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An Iteration Procedure for Solving Integral Equations Related to Optimal Stopping Problems


  • Denis Belomestny
  • Pavel V. Gapeev


A new algorithm for finding value functions of finite horizon optimal stopping problems in one-dimensional diffusion models is presented. It is based on a time discretization of the corresponding integral equation. The proposed iterative procedure for solving the discretized integral equation converges in a finite number of steps and delivers in each step a lower or an upper bound for value of discretized problem on the whole time interval. The remarks on the application of the method for solving integral equations related to some optimal stopping problems are given.

Suggested Citation

  • Denis Belomestny & Pavel V. Gapeev, 2006. "An Iteration Procedure for Solving Integral Equations Related to Optimal Stopping Problems," SFB 649 Discussion Papers SFB649DP2006-043, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  • Handle: RePEc:hum:wpaper:sfb649dp2006-043

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    References listed on IDEAS

    1. Carr, Peter, 1998. "Randomization and the American Put," Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 597-626.
    2. Alexander Novikov & Albert Shiryaev, 2004. "On an Effective Solution of the Optimal Stopping Problem for Random Walks," Research Paper Series 131, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Kim, In Joon, 1990. "The Analytic Valuation of American Options," Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 547-572.
    4. Peter Carr & Robert Jarrow & Ravi Myneni, 2008. "Alternative Characterizations Of American Put Options," World Scientific Book Chapters,in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 5, pages 85-103 World Scientific Publishing Co. Pte. Ltd..
    5. Denis Belomestny & Grigori Milstein, 2006. "Adaptive Simulation Algorithms for Pricing American and Bermudian Options by Local Analysis of Financial Market," SFB 649 Discussion Papers SFB649DP2006-038, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    6. Gapeev, P.V. & Peskir, G., 2006. "The Wiener disorder problem with finite horizon," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1770-1791, December.
    7. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286.
    8. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14.
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    More about this item


    Optimal stopping; finite horizon; diffusion process; upper and lower bounds; Black-Scholes model; American put option; Asian option; Russian option; Bayesian sequential testing problem; disorder detection problem;

    JEL classification:

    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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