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A martingale representation theorem and valuation of defaultable securities

Author

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  • Tahir Choulli
  • Catherine Daveloose
  • Michèle Vanmaele

Abstract

We consider a financial framework with two levels of information: the public information generated by the financial assets, and a larger flow of information that contains additional knowledge about a random time. This random time can represent many economic and financial settings, such as the default time of a firm for credit risk, and the death time of an insured for life insurance. As the random time cannot be seen before its occurrence, the progressive enlargement of filtration seems tailor‐fit to model the larger flow of information that incorporates both the public flow and the information about the random time. In this context, our interest focuses on the following challenges: (a) How to single out the various risks coming from the financial assets, the random time, and their correlations? (b) How these risks interplay and lead to the formation of any risk in the larger flow of information? It is clear that understanding how risks build‐up and interact, when one enlarges the flow of information, is vital for an efficient risk management and derivatives' evaluation in those informational markets. Our answers to these challenges are full and complete no matter what the model for the random time is and no matter how the random time is related to the public flow. In fact, we introduce “pure default” risks, and quantify and classify these risks afterward. Then we elaborate our martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into orthogonal local martingales (i.e., local martingales whose product remains a local martingale). This constitutes our first principal contribution, while our second contribution consists in evaluating various defaultable securities according to the recovery policy, within our financial setting that encompasses any default model, using a martingale “basis.” Our pricing formulas explain the impact of various recovery policies on securities and determine the types of pure default risk they entail.

Suggested Citation

  • Tahir Choulli & Catherine Daveloose & Michèle Vanmaele, 2020. "A martingale representation theorem and valuation of defaultable securities," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1527-1564, October.
    • Handle: RePEc:bla:mathfi:v:30:y:2020:i:4:p:1527-1564
    DOI: 10.1111/mafi.12244
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    Cited by:

    1. Libo Li & Ruyi Liu & Marek Rutkowski, 2022. "Vulnerable European and American Options in a Market Model with Optional Hazard Process," Papers 2212.12860, arXiv.org.
    2. Ferdoos Alharbi & Tahir Choulli, 2022. "Log-optimal portfolio after a random time: Existence, description and sensitivity analysis," Papers 2204.03798, arXiv.org.
    3. Libo Li & Ruyi Liu & Marek Rutkowski, 2022. "Well-posedness and penalization schemes for generalized BSDEs and reflected generalized BSDEs," Papers 2212.12854, arXiv.org.
    4. Safa Alsheyab & Tahir Choulli, 2021. "Reflected backward stochastic differential equations under stopping with an arbitrary random time," Papers 2107.11896, arXiv.org.
    5. Anna Aksamit & Libo Li & Marek Rutkowski, 2021. "Generalized BSDEs with random time horizon in a progressively enlarged filtration," Papers 2105.06654, arXiv.org.
    6. T. Choulli & S. Alsheyab, 2024. "Linear reflected backward stochastic differential equations arising from vulnerable claims in markets with random horizon," Papers 2408.04758, arXiv.org.
    7. Tahir Choulli & Emmanuel Lepinette, 2024. "Super-hedging-pricing formulas and Immediate-Profit arbitrage for market models under random horizon," Papers 2401.05713, arXiv.org.
    8. Tahir Choulli & Sina Yansori, 2022. "Log-optimal and numéraire portfolios for market models stopped at a random time," Finance and Stochastics, Springer, vol. 26(3), pages 535-585, July.
    9. Tahir Choulli & Ferdoos Alharbi, 2022. "Representation for martingales living after a random time with applications," Papers 2203.11072, arXiv.org, revised Nov 2022.
    10. Choulli, Tahir & Yansori, Sina, 2022. "Explicit description of all deflators for market models under random horizon with applications to NFLVR," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 230-264.
    11. El Karoui & Mrad Mohamed & Caroline Hillairet, 2022. "Bi-revealed utilities in a defaultable universe : a new point of view on consumption," Working Papers hal-03919186, HAL.
    12. Tahir Choulli & Catherine Daveloose & Michèle Vanmaele, 2021. "Mortality/Longevity Risk-Minimization with or without Securitization," Mathematics, MDPI, vol. 9(14), pages 1-27, July.

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