IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1206.5393.html
   My bibliography  Save this paper

Numerical methods for the quadratic hedging problem in Markov models with jumps

Author

Listed:
  • Carmine De Franco
  • Peter Tankov
  • Xavier Warin

Abstract

We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by pure jump Markov process. Using the Hamilton-Jacobi-Bellman approach, the value function of the quadratic hedging problem can be related to a triangular system of parabolic partial integro-differential equations (PIDE), which can be shown to possess unique smooth solutions in our setting. The first equation is non-linear, but does not depend on the pay-off of the option to hedge (the pure investment problem), while the other two equations are linear. We propose convergent finite difference schemes for the numerical solution of these PIDEs and illustrate our results with an application to electricity markets, where time-inhomogeneous pure jump Markov processes appear in a natural manner.

Suggested Citation

  • Carmine De Franco & Peter Tankov & Xavier Warin, 2012. "Numerical methods for the quadratic hedging problem in Markov models with jumps," Papers 1206.5393, arXiv.org, revised Dec 2013.
    • Handle: RePEc:arx:papers:1206.5393
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1206.5393
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. St'ephane Goutte & Nadia Oudjane & Francesco Russo, 2013. "Variance optimal hedging for continuous time additive processes and applications," Papers 1302.1965, arXiv.org.
    2. Huyên Pham, 2000. "On quadratic hedging in continuous time," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(2), pages 315-339, April.
    3. Jean-Paul Laurent & Huyen Pham, 1999. "Dynamic programming and mean-variance hedging," Post-Print hal-03675953, HAL.
    4. Friedrich Hubalek & Jan Kallsen & Leszek Krawczyk, 2006. "Variance-optimal hedging for processes with stationary independent increments," Papers math/0607112, arXiv.org.
    5. Fred Espen Benth & Jan Kallsen & Thilo Meyer-Brandis, 2007. "A Non-Gaussian Ornstein-Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(2), pages 153-169.
    6. David Heath & Eckhard Platen & Martin Schweizer, 2001. "A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 11(4), pages 385-413, October.
    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. Jan Kallsen & Richard Vierthauer, 2009. "Quadratic hedging in affine stochastic volatility models," Review of Derivatives Research, Springer, vol. 12(1), pages 3-27, April.
    2. Wanyang Dai, 2014. "Mean-variance hedging based on an incomplete market with external risk factors of non-Gaussian OU processes," Papers 1410.0991, arXiv.org, revised Aug 2015.
    3. Aleš Černý, 2007. "Optimal Continuous‐Time Hedging With Leptokurtic Returns," Mathematical Finance, Wiley Blackwell, vol. 17(2), pages 175-203, April.
    4. René Aïd & Luciano Campi & Nicolas Langrené, 2010. "A structural risk-neutral model for pricing and hedging power derivatives," Working Papers hal-00525800, HAL.
    5. Stephane Goutte & Armand Ngoupeyou, 2012. "Optimization problem and mean variance hedging on defaultable claims," Papers 1209.5953, arXiv.org.
    6. Cl'ement M'enass'e & Peter Tankov, 2015. "Asymptotic indifference pricing in exponential L\'evy models," Papers 1502.03359, arXiv.org, revised Feb 2015.
    7. St'ephane Goutte & Nadia Oudjane & Francesco Russo, 2009. "Variance Optimal Hedging for continuous time processes with independent increments and applications," Papers 0912.0372, arXiv.org.
    8. Koichi Matsumoto, 2009. "Dynamic programming and mean-variance hedging with partial execution risk," Review of Derivatives Research, Springer, vol. 12(1), pages 29-53, April.
    9. Ewald, Christian-Oliver & Nawar, Roy & Siu, Tak Kuen, 2013. "Minimal variance hedging of natural gas derivatives in exponential Lévy models: Theory and empirical performance," Energy Economics, Elsevier, vol. 36(C), pages 97-107.
    10. Terry Lyons & Sina Nejad & Imanol Perez Arribas, 2019. "Nonparametric pricing and hedging of exotic derivatives," Papers 1905.00711, arXiv.org.
    11. Ismail Laachir & Francesco Russo, 2016. "BSDEs, càdlàg martingale problems and orthogonalisation under basis risk," Working Papers hal-01086227, HAL.
    12. Hardy Hulley & Thomas A. McWalter, 2015. "Quadratic Hedging of Basis Risk," JRFM, MDPI, vol. 8(1), pages 1-20, February.
    13. Vicky Henderson, 2002. "Analytical Comparisons of Option prices in Stochastic Volatility Models," OFRC Working Papers Series 2002mf03, Oxford Financial Research Centre.
    14. St'ephane Goutte & Nadia Oudjane & Francesco Russo, 2012. "Variance Optimal Hedging for discrete time processes with independent increments. Application to Electricity Markets," Papers 1205.4089, arXiv.org.
    15. Thorsten Rheinländer, 2005. "An entropy approach to the Stein and Stein model with correlation," Finance and Stochastics, Springer, vol. 9(3), pages 399-413, July.
    16. M. Mania & R. Tevzadze, 2003. "Backward Stochastic PDE and Imperfect Hedging," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 6(07), pages 663-692.
    17. Aleš Černý & Jan Kallsen, 2008. "Mean–Variance Hedging And Optimal Investment In Heston'S Model With Correlation," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 473-492, July.
    18. Okhrati, Ramin & Balbás, Alejandro & Garrido, José, 2014. "Hedging of defaultable claims in a structural model using a locally risk-minimizing approach," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2868-2891.
    19. Alev{s} v{C}ern'y & Stephan Denkl & Jan Kallsen, 2013. "Hedging in L\'evy Models and the Time Step Equivalent of Jumps," Papers 1309.7833, arXiv.org, revised Jul 2017.
    20. Francesca Biagini & Paolo Guasoni & Maurizio Pratelli, 2000. "Mean‐Variance Hedging for Stochastic Volatility Models," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 109-123, April.

    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:1206.5393. 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.