IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1502.07531.html
   My bibliography  Save this paper

Feynman-Kac formula for L\'evy processes with discontinuous killing rate

Author

Listed:
  • Kathrin Glau

Abstract

The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to compute option prices in L\'evy models by solving partial integro differential equations have been developed. In order to provide a solid mathematical foundation for these methods, we derive a Feynman-Kac representation of variational solutions to partial integro differential equations that characterize conditional expectations of functionals of killed time-inhomogeneous L\'evy processes. We allow for a wide range of underlying stochastic processes, comprising processes with Brownian part, and a broad class of pure jump processes such as generalized hyperbolic, multivariate normal inverse Gaussian, tempered stable, and $\alpha$-semi stable L\'evy processes. By virtue of our mild regularity assumptions as to the killing rate and the initial condition of the partial differential equation, our results provide a rigorous basis for numerous applications, not only in financial mathematics but also in probability theory and relativistic quantum mechanics.

Suggested Citation

  • Kathrin Glau, 2015. "Feynman-Kac formula for L\'evy processes with discontinuous killing rate," Papers 1502.07531, arXiv.org, revised Nov 2015.
  • Handle: RePEc:arx:papers:1502.07531
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1502.07531
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ernst Eberlein & Sebastian Raible, 1999. "Term Structure Models Driven by General Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 9(1), pages 31-53, January.
    2. Ernst Eberlein & Kathrin Glau, 2014. "Variational Solutions of the Pricing PIDEs for European Options in Lévy Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(5), pages 417-450, November.
    3. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 471-500, September.
    4. Ernst Eberlein & Jean Jacod & Sebastian Raible, 2005. "Lévy term structure models: No-arbitrage and completeness," Finance and Stochastics, Springer, vol. 9(1), pages 67-88, January.
    5. Albrecher, Hansjörg & Lautscham, Volkmar, 2013. "From Ruin To Bankruptcy For Compound Poisson Surplus Processes," ASTIN Bulletin, Cambridge University Press, vol. 43(2), pages 213-243, May.
    6. Jérémy Poirot & Peter Tankov, 2006. "Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 13(4), pages 327-344, December.
    7. Rama Cont & Ekaterina Voltchkova, 2005. "Integro-differential equations for option prices in exponential Lévy models," Finance and Stochastics, Springer, vol. 9(3), pages 299-325, July.
    8. Andrey Itkin, 2013. "Efficient Solution of Backward Jump-Diffusion PIDEs with Splitting and Matrix Exponentials," Papers 1304.3159, arXiv.org, revised Apr 2014.
    9. Landriault, David & Renaud, Jean-François & Zhou, Xiaowen, 2011. "Occupation times of spectrally negative Lévy processes with applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2629-2641, November.
    10. Rama Cont & Nicolas Lantos & Olivier Pironneau, 2011. "A reduced basis for option pricing," Post-Print hal-00522410, HAL.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kathrin Glau, 2016. "A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates," Finance and Stochastics, Springer, vol. 20(4), pages 1021-1059, October.
    2. Maximilian Ga{ss} & Kathrin Glau, 2016. "A Flexible Galerkin Scheme for Option Pricing in L\'evy Models," Papers 1603.08216, arXiv.org.
    3. Lukas Gonon & Christoph Schwab, 2021. "Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models," Finance and Stochastics, Springer, vol. 25(4), pages 615-657, October.
    4. Kathrin Glau & Daniel Kressner & Francesco Statti, 2019. "Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing," Papers 1902.04367, arXiv.org.
    5. Asmerilda Hitaj & Lorenzo Mercuri & Edit Rroji, 2019. "Lévy CARMA models for shocks in mortality," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 205-227, June.
    6. Lijun Bo & Ying Jiao & Xuewei Yang, 2011. "Credit derivatives pricing with default density term structure modelled by L\'evy random fields," Papers 1112.2952, arXiv.org.
    7. Guérin, Hélène & Renaud, Jean-François, 2017. "On the distribution of cumulative Parisian ruin," Insurance: Mathematics and Economics, Elsevier, vol. 73(C), pages 116-123.
    8. Ernst Eberlein & Christoph Gerhart & Zorana Grbac, 2019. "Multiple curve Lévy forward price model allowing for negative interest rates," Post-Print hal-03898912, HAL.
    9. Ole E. Barndorff-Nielsen & Neil Shephard, 2012. "Basics of Levy processes," Economics Papers 2012-W06, Economics Group, Nuffield College, University of Oxford.
    10. Lukas Gonon & Christoph Schwab, 2021. "Deep ReLU Network Expression Rates for Option Prices in high-dimensional, exponential L\'evy models," Papers 2101.11897, arXiv.org, revised Jul 2021.
    11. Jean Jacod & Philip Protter, 2010. "Risk-neutral compatibility with option prices," Finance and Stochastics, Springer, vol. 14(2), pages 285-315, April.
    12. Jacek Jakubowski & Jerzy Zabczyk, 2007. "Exponential moments for HJM models with jumps," Finance and Stochastics, Springer, vol. 11(3), pages 429-445, July.
    13. Landriault, David & Li, Bin & Wong, Jeff T.Y. & Xu, Di, 2018. "Poissonian potential measures for Lévy risk models," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 152-166.
    14. Ernst Eberlein & Wolfgang Kluge & Antonis Papapantoleon, 2006. "Symmetries In Lévy Term Structure Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(06), pages 967-986.
    15. Hansjörg Albrecher & Jevgenijs Ivanovs, 2013. "A Risk Model with an Observer in a Markov Environment," Risks, MDPI, vol. 1(3), pages 1-14, November.
    16. Lijun Bo & Ying Jiao & Xuewei Yang, 2014. "Credit derivatives pricing with default density term structure modelled by Lévy random fields," Post-Print hal-00651397, HAL.
    17. Wolfgang Kluge & Antonis Papapantoleon, 2009. "On the valuation of compositions in Levy term structure models," Quantitative Finance, Taylor & Francis Journals, vol. 9(8), pages 951-959.
    18. Eckhard Platen & Stefan Tappe, 2011. "Affine Realizations for Levy Driven Interest Rate Models with Real-World Forward Rate Dynamics," Research Paper Series 289, Quantitative Finance Research Centre, University of Technology, Sydney.
    19. Eckhard Platen & Steffan Tappe, 2015. "Real-World Forward Rate Dynamics With Affine Realizations," Published Paper Series 2015-7, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    20. Ernst Eberlein & Fehmi Özkan, 2005. "The Lévy LIBOR model," Finance and Stochastics, Springer, vol. 9(3), pages 327-348, July.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1502.07531. See general information about how to correct material in RePEc.

    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 CitEc recognized a bibliographic reference but did not link an item in RePEc 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 RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.