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Numerical Method for Solving Free Boundary Problem Arising from Fixed Rate Mortgages

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Large-Scale Scientific Computing (LSSC 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8353))

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Abstract

In this paper a mortgage contract with a given duration and a fixed mortgage interest rate is considered. The borrower is allowed to terminate the contract at any time at his choice by paying off the outstanding sum to the issuer. The mathematical model leads to a free boundary problem where the moving boundary is the optimal time of termination. A new numerical method, based on the immersed interface method (IIM) and integral representation of the solution is proposed. Using Thomas algorithm the nonlinear equation for the free boundary position is obtained and solved iteratively. Numerical analysis is presented and discussed.

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References

  1. Buser, S.A., Hendershott, P.H.: Pricing default-free fixed rate mortgages. Hous. Financ. Rev. 3, 405–429 (1984)

    Google Scholar 

  2. Chen, X., Chadam, J., Jiang, L., Zheng, W.: Convexity of the exercise boundary of the American put option on a zero dividend asset. Math. Fin. 18, 185–197 (2008)

    Article  MathSciNet  Google Scholar 

  3. Epperson, J., Kau, J.B., Keenan, D.C., Muller, W.J.: Pricing default risk in mortgages. AREUEA J. 13, 152–167 (1985)

    Article  Google Scholar 

  4. Jiang, L., Bian, B., Yi, F.: A parabolic variational inequality arising from the valuation of fixed rate mortgages. Euro. J. Appl. Math. 16, 361–383 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kandilarov, J.: The immersed interface method for reaction-diffusion equation with a moving concentrated source. In: Dimov, I.T., Lirkov, I., Margenov, S., Zlatev, Z. (eds.) NMA 2002. LNCS, vol. 2542, pp. 506–513. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  6. Kandilarov, J.D., Valkov, R.L.: A numerical approach for the American call option pricing model. In: Dimov, I., Dimova, S., Kolkovska, N. (eds.) NMA 2010. LNCS, vol. 6046, pp. 453–460. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  7. Li, Z., Ito, K.: The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. SIAM, Philadelphia (2006)

    Book  Google Scholar 

  8. Nielsen, B., Skavhaug, O., Tveito, A.: Penalty and front-fixing methods for the numerical solution of American option problems. J. Comp. Fin. 5(4), 69–97 (2002)

    Google Scholar 

  9. Samarskii, A.: The Theory of Difference Schemes. Marcel Dekker, New York (2001)

    Book  MATH  Google Scholar 

  10. Ševčovič, D.: Analysis of the free boundary for the pricing of an American call option. Eur. J. Appl. Math. 12, 25–37 (2001)

    MATH  Google Scholar 

  11. Ševčovič, D.: Transformation methods for evaluating approximations to the optimal exercise boundary for linear and nonlinear Black-Sholes equations. In: Ehrhard, M. (ed.) Nonlinear Models in Mathematical Finance: New Research Trends in Optimal Pricing, pp. 153–198. Nova Sci. Publ., New York (2008)

    Google Scholar 

  12. Stamicar, R., Ševčovič, D., Chadam, J.: The early exercise boundary for the American put near expiry: numerical approximation. Canad. Appl. Math. Quart. 7(4), 427–444 (1999)

    MATH  MathSciNet  Google Scholar 

  13. Vasicek, O.: An equilibrium characterization of the term structure. J. Fin. Econ. 5, 177–188 (1977)

    Article  Google Scholar 

  14. Xie, D.: Fixed rate mortgages: valuation and closed form approximation. IAENG Int. J. Appl. Math. 39(1), 9 (2003)

    Google Scholar 

  15. Xie, D., Chen, X., Chadam, J.: Optimal payment of mortgages. Euro. J. Appl. Math. 18(3), 363–388 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This research was supported by the European Union under Grant Agreement number 304617 (FP7 Marie Curie Action Project Multi-ITN STRIKE - Novel Methods in Computational Finance) and the Bulgarian National Fund of Science under Project DID 02/37-2009.

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Authors and Affiliations

  1. Department of Mathematics, University of Rousse, Rousse, Bulgaria

    Juri D. Kandilarov

Authors
  1. Juri D. Kandilarov
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Correspondence to Juri D. Kandilarov .

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Editors and Affiliations

  1. Inst. of Info. & Comm. Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

  2. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Svetozar Margenov

  3. Dept. of Informatics and Mathematical Modell, Technical University of Denmark, Kongens Lyngby, Denmark

    Jerzy Waśniewski

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Kandilarov, J.D. (2014). Numerical Method for Solving Free Boundary Problem Arising from Fixed Rate Mortgages. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2013. Lecture Notes in Computer Science(), vol 8353. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43880-0_68

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  • DOI: https://doi.org/10.1007/978-3-662-43880-0_68

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-43879-4

  • Online ISBN: 978-3-662-43880-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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