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Pathwise Dynamic Programming

Author

Listed:
  • Christian Bender

    (Department of Mathematics, Saarland University, Postfach 151150, D-66041 Saarbrücken, Germany)

  • Christian Gärtner

    (Department of Mathematics, Saarland University, Postfach 151150, D-66041 Saarbrücken, Germany)

  • Nikolaus Schweizer

    (Department of Econometrics and Operations Research, Tilburg University, NL-5000 LE Tilburg, Netherlands)

Abstract

We present a novel method for deriving tight Monte Carlo confidence intervals for solutions of stochastic dynamic programming equations. Taking some approximate solution to the equation as an input, we construct pathwise recursions with a known bias. Suitably coupling the recursions for lower and upper bounds ensures that the method is applicable even when the dynamic program does not satisfy a comparison principle. We apply our method to three nonlinear option pricing problems, pricing under bilateral counterparty risk, under uncertain volatility, and under negotiated collateralization.

Suggested Citation

  • Christian Bender & Christian Gärtner & Nikolaus Schweizer, 2018. "Pathwise Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 965-965, August.
    • Handle: RePEc:inm:ormoor:v:43:y:2018:i:3:p:965-965
    DOI: 10.1287/moor.2017.0891
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    References listed on IDEAS

    as
    1. Leif Andersen & Mark Broadie, 2004. "Primal-Dual Simulation Algorithm for Pricing Multidimensional American Options," Management Science, INFORMS, vol. 50(9), pages 1222-1234, September.
    2. Christopher Beveridge & Mark Joshi, 2011. "Monte Carlo Bounds for Game Options Including Convertible Bonds," Management Science, INFORMS, vol. 57(5), pages 960-974, May.
    3. David B. Brown & Martin B. Haugh, 2017. "Information Relaxation Bounds for Infinite Horizon Markov Decision Processes," Operations Research, INFORMS, vol. 65(5), pages 1355-1379, October.
    4. repec:dau:papers:123456789/5524 is not listed on IDEAS
    5. John Schoenmakers, 2012. "A pure martingale dual for multiple stopping," Finance and Stochastics, Springer, vol. 16(2), pages 319-334, April.
    6. Roger Lord & Remmert Koekkoek & Dick Van Dijk, 2010. "A comparison of biased simulation schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 177-194.
    7. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    8. Christian Bender & Nikolaus Schweizer & Jia Zhuo, 2017. "A Primal–Dual Algorithm For Bsdes," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 866-901, July.
    9. Stéphane Crépey & Rémi Gerboud & Zorana Grbac & Nathalie Ngor, 2013. "Counterparty Risk And Funding: The Four Wings Of The Tva," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(02), pages 1-31.
    10. David B. Brown & James E. Smith & Peng Sun, 2010. "Information Relaxations and Duality in Stochastic Dynamic Programs," Operations Research, INFORMS, vol. 58(4-part-1), pages 785-801, August.
    11. Martin B. Haugh & Leonid Kogan, 2004. "Pricing American Options: A Duality Approach," Operations Research, INFORMS, vol. 52(2), pages 258-270, April.
    12. repec:dau:papers:123456789/12195 is not listed on IDEAS
    13. Kraft, Holger & Seifried, Frank Thomas, 2014. "Stochastic differential utility as the continuous-time limit of recursive utility," Journal of Economic Theory, Elsevier, vol. 151(C), pages 528-550.
    14. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286, July.
    15. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    16. Joel Vanden, 2006. "Exact Superreplication Strategies for a Class of Derivative Assets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(1), pages 61-87.
    17. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
    18. David B. Brown & James E. Smith, 2011. "Dynamic Portfolio Optimization with Transaction Costs: Heuristics and Dual Bounds," Management Science, INFORMS, vol. 57(10), pages 1752-1770, October.
    19. Tianyang Nie & Marek Rutkowski, 2016. "A BSDE approach to fair bilateral pricing under endogenous collateralization," Finance and Stochastics, Springer, vol. 20(4), pages 855-900, October.
    20. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
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