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

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  • 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
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    1. St'ephane Goutte & Nadia Oudjane & Francesco Russo, 2013. "Variance optimal hedging for continuous time additive processes and applications," Papers 1302.1965, arXiv.org.
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    3. 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.
    4. 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.
    5. 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.
    6. Christian Gourieroux & Jean Paul Laurent & Huyên Pham, 1998. "Mean‐Variance Hedging and Numéraire," Mathematical Finance, Wiley Blackwell, vol. 8(3), pages 179-200, July.
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    Cited by:

    1. 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.
    2. Tetsuya Ishikawa & Scott Robertson, 2017. "Optimal Investment and Pricing in the Presence of Defaults," Papers 1703.00062, arXiv.org.

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