IDEAS home Printed from https://ideas.repec.org/a/spr/jqecon/v16y2018i1d10.1007_s40953-016-0069-9.html
   My bibliography  Save this article

Stabilization of Mortgage Price Dynamics Using a Boundary PDE Feedback Control Approach

Author

Listed:
  • G. Rigatos

    (Industrial Systems Institute)

  • P. Siano

    (University of Salerno)

Abstract

Pricing of mortgages (loans for the purchase of residences, land or farms) is vital for the majority of transactions taking place in financial markets. In this article, a method for stabilization of mortgage price dynamics is developed. It is considered that mortgage prices follow a PDE model which is equivalent to a multi-asset Black–Scholes PDE. Actually it is a diffusion process evolving in a 2D assets space, where the first asset is the house price and the second asset is the interest rate. By applying semi-discretization and a finite differences scheme this multi-asset PDE is transformed into a state-space model consisting of ordinary nonlinear differential equations. For the local subsystems, into which the mortgage PDE is decomposed, it becomes possible to apply boundary-based feedback control. The control system’s design proceeds by showing that the state-space model of the mortgage price PDE stands for a differentially flat system. Next, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the multi-factor mortgage price PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the mortgage price PDE system so as to assure that all its state variables will converge to the desirable setpoints. By showing the feasibility of such a control method it is also proven that through selected modification of the PDE boundary conditions the price of the mortgage can be made to converge and stabilize at specific reference values.

