Advanced Search
MyIDEAS: Login to save this paper or follow this series

Optimal dual martingales, their analysis and application to new algorithms for Bermudan products

Contents:

Author Info

  • John Schoenmakers
  • Junbo Huang
  • Jianing Zhang
Registered author(s):

    Abstract

    In this paper we introduce and study the concept of optimal and surely optimal dual martingales in the context of dual valuation of Bermudan options, and outline the development of new algorithms in this context. We provide a characterization theorem, a theorem which gives conditions for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided by these results we develop a framework of backward algorithms for constructing such a martingale. In turn this martingale may then be utilized for computing an upper bound of the Bermudan product. The methodology is pure dual in the sense that it doesn't require certain input approximations to the Snell envelope. In an It\^o-L\'evy environment we outline a particular regression based backward algorithm which allows for computing dual upper bounds without nested Monte Carlo simulation. Moreover, as a by-product this algorithm also provides approximations to the continuation values of the product, which in turn determine a stopping policy. Hence, we may obtain lower bounds at the same time. In a first numerical study we demonstrate the backward dual regression algorithm in a Wiener environment at well known benchmark examples. It turns out that the method is at least comparable to the one in Belomestny et. al. (2009) regarding accuracy, but regarding computational robustness there are even several advantages.

    Download Info

    If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
    File URL: http://arxiv.org/pdf/1111.6038
    File Function: Latest version
    Download Restriction: no

    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1111.6038.

    as in new window
    Length:
    Date of creation: Nov 2011
    Date of revision: Feb 2012
    Handle: RePEc:arx:papers:1111.6038

    Contact details of provider:
    Web page: http://arxiv.org/

    Related research

    Keywords:

    This paper has been announced in the following NEP Reports:

    References

    References listed on IDEAS
    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
    as in new window
    1. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1.
    2. Denis Belomestny & Anastasia Kolodko & John Schoenmakers, 2009. "Regression methods for stochastic control problems and their convergence analysis," SFB 649 Discussion Papers SFB649DP2009-026, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    3. Mark Joshi & Jochen Theis, 2002. "Bounding Bermudan swaptions in a swap-rate market model," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 2(5), pages 370-377.
    4. Nan Chen & Paul Glasserman, 2007. "Additive and multiplicative duals for American option pricing," Finance and Stochastics, Springer, Springer, vol. 11(2), pages 153-179, April.
    5. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 12(3), pages 271-286.
    6. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management, Anderson Graduate School of Management, UCLA qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    7. Carriere, Jacques F., 1996. "Valuation of the early-exercise price for options using simulations and nonparametric regression," Insurance: Mathematics and Economics, Elsevier, vol. 19(1), pages 19-30, December.
    8. Johnson, Herb, 1987. "Options on the Maximum or the Minimum of Several Assets," Journal of Financial and Quantitative Analysis, Cambridge University Press, Cambridge University Press, vol. 22(03), pages 277-283, September.
    9. Anastasia Kolodko & John Schoenmakers, 2006. "Iterative construction of the optimal Bermudan stopping time," Finance and Stochastics, Springer, Springer, vol. 10(1), pages 27-49, 01.
    10. Leif Andersen & Mark Broadie, 2004. "Primal-Dual Simulation Algorithm for Pricing Multidimensional American Options," Management Science, INFORMS, INFORMS, vol. 50(9), pages 1222-1234, September.
    11. Denis Belomestny & Christian Bender & John Schoenmakers, 2009. "True Upper Bounds For Bermudan Products Via Non-Nested Monte Carlo," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 19(1), pages 53-71.
    12. Christian Bender & Anastasia Kolodko & John Schoenmakers, 2008. "Enhanced policy iteration for American options via scenario selection," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 8(2), pages 135-146.
    Full references (including those not matched with items on IDEAS)

    Citations

    Lists

    This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

    Statistics

    Access and download statistics

    Corrections

    When requesting a correction, please mention this item's handle: RePEc:arx:papers:1111.6038. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (arXiv administrators).

    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 references are entirely missing, you can add them using this form.

    If the full references list an item that is present in RePEc, but the system did not link 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 profile, as there may be some citations waiting for confirmation.

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