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On absolutely continuous compensators and nonlinear filtering equations in default risk models

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  • Çetin, Umut
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    Abstract

    We discuss the pricing of defaultable assets in an incomplete information model where the default time is given by a first hitting time of an unobservable process. We show that in a fairly general Markov setting, the indicator function of the default has an absolutely continuous compensator. Given this compensator we then discuss the optional projection of a class of semimartingales onto the filtration generated by the observation process and the default indicator process. Available formulas for the pricing of defaultable assets are analyzed in this setting and some alternative formulas are suggested.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 122 (2012)
    Issue (Month): 11 ()
    Pages: 3619-3647

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    Handle: RePEc:eee:spapps:v:122:y:2012:i:11:p:3619-3647

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    Related research

    Keywords: Azéma supermartingale; Default indicator; Absolutely continuous compensators; Pricing of default risk; Nonlinear filtering; Zakai equation; Kushner–Stratonovich equation;

    References

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    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    1. Delia Coculescu & Hélyette Geman & Monique Jeanblanc, 2008. "Valuation of default-sensitive claims under imperfect information," Finance and Stochastics, Springer, vol. 12(2), pages 195-218, April.
    2. Svante Janson & Sokhna M'Baye & Philip Protter, 2011. "Absolutely Continuous Compensators," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(03), pages 335-351.
    3. Luciano Campi & Umut Çetin, 2007. "Insider trading in an equilibrium model with default: a passage from reduced-form to structural modelling," Finance and Stochastics, Springer, vol. 11(4), pages 591-602, October.
    4. Duffie, Darrell & Lando, David, 2001. "Term Structures of Credit Spreads with Incomplete Accounting Information," Econometrica, Econometric Society, vol. 69(3), pages 633-64, May.
    5. Rüdiger Frey & Wolfgang Runggaldier, 2010. "Pricing credit derivatives under incomplete information: a nonlinear-filtering approach," Finance and Stochastics, Springer, vol. 14(4), pages 495-526, December.
    6. Umut Cetin & Robert Jarrow & Philip Protter & Yildiray Yildirim, 2004. "Modeling Credit Risk with Partial Information," Papers math/0407060, arXiv.org.
    7. Delia Coculescu & Monique Jeanblanc & Ashkan Nikeghbali, 2012. "Default times, no-arbitrage conditions and changes of probability measures," Finance and Stochastics, Springer, vol. 16(3), pages 513-535, July.
    8. R. J. Elliott & M. Jeanblanc & M. Yor, 2000. "On Models of Default Risk," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 179-195.
    9. Robert Jarrow & Philip Protter & A. Sezer, 2007. "Information reduction via level crossings in a credit risk model," Finance and Stochastics, Springer, vol. 11(2), pages 195-212, April.
    10. Campi, Luciano & Cetin, Umut, 2007. "Insider trading in an equilibrium model with default: a passage from reduced-form to structural modelling," Economics Papers from University Paris Dauphine 123456789/4436, Paris Dauphine University.
    11. Rüdiger Frey & Thorsten Schmidt, 2009. "Pricing Corporate Securities Under Noisy Asset Information," Mathematical Finance, Wiley Blackwell, vol. 19(3), pages 403-421.
    12. Leland, Hayne E, 1994. " Corporate Debt Value, Bond Covenants, and Optimal Capital Structure," Journal of Finance, American Finance Association, vol. 49(4), pages 1213-52, September.
    13. Duffie, Darrell & Singleton, Kenneth J, 1999. "Modeling Term Structures of Defaultable Bonds," Review of Financial Studies, Society for Financial Studies, vol. 12(4), pages 687-720.
    14. Rüdiger Frey & Thorsten Schmidt, 2012. "Pricing and hedging of credit derivatives via the innovations approach to nonlinear filtering," Finance and Stochastics, Springer, vol. 16(1), pages 105-133, January.
    15. P. Collin-Dufresne & R. Goldstein & J. Hugonnier, 2004. "A General Formula for Valuing Defaultable Securities," Econometrica, Econometric Society, vol. 72(5), pages 1377-1407, 09.
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