Suggested Citation

  • G. Rigatos & P. Siano, 2018. "Stabilization of Mortgage Price Dynamics Using a Boundary PDE Feedback Control Approach," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 16(1), pages 37-56, March.
    • Handle: RePEc:spr:jqecon:v:16:y:2018:i:1:d:10.1007_s40953-016-0069-9
    DOI: 10.1007/s40953-016-0069-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s40953-016-0069-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s40953-016-0069-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Andrew Kalotay & Deane Yang & Frank J. Fabozzi, 2004. "An Option-Theoretic Prepayment Model For Mortgages And Mortgage-Backed Securities," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(08), pages 949-978.
    2. Yong Chen & Michael Connolly & Wenjin Tang & Tie Su, 2009. "The value of mortgage prepayment and default options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 29(9), pages 840-861, September.
    3. Pierre Bernhard, 2005. "The Robust Control Approach to Option Pricing and Interval Models: An Overview," Springer Books, in: Michèle Breton & Hatem Ben-Ameur (ed.), Numerical Methods in Finance, chapter 0, pages 91-108, Springer.
    4. Landini, Simone & Uberti, Mariacristina & Casellina, Simone, 2015. "Italian mortgage markets and their dynamics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 108(C), pages 245-259.
    5. Michel Fliess & C'edric Join, 2010. "Delta Hedging in Financial Engineering: Towards a Model-Free Approach," Papers 1005.0194, arXiv.org.
    6. Michel Fliess & Cédric Join, 2010. "Delta Hedging in Financial Engineering: Towards a Model-Free Approach," Post-Print inria-00479824, HAL.
    7. Michel Fliess & Cédric Join, 2012. "Preliminary remarks on option pricing and dynamic hedging," Post-Print hal-00705373, HAL.
    8. Michel Fliess & C'edric Join, 2012. "Preliminary remarks on option pricing and dynamic hedging," Papers 1206.1504, arXiv.org.
    9. Kyoung-Sook Moon & Hongjoong Kim, 2013. "A multi-dimensional local average lattice method for multi-asset models," Quantitative Finance, Taylor & Francis Journals, vol. 13(6), pages 873-884, May.
    10. Nicholas Sharp & David Newton & Peter Duck, 2008. "An Improved Fixed-Rate Mortgage Valuation Methodology with Interacting Prepayment and Default Options," The Journal of Real Estate Finance and Economics, Springer, vol. 36(3), pages 307-342, April.
    11. Stokes, Jeffrey R. & Brinch, Brian M., 2001. "Valuing Agricultural Mortgage-Backed Securities," Journal of Agricultural and Applied Economics, Southern Agricultural Economics Association, vol. 33(3), pages 1-19, December.
    12. Stanton, Richard, 1995. "Rational Prepayment and the Valuation Mortgage-Backed Securities," The Review of Financial Studies, Society for Financial Studies, vol. 8(3), pages 677-708.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Gerasimos G. Rigatos, 2016. "Boundary Control Of The Black–Scholes Pde For Option Dynamics Stabilization," Annals of Financial Economics (AFE), World Scientific Publishing Co. Pte. Ltd., vol. 11(02), pages 1-29, June.
    2. Werner Hürlimann, 2012. "Valuation of fixed and variable rate mortgages: binomial tree versus analytical approximations," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 35(2), pages 171-202, November.
    3. Nicholas Sharp & Paul Johnson & David Newton & Peter Duck, 2009. "A New Prepayment Model (with Default): An Occupation-time Derivative Approach," The Journal of Real Estate Finance and Economics, Springer, vol. 39(2), pages 118-145, August.
    4. Yerkin Kitapbayev & Scott Robertson, 2020. "Mortgage Contracts and Underwater Default," Papers 2005.03554, arXiv.org, revised May 2022.
    5. Matteo Bissiri & Riccardo Cogo, 2017. "Behavioral Value Adjustments," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(08), pages 1-37, December.
    6. Scott Robertson & Zhe Cheng, 2015. "Endogenous Current Coupons," Papers 1510.02010, arXiv.org.
    7. Giulia De Rossi & Tiziano Vargiolu, 2010. "Optimal prepayment and default rules for mortgage-backed securities," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 33(1), pages 23-47, May.
    8. Patel, Nimesh & Daglish, Toby, 2011. "Fixed come hell or high water? Selection and prepayment of fixed rate mortgages outside the US," Working Paper Series 19215, Victoria University of Wellington, The New Zealand Institute for the Study of Competition and Regulation.
    9. Eickholt, Mathias & Entrop, Oliver & Wilkens, Marco, 2014. "Individual investors and suboptimal early exercises in the fixed-income market," Passauer Diskussionspapiere, Betriebswirtschaftliche Reihe 14, University of Passau, Faculty of Business and Economics.
    10. Zhe Cheng & Scott Robertson, 2017. "Endogenous current coupons," Finance and Stochastics, Springer, vol. 21(4), pages 1027-1071, October.
    11. Sumit Agarwal & John C. Driscoll & David I. Laibson, 2013. "Optimal Mortgage Refinancing: A Closed‐Form Solution," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 45(4), pages 591-622, June.
    12. Xiaoxia Wu & Dejun Xie & David A. Edwards, 2019. "An Optimal Mortgage Refinancing Strategy with Stochastic Interest Rate," Computational Economics, Springer;Society for Computational Economics, vol. 53(4), pages 1353-1375, April.
    13. Michel Fliess & Cédric Join, 2013. "Systematic and multifactor risk models revisited," Post-Print hal-00920175, HAL.
    14. Emanuele Casamassima & Lech A. Grzelak & Frank A. Mulder & Cornelis W. Oosterlee, 2021. "Pricing and Hedging Prepayment Risk in a Mortgage Portfolio," Papers 2109.14977, arXiv.org, revised Oct 2021.
    15. repec:vuw:vuwscr:19215 is not listed on IDEAS
    16. Sven Balder & Wolf Christoph Gramatke & Antje Mahayni & Gordon Müller-Seitz & Jörg Sydow, 2014. "Bewertung von Kündigungsrechten in der privaten Wohnungsbaufinanzierung — Über den separaten Ausweis von Margen- und Kursschäden," Schmalenbach Journal of Business Research, Springer, vol. 66(1), pages 3-36, February.
    17. Bilgi Yilmaz & A. Sevtap Selcuk-Kestel, 2019. "Computation of Hedging Coefficients for Mortgage Default and Prepayment Options: Malliavin Calculus Approach," The Journal of Real Estate Finance and Economics, Springer, vol. 59(4), pages 673-697, November.
    18. Michel Fliess & C'edric Join, 2013. "Systematic and multifactor risk models revisited," Papers 1312.5271, arXiv.org.
    19. Patel, Nimesh & Daglish, Toby, 2011. "Fixed come hell or high water? Selection and prepayment of fixed rate mortgages outside the US," Working Paper Series 4107, Victoria University of Wellington, The New Zealand Institute for the Study of Competition and Regulation.
    20. Christopoulos, Andreas D. & Jarrow, Robert A., 2018. "CMBS market efficiency: The crisis and the recovery," Journal of Financial Stability, Elsevier, vol. 36(C), pages 159-186.
    21. Agarwal, Sumit & Driscoll, John D. & Laibson, David I., 2012. "Optimal Mortgage Reï¬ nancing: A Closed Form Solution," Scholarly Articles 9918811, Harvard University Department of Economics.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jqecon:v:16:y:2018:i:1:d:10.1007_s40953-016-0069-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.