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A primal-dual algorithm for BSDEs

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  • Christian Bender
  • Nikolaus Schweizer
  • Jia Zhuo
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    Abstract

    We generalize the primal-dual methodology, which is popular in the pricing of early-exercise options, to a backward dynamic programming equation associated with time discretization schemes of (reflected) backward stochastic differential equations (BSDEs). Taking as an input some approximate solution of the backward dynamic program, which was pre-computed, e.g., by least-squares Monte Carlo, our methodology allows to construct a confidence interval for the unknown true solution of the time discretized (reflected) BSDE at time 0. We numerically demonstrate the practical applicability of our method in two five-dimensional nonlinear pricing problems where tight price bounds were previously unavailable.

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    File URL: http://arxiv.org/pdf/1310.3694
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    Paper provided by arXiv.org in its series Papers with number 1310.3694.

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    Date of creation: Oct 2013
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    Handle: RePEc:arx:papers:1310.3694

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    1. Fahim, Arash & Touzi, Nizar & Warin, Xavier, 2011. "A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs," Economics Papers from University Paris Dauphine 123456789/5524, Paris Dauphine University.
    2. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 12(3), pages 271-286.
    3. 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.
    4. Andrea Pallavicini & Daniele Perini & Damiano Brigo, 2012. "Funding, Collateral and Hedging: uncovering the mechanics and the subtleties of funding valuation adjustments," Papers 1210.3811, arXiv.org, revised Dec 2012.
    5. 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., World Scientific Publishing Co. Pte. Ltd., vol. 16(02), pages 1350006-1-1.
    6. Bouchard, Bruno & Elie, Romuald, 2008. "Discrete-time approximation of decoupled Forward-Backward SDE with jumps," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 118(1), pages 53-75, January.
    7. 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.
    8. Bender, Christian & Denk, Robert, 2007. "A forward scheme for backward SDEs," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 117(12), pages 1793-1812, December.
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