IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v125y2015i4p1323-1351.html
   My bibliography  Save this article

The use of BSDEs to characterize the mean–variance hedging problem and the variance optimal martingale measure for defaultable claims

Author

Listed:
  • Goutte, Stéphane
  • Ngoupeyou, Armand

Abstract

In this paper, we consider the problem of mean–variance hedging of a defaultable claim. We assume the underlying assets are jump processes driven by Brownian motion and default processes. Using the dynamic programming principle, we link the existence of the solution of the mean–variance hedging problem to the existence of solution of a system of coupled backward stochastic differential equations (BSDEs). First we prove the existence of a solution to this system of coupled BSDEs. Then we give the corresponding solution to the mean–variance hedging problem. Finally, we give some existence conditions and characterize the well known variance optimal martingale measure (VOMM) using the solution to the first quadratic BSDE with jumps that we derived from the previous stochastic control problem. We conclude with an explicit example of our credit risk model giving a numerical application in a two defaults case.

Suggested Citation

  • Goutte, Stéphane & Ngoupeyou, Armand, 2015. "The use of BSDEs to characterize the mean–variance hedging problem and the variance optimal martingale measure for defaultable claims," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1323-1351.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:4:p:1323-1351
    DOI: 10.1016/j.spa.2014.10.017
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414914002658
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2014.10.017?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    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. Christian Gourieroux & Jean Paul Laurent & Huyên Pham, 1998. "Mean‐Variance Hedging and Numéraire," Mathematical Finance, Wiley Blackwell, vol. 8(3), pages 179-200, July.
    3. Merton, Robert C, 1974. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 29(2), pages 449-470, May.
    4. El Karoui, Nicole & Jeanblanc, Monique & Jiao, Ying, 2010. "What happens after a default: The conditional density approach," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1011-1032, July.
    5. Michael Kohlmann & Dewen Xiong & Zhongxing Ye, 2010. "Mean Variance Hedging in a General Jump Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(1), pages 29-57.
    6. Martin Schweizer & Christophe Stricker & Freddy Delbaen & Pascale Monat & Walter Schachermayer, 1997. "Weighted norm inequalities and hedging in incomplete markets," Finance and Stochastics, Springer, vol. 1(3), pages 181-227.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Tetsuya Ishikawa & Scott Robertson, 2017. "Optimal Investment and Pricing in the Presence of Defaults," Papers 1703.00062, arXiv.org.
    2. Abid, Ilyes & Dhaoui, Abderrazak & Goutte, Stéphane & Guesmi, Khaled, 2019. "Contagion and bond pricing: The case of the ASEAN region," Research in International Business and Finance, Elsevier, vol. 47(C), pages 371-385.

    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. Stephane Goutte & Armand Ngoupeyou, 2012. "Optimization problem and mean variance hedging on defaultable claims," Papers 1209.5953, arXiv.org.
    2. 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.
    3. Cheikh Mbaye & Fr'ed'eric Vrins, 2019. "An arbitrage-free conic martingale model with application to credit risk," Papers 1909.02474, arXiv.org.
    4. Salmerón Garrido, José Antonio & Nunno, Giulia Di & D'Auria, Bernardo, 2022. "Before and after default: information and optimal portfolio via anticipating calculus," DES - Working Papers. Statistics and Econometrics. WS 35411, Universidad Carlos III de Madrid. Departamento de Estadística.
    5. Claudio Fontana & Thorsten Schmidt, 2016. "General dynamic term structures under default risk," Papers 1603.03198, arXiv.org, revised Nov 2017.
    6. Fontana, Claudio & Schmidt, Thorsten, 2018. "General dynamic term structures under default risk," Stochastic Processes and their Applications, Elsevier, vol. 128(10), pages 3353-3386.
    7. Leitner, Johannes, 2000. "Mean-Variance Efficiency and Intertemporal Price for Risk," CoFE Discussion Papers 00/35, University of Konstanz, Center of Finance and Econometrics (CoFE).
    8. Kohlmann, Michael & Tang, Shanjian, 2000. "Global Adapted Solution of One-Dimensional Backward Stochastic Riccati Equations, with Application to the Mean-Variance Hedging," CoFE Discussion Papers 00/26, University of Konstanz, Center of Finance and Econometrics (CoFE).
    9. Mohamed N. Abdelghani & Alexander V. Melnikov, 2017. "Optional Defaultable Markets," Risks, MDPI, vol. 5(4), pages 1-21, October.
    10. Jos'e A. Salmer'on & Giulia Di Nunno & Bernardo D'Auria, 2022. "Before and after default: information and optimal portfolio via anticipating calculus," Papers 2208.07163, arXiv.org, revised May 2023.
    11. Michael Mania & Revaz Tevzadze, 2008. "Backward Stochastic PDEs Related to the Utility Maximization Problem," ICER Working Papers - Applied Mathematics Series 07-2008, ICER - International Centre for Economic Research.
    12. Thorsten Rheinländer & Jenny Sexton, 2011. "Hedging Derivatives," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8062, August.
    13. Vicky Henderson & David Hobson & Sam Howison & Tino Kluge, 2003. "A Comparison of q-optimal Option Prices in a Stochastic Volatility Model with Correlation," OFRC Working Papers Series 2003mf02, Oxford Financial Research Centre.
    14. Gady Jacoby & Chuan Liao & Jonathan A. Batten, 2007. "A Pure Test for the Elasticity of Yield Spreads," The Institute for International Integration Studies Discussion Paper Series iiisdp195, IIIS.
    15. Andrey Krishenik & Andreea Minca & Johannes Wissel, 2015. "When do creditors with heterogeneous beliefs agree to run?," Finance and Stochastics, Springer, vol. 19(2), pages 233-259, April.
    16. Kroencke, Tim A. & Schmeling, Maik & Schrimpf, Andreas, 2021. "The FOMC Risk Shift," Journal of Monetary Economics, Elsevier, vol. 120(C), pages 21-39.
    17. ilya, gikhman, 2006. "Fixed-income instrument pricing," MPRA Paper 1449, University Library of Munich, Germany.
    18. Gordian Rättich & Kim Clark & Evi Hartmann, 2011. "Performance measurement and antecedents of early internationalizing firms: A systematic assessment," Working Papers 0031, College of Business, University of Texas at San Antonio.
    19. C, Loran & Eckbo, Espen & Lu, Ching-Chih, 2014. "Does Executive Compensation Reflect Default Risk?," UiS Working Papers in Economics and Finance 2014/11, University of Stavanger.
    20. Güntay, Levent & Hackbarth, Dirk, 2010. "Corporate bond credit spreads and forecast dispersion," Journal of Banking & Finance, Elsevier, vol. 34(10), pages 2328-2345, October.

    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:eee:spapps:v:125:y:2015:i:4:p:1323-1351. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.