay in prepayment by the borrower (beyond the time simple ruthless prepayment dictates), thus increasing the value of the mortgage to the lender, compared to the value gained using more basic models. This allows for a more advanced borrower decision process, where a rational exercise structure is retained in a modified form. Empirical evidence supports this theory, which should be beneficial for accurate mortgage-backed security pricing. The results in this paper explore thoroughly the effect on the mortgage value of a delay in prepayment by the borrower on the embedded options held and on the insurance component.

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Notes

  1. This is the standard PDE (7) for the valuation of any asset F(H,r,τ) whose value is a function only of house price H, interest rate r, and time τ, according to Cox et al. (1985a, b), Kau et al. (1995), Azevedo-Pereira et al. (2002, 2003), Sharp (2006), Sharp et al. (2008).

  2. Haber et al. (1999) define the dynamics of barrier time \(\bar{t}\) so that it increases at the same rate as the real time t, therefore \(d\bar{t}=dt\), if the underlying S is beyond the barrier. The barrier time \(\bar{t}\) is reset to zero if S hits the barrier \(S=\bar{S}\), and does not change if \(S<\bar{S}\).

  3. Convergence of the boundary location procedure is guaranteed due to the way the discrete approximation is found. As the check for V ≥ FV is performed for increasing values of interest rate from r = 0 and it is known that prepayment region expands as we move away from the maturity of the mortgage (the same direction as the valuation procedure), the discrete approximation of the free boundary is always captured.

  4. When the mortgage value referred to is not the true value inverted commas are used to mark the distinction.

  5. Results were taken for longer maturities but were not qualitatively different.

  6. The results were obtained using a 2412 MHz AMD Athlon computer.

References

  • Ambrose, B. W., Buttimer, R. J., & Capone, C. A. (1997). Pricing mortgage default and foreclosure delay. Journal of Money, Credit and Banking, 29, 314–325.

    Article  Google Scholar 

  • Avellaneda, M., & Wu, L. (1999). Pricing Parisian-style options with a lattice method. International Journal of Theoretical and Applied Finance, 2, 1–16.

    Article  Google Scholar 

  • Azevedo-Pereira, J. A. (1997). Fixed rate mortgage valuation using a contingent claims approach. PhD thesis: Manchester Business School, The University of Manchester.

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Boudoukh, J., Whitelaw, R. F., Richardson, M., & Stanton, R. (1997). Pricing mortgage-backed securities in a multifactor interest rate environment: A multivariate density estimation approach. Review of Financial Studies, 10, 405–446.

    Article  Google Scholar 

  • Brennan, M. J., & Schwartz, E. S. (1985). Determinants of GNMA mortgage prices. AREUEA Journal, 13, 209–228.

    Google Scholar 

  • Chesney, M., Jeanblanc-Picque, M., & Yor, M. (1997). Brownian excursions and Parisian barrier options. Advances in Applied Probability, 29, 165–184.

    Article  Google Scholar 

  • Clapp, J. M., Deng, Y., & An, X. (2006). Unobserved heterogeneity in models of competing mortgage termination risks. Real Estate Economics, 34, 243–273.

    Article  Google Scholar 

  • Cornwall, J., & Kentwell, G. (1995). A quasi-analytical approach to occupation time barrier. Bankers Trust Working Paper.

  • Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985a). An inter-temporal general equilibrium model of asset prices. Econometrica, 53, 363–384.

    Article  Google Scholar 

  • Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985b). A theory of the term structure of interest rates. Econometrica, 53, 385–407.

    Article  Google Scholar 

  • Crank, J. (1984). Free and moving boundary problems. Oxford, UK: Clarendon Press.

    Google Scholar 

  • Deng, Y., & Quigley, M. (2006). Irrational borrowers and the pricing of residential mortgages. Working paper presented at the AEA 2007 Annual Conference, Chicago.

  • Deng, Y., Quigley, M., & Van Order, R. (2000). Mortgage terminations, heterogeneity and the exercise of mortgage options. Econometrica, 68, 275–307.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Dunn, K. B., & McConnell, J. J. (1981a). A comparison of alternative models for pricing GNMA mortgage-backed securities. Journal of Finance, 36, 471–484.

    Article  Google Scholar 

  • Dunn, K. B., & McConnell, J. J. (1981b). Valuation of GNMA mortgage-backed securities. Journal of Finance, 36, 599–616.

