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Branching Diffusions with Jumps and Valuation with Systemic Counterparties

Author

Listed:
  • Christoph Belak
  • Daniel Hoffmann
  • Frank T. Seifried

Abstract

We extend the branching diffusion Monte Carlo method of Henry-Labordere e.a.[11] to the case of parabolic PDEs with mixed local-nonlocal analytic nonlinearities. We investigate branching diffusion representations of classical solutions, and we provide suficient conditions under which the branching diffusion representation solves the PDE in the viscosity sense. Our theoretical setup directly leads´to a Monte Carlo algorithm, whose applicability is showcased in a stylized high-dimensional example. As our main application, we demonstrate how the methodology can be used to value financial positions with defaultable, systemically important counterparties.

Suggested Citation

  • Christoph Belak & Daniel Hoffmann & Frank T. Seifried, 2020. "Branching Diffusions with Jumps and Valuation with Systemic Counterparties," Working Paper Series 2020-04, University of Trier, Research Group Quantitative Finance and Risk Analysis.
    • Handle: RePEc:trr:qfrawp:202004
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    References listed on IDEAS

    as
    1. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    2. Duffie, Darrell & Singleton, Kenneth J, 1999. "Modeling Term Structures of Defaultable Bonds," The Review of Financial Studies, Society for Financial Studies, vol. 12(4), pages 687-720.
    3. Henry-Labordère, Pierre & Tan, Xiaolu & Touzi, Nizar, 2014. "A numerical algorithm for a class of BSDEs via the branching process," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1112-1140.
    4. Kim, Jinbeom & Leung, Tim, 2016. "Pricing derivatives with counterparty risk and collateralization: A fixed point approach," European Journal of Operational Research, Elsevier, vol. 249(2), pages 525-539.
    5. Damiano Brigo & Nicola Pede & Andrea Petrelli, 2015. "Multi Currency Credit Default Swaps Quanto effects and FX devaluation jumps," Papers 1512.07256, arXiv.org, revised Jan 2018.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Branching Diffusion; Mixed Local-Nonlocal PDE; Nonlinear Jumps; Monte Carlo Simulation; Credit Valuation Adjustment;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

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