    Article  Google Scholar 

  • Foster, C., & Van Order, R. (1984). An option-based model of mortgage default. Housing Finance Review, 3, 351–372.

    Google Scholar 

  • Foster, C., & Van Order, R. (1985). FHA terminations: A prelude to rational mortgage pricing. AREUEA Journal, 13, 273–291.

    Google Scholar 

  • Gauthier, L. (2002). Excursion height- and length-related stopping times, and applications to finance. Advances in Applied Probability, 34, 846–868.

    Article  Google Scholar 

  • Gauthier, L., & Morellec, E. (2001). Investment under uncertainty with implementation delay. In: E. S. Schwartz, & L. Trigeorgis (Eds.), Real options and investment under uncertainty. Cambridge, MA: MIT Press.

    Google Scholar 

  • Haber, R. J., Schönbucher, P. J., & Wilmott, P. (1999). Pricing Parisian options. Journal of Derivatives, 6, 71–79.

    Article  Google Scholar 

  • Hall, A. (2000). Controlling for burnout in estimating mortgage prepayment models. Journal of Housing Economics, 9, 215–232.

    Article  Google Scholar 

  • Hugonnier, J. (1999). The Feynman–Kac formula and pricing occupation time derivatives. International Journal of Theoretical and Applied Finance, 2, 153–178.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Kelly, A., & Slawson Jr., V. C. (2001). Time-varying mortgage prepayment penalties. Journal of Real Estate Finance and Economics, 23, 235–254.

    Article  Google Scholar 

  • Kwok, Y., & Lau, K. (2001). Pricing algorithms for options with exotic path-dependence. Journal of Derivatives, 9, 28–38.

    Article  Google Scholar 

  • Longstaff, F. A. (2005). Borrower credit and the valuation of mortgage-backed securities. Real Estate Economics, 33, 619–661.

    Article  Google Scholar 

  • Merton, R. C. (1973). The theory of rational option pricing. Bell Journal of Economics and Management Science, 4, 141–183.

    Article  Google Scholar 

  • Moraux, F. (2002). On cumulative parisian options. Finance, 23, 127–132.

    Google Scholar 

  • Richard, S. F., & Roll, R. (1989). Prepayments on fixed rate mortgage-backed securities. Journal of Portfolio Management, 15, 73–82.

    Article  Google Scholar 

  • Schwartz, E. S., & Torous, W. N. (1989). Prepayment and the valuation of mortgage backed securities. Journal of Finance, 44, 375–392.

    Article  Google Scholar 

  • Schwartz, E. S., & Torous, W. N. (1992). Prepayment, default and the valuation of mortgage pass-through securities. Journal of Business, 65, 221–239.

    Article  Google Scholar 

  • Schwartz, E. S., & Torous, W. N. (1993). Mortgage prepayment and default decisions: A Poisson regression approach. Journal of the American Real Estate and Urban Economics Association, 21, 431–449.

    Article  Google Scholar 

  • Sharp, N. J. (2006). Advances in mortgage valuation: An option-theoretic approach. PhD thesis, University of Manchester.

  • Sharp, N. J., Newton, D. P., & Duck, P. W. (2008). An improved fixed-rate mortgage valuation methodology with interacting prepayment and default options. Journal of Real Estate Finance and Economics, 36(4) (in press)

  • Stanton, R. (1995). Rational prepayment and the valuation of mortgage-backed securities. The Review of Financial Studies, 8, 677–708.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Vetzal, K. R., & Forsyth, P. A. (1999). Discrete Parisian and delayed barrier options: A general numerical approach. Advances in Futures and Options Research, 10, 1–15.

    Google Scholar 

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Acknowledgements

The research of the first and second authors was supported by the EPSRC. The authors acknowledge a number of useful and insightful comments raised by the reviewers of a previous version of this paper.

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Authors and Affiliations

  1. Nottingham University Business School, Nottingham, UK

    Nicholas J. Sharp & David P. Newton

  2. School of Mathematics, The University of Manchester, Manchester, UK

    Paul V. Johnson & Peter W. Duck

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  1. Nicholas J. Sharp
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  2. Paul V. Johnson
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  3. David P. Newton
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Correspondence to Nicholas J. Sharp.

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Sharp, N.J., Johnson, P.V., Newton, D.P. et al. A New Prepayment Model (with Default): An Occupation-time Derivative Approach. J Real Estate Finan Econ 39, 118–145 (2009). https://doi.org/10.1007/s11146-008-9105-7

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